A method and system for calculating the thickness of a large-span tunnel anchoring bearing arch

Abstract

The application provides a giant-span tunnel anchoring bearing arch thickness analytical calculation method and system, and relates to the technical field of underground engineering rock mechanics and support design. The existing method cannot adapt to the requirements of the environment, resulting in a serious deviation of the actual calculation results from the reality. The existing method does not consider the influence of the actual environment, nor does it solve the problems caused by the difference between the results by a suitable method. In view of the above problems, the anchoring effect is coupled into the control equation in the form of continuous volume force, and the corresponding equivalent strength parameter system of surrounding rock is created. The limitation of the existing calculation method is broken through, the mechanism of the anchoring effect is explained from the mechanical essence, and the quantitative and refined support of the anchoring effect is directly supported.

Description

Technical Field

[0001] This invention relates to the field of rock mechanics and support design technology for underground engineering, and in particular to an analytical calculation method and system for the thickness of the anchorage bearing arch in a mega-span tunnel. Background Technology

[0002] As my country's infrastructure extends into the complex mountainous regions of the west, the number of deep-buried mega-span tunnel projects is increasing. Under the coupled effect of high ground stress and ultra-large spans, the redistribution of surrounding rock stress exhibits significant nonlinear characteristics, with the loosened zone around the tunnel reaching depths of over 5 meters. This renders traditional support theories based on the loose mass assumption inadequate for the needs of such projects. Therefore, existing methods for determining the thickness of the anchor bolt bearing arch all have significant limitations. The empirical analogy method, relying mainly on statistical databases of small- and medium-span projects, suffers errors of 30%–50% when extrapolated to mega-span sections, and it cannot establish a quantitative relationship between anchor bolt design parameters and the bearing arch range. While numerical simulation can consider complex geological conditions, it is constrained by high sensitivity to rock mass parameters, high computational costs, and insufficient mechanistic explanations, making it difficult to meet the needs of rapid engineering design. The classical analytical method neglects the cohesive effect of deeply buried hard rock and the volume force strengthening mechanism generated by active anchor bolt support, and lacks scale correction for the weakening of the arch effect due to increased span-to-height ratio, resulting in calculation results that deviate significantly from reality under mega-span conditions.

[0003] The fundamental flaw in the aforementioned methods for determining the load-bearing arch thickness lies in their failure to establish a coupled analytical model of the anchor prestressing field and the surrounding rock stress redistribution field based on the fundamental equations of continuum mechanics. In particular, they consistently fail to consider an explicit algorithm for determining the load-bearing arch thickness that takes into account the synergistic strengthening effect of the anchor and surrounding rock, as well as the geometric scale effect of large-span tunnels. Therefore, there is an urgent need to develop a method for determining the load-bearing arch thickness that is based on the limit equilibrium-stress redistribution coupling theory, applicable to large-span tunnels, and balances theoretical rigor with computational efficiency, to fill the gaps in existing technologies regarding theoretical completeness and methodological applicability.

[0004] Therefore, this application provides an analytical calculation method and system for the thickness of the anchorage bearing arch in mega-span tunnels. By rigorously deriving the equivalent volume force of the anchor bolts, establishing an equivalent strength parameter system for the surrounding rock, and introducing a mega-span scale correction factor, it provides stability analysis and support optimization design for specific types of underground structures. Summary of the Invention

[0005] Existing methods in current technologies are inadequate for adapting to real-world conditions, resulting in calculation results that significantly deviate from reality. These methods fail to consider the influence of various environmental factors and lack appropriate solutions to address the discrepancies. This invention first proposes an analytical calculation method for the thickness of the anchorage bearing arch in mega-span tunnels to solve these problems. The method includes the following steps: Step S1: Obtain the geometric parameters and surrounding rock mechanical parameters of the mega-span tunnel, and convert the non-circular tunnel cross section into a circular cross section with a radius of r0. Step S2: Determine the anchor bolt support parameters, and based on the anchor bolt support parameters, convert the support stress field provided by the discrete anchor bolt system into an equivalent radially decaying continuous volume force. ; Step S3: Based on the strengthening effect of the anchor bolt support parameters on the surrounding rock, establish the equivalent cohesion of the surrounding rock. With equivalent internal friction angle Thus, an equivalent strength parameter system for the surrounding rock-anchor composite is constructed; Step S4, in the equivalent strength parameter system of the surrounding rock-anchor composite, combined with continuous volume force Based on the Mohr-Coulomb strength criterion, the radial force balance equation of the bearing arch is established, and the initial bearing arch thickness h is obtained by solving based on the limit equilibrium condition at the outer boundary of the bearing arch. Step S5: Based on the span-to-depth ratio and absolute span of the mega-span tunnel, construct the scale correction factor. The initial bearing arch thickness h is corrected to obtain the corrected bearing arch thickness. .

[0006] Furthermore, in step S1, the radius r0 of the equivalent circular cross-section is calculated using the formula: ; Where B is the tunnel span and H is the tunnel height.

