A calculation method for equivalent modulus of double-layer support in tunnels

Abstract

The present invention discloses a method for calculating the equivalent modulus of a double-layer support for a tunnel, which belongs to the technical field of highway traffic engineering. The method comprises the following steps: S1, analyzing the environmental conditions, geological factors and construction conditions of the tunnel through on-site investigation, obtaining the influencing factors generated at the tunnel collapse site, and performing sensitivity analysis on these influencing factors; S2, studying the common influencing factors of the elastic modulus of the double-layer initial support through a refined numerical simulation method, and calculating the equivalent elastic modulus of the double-layer support structure; S3, verifying the reliability of the refined numerical simulation method and proposing a modulus correction formula; the method for calculating the equivalent modulus of the double-layer support for a tunnel provided by the present invention can fully consider the interaction between steel bars and concrete materials, the interaction between the double-layer initial support structure and the stratum, and the influence of different spatial structures of steel bar components on the elastic modulus of the double-layer initial support structure through a refined numerical simulation method.

Description

Technical Field

[0001] The present invention relates to the technical field of highway traffic engineering, and in particular to a method for calculating the equivalent modulus of a double-layer support of a tunnel. Background Art

[0002] The terrain in western China is primarily mountainous and hilly, making tunnel engineering a crucial role. Tunnel construction and operation are directly linked to smooth urban transportation, leading to growing concern for tunnel safety and stability. Current statistics show that my country currently leads the world in tunnel construction mileage and the number of tunnel projects, with an increasing number of tunnel projects being constructed in sandy loess regions. Loess, a common yet unique stratum with characteristics such as collapsibility and susceptibility to sedimentation, is widely distributed in northwestern my country, accounting for approximately 6.3% of the country's land area. Loess in my country can be subdivided into sandy, silty, and clayey loess based on its characteristics. In central and western China, it is primarily distributed in a zonal pattern, with the sand content decreasing and the clay content increasing from northwest to southeast. Sandy loess refers to loess with a fine sand content greater than 30%. The sensitivity of sandy loess means that during tunnel construction, changes in groundwater and soil mass can cause significant changes in the strata surrounding the tunnel, potentially leading to landslides. Tunnel collapse may cause engineering projects to stall, damage existing tunnel structures, and even cause serious harm to the surrounding environment. Understanding the corresponding causes of engineering disasters and taking targeted measures to address engineering problems are crucial to ensuring the stability and safety of tunnels during construction and operation.

[0003] To effectively prevent tunnel collapse, initial support is crucial in tunnel construction. The rationality and effectiveness of initial support directly impacts the overall stability of the entire project. Given the unique characteristics of sandy loess regions, double-layer support is particularly worthy of research as a means of enhancing support. Optimizing the design of double-layer support can better adapt to complex geological conditions, improve the overall stability of the tunnel, ensure stability during excavation, and prevent collapse.

[0004] A highway tunnel traverses sandy loess strata, with the shallow-buried section of the tunnel portal underpassing existing highways, drainage channels, irrigation canals, and other important structures. Construction was challenging and risky, and during construction, a large-scale landslide occurred in the shallow-buried section of the tunnel portal, leading to engineering problems such as deformation and encroachment of the primary support structure, cracking of the double-layer primary support, and twisting of the steel arch. Field research and analysis identified insufficient primary support strength as the primary cause of tunnel collapse, necessitating research and analysis of primary support design methods. Existing primary support design systems, including single-layer support, double-layer support, and advanced support, often employ engineering analogy, with the elastic modulus of the primary support structure determined using a conversion method. However, this conversion method has drawbacks and fails to account for the interaction between the support structure and the surrounding rock, or the mechanical response between steel and concrete. This paper proposes a refined numerical simulation method for the design of double-layer tunnel support structures. Taking into account the interaction between the initial support and the surrounding rock, the bond between the steel components and the concrete, and the spatial distribution of the steel bars, the variation pattern of the equivalent elastic modulus of the double-layer initial support structure under different influencing factors is studied. Finally, an empirical formula considering these influencing factors is proposed based on the conversion method. This provides a scientific reference for future support structure design and has certain reference significance for solving practical problems in similar projects. Summary of the Invention

[0005] The purpose of the present invention is to provide a method for calculating the equivalent modulus of double-layer support of a tunnel to solve the problems existing in the above-mentioned background technology.

[0006] To achieve the above object, the present invention provides a method for calculating the equivalent modulus of a double-layer tunnel support, comprising the following steps:

[0007] S1. Analyze the environmental conditions, geological factors, and construction conditions of the tunnel through on-site investigations to identify the factors that may affect the tunnel collapse and conduct sensitivity analysis on these factors.

[0008] S2. Use refined numerical simulation methods to study the common influencing factors on the elastic modulus of the double-layer initial support and calculate the equivalent elastic modulus of the double-layer support structure;

[0009] S3. Propose a refined numerical simulation method for reliability verification and a modulus correction formula.

[0010] Preferably, the influencing factors in step S1 include rainfall infiltration depth, excavation cycle footage, and initial support strength, specifically including:

[0011] a. Analysis of the impact of rainfall infiltration depth on tunnel stability, including

[0012] A stratum-tunnel model was established using the numerical simulation software MIDAS GTS / NX. First, the phenomenon of tunnel engineering collapse involves multiple complex factors, such as geological conditions, hydrogeology, support structures, etc. Numerical simulation can simulate these complex engineering systems to help understand the overall behavior of the project. Secondly, numerical simulation predicts potential risks in advance, allowing engineers to take necessary measures before design and construction to reduce the risk of tunnel collapse. In addition, by simulating the impact of different design schemes, the engineering design can be optimized and the stability and safety of the tunnel can be improved. Numerical simulation can also detect problems before actual construction, reduce engineering costs, and improve construction efficiency, thereby achieving cost and time benefits. Most importantly, this method deeply analyzes the mechanism of tunnel collapse, providing strong support for scientifically explaining the causes of collapse and providing valuable lessons for future projects;

[0013] The tunnel collapse occurred in the shallow buried section of the tunnel entrance, and the secondary lining had not yet been constructed. Therefore, the secondary lining structure was not considered in the numerical model. The double-layer primary support structure and the temporary middle wall are both composite double-layer primary support structures. The elastic modulus value is determined using the conversion method. The conversion method is based on the principle that the strength of the primary support structure before and after is equal. The strength of the steel component is converted to concrete. The calculation formula is:

[0014]

[0015] Among them, E is the initial support elastic modulus of the conversion method; E c is the elastic modulus of plain concrete; E s is the elastic modulus of steel; A w is the cross-sectional area of ​​the steel mesh; A g is the cross-sectional area of ​​the I-beam; A gs is the cross-sectional area of ​​the grid steel frame; A c is the concrete cross-sectional area;

[0016] The elastic modulus of the temporary support is:

[0017]

[0018] Among them, E 临 is the converted elastic modulus of temporary support;

[0019] b. Analysis of the impact of excavation cycle footage on tunnel stability, by simulating the ground stress state and soil deformation at different footages to understand the risk of geological disasters;

[0020] c. Analysis of the influence of initial support strength on tunnel stability. The elastic modulus E of the single-layer support is calculated by the conversion method. d .

[0021] Preferably, a sensitivity analysis is performed on the factors affecting the rainfall infiltration depth, excavation cycle footage, and initial support strength in S1, specifically including:

[0022] Since the unit dimensions of the three influencing factors, namely rainfall infiltration depth, excavation cycle advance, and initial support strength, are inconsistent and cannot be directly compared, a variable sensitivity coefficient M with a dimension of 1 is defined to quantitatively describe their impact on tunnel collapse. The sensitivity coefficient formula of each influencing factor is:

[0023] M l =(Δσ / σ) / (ΔH / H);

[0024] M m =(Δσ / σ) / (ΔL / L);

[0025] M n =(Δσ / σ) / (ΔE / E);

[0026] Among them, M>0 means that the tunnel collapse is positively correlated with the influencing factors, and M<0 means that the tunnel collapse is negatively correlated with the influencing factors; M l M is the sensitivity coefficient of rainfall infiltration depth to tunnel collapse; m M is the sensitivity coefficient of excavation cycle advance to tunnel collapse; n is the sensitivity coefficient of initial support strength to tunnel collapse; Δσ is the difference between the maximum tensile stress of the initial support vault of tunnel excavation and the maximum tensile stress of the reference initial support vault; σ is the maximum tensile stress of the tunnel vault of the reference tunnel excavation; ΔH is the difference between the rainfall infiltration depth and the reference rainfall infiltration depth; H is the reference rainfall infiltration depth; ΔL is the difference between the tunnel excavation cycle footage and the reference tunnel excavation footage; L is the reference tunnel excavation footage; ΔE is the difference between the elastic modulus of the initial support strength and the elastic modulus of the reference initial support strength; E is the elastic modulus of the reference initial support strength.

[0027] Preferably, the refined numerical simulation method in S2 specifically includes:

[0028] S21. Determination of constitutive parameters of double-layer support structure units, including:

[0029] A double-layer initial support model was established using the numerical simulation software ABAQUS. The concrete damage plasticity model (CDP) in ABAQUS was used to simulate the interaction between the double-layer initial support structure and the ground, the interaction between the steel bars and the concrete, and the spatial structure of the steel components.

[0030] Establish a finite element model. When establishing the double-layer primary support finite element model, separate the structure into concrete, steel mesh, I-beam arch, and steel grating, and then assemble them into primary support units in the assembly module. Based on the unit library provided by ABAQUS, solid elements are used for concrete, while linear elements are used for steel, as steel primarily acts in tension. The constitutive model used for concrete is a plastic damage model, while the steel uses an elastic model.

