Tunnel face anchor rod design method based on generalized logarithmic spiral failure model

Abstract

The invention belongs to the technical field of rock soil and tunnel engineering, and particularly discloses a tunnel face anchor rod design method based on a generalized logarithmic spiral failure model. According to the method, by introducing a friction weight coefficient, a generalized logarithmic helix failure model capable of continuously transitioning along with rock-soil mass cohesion and an internal friction angle is constructed; establishing a three-dimensional geometric model of a potential failure body in front of the tunnel face by combining a hemispherical collapse body, a dome rectangular bottom silo and a generalized logarithmic spiral slip body; and finally, traversing and solving the ultimate constraint force and the corresponding fracture depth of the tunnel face based on an overall moment balance equation, further providing an anchor rod reinforcement density calculation method, providing a tunnel face core soil fracture surface curve, and realizing the anchor rod length partition design with equal length of the upper half section and gradual change of the lower half section. According to the method, the defects that a traditional model is single in form, the spatial form of the loose area cannot be accurately represented, and anchor rod design depends on experience are overcome, and the theoretical property, accuracy and economical efficiency of supporting parameters are remarkably improved.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of geotechnical engineering and tunnel engineering, and more particularly to a tunnel face anchor design method based on a generalized logarithmic spiral failure model. BACKGROUND

[0002] Face instability is the most common and most harmful form of engineering disaster in soft surrounding rock tunnel construction. A large number of engineering practices show that most of the surrounding rock instability starts from the adjacent area of the face, and its stability directly restricts the overall construction safety and progress of the tunnel. In order to effectively control the deformation of the face, a geotechnical deformation control analysis method is proposed, which emphasizes that the deformation of the advanced core soil in front of the face is the root cause of the overall deformation of the tunnel surrounding rock, and the face must be actively constrained by systematic advanced support to maintain the self-stability of the core soil.

[0003] Under this background, accurately determining the limit constraint force required for face stability and scientifically formulating the anchor design parameters accordingly have become an urgent need in current soft surrounding rock tunnel engineering. At present, the stability analysis of the face mainly uses the limit equilibrium method and the limit analysis method, among which the limit equilibrium method is most widely used in engineering practice due to its relatively simple calculation. This method generally uses the classical "prism-wedge" failure model, simplifies the potential loose body into a combination of lower wedge and upper prism, simplifies the failure surface into a single logarithmic spiral or circular arc curve, and calculates the limit constraint force of the face based on the static equilibrium condition.

[0004] However, such traditional simplified models have the following significant limitations: First, the failure surface shape assumption is highly idealized, and the calculation model shape is single and has insufficient universality.

[0005] The existing stability analysis of the advanced core soil of the tunnel face usually simplifies the failure surface into a single logarithmic spiral or circular arc curve, which simplifies the calculation process but has significant physical limitations: the former is mainly suitable for sandy soil dominated by internal friction angle, and the latter is more suitable for cohesive soil dominated by cohesion. However, most rock-soil bodies exhibit complex characteristics of cohesion and internal friction angle, and their strength characteristics and failure mechanisms are jointly governed by both, making it difficult for traditional methods to accurately simulate the true failure trajectory of rock-soil bodies.

[0006] Second, the existing research still has obvious limitations in characterizing the three-dimensional shape of the loose zone.

[0007] Current theoretical analyses are mostly based on the assumption of two-dimensional plane strain, which makes it difficult to accurately characterize the three-dimensional spatial effects during the instability process at the tunnel face. Although some studies have attempted to construct three-dimensional models, their geometry is still overly idealized: the upper part of the failure body is usually simplified to a cylinder or ellipsoid, and the lower part to a wedge. This combination leads to abnormal curvature continuity at the junction of the upper and lower parts, ignoring the continuous deformation and progressive failure process of the failure body in three-dimensional space. This results in biases in the judgment of the spatial range of the loosening zone, the failure volume, and the instability mechanism, directly weakening the reliability of subsequent support parameter design.

[0008] Third, the design of anchor bolts at the tunnel face is highly dependent on engineering experience, and the theoretical support is still incomplete.

[0009] Currently, key design parameters such as anchor spacing and density are mainly determined by engineering experience, lacking systematic theoretical calculation methods. This leads to highly subjective design results and is prone to imbalances between safety and economy, such as insufficient or excessive support. Regarding length configuration, current designs generally use uniform lengths for homogeneous placement, failing to fully consider the spatial variation in fracture depth at different elevations of the tunnel face. Especially in the lower tunnel region, influenced by ground stress distribution and surrounding rock characteristics, the actual fracture depth is usually shallower; using the same anchor length as in the upper region would result in significant material redundancy.