[0007] Furthermore, in step S2, continuous volume force Determined in the following ways: The radial support stress field of the anchor bolt Defined as a model that exhibits exponential decay from the cave wall to the depth of the surrounding rock: ; Where r is the radial coordinate in polar coordinates, representing the distance from the equivalent center of the tunnel to any point within the surrounding rock, and P0 is the equivalent radial support stress at the tunnel wall, determined by the prestressing of a single anchor bolt. The spacing between anchor bolts, a×b, is determined by the following: a is the anchor bolt spacing between rows, representing the area supported by a single anchor bolt; a is the anchor bolt spacing, the horizontal distance between two adjacent anchor bolts in the same row; b is the anchor bolt row spacing, the longitudinal distance between two adjacent rows of anchor bolts. λ is the support stress attenuation coefficient, which is related to the interaction between the anchor bolt and the rock mass. , The diameter of the anchor rod. The average bond strength of the anchorage section; then the continuous volume force Determined as radial support stress field Negative gradient: .

[0008] Furthermore, in step S3, the equivalent cohesion of the surrounding rock... With equivalent internal friction angle By introducing anchor bolt design parameters, a functional relationship is established between the anchor bolt design parameters and the original equivalent strength parameters of the surrounding rock; equivalent cohesion The established functional relationship is as follows: ; Equivalent internal friction angle The established functional relationship is as follows: ; in, , These are the equivalent cohesion and equivalent internal friction angle of the surrounding rock after anchor bolt reinforcement. , These are the original equivalent strength parameters of the surrounding rock, and L is the length of the anchor bolt. For the unit weight of the surrounding rock, , These are the strengthening coefficients for cohesion and internal friction angle, respectively.

[0009] Furthermore, step S4 also includes: S401: Consider a small element within the bearing arch and establish the radial force balance equation in polar coordinates: ; in, , These are radial stress and circumferential stress, respectively. The equivalent volume force determined in step S2; S402: Utilize any point within the bearing arch to satisfy the equivalent strength parameters obtained in step S3. , The defined Mohr-Coulomb strength criterion is used to establish circumferential stress. With radial stress Relationship: ; S403: Substituting the relationship between circumferential stress and radial stress into the radial force balance equation, we obtain the following... Solve the first-order linear differential equation and find the general solution; S404: Based on the boundary conditions of the tunnel wall and the limit equilibrium conditions at the outer boundary of the bearing arch, determine the integral constant and the outer radius r1 of the bearing arch in the general solution; S405: The initial load-bearing arch thickness h is calculated using the formula h=r1-r0.

[0010] Furthermore, in step S4, when solving the differential equation in step S403, the exponential function term e in the integral term is... λ(r r 0 ) A first-order Taylor expansion is performed at r=r0 to obtain the iterative solution scheme for the bearing arch thickness h, and the numerical solution is obtained by iterative method.

[0011] Furthermore, in step S5, the scale correction factor The construction formula is: ; Where B is the span and H is the height, B ref The reference span is 10m; δ, η, and ω are scale effect coefficients. Corrected bearing arch thickness Through formula calculate.

[0012] According to another aspect of the present invention, an analytical calculation system for the thickness of the anchorage bearing arch in a giant-span tunnel is also proposed, applied to the above-mentioned analytical calculation method for the thickness of the anchorage bearing arch in a giant-span tunnel, comprising: The data acquisition module is used to acquire the geometric parameters and surrounding rock mechanical parameters of the mega-span tunnel; The cross-section equivalence module, connected to the data acquisition module, is used to convert the non-circular tunnel cross-section into a circular cross-section with a radius of r0. The volume force determination module, connected to the cross-section equivalent module, is used to determine the equivalent continuous volume force field provided by the anchor system based on the input anchor support parameters. The composite modeling module, connected to the volume force determination module, is used to calculate the equivalent strength parameters of the surrounding rock based on the anchor bolt strengthening effect, and to construct the equivalent strength parameter system of the surrounding rock-anchor bolt composite. The initial thickness calculation module, connected to the composite modeling module, is used to solve for the initial load-bearing arch thickness h based on the radial force balance equation and the limit equilibrium condition. The scale correction module, connected to the initial thickness calculation module, is used to calculate the scale correction factor based on the scale characteristics of the mega-span tunnel. and output the corrected bearing arch thickness. .

[0013] Furthermore, the system also includes: an equation building unit, used to establish and combine the radial force balance equation of the bearing arch micro-element with the Mohr-Coulomb strength criterion relationship based on the equivalent strength parameter, to form a differential control equation for radial stress.

[0014] Furthermore, the system also includes an analytical solving unit, used to integrate the differential control equations and obtain explicit approximate solutions using Taylor expansion; The iterative calculation unit is used to numerically solve the explicit approximate solution using an iterative algorithm until the convergence condition is met, and outputs the initial bearing arch thickness h.