[0031] The elastic modulus E of the double-layer initial support elastic stage obtained by the refined numerical simulation method eq ;

[0032] The factors affecting the stiffness of the double-layer initial support structure were studied to find the variation pattern of the equivalent elastic modulus of the double-layer initial support structure with different influencing factors. As a reinforced concrete structure, the elastic modulus of the double-layer initial support structure mainly includes the following factors: concrete strength, concrete thickness, steel mesh diameter, steel mesh spacing, grid steel frame diameter, grid steel frame spacing, I-beam arch frame type, and I-beam arch frame spacing.

[0033] S22. The equivalent elastic modulus of the double-layer support structure is studied using the stratum-structure method.

[0034] Preferably, the influence of different influencing factors on the initial support elastic modulus is as follows:

[0035] A. The influence of concrete strength q on the elastic modulus of initial support. By changing the strength of concrete material, under the same load conditions, the concrete strength and the elastic modulus E calculated by the refined numerical simulation method are obtained. eq The change law of elastic modulus E calculated by the conversion method and the refined numerical simulation method are both positively correlated with the elastic modulus of the double-layer initial support and the concrete strength, thereby obtaining the relative difference δ = |E eq -E| / E×100%, the linear relationship between the relative difference and concrete strength is: δ=-0.0179q+0.7888;

[0036] Concrete strength and E eq The linear function relationship is: E eq =1.3679q+3.0728;

[0037] The linear functional relationship between concrete strength and E is E = q + 25.17;

[0038] B. The influence of concrete thickness d on the initial support elastic modulus. The elastic modulus of double-layer support is negatively correlated with the concrete thickness. The fitting formula of relative difference and concrete thickness is: δ = 0.0002d + 0.2029;

[0039] Concrete thickness and E eq The linear function relationship is: E eq =-0.0375d+59.171;

[0040] The linear function relationship between concrete thickness and E is: E = -0.0418d + 73.082;

[0041] C. The influence of steel mesh diameter s1 on the initial support elastic modulus. The initial support equivalent elastic modulus is positively correlated with the steel mesh diameter. The fitting formula between the relative difference and the steel mesh diameter is: δ = 0.0017s1 + 0.2744;

[0042] Steel mesh diameter and E eq The linear function relationship is: E eq =0.01s1+41.22;

[0043] The linear function relationship between the steel mesh diameter and E is: E = 0.0824s1 + 52.529;

[0044] D. The influence of the steel mesh spacing l1 on the initial support elastic modulus. The double-layer initial support elastic modulus is negatively correlated with the steel mesh spacing. The fitting formula is: δ = -0.000017l1 + 0.289680;

[0045] Steel mesh spacing and E eq The linear function relationship is: E eq =-0.0002l1+41.33;

[0046] The linear function relationship between the steel mesh spacing and E is: E = -0.001l1 + 53.302;

[0047] E. The influence of grid steel frame diameter s2 on the initial support elastic modulus. The initial support elastic modulus is positively correlated with the grid steel frame diameter, and the fitting relationship is:

[0048] δ=-0.0034s2+0.3744;

[0049] Diameter of grid steel frame and E eq The linear function relationship is: E eq =0.2206s2+35.713;

[0050] The linear function relationship between the diameter of the grid steel frame and E is: E = 0.1484s2 + 49.443;

[0051] F. The influence of grid steel frame spacing l2 on the initial support elastic modulus. The initial support elastic modulus is negatively correlated with the grid steel frame spacing, and the fitting relationship is:

[0052] δ=-0.0003l2+0.3082;

[0053] Grid steel frame spacing and E eq The linear function relationship is: E eq =-0.0252l2+53.51;

[0054] The linear function relationship between the grid steel frame spacing and E is: E = -0.0417l2 + 68.648;

[0055] G. The influence of I-beam model a on the initial support elastic modulus. The double-layer initial support elastic modulus is positively correlated with the I-beam model. The fitting formula is:

[0056] δ=-0.0043a+0.4798;

[0057] I-beam model and E eq The linear function relationship is: E eq =0.2212a+31.325;

[0058] The linear function relationship between the I-beam model and E is: E = 0.1094a + 48.081;

[0059] H. The influence of the spacing between I-steel arches on the elastic modulus of the initial support. The elastic modulus of the initial support is negatively correlated with the spacing between I-steel arches. The fitting formula is:

[0060] δ=0.0002l3+0.1651;

[0061] I-beam arch spacing and E eq The linear function relationship is: E eq =-0.0108l3+47.26;

[0062] The linear function relationship between the I-beam arch spacing and E is: E = -0.0068l3 + 55.375.

[0063] Preferably, step S3 is specifically as follows:

[0064] The calculation formula for the reinforcement ratio of reinforced concrete structure is:

[0065] Among them, A s is the cross-sectional area of ​​the steel bars in reinforced concrete structures;

[0066] The calculation formula for the double-layer initial support reinforcement ratio is:

[0067] The calculation formula for the change rate of the double-layer support elastic modulus is:

[0068] Among them, Eeq is the elastic modulus of the double-layer support structure under different reinforcement ratios;

[0069] Determine the equivalent elastic modulus of double-layer primary support with different reinforcement ratios using a refined numerical simulation method;

[0070] The elastic modulus E of the double-layer initial support structure under different reinforcement ratios ρ obtained by the refined numerical simulation method eq The following relationship is established between the converted elastic modulus E:

[0071]

[0072] Therefore, the present invention adopts the above-mentioned calculation method of the equivalent modulus of the double-layer support of a tunnel, which has the following beneficial effects:

[0073] (1) Based on the damage plasticity model (CDP) and the ground structure method, a refined numerical simulation method for the mechanical response of the double-layer support structure in the tunnel is proposed. This method can consider the interaction between the steel mesh, I-beams, grid steel frame and concrete in the double-layer support structure, as well as the interaction between the double-layer support structure and the surrounding rock, and other complex factors such as the timing of support.

[0074] (2) By studying the changing laws of the equivalent elastic modulus of the double-layer initial support structure due to different influencing factors, an empirical formula considering these influencing factors is proposed based on the conversion method, which provides a scientific reference for future support structure design and a certain reference significance for solving practical problems in similar projects;

[0075] (3) By proposing a modified conversion method for the elastic modulus of the double-layer support structure, and consulting relevant literature, the rates of change of the elastic modulus with the reinforcement ratio calculated by the refined numerical simulation method are all within the range of the measured values. Therefore, it can be considered that the refined numerical simulation method can effectively calculate the equivalent elastic modulus of the double-layer support.

[0076] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0077] Figure 1 This is a flow chart of a method for calculating the equivalent modulus of double-layer support for a tunnel according to the present invention;

[0078] Figure 2 Schematic diagram of sensitivity analysis of factors affecting tunnel collapse according to an embodiment of the present invention;

[0079] Figure 3Schematic diagram of the CDP damage parameter change curve of an embodiment of the present invention, wherein (a) is a schematic diagram of the concrete compressive stress-inelastic strain curve, (b) is a schematic diagram of the concrete tensile stress-cracking strain curve, (c) is a schematic diagram of the concrete compressive stress-inelastic strain curve, and (d) is a schematic diagram of the concrete tensile stress-inelastic strain curve;

[0080] Figure 4 Schematic diagram of stress-strain curve of initial support according to an embodiment of the present invention;

[0081] Figure 5 Schematic diagram of the relationship between elastic modulus and concrete strength according to an embodiment of the present invention;

[0082] Figure 6 Schematic diagram of the relationship between elastic modulus and concrete thickness according to an embodiment of the present invention;

[0083] Figure 7 Schematic diagram of the relationship between elastic modulus and steel mesh diameter according to an embodiment of the present invention;

[0084] Figure 8 Schematic diagram of the relationship between elastic modulus and steel mesh spacing according to an embodiment of the present invention;

[0085] Figure 9 Schematic diagram of the relationship between the elastic modulus and the diameter of the grid steel frame according to an embodiment of the present invention;

[0086] Figure 10 Schematic diagram of the relationship between the elastic modulus and the grid steel frame spacing according to an embodiment of the present invention;

[0087] Figure 11 Schematic diagram of the relationship between elastic modulus and I-beam model according to an embodiment of the present invention;

[0088] Figure 12 Schematic diagram of the relationship between the elastic modulus and the spacing between I-beam arches according to an embodiment of the present invention;

[0089] Figure 13 Schematic diagram of the relationship between the reinforcement ratio and the measured elastic modulus change rate according to an embodiment of the present invention;

[0090] Figure 14 Schematic diagram comparing the elastic modulus and fitting value of the refined numerical simulation method according to an embodiment of the present invention. DETAILED DESCRIPTION

[0091] The following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but rather merely represents selected embodiments of the present invention. All other embodiments derived by persons of ordinary skill in the art based on the embodiments of the present invention without inventive effort shall fall within the scope of protection of the present invention.

[0092] See also Figure 1 A calculation method for the equivalent modulus of double-layer tunnel support is proposed. This paper takes a highway tunnel as the research background and focuses on the collapse accidents that occurred during the construction of this shallow sandy loess tunnel. Through field research, literature research, and numerical analysis, the main causes of tunnel collapse are analyzed and studied, including:

[0093] (1) Through on-site investigation, the environmental conditions, geological factors, construction conditions, etc. of the tunnel are analyzed to obtain the main influencing factors of tunnel collapse. Sensitivity analysis is performed on these influencing factors to find the most unfavorable influencing factors.