[0010] Therefore, there is an urgent need to develop an analytical method that can be compatible with the properties of various soil and rock masses, accurately characterize the spatial morphology of loosened areas, and directly guide the quantitative design of anchor bolt parameters. Summary of the Invention

[0011] In view of the above problems, the present invention is proposed to provide a tunnel face anchor bolt design method based on a generalized logarithmic spiral failure model to overcome or at least partially solve the above problems.

[0012] To achieve the above objectives, the present invention adopts the following technical solution: In a first aspect, embodiments of the present invention provide a method for designing anchor bolts at the tunnel face based on a generalized logarithmic spiral failure model, including: S1: Construct a generalized logarithmic spiral failure model. The generalized logarithmic spiral failure model introduces a friction weight coefficient to characterize the contribution weight of the friction angle in the overall strength of the soil and rock mass, and establishes a functional relationship between the friction weight coefficient and the cohesion and friction angle in the soil and rock mass. S2: Construct a three-dimensional geometric model of the potential failure body in front of the tunnel face, optimize the upper loosening zone into a combined structure consisting of a hemispherical collapse body and a domed rectangular bottom silo, and describe the lower slip body using the generalized logarithmic spiral failure model. S3: Based on the three-dimensional geometric model and the generalized logarithmic spiral failure model, with the pole of the generalized logarithmic spiral sliding surface as the torque equilibrium point, establish the overall torque equilibrium equation of the sliding body in front of the tunnel face, and derive the functional expression of the horizontal constraint force of the tunnel face with respect to the spiral angle. S4: Iterate through the spiral angle interval to solve the horizontal constraint force function of the working face, take its maximum value as the ultimate constraint force of the working face, and simultaneously obtain the fracture depth under the corresponding spiral angle. S5: Calculate the anchor reinforcement density based on the ultimate constraint force of the tunnel face, and determine the anchor spacing in combination with the anchor arrangement; at the same time, utilize the spatial distribution characteristics of the fracture depth as a function of the tunnel face elevation to construct the fracture envelope of the core soil at the tunnel face, wherein equal-length anchors are used in the area above the intersection of the envelope, and gradually changing length anchors are used in the area below the intersection.

[0013] Preferably, in S1, the friction weighting coefficient is introduced. Based on the Mohr-Coulomb strength criterion, the cohesion c between the soil and rock mass and the internal friction angle of the soil and rock mass were established. Functional relationship:

[0014] in, The tunnel height; A generalized logarithmic spiral curve is constructed using the aforementioned friction weighting coefficient:

[0015] in, Polar angle, The initial radius of the generalized logarithmic spiral curve. The corresponding polar angle on the generalized logarithmic spiral curve The radius at that location; When the friction angle within the rock and soil When =0, =0, the generalized logarithmic spiral curve degenerates into a circular arc; When the cohesion of the soil and rock mass is c=0 =1, the generalized logarithmic spiral curve is represented by the standard logarithmic spiral curve; When 0 < When <1, the generalized logarithmic spiral curve transitions smoothly between the circular arc and the standard logarithmic spiral curve.

[0016] Preferably, in S2, the diameter of the hemispherical collapse body is the diameter of the upper bottom of the dome-shaped rectangular bottom silo, which is equal to the width of the lower bottom of the dome-shaped rectangular bottom silo; the line connecting the geometric centers of the upper and lower bottom surfaces of the dome-shaped rectangular bottom silo is perpendicular to the horizontal plane; the lower slip body is described by the generalized logarithmic spiral failure model to obtain a generalized logarithmic spiral slip body, and the height of the generalized logarithmic spiral slip body is the same as the tunnel excavation height.

[0017] Preferably, in step S3, the overall torque balance equation is: 0 in, The torque generated by the weight of the collapsing hemispherical body. The torque generated by the self-weight of the dome-shaped rectangular bottom silo The torque generated by the weight of the generalized logarithmic spiral sliding body, For the tangential frictional torque of the generalized logarithmic spiral sliding surface, For the generalized logarithmic spiral sliding body side friction torque, The moment corresponding to the support force at the tunnel face is calculated using the following formulas:

[0018] in, The radius of the hemispherical collapse body is... This represents the horizontal offset from the poles of the generalized logarithmic spiral curve to the tunnel face. The weight of the surrounding rock;

[0019] in, The width of the tunnel;

[0020] in, The corresponding polar angle on the generalized logarithmic spiral curve The radius at that point, The spiral angle of the working face;

[0021] in, The internal friction angle of the rock and soil. The initial radius of the generalized logarithmic spiral curve;

[0022] in, The initial radius of the generalized logarithmic spiral curve is the angle between the radial direction and the horizontal plane. The active earth pressure coefficient of the soil and rock mass. ; For the vertical earth pressure above, ;

[0023] in, Let A be the horizontal constraint force at the tunnel face, A be the area of ​​the tunnel face, and D be the tunnel height.