[0015] Compared with the prior art, the beneficial effects of the present invention are as follows: Firstly, this invention establishes an analytical determination method based on continuum mechanics and strength enhancement theory. Abandoning the traditional loose body assumption, this invention starts from the fundamental framework of solid mechanics—axisymmetric equilibrium differential equations and the Mohr-Coulomb criterion—and rigorously derives the stress field of the bearing arch, providing a scientific basis for practical engineering needs. Furthermore, it is the first to couple the anchor support effect into the governing equations as a continuous volume force, and simultaneously creates a corresponding equivalent strength parameter system for the surrounding rock, achieving a complete and self-consistent theoretical construction from basic mechanical principles to the solution of the bearing arch thickness, as well as a clear physical mechanism. Secondly, an equivalent strength parameter system for the surrounding rock-anchor composite is proposed. The equivalent strength parameters defined in this invention establish a quantitative functional relationship between anchor design parameters and the improvement of the macroscopic mechanical properties of the surrounding rock. This breaks through the limitation of analyzing support separately from the surrounding rock, explaining from a mechanical perspective the mechanism by which anchors strengthen the rock mass by providing additional confining pressure, thus providing a direct theoretical tool for the quantitative and refined design of anchor support. Third, a targeted correction model for the scale effect of mega-span tunnels was constructed. Addressing the unique engineering challenges of mega-span tunnels, such as large span-to-height ratios and weakened arch effects, this invention innovatively proposes a scale correction factor composed of the superposition of logarithmic and power function terms. This model can simultaneously reflect the flattening effect of the tunnel shape and the nonlinear scaling effect caused by the ultra-large span, effectively solving the core problem of the traditional method's poor applicability under the condition of large span due to its reliance on experience with small and medium spans, and significantly improving the engineering relevance and the accuracy of actual calculation results. Attached Figure Description

[0016] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0017] Figure 1 Flowchart of analytical calculation method for the thickness of anchorage bearing arch in mega-span tunnels; Figure 2 Flowchart for analytical calculation of the thickness of the anchorage bearing arch in a mega-span tunnel; Figure 3 Schematic diagram of the thickness of the load-bearing arch in a mega-span tunnel; Figure 4 This is a schematic diagram of the force analysis of a micro-element. Figure 5 A schematic diagram of the structure of the analytical calculation system for the thickness of the anchorage bearing arch in a mega-span tunnel. Detailed Implementation

[0018] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below. Obviously, the described embodiments are only a part of the embodiments of this invention, and not all of them. Based on the embodiments of this invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this invention.

[0019] The specific embodiments of the present invention will be described below.

[0020] Existing methods in current technologies are inadequate for environmental requirements, leading to significant deviations from reality in calculation results. These methods fail to consider the impact of the actual environment and lack appropriate solutions to address the discrepancies. This invention couples the anchor bolt support effect into the governing equations as a continuous volume force and creates a corresponding system of equivalent strength parameters for the surrounding rock. This achieves a complete and self-consistent theoretical construction from basic mechanical principles to the solution of the bearing arch thickness, with a clear physical mechanism. It overcomes the limitations of analyzing support separately from the surrounding rock, explaining from a mechanical perspective the mechanism by which anchor bolts strengthen the rock mass by providing additional confining pressure. This provides a direct theoretical tool for the quantitative and refined design of anchor bolt support.

[0021] Example 1 like Figure 1 and Figure 2 As shown, this invention proposes an analytical calculation method for the thickness of the anchorage bearing arch in mega-span tunnels. Starting from the basic equilibrium equations of continuum mechanics, the discrete anchor system is equivalent to a continuous volume force acting on the surrounding rock, constructing an equivalent strength parameter system for the surrounding rock-anchor composite. Specifically, it includes the following steps: Step S1: Obtain the geometric parameters and surrounding rock mechanical parameters of the mega-span tunnel, and equate the non-circular tunnel cross-section to a circular cross-section with radius r0. The actual tunnel cross-section is simplified by equivalence, converting the non-circular tunnel cross-section to a circular cross-section. In reality, the tunnel cross-section may be a straight-wall arch, horseshoe shape, etc. For mega-span tunnels where the crown arch is the primary load-bearing element, using an equivalent circular radius can better maintain the consistency of the area of ​​the key load-bearing arch region. Therefore, the equivalent circular cross-section radius is calculated using the formula: ; Where r0 is the radius of the equivalent circular cross-section, B is the tunnel span, and H is the tunnel height. Since the surrounding rock is a homogeneous, isotropic elastoplastic medium, it obeys the Mohr-Coulomb strength criterion, thus allowing for equivalent calculations. Furthermore, the support effect of the anchor bolts can be equivalent to a continuously distributed radial volume force within the surrounding rock. In this embodiment, the bearing arch refers to the rock mass region where, under the active support of the anchor bolts, the stress state of the surrounding rock is reshaped, the bearing capacity is enhanced, and it can work collaboratively with the support structure. Its outer boundary is the ultimate equilibrium surface where the peak strength of the surrounding rock is realized.