[0094] (2) Aiming at the problem of determining the elastic modulus of the double-layer initial support structure, the interaction between the steel bars and concrete in the double-layer initial support structure was simulated using ABAQUS finite element software. The interaction between the double-layer initial support structure and the stratum was considered, and the spatial structure influence of the steel bar components was considered. A refined numerical simulation method was proposed to study the variation of the elastic modulus of the double-layer initial support structure under different influencing factors. The literature on the determination of the elastic modulus of reinforced concrete structures was consulted, and the reliability of the refined numerical simulation method for the design of the double-layer support structure of the tunnel was verified by the measured values ​​of the elastic modulus. A correction conversion method for the elastic modulus of the double-layer support structure was proposed.

[0095] (3) In order to ensure the construction safety of shallow sandy loess tunnels, the elastic modulus of the double-layer initial support structure was determined based on a refined numerical simulation method. The numerical simulation software MIDAS GTS / NX was used to study the influence of different excavation schemes, the timing of double-layer initial support application, different excavation advances and other factors on tunnel excavation. The surrounding rock displacement, the mechanical response of double-layer initial support and the surface structure channel under different working conditions were analyzed. A construction process for rapid entry of shallow sandy loess tunnels and prevention of surface channel cracking was proposed.

[0096] 1. Analysis of the collapse mechanism of a highway tunnel

[0097] Tunnel Design Overview

[0098] A highway tunnel, a key control element for the entire project, was located on a Class I terrace. The exit section featured relatively flat terrain, lush surface vegetation consisting primarily of crops and low shrubs, and poor slope stabilization. The tunnel's exit, excavated underground, passed beneath a highway, drainage channels, and irrigation canals. The surrounding rock was unstable, making construction difficult and risky, and prone to landslides during tunnel construction.

[0099] In principle, the tunnel plan design follows the route direction, but it also fully considers the engineering geology and hydrogeology conditions of the tunnel site, avoids large unfavorable geological sections, and considers the difficulty of tunnel opening construction.

[0100] The highway grade of a certain highway tunnel is a six-lane, two-way, first-class highway. The design speed of the tunnel is 60 km / h, the tunnel construction limit width is 14.00 m, and the height is 5 m. The cross slope of the tunnel pavement is 2% in one direction, and the superelevation is not more than ±3%. The maximum longitudinal slope in the tunnel is allowed to be ±3%, and the minimum longitudinal slope is allowed to be ±0.5%. The design load of the tunnel is highway grade I, the tunnel waterproof grade is grade II, and the secondary double-layer initial support concrete anti-seepage grade is not less than P8. The main structure of the tunnel is designed for a service life of 100 years, and the design service life of replaceable and repairable components is 30 years.

[0101] Analysis on the Causes of Sandy Loess Tunnel Collapse

[0102] During the construction of this large-section shallow sandy loess tunnel, landslides and other engineering disasters occurred in the shallow buried section of the tunnel entrance (the tunnel arch is buried at a depth of about 14m). Multiple surface collapses, roof falls, cracks, and structural damage occurred in the exit section. Specifically, the exit section surface collapsed, the surface ramp cracked and sunken, the surface cracked, and the surface water channel failed (structural cracks and foundation voids).

[0103] Field investigations revealed that the tunnel exit section experienced extensive rainfall during the rainy season, resulting in severe waterlogging on the slopes. The sandy loess soil at the tunnel site experienced collapsing and softening, and poor drainage from the tunnel's overhead channel resulted in infiltration. These environmental factors may have contributed to the tunnel collapse and roof caving. Furthermore, the construction team, driven by a tight deadline, implemented a rapid on-site construction schedule, and insufficient initial support strength also negatively impacted tunnel stability. In summary, the wetting and deformation of the sandy loess soil after rainfall, coupled with inappropriate excavation methods and support timing, led to engineering problems such as localized instability of the tunnel surrounding rock and damage to surface structures. Therefore, further analysis of the causes of tunnel collapse is necessary to ensure the stability of the support structure and cavern during tunnel construction, prevent structural deformation and damage, and mitigate tunnel disasters and operational hazards, thereby providing effective guidance for the construction of this sandy loess tunnel. A numerical simulation study plan was developed for the main causes of tunnel collapse: rainfall infiltration, excavation cycle length, and initial support strength. See Table 1 for details.

[0104] Table 1 Numerical simulation scheme

[0105]

[0106]

[0107] Analysis of the influence of rainfall infiltration on the stability of shallow buried sandy loess tunnels

[0108] 1. Numerical analysis methods

[0109] To investigate the causes of tunnel collapse in shallow sandy loess, a stratum-tunnel model was constructed using the numerical simulation software MIDAS GTS / NX. First, tunnel collapse involves multiple complex factors, such as geological conditions, hydrogeology, and support structures. Numerical simulation can simulate these complex engineering systems, helping to understand the overall behavior of the project. Second, numerical simulation can predict potential risks in advance, enabling engineers to take necessary measures to mitigate the risk of tunnel collapse before design and construction. Furthermore, by simulating the impact of different design options, engineering designs can be optimized, improving tunnel stability and safety. Numerical simulation can also identify problems before actual construction, reducing project costs and improving construction efficiency, thereby achieving both cost and time benefits. This approach provides in-depth insights into the mechanisms of tunnel collapse, providing strong support for scientifically explaining the causes of collapse and providing valuable lessons for future projects.

[0110] (1) Calculation parameters

[0111] The parameters of the surrounding rock and support structure in the numerical model are based on the "Highway Tunnel Design Code" and the tunnel engineering investigation report. The tunnel collapse occurred in the shallow buried section of the tunnel entrance, and the secondary lining had not yet been applied. Therefore, the secondary lining structure was not considered in the numerical model. The double-layer initial support structure and the temporary middle wall are both composite double-layer initial support structures. The elastic modulus value is determined using the conversion method. The conversion method is based on the principle of equality before and after the initial support structure is equivalent. The strength of the steel component is converted to concrete. The specific calculation formula is:

[0112]

[0113] The initial support for the shallow buried tunnel entrance is a composite double-layer initial support. The shotcrete is 48cm thick and the concrete design grade is C25. The steel mesh is double-layered, φ8 in size, with a spacing of @15*15. The steel arch frame is I22b I-beam and H18 grid, with a spacing of 60cm. Based on the above formula, the initial support elastic modulus is:

[0114]

[0115] The temporary support of the shallow buried section of the tunnel entrance is made of 20cm thick C25 sprayed concrete, with I18 I-beams installed inside at a spacing of 60cm. The elastic modulus of the temporary support is:

[0116]

[0117] According to engineering experience and relevant research results, the grouting effect is mainly achieved by enhancing the E, C, reflect.

[0118] (2) Model size and boundary conditions

[0119] A six-lane, bidirectional highway tunnel with a 17m wide and 12m high tunnel bore is designed. Based on Saint-Venant's principle and tunnel excavation simulations conducted by researchers both domestically and internationally, it is assumed that tunnel excavation disturbs the surrounding rock by 3 to 5 times the tunnel bore. Therefore, the final model dimensions were determined to be 84m × 30m × 61m (length × width × height). The numerical model's left and right boundary conditions were X-displacement constraints, the longitudinal boundary condition was Y-displacement constraints, and the lower boundary condition was Z-displacement constraints. The mesh size for the tunnel excavation was set to 1m, the surface stratum mesh size was 2m, and the underlying sandstone stratum mesh size was 4m. The mesh properties were set to a hybrid mesh, primarily using hexahedral elements. Initial support and temporary supports were generated by extracting the tunnel's geometric surface to ensure mesh node coupling. During the initial stress phase, displacements were reset to zero, and self-weight stress was applied.

[0120] (3) Construction process simulation

[0121] The surrounding rock of a tunnel collapse section is Class V, primarily sandy loess, with low strength and poor stability. This makes it prone to collapse during construction. Furthermore, tunnel excavation would adversely impact the underpass, surface structures such as the highway, drainage channels, and irrigation canals. Therefore, the design utilizes a double-sidewall pilot tunnel method to minimize surrounding rock deformation. This excavation method involves two steps, upper and lower, and three pilot tunnels, left, center, and right, with excavation and support carried out along the tunnel's longitudinal direction.

[0122] In the finite element software MIDAS GTS / NX, mesh groups and boundary groups are operated through the "Activate" and "Remove" functions. The specific construction steps are as follows:

[0123] ①STEP 1: Activate all mesh elements in the initial stage, including the original soil and channel structure before tunnel excavation, activate boundary conditions and gravity, and reset displacements;

[0124] ②STEP2: Passivate one part of the upper step of the left pilot pit and activate one part of the initial support and temporary support;

[0125] ③STEP3: Passivate two units of the lower step of the left pilot pit and activate the initial support and temporary support of the two parts;

[0126] ④STEP4: Passivate the three units on the upper step of the right pilot pit and activate the initial support and temporary support of the three parts;

[0127] ⑤STEP5: Passivate the four units of the lower step of the right pilot pit and activate the initial support and temporary support of the four parts;

[0128] ⑥STEP6: Passivate the 5 units on the upper step of the middle pilot pit and activate the initial support of the 5 units;

[0129] ⑦STEP7: Passivate the 6 units of the lower step of the middle pilot pit and activate the initial support of the 6 units;

[0130] ⑧ Repeat the excavation steps until the entire model tunnel is excavated.