[0022] Step S2: Determine the anchor bolt support parameters, and based on the anchor bolt support parameters, convert the support stress field provided by the discrete anchor bolt system into an equivalent radially decaying continuous volume force. Among them, continuous volume forces The radial support stress field of the anchor bolt is determined in the following way: Defined as a model that exhibits exponential decay from the cave wall to the depth of the surrounding rock: ; Where r is the radial coordinate in polar coordinates, representing the distance from the equivalent center of the tunnel to any point in the surrounding rock; P0 is the equivalent radial support stress at the tunnel wall, determined by the prestress of a single anchor bolt and the row spacing a×b; a×b is the row spacing between anchor bolts, representing the area of ​​the support area borne by a single anchor bolt; a is the anchor bolt spacing, the horizontal distance between two adjacent anchor bolts in the same row; and b is the row spacing between anchor bolts, the longitudinal distance between two adjacent rows of anchor bolts. λ is the support stress attenuation coefficient, which is related to the interaction between the anchor bolt and the rock mass. , The diameter of the anchor rod. The average bond strength of the anchorage section; then the continuous volume force Determined as radial support stress field Negative gradient: .

[0023] Step S3: Based on the strengthening effect of the anchor bolt support parameters on the surrounding rock, establish the equivalent cohesion of the surrounding rock. With equivalent internal friction angle This leads to the construction of an equivalent strength parameter system for the surrounding rock-anchor composite. This system comprises equivalent cohesion, equivalent internal friction angle, and other elements. The equivalent strength parameters establish a quantitative functional relationship between anchor design parameters and the improvement in the macroscopic mechanical properties of the surrounding rock. The radial constraint effect of the anchor is macroscopically equivalent to increasing the confining pressure of the rock mass, thereby improving its equivalent cohesion and frictional strength, overcoming the limitations of analyzing support separately from the surrounding rock. Macroscopic mechanical properties refer to the equivalent mechanical parameters exhibited by the composite in mechanical tests or theoretical analyses after considering discrete anchors and surrounding rock as continuous homogeneous media, distinguishing them from the microscopic mechanical behavior of a single anchor or the microstructural characteristics of the surrounding rock. The prestress of a single anchor refers to the axial tensile force applied and locked by a tensioning device after installation. Anchor design parameters refer to the designable and adjustable engineering parameters of the anchor support system itself. It includes: P0 prestressing of a single anchor bolt, in kN; a. anchor bolt spacing, in meters; b. anchor bolt row spacing, in meters; L. anchor bolt length, in meters; d. b Anchor bolt diameter, in meters; τ b Average bond strength of the anchorage section, in MPa.

[0024] Within the bearing arch range, the stress state satisfies the Mohr-Coulomb strength criterion, but the macroscopic strength enhancement of the surrounding rock caused by anchor reinforcement must be considered. Therefore, the radial constraint effect of the anchor is macroscopically equivalent to increasing the confining pressure of the rock mass, thereby improving its cohesion and frictional strength. Specifically, the equivalent cohesion of the surrounding rock... With equivalent internal friction angle By introducing anchor bolt design parameters, a functional relationship is established between the anchor bolt design parameters and the original equivalent strength parameters of the surrounding rock; equivalent cohesion The established functional relationship is as follows: ; Equivalent internal friction angle The established functional relationship is as follows: ; in, , These are the equivalent cohesion and equivalent internal friction angle of the surrounding rock after anchor bolt reinforcement. , These are the original equivalent strength parameters of the surrounding rock, and L is the length of the anchor bolt. For the unit weight of the surrounding rock, , These are the strengthening coefficients for cohesion and the angle of internal friction, respectively. Their value ranges are as follows: =0.5~0.9, =0.2~0.4.

[0025] like Figure 3 and Figure 4 As shown, in step S4, in the equivalent strength parameter system of the surrounding rock-anchor composite, continuous volume force is considered. Using the Mohr-Coulomb strength criterion, the radial force balance equation of the bearing arch region is established, and the initial bearing arch thickness h is obtained by solving based on the limit equilibrium condition at the outer boundary of the bearing arch.

[0026] Specifically, it also includes: S401: Consider a small element within the bearing arch and establish the radial force balance equation in polar coordinates: ; in, , These are radial stress and circumferential stress, respectively. The equivalent volume force is determined in step S2.

[0027] S402: Utilize any point within the bearing arch to satisfy the equivalent strength parameters obtained in step S3. , The defined Mohr-Coulomb strength criterion is used to establish circumferential stress. With radial stress Relationship: .

[0028] S403: Circumferential stress With radial stress Substituting the relational expression into the radial force balance equation, we obtain the following: Solve the first-order linear differential equation and find the general solution; when solving the differential equation, consider the exponential function term e in the integral term. λ(r r 0 ) A first-order Taylor expansion is performed at r=r0 to obtain the iterative solution scheme for the thickness h of the bearing arch, and the numerical solution is obtained by iterative method; the limit equilibrium condition at the outer boundary of the bearing arch is that the surrounding rock reaches the peak strength limit equilibrium state when r=r0.