[0131] (4) Tunnel stability analysis under rainfall infiltration conditions

[0132] This section focuses on the impact of rainfall infiltration on tunnel excavation. Rainfall infiltration can cause the sandy loess to saturate and soften, which in turn weakens the tunnel's arching effect. This can lead to deformation and collapse of underground engineering structures, induce surface subsidence, and damage surface buildings. Furthermore, rising water levels directly lift supporting structures, adversely affecting their mechanical response.

[0133] When simulating the effects of rainfall infiltration using the finite element software MIDAS GTS / NX, the ground layers at 2, 6, 10, and 14 meters below the surface were considered saturated, until the entire loess layer within the tunnel was saturated. To simulate the effects of rainfall infiltration, the original ground parameters for the corresponding thicknesses were directly replaced with those of the saturated layers. To analyze the stability of the tunnel excavation under different rainfall infiltration conditions, the section at Y = 26 meters in the model was selected as the research object.

[0134] The maximum tensile stress on the double-layer primary support structure increases with the depth of rainfall infiltration. As the rainfall infiltration depth increases from 0m to 14m, the maximum tensile stress at the dome at Y = 26m increases from 2.3MPa to 2.8MPa, an increase of 21.7%.

[0135] Analysis of the influence of excavation cycle advance on the stability of shallow sandy loess tunnels

[0136] This section studies the impact of different cyclic advances on tunnel excavation in order to gain a deeper understanding of the changes and challenges in the construction process. By simulating the stratum stress state and soil deformation at different advances, we can better understand the risks of possible geological disasters, such as landslides and caving. In addition, the simulation can also help evaluate the adaptability of the support structure at different advances, optimize the design, and improve the support effect. Studying the numerical simulation of different cyclic advances can also provide a scientific basis for planning the construction process, adjusting construction methods, and selecting support measures to reduce tunnel construction risks and improve efficiency and safety. This study studied the impact of cyclic advances of 1.5m, 2m, and 3m on tunnel excavation.

[0137] When the tunnel excavation cycle advance is 1.5m, the maximum tensile stress of the double-layer initial support structure is 17.2MPa, located at the arch spandrel position of the shallow buried section of the tunnel entrance, and the maximum compressive stress is 19.3MPa, located at the arch foot position of the shallow buried section of the tunnel entrance; when the tunnel excavation cycle advance is 2m, the maximum tensile stress of the double-layer initial support structure is 16.2MPa, and the maximum compressive stress is 17.7MPa, which are reduced by 5.8% and 8.3% respectively compared with the advance of 1.5m; when the tunnel excavation cycle advance is 3m, the maximum tensile stress of the double-layer initial support structure is 15.8MPa, and the maximum compressive stress is 16.7MPa, which are reduced by 8.1% and 13.5% respectively compared with the advance of 1.5m.

[0138] Analysis of the influence of initial support strength on the stability of shallow loess tunnels

[0139] This section focuses on the impact of different initial support strengths on tunnel excavation stability. Studying the impact of initial support strength on tunnels is crucial. This research can directly guide and optimize the design and construction of support structures, ensuring that problems such as ground deformation and collapse are effectively mitigated or resisted in the early stages of construction. Reasonable initial support is directly related to the subsequent construction process. If the strength is insufficient, it may cause excessive displacement, affect subsequent processes, and even cause accidents. Therefore, in-depth research on the impact of initial support strength will help to identify problems in advance, take technical measures to adjust and reinforce, and ensure the continuity and efficiency of construction. In addition, a systematic study of initial support strength can also provide experience and reference for similar projects in the future, summarize lessons learned, promote the continuous improvement and innovation of engineering technology, and ensure the quality and safety of tunnel projects.

[0140] The design parameters for single-layer support are: shotcrete thickness 28cm, concrete design grade C25; single-layer steel mesh, size φ8, spacing @15*15; steel arch frame is I22b I-beam, spacing 60cm. Therefore, the elastic modulus of the single-layer support calculated by the conversion method is:

[0141]

[0142] The double-layer support elastic modulus E=53.17GPa calculated by the conversion method.

[0143] The initial support structure of the tunnel was changed from a single-layer support to a double-layer support. The maximum tensile stress of the initial support was reduced from 16.3MPa to 15.8MPa, a decrease of 3.1%. This shows that the double-layer support structure can improve the stress condition of the support structure.

[0144] Sensitivity analysis of different influencing factors on tunnel collapse

[0145] Based on the numerical simulation analysis of the causes of tunnel collapse in sandy loess soil, a sensitivity analysis of three factors influencing the collapse was conducted to identify the primary cause. Because the units of rainfall infiltration depth, excavation cycle length, and initial support strength differ, direct comparison is impossible. Therefore, a variable sensitivity coefficient, M, with a dimension of 1, was defined to quantitatively describe its impact on tunnel collapse.

[0146] M l =(Δσ / σ) / (ΔH / H);

[0147] M m =(Δσ / σ) / (ΔL / L);

[0148] M n =(Δσ / σ) / (ΔE / E);

[0149] In the formula, M>0 indicates that there is a positive correlation between tunnel collapse and influencing factors, and M<0 indicates that there is a negative correlation between tunnel collapse and influencing factors.

[0150] The calculation results show that the sensitivity coefficient of the design parameter initial support strength to tunnel collapse is 5.547, the sensitivity coefficient of the external condition rainfall infiltration to tunnel collapse is 0.023, and the sensitivity coefficient of the construction influencing factor excavation advance to tunnel collapse is 0.003. The relationship between the three is as follows: Figure 2 As shown in the figure, initial support strength has the greatest impact on tunnel collapse, followed by rainfall infiltration, and finally by construction excavation progress. Therefore, it is necessary to analyze and study the value of initial support strength in tunnels to prevent tunnel collapse from occurring through design.

[0151] A sensitivity analysis of the factors influencing tunnel collapse in sandy loess soil reveals that the sensitivity index, M, follows the order: initial support strength > rainfall infiltration > excavation advance. Therefore, it is necessary to analyze and discuss the value of the strength, or elastic modulus, of the double-layer initial support.

[0152] Refined numerical simulation method for the design of double-layer tunnel support structure

[0153] A large-span, six-lane, twin-hole highway tunnel traverses the sandy loess strata of Qinghai Province. The shallowly buried section of the tunnel portal underpasses existing highways, power transmission lines, and other critical structures. Construction is challenging and risky, so the tunnel's support structure utilizes a composite double-layer primary support. In this composite double-layer primary support structure, the primary support bears the majority of the load during tunnel construction and operation. In this tunnel, double-layer primary support ensures surrounding rock stability. The elastic modulus is the most important indicator of the safety and strength of the primary support structure, determining the surrounding rock pressure that the primary support is designed to bear.

[0154] The initial support of the composite double-layer initial support is a reinforced concrete structure. Currently, the commonly used calculation methods for the elastic modulus of reinforced concrete structures include: elastic theory method, empirical formula method, and numerical simulation method.

[0155] Elasticity theory methods. This method primarily calculates material deformation based on the material's elastic modulus and the stress conditions, using the stress-strain relationship. For reinforced concrete structures, the elastic modulus can be calculated using parameters such as the elastic modulus of concrete and steel, the volume fraction of steel, and the arrangement of steel bars.

[0156] Empirical formula method. This method mainly uses formulas derived from a large amount of experimental data to calculate the elastic modulus of the material. For reinforced concrete structures, commonly used empirical formulas include the Euler-Bernoulli beam theory formula and the Timoshenko beam theory formula.

[0157] Numerical simulation methods use numerical calculation methods such as finite element analysis to simulate the deformation of materials after being subjected to stress and calculate the material's elastic modulus. This method can more accurately calculate the elastic modulus of reinforced concrete.

[0158] In previous numerical simulation analyses of tunnel initial support, researchers often used conversion methods to simplify calculations, converting the elastic modulus of steel components such as I-beams, grid steel frames, and steel meshes to the elastic modulus of shotcrete. This method only considers the steel structure as a rigid body, without considering the bonding effect between the steel bars and concrete and the influence of the layout, and has certain shortcomings. This embodiment proposes a refined numerical simulation and auxiliary design method for a double-layer support structure for a large-section collapsible loess tunnel similar to a highway tunnel, explores the differences between the conversion method of the double-layer support elastic modulus and the refined numerical simulation method, and further applies it to tunnel construction and support scheme design.

[0159] Structural design concept of double-layer support structure

[0160] The previous analysis of the effects of rainfall infiltration, different cyclical advances, and initial support strength on tunnel primary support structures indicates that insufficient initial support strength is the most important cause of tunnel collapse. Large deformation in shallow, large-section sandy loess tunnels is a challenging issue during tunnel construction. Sandy loess, with its low shear strength and high collapsibility, is highly susceptible to large deformation during tunnel excavation, leading to common engineering problems such as initial support structure deformation encroachment, cracking of double-layer primary support, and distortion of steel arches.

[0161] To address these engineering challenges, double-layer primary support, a commonly used construction method, plays a crucial role in stabilizing tunnel construction and controlling surrounding rock deformation. Therefore, it is necessary to discuss the design methods for double-layer support structures. Material selection and sizing are key design parameters in double-layer primary support structure design, requiring rational determination based on specific geological conditions, project requirements, and construction techniques. There is no single standard design approach. Design primarily draws on engineering experience and relevant standards and specifications. However, empirical methods are often a one-size-fits-all approach that struggles to address the unique characteristics of each project. They may fail to account for specific geological conditions or engineering environments, resulting in inaccurate designs and difficulty accurately reflecting the risks of a specific project. Therefore, when using empirical methods to design double-layer primary support structures, engineering designers often employ numerical simulation methods to enhance the scientific nature and accuracy of their designs.

[0162] Based on the relevant literature on double-layer support structures and reference to the "Highway Tunnel Design Code", it can be seen that the design parameters of double-layer support do not change much. The general parameters of double-layer support are: shotcrete strength is C20, C25, C30, C35; shotcrete thickness is 480, 510, 550, 600mm; steel mesh diameter is 6, 8, 10, 12mm; steel mesh spacing is 150, 200, 250, 300mm; grid steel frame diameter is 18, 20, 22, 25mm; grid steel frame spacing is 300, 400, 500, 600mm; I-beam arch frame model is I18, I20b, I22b, I25b; I-beam arch frame spacing is 300, 400, 500, 600mm.

[0163] Tunnel excavation progress is strictly controlled according to the designed steel frame spacing. After each step and cycle of excavation, the face is immediately sealed and initial support construction is carried out. After tunnel excavation is completed, the cross-sectional dimensions of the excavated face are inspected. High-pressure water or air is used for flushing before the initial spraying. Pre-buried rebar controls the thickness of the shotcrete. Shotcrete operations should closely follow the excavation working face, with an initial spraying thickness of 40mm. This is followed by the installation of the steel mesh, which should be centrally processed in the factory or on-site. The second layer of steel mesh is laid together with the I-beam arch frame. The arch foot displacement is fixed with locking anchor rods. U-shaped clamps are welded to the I-beam arch frame and filled with mortar. Adjacent steel frames are connected with longitudinal reinforcement to ensure rigidity and stability along the tunnel's longitudinal direction. The grid steel frame is then installed and the concrete is sprayed again.

[0164] Refined numerical simulation method for double-layer support structure

[0165] Elastic modulus is an important mechanical property of reinforced concrete structures, and it is known that the strength of the initial support is the most sensitive to tunnel collapse. In previous studies, model test methods or theoretical calculation methods were generally used to determine the elastic modulus of similar reinforced concrete structures. This embodiment aims to propose a refined numerical simulation method for double-layer support of shallow loess tunnels. The numerical simulation software ABAQUS is used to simulate the actual structure of the initial support structure in the project, including the spatial distribution of steel mesh, I-beams, and grid steel frames, the mutual bonding between steel bars and concrete, and the boundary constraints of the initial support. The common influencing factors of the elastic modulus of the double-layer initial support: concrete strength and thickness, diameter and spacing of the steel mesh, diameter and spacing of the grid steel frame, I-beam arch frame and spacing and the relationship between the elastic modulus are discussed and studied. The following simulation methods are mainly used as refined numerical simulation methods: double-layer initial support unit method, load-structure method, stratum-structure method, and theoretical method.

[0166] Determination of constitutive parameters of double-layer support structure unit

[0167] Previous research on composite materials similar to reinforced concrete has primarily used homogenization theory, using the unit cell that makes up the material's microstructure as the basic unit. Assuming the unit cell has periodicity, the macrostructural properties are described through the microstructure. To determine the equivalent elastic modulus of the double-layer initial support, a unit cell with dimensions of 1000mm×1000mm×480mm (length×width×height) was selected, and the elastic modulus of this unit cell was used to predict the equivalent elastic modulus of the initial support as a whole. In the finite element analysis of reinforced concrete structures, ABAQUS can effectively simulate the mechanical response characteristics of reinforced concrete structures, so a double-layer initial support model was established using ABAQUS.

[0168] 1. Numerical simulation parameters

[0169] The concrete uses the unique plastic damage model (CDP) in ABAQUS to simulate the interaction between the double-layer initial support structure and the ground, the interaction between steel bars and concrete, and the spatial structure of steel components, such as Figure 3 The specific values ​​of the simulation parameters are shown in Table 2-Table 3.

[0170] Table 2 Model parameters

[0171] Material name Density / kg.m-3 Elastic modulus / GPa μ Constitutive model C25 2500 28 0.2 CDP steel 7850 210 0.25 elasticity

[0172] Table 3 CDP parameters

[0173] Dilation Ecc fb0 / fc0 K Vis 40 0.1 1.16 0.6667 0

[0174] Note: Dilation is the expansion angle; Ecc is the eccentricity; Vis is the viscosity coefficient.

[0175] 2. Establishment of finite element model

[0176] Based on the above considerations, the finite element model dimensions were determined to be 1000mm × 1000mm × 480mm (length × width × height). In the process of establishing the double-layer primary support finite element model, the structure was divided into concrete, steel mesh, I-beam arch frame, and steel grating, which were modeled separately and assembled into primary support units in the assembly module. Based on the element library provided by ABAQUS, solid elements were used for concrete, while linear elements were used for steel, as steel bars primarily play a tensile role in the component. The constitutive model used for concrete was a plastic damage model, and for steel bars, an elastic model was used.

[0177] The specific modeling process is as follows:

[0178] In the part module, create component models. First, create models for concrete, steel mesh, grid steel frame, and I-beam arch frame. When modeling the concrete component, select "Deformable" for type, "Solid" for shape, and "Stretch" for type. Draw the cross-sectional dimensions of the concrete structure in the drawing interface, and use the stretch command to form the component. For the steel mesh, grid steel frame, and I-beam arch frame components, only one component needs to be created. Select "Deformable" for type, "Line" for shape, and "Plane" for type. Create a 1000mm long line component.

[0179] In the Property module, first create the material properties for the steel and concrete based on the selected constitutive models for concrete and steel. The concrete is C25, and the constitutive model is plastic damage. Steel components such as the steel mesh, grid steel frame, and I-beam arch are modeled using elasticity, without considering plasticity.

[0180] After defining the materials, define the required cross-sectional shapes for the rebar components, including circular sections with diameters of 8mm, 10mm, 12mm, and 25mm, as well as I-beam sections. Next, create the sections. For the concrete section type, select "Solid," "Homogeneous," and "Concrete" as the material. For the rebar component, select "Beam," "Beam" as the type, select the corresponding rebar size for the section, and select the defined rebar material as the material. Finally, assign the corresponding sections to the components and the orientation of the beam elements, then assign the cross-sectional properties to the rebar components.

[0181] In the Assembly module, the actual spatial structure of the initial support is created by using functions such as translation, rotation, and array. The steel mesh, I-beam arch, and grid steel frame components are then formed separately through the merge function. This ensures that common nodes can be formed at the connections of the steel components, ensuring that the deformation of the components is coordinated and consistent. Finally, the steel components and concrete components are assembled into a complete initial support unit.

[0182] The Step module defines analysis steps. The primary purpose of creating these steps is to load the double-layer initial support unit components in stages, obtaining the stress-strain curves of the reinforced concrete components and, therefore, determining the equivalent elastic modulus during the initial support elastic phase. Two analysis steps are set up. The Initial analysis step considers the effects of the structure's deadweight. Building upon the existing Initial analysis step, a second analysis step (Step-1) is created to apply a surface load to the top surface of the structure.

[0183] In the Interaction module, in order to simulate the interaction between steel bars and concrete structures, the steel bar components are embedded into the concrete components using the Embedded region constraint function.

[0184] In the load module, boundary conditions are first created. Because the initial support in the tunnel structure is under the condition of constraints on all sides, fixed constraints are applied on all sides of the model in the Initial analysis step, and no settings are made on the upper and lower sides. Then, the load conditions of the initial support are simulated. Because the elastic modulus is taken in the elastic stage of the structure, the applied load should not be too large to ensure that the structure is in the elastic stage. After consulting relevant literature and specifications, the load size is finally determined to be 1MPa. The load is applied to the upper top surface to simulate the surrounding rock pressure on the initial support.

[0185] In the Mesh module, mesh quality is crucial to the accuracy of calculation results. Meshing is performed based on component size. To ensure convergence, the approximate mesh size for concrete elements is 50 mm, and the approximate mesh size for steel element components is 25 mm. The concrete element type is a three-dimensional eight-node linear hexahedron element (C3D8R), and the steel element type is a two-node spatial linear beam element (B31).

[0186] 3. Obtaining the elastic modulus obtained by refined numerical simulation methods

[0187] In the job module, create a new job. After the calculation is completed, enter the visualization module to view the results. By extracting the vertical stress (σ 22 ) and strain (ε 22 ), the stress-strain curve of the initial support under 1MPa surface load is obtained, as shown in Figure 4 As shown in the figure, the elastic modulus of the double-layer initial support elastic stage is 41.11 GPa according to this curve, while the elastic modulus calculated by the conversion method is 55.63 GPa. The elastic modulus obtained by the refined numerical simulation method is reduced by nearly 35% compared with it.

[0188] 4. Numerical simulation scheme

[0189] After establishing a preliminary calculation method for the equivalent elastic modulus of double-layer primary support, the main factors influencing the stiffness of the double-layer primary support structure were studied to identify the variation patterns between these factors and the equivalent elastic modulus of the double-layer primary support structure. As a reinforced concrete structure, the elastic modulus of the double-layer primary support structure is primarily influenced by factors such as concrete strength and thickness, steel mesh diameter and spacing, grid steel frame diameter and spacing, and I-beam type and spacing. The values ​​of these factors were primarily determined based on the Highway Tunnel Design Code, the Highway Tunnel Construction Technical Code, and commonly used specifications in construction. The numerical simulation scheme is shown in Table 4.