[0029] S404: Based on the boundary conditions of the tunnel wall and the limit equilibrium conditions at the outer boundary of the bearing arch, determine the integral constant and the outer radius r1 of the bearing arch in the general solution. Substituting the strength relationship into the equilibrium equation, the strength relationship is the same as the relationship between the circumferential stress and the radial stress mentioned above, yielding the relationship regarding... First-order linear differential equations: ; in, , When the boundary condition is r=r0, When p0 and r=r1 bear the outer boundary of the arch, the surrounding rock reaches the peak strength ultimate equilibrium state, then the tunnel wall conditions apply. and outer boundary conditions To obtain a practical explicit solution and ensure a safety margin, the circumferential stress at this location is defined. To reach an equivalent peak intensity Therefore, the boundary conditions can be derived: ; After solving using the integral factor method, the standard form of the equation is: ; The general solution is: ; Will , Substituting into the above equation and using the boundary conditions to determine the integration constant C, the radial stress distribution can be obtained after simplification. The complete parsed expression.

[0030] S405: Substituting the solution h=r1-r0 into the boundary condition r=r1, we obtain the determining equation for the bearing arch radius r1 or thickness h=r1-r0: .

[0031] Step S5: Based on the span-to-depth ratio and absolute span of the mega-span tunnel, construct the scale correction factor. The initial bearing arch thickness h is corrected to obtain the corrected bearing arch thickness. Typically, for mega-span tunnels with a span (B) greater than 20m, their geometric characteristics significantly influence the arching mechanism of the surrounding rock. Increasing the absolute span size and the span-to-height ratio weakens the natural arch effect and alters the stress transfer path. Therefore, a scale correction factor is constructed to mitigate these effects. The construction formula is: ; Where B is the span and H is the height, B ref The reference span is 10m; δ, η, and ω are scale effect coefficients, with values ​​ranging from δ=0.15 to 10m. 0.25, η=0.05 0.12, ω=1.2 1.5. Corrected bearing arch thickness Through formula calculate. This indicates the span-to-height ratio effect. When the value is greater than 1, the tunnel is a flat tunnel. At this time, the natural arch effect is weakened, and a thicker load-bearing arch is required. Therefore, the logarithmic form is used to reflect the attenuation of its effect. Represented as the absolute span effect, when the span far exceeds the conventional size, the failure and instability modes of the surrounding rock may undergo nonlinear and dramatic changes. Therefore, a power function is used to capture this characteristic.

[0032] To facilitate practical engineering applications, the thickness is set to determine the exponential term in the equation. Perform a first-order Taylor expansion at r=r0 Linearization in It possesses high engineering precision. In actual tunnel anchoring projects, the radial attenuation coefficient λ of the anchor bolt prestress field with increasing rock depth mainly depends on the degree of rock fragmentation and the shear stiffness of the anchoring agent, typically ranging from 0.1 to 0.4 m⁻¹. Meanwhile, the reasonable anchoring bearing arch thickness h for mega-span tunnels is usually within the range of 1.5 to 5.0 m. The product of these two factors falls between 0.15 and 2.0 under most real-world conditions, falling within the high-precision control range of λh ≤ 2, which is sufficient to meet the actual working conditions of most anchor bolts.

[0033] The simplified steps are to Substituting these into the first-order linear differential equation, we obtain particular solutions: correspond Special solution: ; correspond Special solution: ; correspond Special solution: ; The combined general solution yields: ; Apply boundary conditions to determine C and derive h, then apply the tunnel wall conditions. Substitute the values ​​to determine the integration constant C, and then apply the outer boundary conditions. Substituting the values, simplifying algebraically, combining like terms, and ignoring higher-order minor terms, we obtain the iterative solution format for h: ; Perform initialization. For the 0th approximation of the initial value during iteration , The calculated value of the bearing arch thickness in the k-th iteration is substituted into the right side of the iteration scheme as a known quantity; The calculated value of the bearing arch thickness for the (k+1)th iteration is the output result on the left side of the iteration format; The equivalent radius of influence is used for calculation. The feature length parameter is updated during iteration, let... It can fine-tune the equivalent cohesion. Next, update the thickness, first by initializing it, letting... Input all basic parameters; calculate the equivalent parameters using the initial r0. Next, substitute the solution into the iterative solution format, and... Substituting into the right side, we get Perform convergence judgment If yes, proceed to the next step; otherwise, let ,renew Return to the equivalent parameter calculation and recalculate; finally, calculate the scale correction factor. Output the final load-bearing arch thickness .

[0034] In this embodiment, the present invention establishes an analytical determination method based on continuum mechanics and strength strengthening theory. This invention abandons traditional methods such as the loose body assumption, and rigorously derives the stress field of the bearing arch from the fundamental framework of solid mechanics—axisymmetric equilibrium differential equations and the Mohr-Coulomb criterion. For the first time, the anchor support effect is coupled into the governing equations in the form of continuous volume forces, and a corresponding equivalent strength parameter system for the surrounding rock is created. This achieves a complete and self-consistent theoretical construction from basic mechanical principles to the solution of the bearing arch thickness, with a clear physical mechanism. The equivalent strength parameters defined in this invention establish a quantitative functional relationship between anchor design parameters and the improvement of the macroscopic mechanical properties of the surrounding rock. This breaks through the limitations of existing analyses that separate support from the surrounding rock, explaining from a mechanical perspective the mechanism by which anchors strengthen the rock mass by providing additional confining pressure, providing a direct theoretical tool for the quantitative and refined design of anchor support.

[0035] Example 2 like Figure 5 As shown, this invention also proposes an analytical calculation system for the thickness of the anchorage bearing arch in a giant-span tunnel, using an analytical calculation method for the thickness of the anchorage bearing arch in a giant-span tunnel as described in Example 1, including the following: The data acquisition module is used to acquire the geometric parameters and surrounding rock mechanical parameters of the mega-span tunnel; The cross-section equivalence module, connected to the data acquisition module, is used to convert the non-circular tunnel cross-section into a circular cross-section with a radius of r0. The volume force determination module, connected to the cross-section equivalent module, is used to determine the equivalent continuous volume force field provided by the anchor system based on the input anchor support parameters. The composite modeling module, connected to the volume force determination module, is used to calculate the equivalent strength parameters of the surrounding rock based on the anchor bolt strengthening effect, and to construct the equivalent strength parameter system of the surrounding rock-anchor bolt composite. The initial thickness calculation module, connected to the composite modeling module, is used to solve for the initial load-bearing arch thickness h based on the radial force balance equation and the limit equilibrium condition. The scale correction module, connected to the initial thickness calculation module, is used to calculate the scale correction factor based on the scale characteristics of the mega-span tunnel. and output the corrected bearing arch thickness. .

[0036] Specifically, this embodiment also includes: an equation construction unit, used to establish and combine the radial force balance equation of the bearing arch micro-element with the Mohr-Coulomb strength criterion relationship based on equivalent strength parameters to form a differential governing equation for radial stress; an analytical solution unit, used to integrate the differential governing equation and obtain an explicit approximate solution using Taylor expansion; and an iterative calculation unit, used to numerically solve the explicit approximate solution using an iterative algorithm until the convergence condition is met, and output the initial bearing arch thickness h.

[0037] In one specific embodiment, the data acquisition module first receives the geometric parameters and surrounding rock mechanical parameters of the mega-span tunnel input by the user. This data includes the tunnel's span and height, the original cohesion of the surrounding rock, the internal friction angle, and the unit weight. This module converts these parameters into a unified data format within the system and transmits them to the cross-section equivalence module. Upon receiving the geometric parameters, the cross-section equivalence module converts the non-circular cross-section into a circular cross-section based on the span and height values, using the principle of equal area equivalence, and calculates the equivalent radius. This equivalent radius, along with the surrounding rock mechanical parameters, is then transmitted to the volume force determination module. The volume force determination module simultaneously receives the equivalent radius from the cross-section equivalence module and the anchor bolt support parameters input by the user. This module first calculates the equivalent radial support stress at the tunnel wall based on the anchor bolt prestress and spacing, and then, combined with the anchor bolt diameter and the bond strength of the anchored section, determines the radial attenuation coefficient of the support stress. Based on this, the module constructs a radial support stress field that decreases exponentially from the tunnel wall to the depth of the surrounding rock, and further obtains an equivalent continuous volumetric force field through radial differentiation. This volumetric force field data is then transferred to the composite modeling module.

[0038] The composite modeling module receives raw surrounding rock parameters from the data acquisition module and anchor bolt parameters from the volume force determination module. Based on the anchor bolt length, surrounding rock unit weight, and a preset strengthening coefficient, this module calculates the equivalent cohesion and equivalent internal friction angle of the reinforced surrounding rock. These equivalent strength parameters, together with the aforementioned volume force field, constitute the equivalent strength parameter system of the surrounding rock-anchor bolt composite and are transmitted as input data to the initial thickness calculation module. Upon receiving the equivalent strength parameter system, the initial thickness calculation module first establishes the radial force balance equation of the micro-element in the bearing arch region using the equation construction unit. Then, combining this with the Mohr-Coulomb strength criterion based on the equivalent strength parameters, it forms a first-order linear differential equation regarding radial stress. Subsequently, the analytical solution unit integrates this differential equation and performs a first-order Taylor expansion on the exponential function terms appearing in the equation at the tunnel wall to obtain the iterative solution format for the bearing arch thickness. Since the expression still involves the implicit relationship of the bearing arch radius, the iterative calculation unit takes over the calculation process: This unit first sets the initial iterative value of the initial bearing arch thickness, substitutes it into the iterative solution format to calculate the new thickness value, and compares it with the result of the previous iteration. If the absolute value of the difference between the two results is greater than the preset tolerance, the equivalent radius is updated with the new thickness value, and the equivalent strength parameters and thickness value are recalculated until the convergence condition is met. After the iteration is completed, the initial thickness calculation module outputs the initial bearing arch thickness to the scale correction module.