[0190] Table 4 Numerical simulation research plan

[0191]

[0192] Stratigraphic-structural method

[0193] The load-structure method simplified the load and boundary conditions in the analysis of double-layer support structures. Therefore, based on this research, the equivalent elastic modulus of the double-layer support structure was investigated using the stratum-structure method. The finite element model dimensions were 100m × 80m × 1.2m (length × width × height). To ensure mesh quality, the model was divided into four sections at the center. The analysis was conducted in two steps: the first step was for ground stress equilibrium, and the second step was for tunnel excavation.

[0194] (1) Influence of concrete strength on initial support elastic modulus

[0195] During the initial support construction phase of shallow loess tunnels, most of the surrounding rock pressure is borne. A composite double-layer initial support is generally used, which is a reinforced concrete composite structure. The equivalent elastic modulus of reinforced concrete structures is greatly affected by the mechanical properties of both steel and concrete. The elastic modulus of steel is generally fixed at 210GPa, so only the elastic modulus change of concrete is considered. The higher the concrete grade, the greater the equivalent elastic modulus, and the greater the elastic modulus of the double-layer initial support. By changing the strength of the concrete material, under the same load conditions, the concrete strength and the elastic modulus E calculated by the refined numerical simulation method are obtained. eq The change law of elastic modulus E is calculated by the conversion method, such as Figure 5 shown.

[0196] The elastic modulus of the double-layer initial support obtained by the refined numerical simulation method and the conversion method are positively correlated with the concrete strength. The higher the concrete grade, the greater the initial support elastic modulus. The elastic modulus of the refined numerical simulation method is smaller than the elastic modulus of the conversion method. The relative difference δ = |E eq-E| / E×100%, the relative difference is 28% to 31% within the range of concrete strength, and the average error is 27.46%. The relative difference gradually converges with the increase of concrete strength, and the linear relationship between it and concrete strength is:

[0197] δ=-0.0179q+0.7888;

[0198] For the refined numerical simulation method, when the concrete grade increases from C20 to C25, the elastic modulus increases from 38 GPa to 41.3 GPa, an increase of 8.68%; when the concrete grade increases from C25 to C30, the elastic modulus increases from 41.3 GPa to 44.1 GPa, an increase of 6.78%; when the concrete grade increases from C30 to C35, the elastic modulus increases from 44.1 GPa to 46.2 GPa, an increase of 4.76%. The curves of concrete grade and elastic modulus show that the two conform to a linear function relationship:

[0199] E eq =1.3679q+3.0728;

[0200] For the conversion method, when the concrete grade increases from C20 to C25, the elastic modulus increases from 50.67 GPa to 53.17 GPa, an increase of 4.93%; when the concrete grade increases from C25 to C30, the elastic modulus increases from 53.17 GPa to 55.17 GPa, an increase of 3.76%; when the concrete grade increases from C30 to C35, the elastic modulus increases from 55.17 GPa to 56.67 GPa, an increase of 2.72%. The curves of concrete grade and elastic modulus show that the two conform to a linear function relationship:

[0201] E = q + 25.17;

[0202] (2) Influence of concrete thickness on initial support elastic modulus

[0203] In the initial stage of shallow loess tunnel excavation, the surrounding rock pressure is mainly borne by the I-steel arch frame and the grid arch frame. After a period of time after the initial support is completed, the strength of the concrete gradually increases. At this time, the arch frame and concrete act as a composite to provide surrounding rock support pressure. The thickness of the shotcrete is directly related to the size of the support pressure. Its thickness is almost equal to the cross-sectional height of the steel arch frame. The equivalent elastic modulus of the double-layer initial support and the thickness of the concrete change law are as follows: Figure 6 shown.

[0204] The elastic modulus calculated by both methods decreases with the thickness of the concrete. The equivalent elastic modulus calculated by the refined numerical simulation method is generally smaller than that calculated by the conversion method. The elastic modulus of the double-layer initial support shows a negative correlation with the concrete thickness. The thicker the concrete, the smaller the initial support elastic modulus. The relative difference δ varies from 23% to 26%, with an average error of 29.72%. It gradually converges with the increase of concrete thickness. The fitting formula for the concrete thickness is:

[0205] δ=0.0002d+0.2029;

[0206] For the refined numerical simulation method, when the concrete thickness increases from 480mm to 510mm, the elastic modulus decreases from 41.3GPa to 40GPa, a decrease of 3.15%; when the concrete grade increases from 510mm to 550mm, the elastic modulus decreases from 40GPa to 38.4GPa, a decrease of 4.00%; when the concrete grade increases from 550mm to 600mm, the elastic modulus decreases from 38.4GPa to 36.8GPa, a decrease of 4.17%. The curves of concrete grade and elastic modulus show that the two conform to a linear function relationship:

[0207] E eq =-0.0375d+59.171;

[0208] For the conversion method, when the concrete thickness increases from 480mm to 510mm, the elastic modulus increases from 53.17GPa to 51.69GPa, a decrease of 2.78%; when the concrete grade increases from 510mm to 550mm, the elastic modulus decreases from 51.69GPa to 49.97GPa, a decrease of 3.33%; when the concrete grade increases from 550mm to 600mm, the elastic modulus decreases from 49.97GPa to 48.14GPa, a decrease of 3.66%. The curves of concrete grade and elastic modulus show that the two conform to a linear function relationship:

[0209] E = -0.0418d + 73.082;

[0210] From the stiffness of the steel bars and concrete materials in the composite double-layer primary support structure, it can be seen that the steel bar structure contributes more to the overall elastic modulus of the composite, and the concrete material mainly plays the role of protecting the steel bar structure. Therefore, the thickness of the concrete only needs to leave a sufficient protective layer thickness for the steel bar structure and meet the requirements of the specification.

[0211] (3) Influence of steel mesh diameter on initial support elastic modulus

[0212] The factors that affect the equivalent elastic modulus of double-layer initial support include the diameter of the steel mesh and the spacing between the steel bars in the steel mesh. This section mainly studies the influence of the steel mesh diameter on the elastic modulus of the initial support. The change pattern between the two is as follows: Figure 7 shown.

[0213] The initial support equivalent elastic modulus shows a positive correlation with the steel mesh diameter. The larger the steel mesh diameter, the larger the initial support elastic modulus. This is consistent with construction experience and the researchers' cognition. The relative difference δ varies from 28% to 30%, with an average error of 28.95%. The fitting formula is:

[0214] δ=0.0017s1+0.2744;

[0215] For the refined numerical simulation method, when the steel mesh diameter increases from 6mm to 8mm, the elastic modulus increases from 41.28GPa to 41.3GPa, an increase of 0.05%; when the concrete grade increases from 8mm to 10mm, the elastic modulus increases from 41.3GPa to 41.32GPa, an increase of 0.05%; when the concrete grade increases from 10mm to 12mm, the elastic modulus increases from 41.32GPa to 41.34GPa, an increase of 0.05%. The curves of the equivalent elastic modulus of the double-layer initial support and the steel mesh diameter show that the two conform to a linear function relationship:

[0216] E eq =0.01s1+41.22;

[0217] For the conversion method, when the diameter of the steel mesh increases from 6mm to 8mm, the elastic modulus increases from 53.04GPa to 53.17GPa, an increase of 0.24%; when the concrete grade increases from 8mm to 10mm, the elastic modulus increases from 53.17GPa to 53.34GPa, an increase of 0.31%; when the concrete grade increases from 10mm to 12mm, the elastic modulus increases from 53.34GPa to 53.54GPa, an increase of 0.38%. The curves of the equivalent elastic modulus of the double-layer initial support and the steel mesh diameter show that the two conform to a linear function relationship:

[0218] E=0.0824s1+52.529;

[0219] (4) Influence of steel mesh spacing on initial support elastic modulus

[0220] The previous section studied the effect of mesh diameter on the equivalent elastic modulus of double-layer initial support. This section mainly studies another effect of mesh spacing on the structure of the mesh. The relationship between mesh spacing and equivalent elastic modulus is shown in the following figure: Figure 8 shown.

[0221] The elastic modulus of the double-layer initial support is negatively correlated with the spacing between the steel meshes. The larger the spacing between the steel meshes, the smaller the initial support elastic modulus. The relative difference δ ranges from 28% to 29%, with an average error of 28.59%. It gradually becomes discrete with increasing concrete thickness. The fitting formula for the concrete thickness is:

[0222] δ=-0.000017l1+0.289680;

[0223] For the refined numerical simulation method, when the steel mesh spacing increases from 150mm to 200mm, the elastic modulus decreases from 41.3GPa to 41.29GPa, a decrease of 0.02%; when the steel mesh spacing increases from 200mm to 250mm, the elastic modulus decreases from 41.29GPa to 41.28GPa, a decrease of 0.02%; when the steel mesh spacing increases from 250mm to 300mm, the elastic modulus decreases from 41.28GPa to 41.27GPa, a decrease of 0.02%. The curves of the steel mesh spacing and the equivalent elastic modulus of the double-layer initial support show that the two conform to a linear function relationship:

[0224] E eq =-0.0002l1+41.33;

[0225] For the conversion method, when the steel mesh spacing increases from 150mm to 200mm, the elastic modulus decreases from 53.17GPa to 53.10GPa, a decrease of 2.5%; when the steel mesh spacing increases from 200mm to 250mm, the elastic modulus decreases from 53.10GPa to 53.06GPa, a decrease of 0.07%; when the steel mesh spacing increases from 250mm to 300mm, the elastic modulus decreases from 53.06GPa to 53.02GPa, a decrease of 0.07%. The curves of the steel mesh spacing and the equivalent elastic modulus of the double-layer initial support show that the two conform to a linear function relationship:

[0226] E=-0.001l1+53.302;

[0227] (5) Influence of grid steel frame diameter on initial support elastic modulus

[0228] When the surrounding rock conditions of the tunnel are relatively good, the initial support is generally single-layer support, and the I-steel arch frame mainly bears the surrounding rock pressure. However, the surrounding rock conditions of a certain highway tunnel are poor, and the surface structure settlement requirements are high. Therefore, double-layer initial support is adopted. The I-steel arch frame and the grid steel frame jointly bear the surrounding rock pressure. The two provide most of the support resistance. Therefore, it is necessary to discuss and study the influence of the grid steel frame on the equivalent elastic modulus of the double-layer initial support. This section mainly studies the steel bar diameter among the influencing factors of the grid steel frame. The relationship between the equivalent elastic modulus and the grid steel frame diameter is as follows: Figure 9 shown.