[0039] The scale correction module receives the initial bearing arch thickness and tunnel span and height data from the data acquisition module. Based on the span-to-height ratio and absolute span values, this module constructs a scale correction factor. This factor reflects the weakening effect of the span-to-height ratio on the arch effect through a logarithmic term and the nonlinear scale effect caused by the ultra-large span through a power function term. The module multiplies the initial bearing arch thickness by the scale correction factor, ultimately outputting the corrected bearing arch thickness as the final result of the anchored bearing arch design. Throughout the workflow, data is transferred unidirectionally or iteratively between modules, with the calculation results of each module serving as the input for the next, ultimately forming a complete analytical calculation chain from original parameters to design output. The system can quickly and quantitatively determine the thickness of the anchored bearing arch in mega-span tunnels without relying on empirical databases or large-scale numerical simulation software, while maintaining theoretical consistency.

[0040] A targeted model for correcting the scale effect of mega-span tunnels was constructed. Addressing the unique engineering challenges of mega-span tunnels, such as large span-to-depth ratios and weakened arch effects, this invention innovatively proposes a scale correction factor composed of the superposition of logarithmic and power function terms. This model can simultaneously reflect the flattening effect of the tunnel shape and the nonlinear scaling effect caused by the ultra-large span, effectively solving the core problem of the traditional method's poor applicability under the condition of large span due to its reliance on experience with small and medium spans, and significantly improving the method's engineering relevance and computational accuracy.

[0041] Example 3 An electronic device, comprising: Processor and memory; The processor executes the steps of the analytical calculation method for the thickness of the anchorage bearing arch of a giant-span tunnel, as described in any of Embodiment 1, by calling programs or instructions stored in memory.

[0042] Example 4 A computer-readable storage medium includes computer program instructions that cause a computer to perform the steps of the analytical calculation method for the thickness of the anchorage bearing arch of a mega-span tunnel as described in any of Embodiment 1.

[0043] Computer-readable storage media may take the form of any combination of one or more readable media. A readable medium may be a readable signal medium or a readable storage medium. A readable storage medium may, for example, include, but is not limited to, electrical, magnetic, optical, electromagnetic, infrared, or semiconductor systems, apparatuses, or devices, or any combination thereof. More specific examples of readable storage media (a non-exhaustive list) include: electrical connections having one or more wires, portable disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fibers, portable compact disk read-only memory (CD-ROM), optical storage devices, magnetic storage devices, or any suitable combination thereof.

[0044] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the technical solutions of the embodiments of the present invention.

Claims

1. An analytical calculation method for the thickness of the anchorage bearing arch in a mega-span tunnel, characterized in that, Includes the following steps: Step S1: Obtain the geometric parameters and surrounding rock mechanical parameters of the mega-span tunnel, and convert the non-circular tunnel cross section into a circular cross section with a radius of r0. Step S2: Determine the anchor bolt support parameters, and based on the anchor bolt support parameters, convert the support stress field provided by the discrete anchor bolt system into an equivalent radially decaying continuous volume force. ; Step S3: Based on the strengthening effect of the anchor bolt support parameters on the surrounding rock, establish the equivalent cohesion of the surrounding rock. With equivalent internal friction angle Thus, an equivalent strength parameter system for the surrounding rock-anchor composite is constructed; Step S4, in the equivalent strength parameter system of the surrounding rock-anchor composite, combined with the continuous volume force Based on the Mohr-Coulomb strength criterion, the radial force balance equation of the bearing arch is established, and the initial bearing arch thickness h is obtained by solving based on the limit equilibrium condition at the outer boundary of the bearing arch. Step S5: Based on the span-to-depth ratio and absolute span of the mega-span tunnel, construct the scale correction factor. The initial load-bearing arch thickness h is corrected to obtain the corrected load-bearing arch thickness. .

2. The analytical calculation method for the thickness of the anchorage bearing arch in a mega-span tunnel according to claim 1, characterized in that, In step S1, the radius r0 of the equivalent circular cross-section is calculated using the formula: ； Where B is the tunnel span and H is the tunnel height.

3. The analytical calculation method for the thickness of the anchorage bearing arch in a mega-span tunnel according to claim 2, characterized in that, In step S2, the continuous volume force Determined in the following ways: The radial support stress field of the anchor bolt Defined as a model that exhibits exponential decay from the cave wall to the depth of the surrounding rock: ； Where r is the radial coordinate in polar coordinates, representing the distance from the equivalent center of the tunnel to any point within the surrounding rock, and P0 is the equivalent radial support stress at the tunnel wall, determined by the prestressing of a single anchor bolt. The spacing between anchor bolts, a×b, is determined by the following: a is the anchor bolt spacing between rows, representing the area supported by a single anchor bolt; a is the anchor bolt spacing, the horizontal distance between two adjacent anchor bolts in the same row; b is the anchor bolt row spacing, the longitudinal distance between two adjacent rows of anchor bolts. λ is the support stress attenuation coefficient, which is related to the interaction between the anchor bolt and the rock mass. , The diameter of the anchor rod. The average bond strength of the anchorage section; then the continuous volume force The radial support stress field was determined to be... Negative gradient: 。 4. The analytical calculation method for the thickness of the anchorage bearing arch in a mega-span tunnel according to claim 3, characterized in that, In step S3, the equivalent cohesion of the surrounding rock With equivalent internal friction angle A functional relationship between the anchor bolt design parameters and the original equivalent strength parameters of the surrounding rock is established; the equivalent cohesion The established functional relationship is as follows: ； Equivalent internal friction angle The established functional relationship is as follows: ； in, , These are the equivalent cohesion and equivalent internal friction angle of the surrounding rock after anchor bolt reinforcement. , These are the original equivalent strength parameters of the surrounding rock, and L is the length of the anchor bolt. For the unit weight of the surrounding rock, , These are the strengthening coefficients for cohesion and internal friction angle, respectively.

5. The analytical calculation method for the thickness of the anchorage bearing arch in a mega-span tunnel according to claim 1, characterized in that, Step S4 further includes: S401: Consider a small element within the bearing arch and establish the radial force balance equation in polar coordinates: ； in, , These are radial stress and circumferential stress, respectively. The equivalent volume force determined in step S2; S402: Utilize any point within the bearing arch to satisfy the equivalent strength parameters obtained in step S3. , The defined Mohr-Coulomb strength criterion is used to establish circumferential stress. With radial stress Relationship: ； S403: Substituting the relationship between the circumferential stress and the radial stress into the radial force balance equation, we obtain the following: The first-order linear differential equation is solved and the general solution is obtained. S404: Based on the boundary conditions of the tunnel wall and the limit equilibrium conditions at the outer boundary of the bearing arch, determine the integral constant and the outer radius r1 of the bearing arch in the general solution; S405: The initial load-bearing arch thickness h is calculated using the formula h=r1-r0.

6. The analytical calculation method for the thickness of the anchorage bearing arch in a mega-span tunnel according to claim 1, characterized in that, In step S4, when solving the differential equation in step S403, the exponential function term e in the integral term is... λ(r r 0 ) A first-order Taylor expansion is performed at r=r0 to obtain the iterative solution scheme for the bearing arch thickness h, and the numerical solution is obtained by iterative method.

7. The analytical calculation method for the thickness of the anchorage bearing arch in a mega-span tunnel according to claim 1, characterized in that, In step S5, the scale correction factor The construction formula is: ； Where B is the span and H is the height, B ref The reference span is 10m; δ, η, and ω are scale effect coefficients. The corrected load-bearing arch thickness Through formula calculate.

8. A system for analytically calculating the thickness of the anchorage bearing arch in a mega-span tunnel, applicable to the analytical calculation method for the thickness of the anchorage bearing arch in a mega-span tunnel as described in any one of claims 1 to 7, characterized in that, include: The data acquisition module is used to acquire the geometric parameters and surrounding rock mechanical parameters of the mega-span tunnel; The cross-section equivalence module, connected to the data acquisition module, is used to convert the non-circular tunnel cross-section into a circular cross-section with a radius of r0. The volume force determination module, connected to the cross-section equivalent module, is used to determine the equivalent continuous volume force field provided by the anchor system based on the input anchor support parameters. The composite modeling module, connected to the volume force determination module, is used to calculate the equivalent strength parameters of the surrounding rock based on the anchor bolt strengthening effect, and to construct the equivalent strength parameter system of the surrounding rock-anchor bolt composite. The initial thickness calculation module, connected to the composite modeling module, is used to solve for the initial bearing arch thickness h based on the radial force balance equation and the limit balance condition; The scale correction module, connected to the initial thickness calculation module, is used to calculate the scale correction factor based on the scale characteristics of the mega-span tunnel. It outputs the corrected bearing arch thickness. .

9. The analytical calculation system for the thickness of the anchorage bearing arch in a mega-span tunnel according to claim 8, characterized in that, The system also includes an equation building unit, used to establish and combine the radial force balance equation of the bearing arch micro-element with the Mohr-Coulomb strength criterion relationship based on the equivalent strength parameter, to form a differential control equation for radial stress.

10. The analytical calculation system for the thickness of the anchorage bearing arch in a mega-span tunnel according to claim 9, characterized in that, The system further includes: an analytical solution unit, used to integrate the differential control equation and obtain an explicit approximate solution using Taylor expansion; The iterative calculation unit is used to numerically solve the explicit approximate solution using an iterative algorithm until the convergence condition is met, and outputs the initial bearing arch thickness h.

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