[0229] As can be seen from the figure, the initial support elastic modulus is positively correlated with the grid steel frame diameter. The larger the grid steel frame diameter, the larger the initial support elastic modulus. The relative difference δ varies from 28% to 31%, with an average error of 30.20%. The fitting formula is:

[0230] δ=-0.0034s2+0.3744;

[0231] For the refined numerical simulation method, when the grid steel frame diameter increases from 18mm to 20mm, the elastic modulus increases from 39.8GPa to 40GPa, an increase of 0.5%; when the concrete grade increases from 20mm to 22mm, the elastic modulus increases from 40GPa to 40.5GPa, an increase of 1.25%; when the concrete grade increases from 22mm to 25mm, the elastic modulus increases from 40.5GPa to 41.3GPa, an increase of 1.98%. The curves of concrete grade and elastic modulus show that the two conform to a linear function relationship:

[0232] E eq =0.2206s2+35.713;

[0233] For the conversion method, when the grid steel frame diameter increases from 18mm to 20mm, the elastic modulus increases from 52.13GPa to 52.40GPa, an increase of 0.50%; when the concrete grade increases from 20mm to 22mm, the elastic modulus increases from 52.40GPa to 52.68GPa, an increase of 0.55%; when the concrete grade increases from 22mm to 25mm, the elastic modulus increases from 52.68GPa to 53.17GPa, an increase of 0.92%. The curves of concrete grade and elastic modulus show that the two conform to a linear function relationship:

[0234] E=0.1484s2+49.443;

[0235] (6) Influence of grid steel frame spacing on initial support elastic modulus

[0236] Based on the above-mentioned influence of grid steel frame diameter on the equivalent elastic modulus of double-layer initial support, this section mainly studies the relationship between grid steel frame spacing and elastic modulus. The changing rules between the two are as follows: Figure 10 shown.

[0237] As can be seen from the figure, the initial support elastic modulus is negatively correlated with the grid steel frame spacing. The larger the grid steel frame spacing, the smaller the initial support elastic modulus. The relative difference δ varies from 14% to 22%, with an average error of 18.98%. The fitting formula is:

[0238] δ=-0.0003l2+0.3082;

[0239] For the refined numerical simulation method, when the grid steel frame spacing increases from 300mm to 400mm, the elastic modulus decreases from 44.4GPa to 44.3GPa, a decrease of 0.23%; when the grid steel frame spacing increases from 400mm to 500mm, the elastic modulus decreases from 44.3GPa to 41.5GPa, a decrease of 6.32%; when the grid steel frame increases from 500mm to 600mm, the elastic modulus decreases from 41.5GPa to 41.3GPa, a decrease of 0.48%. The curves of concrete grade and elastic modulus show that the two conform to a linear function relationship:

[0240] E eq =-0.0252l2+53.51;

[0241] For the conversion method, when the grid steel frame spacing increases from 300mm to 400mm, the elastic modulus does not change; when the grid steel frame spacing increases from 400mm to 500mm, the elastic modulus decreases from 53.17GPa to 50.19GPa, a decrease of 5.60%; when the grid steel frame increases from 500mm to 600mm, the elastic modulus decreases from 50.19GPa to 47.22GPa, a decrease of 5.93%. The curves of concrete grade and elastic modulus show that the two conform to a linear function relationship:

[0242] E=-0.0417l2+68.648;

[0243] (7) Influence of I-beam type on initial support elastic modulus

[0244] I-beam is the most commonly used steel arch in tunnel primary support. The selection of I-beam type is related to the safety of tunnel construction quality, as well as the construction cost. It also determines the size of the equivalent elastic modulus of the double-layer primary support. The relationship between the equivalent elastic modulus of the two methods and the I-beam type (i.e. the size of the cross-sectional area) is as follows: Figure 11 shown.

[0245] As can be seen from the figure, the elastic modulus of the double-layer initial support is positively correlated with the type of I-beam (cross-sectional area). The larger the cross-sectional area of ​​the I-beam, the greater the initial support elastic modulus. The relative difference δ varies from 23% to 34%, with an average error of 29.60%. The fitting formula is:

[0246] δ=-0.0043a+0.4798;

[0247] For the refined numerical simulation method, when the I-beam model changes from I18 to I20b, the elastic modulus increases from 38.4 GPa to 39.8 GPa, an increase of 3.65%; when the I-beam model changes from I20b to I22b, the elastic modulus increases from 39.8 GPa to 41.3 GPa, an increase of 3.77%; when the I-beam model changes from I22b to I25b, the elastic modulus increases from 41.3 GPa to 43.5 GPa, an increase of 5.33%. The curves of the I-beam model (cross-sectional area) and the elastic modulus show that the two conform to a linear function relationship:

[0248] E eq =0.2212a+31.325;

[0249] For the conversion method, when the I-beam model changes from I18 to I20b, the elastic modulus increases from 51.45GPa to 52.41GPa, an increase of 1.88%; when the I-beam model changes from I20b to I22b, the elastic modulus increases from 52.41GPa to 53.17GPa, an increase of 1.45%; when the I-beam model changes from I22b to I25b, the elastic modulus increases from 53.17GPa to 56.67GPa, an increase of 1.44%. The curves of the I-beam model (cross-sectional area) and the elastic modulus show that the two conform to a linear function relationship:

[0250] E = 0.1094a + 48.081;

[0251] (8) Influence of I-beam arch spacing on the initial support elastic modulus

[0252] The previous section studied the relationship between the I-steel type and the equivalent elastic modulus of the double-layer initial support. This section mainly studies the influence of the I-steel arch spacing on the elastic modulus of the initial support. The specific curve law is as follows: Figure 12 shown.

[0253] As can be seen from the figure, the initial support elastic modulus is negatively correlated with the spacing between the I-steel arches. The larger the spacing between the I-steel arches, the smaller the initial support elastic modulus. The relative difference δ ranges from 21% to 26%, with an average error of 22.93%. The fitting formula is:

[0254] δ=0.0002l3+0.1651;

[0255] For the refined numerical simulation method, when the I-beam arch spacing increases from 300mm to 400mm, the elastic modulus decreases from 43.8GPa to 43.6GPa, a decrease of 0.46%; when the I-beam arch spacing increases from 400mm to 500mm, the elastic modulus decreases from 43.6GPa to 41.2GPa, a decrease of 5.50%; when the I-beam arch spacing increases from 500mm to 600mm, the elastic modulus decreases from 41.2GPa to 41GPa, a decrease of 0.49%. The curves of the I-beam arch spacing and the elastic modulus show that the two conform to a linear function relationship:

[0256] E eq =-0.0108l3+47.26;

[0257] For the conversion method, when the I-beam arch spacing increases from 300mm to 400mm, the elastic modulus does not change; when the I-beam arch spacing increases from 400mm to 500mm, the elastic modulus decreases from 53.17GPa to 51.47GPa, a decrease of 3.19%; when the I-beam arch spacing increases from 500mm to 600mm, the elastic modulus also does not change. The curves of the I-beam arch spacing and the elastic modulus show that the two conform to a linear function relationship:

[0258] E=-0.0068l3+55.375;

[0259] Reliability verification of refined numerical simulation method and modulus correction formula

[0260] To verify the reliability of the refined numerical simulation method for double-layer support in shallow loess tunnels, research literature on the elastic modulus of reinforced concrete structures was reviewed. Existing literature 1 used experimental methods to study the elastic modulus of reinforced concrete components and specimens with different reinforcement ratios. The experimental results showed that the elastic modulus of reinforced concrete structures increased with increasing reinforcement ratio. Literature 2 used model testing to study the equivalent elastic modulus of reinforced concrete structures under different load states, boundary conditions, and reinforcement ratios. The conclusions showed a positive correlation between the equivalent elastic modulus of reinforced concrete and the reinforcement ratio. Previous analyses of reinforced concrete structures often ignored the effect of reinforcement and directly used the elastic modulus of plain concrete as the elastic modulus of the entire component. Literature 3 studied the elastic moduli of 11 groups of reinforced concrete specimens with different reinforcement ratios. The results showed that the reinforcement ratio has a significant impact on the equivalent elastic modulus of reinforced concrete structures. Ignoring the effect of reinforcement in internal force calculations can lead to significant errors in the calculated results. Reference 4 draws on the calculation method of the composite elastic modulus of composite materials, uses the numerical simulation software ANSYS combined with experimental data, obtains the elastic modulus and reinforcement ratio law of reinforced concrete structures during the construction period, and accurately calculates the elastic modulus of reinforced concrete structures.

[0261] The calculation formula for the reinforcement ratio of reinforced concrete structure is:

[0262]

[0263] The calculation formula for the double-layer initial support reinforcement ratio is:

[0264]

[0265] The calculation formula for the change rate of the double-layer support elastic modulus is:

[0266]

[0267] The specific values ​​of the equivalent elastic modulus of double-layer primary support with different reinforcement ratios under the refined numerical simulation method are shown in Table 5:

[0268] Table 5 Equivalent elastic modulus of double-layer primary support with different reinforcement ratios under refined numerical simulation method

[0269]

[0270]

[0271] The relationship between the measured elastic modulus and reinforcement ratio of reinforced concrete structures and the measured elastic modulus and reinforcement ratio of double-layer primary support structures using a refined numerical simulation method is as follows: Figure 13 shown.

[0272] Depend on Figure 13 It can be seen that the values ​​of the reinforcement ratio and the elastic modulus change rate of different numerical simulation schemes calculated by the refined numerical simulation method are within the range of model test data, and the refined numerical simulation method can be considered feasible.

[0273] In order to facilitate the design, the elastic modulus E of the double-layer initial support structure under different reinforcement ratios ρ obtained by the refined numerical simulation is eq The following empirical relationship is established between the elastic modulus E calculated using the traditional calculation method (conversion method):

[0274]

[0275] Compare the calculated value obtained by the above formula with the elastic modulus value calculated by the refined numerical simulation method, such as Figure 14 As shown in the figure, the correlation coefficient between the calculated value obtained by the empirical formula and the calculated value obtained by the refined numerical simulation reaches 90%, which can basically meet the engineering accuracy.

[0276] Therefore, the present invention adopts the above-mentioned calculation method of the equivalent modulus of the double-layer tunnel support. Through a refined numerical simulation method, it can fully consider the interaction between the steel bars and the concrete material, the interaction between the double-layer initial support structure and the stratum, and the influence of different spatial structures of the steel bar components on the elastic modulus of the double-layer initial support structure.

[0277] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit the same. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solutions of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for calculating the equivalent modulus of double-layer support in a tunnel, characterized in that: The following steps are involved: S1. Analyze the environmental conditions, geological factors, and construction conditions of the tunnel through on-site investigations to identify the factors that may affect the tunnel collapse and conduct sensitivity analysis on these factors. Influencing factors include rainfall infiltration depth, excavation cycle length and initial support strength, including: a. Analysis of the impact of rainfall infiltration depth on tunnel stability, including Establishing the stratum-tunnel model using the numerical simulation software MIDAS GTS / NX; The double-layer initial support structure and the temporary middle wall are both composite double-layer initial support structures. The elastic modulus value is determined using the conversion method. The conversion method is based on the principle that the initial support structure is equal before and after, and the strength of the steel component is converted to concrete. The calculation formula is: ； in, is the initial support elastic modulus of the conversion method; is the elastic modulus of plain concrete; is the elastic modulus of steel; is the cross-sectional area of ​​the steel mesh; is the cross-sectional area of ​​the I-beam; is the cross-sectional area of ​​the grid steel frame; is the concrete cross-sectional area; The elastic modulus of the temporary support is: ； in, is the converted elastic modulus of temporary support; b. Analysis of the impact of excavation cycle footage on tunnel stability, by simulating the ground stress state and soil deformation at different footages to understand the risk of geological disasters; c. Analysis of the influence of initial support strength on tunnel stability, and calculation of the elastic modulus of single-layer support by conversion method ; A sensitivity analysis was conducted on the factors affecting rainfall infiltration depth, excavation cycle advance, and initial support strength, including: Since the unit dimensions of the three influencing factors, namely rainfall infiltration depth, excavation cycle advance, and initial support strength, are inconsistent and cannot be directly compared, a variable sensitivity coefficient M with a dimension of 1 is defined to quantitatively describe their impact on tunnel collapse. The sensitivity coefficient formula of each influencing factor is: ； ； ； Among them, M>0 means that the tunnel collapse has a positive correlation with the influencing factors, and M<0 means that the tunnel collapse has a negative correlation with the influencing factors; is the sensitivity coefficient of rainfall infiltration depth to tunnel collapse; is the sensitivity coefficient of excavation cycle advance to tunnel collapse; is the sensitivity coefficient of initial support strength to tunnel collapse; The difference between the maximum tensile stress of the initial support arch of the tunnel excavation and the maximum tensile stress of the initial support arch of the reference; is the maximum tensile stress in the tunnel crown during the reference tunnel excavation; is the difference between the rainfall infiltration depth and the reference rainfall infiltration depth; is the reference rainfall infiltration depth; The difference between the tunnel excavation cycle footage and the reference tunnel excavation footage; For reference tunnel excavation footage; is the difference between the elastic modulus of initial support strength and the elastic modulus of reference initial support strength; The elastic modulus of the initial support strength is used as a reference. Through sensitivity analysis of influencing factors, the sensitivity index M is obtained as follows: initial support strength > rainfall infiltration > influence of excavation advance. Therefore, the value of the double-layer initial support strength should be analyzed and discussed. S2. Use refined numerical simulation methods to study the common factors affecting the elastic modulus of the double-layer initial support and calculate the equivalent elastic modulus of the double-layer support structure; the refined numerical simulation method specifically includes: S21. Determination of constitutive parameters of double-layer support structure units, including: A double-layer initial support model was established using the numerical simulation software ABAQUS. The plastic damage model of concrete in ABAQUS was used to simulate the interaction between the double-layer initial support structure and the ground, the interaction between the steel bars and concrete, and the spatial structure of the steel components. Establish a finite element model, divide the structure into concrete, steel mesh, I-beam arch frame, and steel grille, and then assemble them into initial support units in the assembly module; Obtain the elastic modulus of the double-layer initial support elastic stage obtained by the refined numerical simulation method ; The factors affecting the stiffness of the double-layer initial support structure were studied to find the variation pattern of the equivalent elastic modulus of the double-layer initial support structure with different influencing factors. As a reinforced concrete structure, the elastic modulus of the double-layer initial support structure mainly includes the following factors: concrete strength, concrete thickness, steel mesh diameter, steel mesh spacing, grid steel frame diameter, grid steel frame spacing, I-beam arch frame type, and I-beam arch frame spacing. S22. Study the equivalent elastic modulus of the double-layer support structure using the stratum-structure method; S3. Propose a refined numerical simulation method for reliability verification and a modulus correction formula.

2. The method for calculating the equivalent modulus of double-layer support for a tunnel according to claim 1, characterized in that: The influence of different factors on the elastic modulus of initial support is as follows: A. Concrete strength The influence of the initial support elastic modulus is obtained by changing the strength of the concrete material and calculating the elastic modulus by the refined numerical simulation method under the same load conditions. Calculate elastic modulus using the conversion method The change law of the double-layer initial support elastic modulus obtained by the refined numerical simulation method and the conversion method shows a positive correlation with the concrete strength, thereby obtaining the relative difference , the linear relationship between the relative difference and the concrete strength is: ; Concrete strength and The linear function relationship is: ; Concrete strength and The linear function relationship is ; B. Concrete thickness Regarding the influence of the initial support elastic modulus, the double-layer support elastic modulus is negatively correlated with the concrete thickness, and the fitting formula of the relative difference and the concrete thickness is: ; Concrete thickness and The linear function relationship is: ; Concrete thickness and The linear function relationship is: ; C. Steel mesh diameter Regarding the influence of the initial support elastic modulus, the initial support equivalent elastic modulus is positively correlated with the steel mesh diameter. The fitting formula between the relative difference and the steel mesh diameter is: ; Steel mesh diameter and The linear function relationship is: ; Steel mesh diameter and The linear function relationship is: ; D. Spacing of steel mesh The influence of the initial support elastic modulus, the double-layer initial support elastic modulus is negatively correlated with the spacing between steel meshes, and the fitting formula is: ； Steel mesh spacing and The linear function relationship is: ; Steel mesh spacing and The linear function relationship is: ; E. Diameter of grid steel frame The influence of the initial support elastic modulus is positively correlated with the grid steel frame diameter, and the fitting relationship is: ； Diameter of grid steel frame The linear function relationship is: ; Diameter of grid steel frame The linear function relationship is: ; F. Grid steel frame spacing The influence of the initial support elastic modulus is that the initial support elastic modulus is negatively correlated with the grid steel frame spacing, and the fitting relationship is: ； Grid steel frame spacing and The linear function relationship is: ; Grid steel frame spacing and The linear function relationship is: ; G. I-beam model Regarding the influence of the initial support elastic modulus, the double-layer initial support elastic modulus is positively correlated with the I-beam model, and the fitting formula is: ； I-beam model and The linear function relationship is: ; I-beam model and The linear function relationship is: ; H. The influence of the spacing between I-steel arches on the elastic modulus of the initial support. The elastic modulus of the initial support is negatively correlated with the spacing between I-steel arches. The fitting formula is: ； I-beam arch spacing and The linear function relationship is: ; I-beam arch spacing and The linear function relationship is: .

3. The method for calculating the equivalent modulus of double-layer support for a tunnel according to claim 1, characterized in that: Step S3 is specifically as follows: The calculation formula for the reinforcement ratio of reinforced concrete structure is: ; in, is the cross-sectional area of ​​the steel bars in reinforced concrete structures; The calculation formula for the double-layer initial support reinforcement ratio is: ; The calculation formula for the change rate of the double-layer support elastic modulus is: ; in, is the elastic modulus of the double-layer support structure under different reinforcement ratios; Determine the equivalent elastic modulus of double-layer primary support with different reinforcement ratios using a refined numerical simulation method; The different reinforcement ratios obtained by the refined numerical simulation method Elastic modulus of the double-layer initial support structure under Equivalent elastic modulus The following relationship is established between them: 。

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