[0024] Preferably, in S3, the function expression of the horizontal constraint force at the working face with respect to the helix angle is as follows:

[0025] in, The horizontal constraint force is at the working face.

[0026] Preferably, S4 includes: Spiral angle of the working face Let be a variable, and traverse it within its interval (0, π / 2) with a preset step size: The friction weighting coefficient is calculated based on the flow law, the cohesion of the soil and rock mass, the internal friction angle of the soil and rock mass, the weight of the surrounding rock, and the tunnel height. The initial radius of the generalized logarithmic spiral curve and the angle between the radial direction and the horizontal plane are also determined. ; Solving the initial radius of the generalized logarithmic spiral curve based on the generalized logarithmic spiral relation and geometric boundary conditions. Termination radius of generalized logarithmic spiral curve ; The radius of the hemispherical collapse body is calculated using the initial radius of the generalized logarithmic spiral curve, the radial angle between the generalized logarithmic spiral curve and the horizontal plane, the initial radius of the generalized logarithmic spiral curve, and the termination radius of the generalized logarithmic spiral curve. ; Judgment 5 Is it greater than or equal to the tunnel burial depth Q? If so, then let... =0, and will Replace with corrected torque Based on Calculate the horizontal constraint force at the working face corresponding to the current helical angle; otherwise, calculate based on the functional expression of the horizontal constraint force at the working face with respect to the helical angle. After traversing, select all The maximum value in the range is used as the limit constraint force at the working face. And record the corresponding spiral angle and fracture depth.

[0027] Preferably, in step S5, calculating the anchor reinforcement density based on the ultimate constraint force at the tunnel face and determining the anchor spacing in conjunction with the anchor arrangement includes: Calculate the anchor bolt reinforcement density based on the ultimate constraint force at the tunnel face:

[0028]

[0029] in, To increase the density of anchor bolts, For limit constraint force, The diameter of the anchor bolt borehole. For the effective anchorage length, This refers to the standard value of the bond strength. The anchor bolts are arranged in a square grid, with the anchor bolt spacing as follows:

[0030] The anchor bolts are arranged in a quincunx pattern, with the following spacing:

[0031] The anchor bolts are arranged in a ring, and the spacing between them is:

[0032] in, Indicates the anchor bolt spacing.

[0033] Preferably, in S5, the core soil fracture envelope at the working face consists of two segments, including: Draw a perpendicular line from the point where the top of the generalized logarithmic spiral curve intersects with the tunnel excavation outline, and extend it downwards toward the tunnel axis. Draw a tangent line at the intersection of the bottom of the generalized logarithmic spiral curve and the working face, and this tangent line intersects the perpendicular line. The boundary formed by the perpendicular line and the tangent line is the envelope of the core soil fracture at the working face.

[0034] Preferably, the formula for calculating the anchor bolt length is:

[0035] in, The length of the anchor bolt. To determine the height of the anchor bolts, For the effective anchorage length, This refers to the planned excavation length.

[0036] As can be seen from the above technical solution, compared with the prior art, the present invention discloses a method for designing anchor bolts at the tunnel face based on a generalized logarithmic spiral failure model, which has the following advantages: (1) This invention proposes a generalized logarithmic spiral failure model, which innovatively introduces a friction weighting coefficient. This coefficient is used to characterize the relative contribution weight of the internal friction angle in the overall strength of soil and rock. Based on the Mohr-Coulomb strength criterion, it is established with respect to the relationship between cohesion c and the internal friction angle. The model dynamically adjusts the geometric morphology of the fracture surface through a functional relationship, using its value. This model overcomes the limitations of traditional methods in terms of the singular fracture surface morphology and narrow applicability, significantly improving the ability to characterize the failure mechanisms of different soil and rock masses. It unifies the mathematical description rules of the spatial morphology of fracture surfaces for various soil and rock masses, and can accurately reproduce the continuous failure process and fracture surface morphology characteristics of various soil and rock materials, from pure cohesive soil to pure sand, and those in between.

[0037] (2) This invention optimizes the traditional "prism-wedge" model by transforming the upper loosening zone of the failure body in front of the tunnel face into a combined structure of a dome-shaped rectangular bottom silo and a hemispherical collapse body, thus achieving a smooth and continuous transition in morphology between the upper loosening zone and the lower generalized logarithmic spiral sliding body. This invention effectively solves the problem of morphological discontinuity at the connection between the upper and lower parts of the original model, and can more realistically reflect the continuous deformation and progressive failure mechanism of the failure body in three-dimensional space, thereby accurately characterizing the spatial morphology of the loosening zone in front of the tunnel face and restoring the actual physical process of instability and failure.

[0038] (3) Based on the proposed generalized logarithmic spiral failure model, and combined with an optimized three-dimensional analysis model that accurately characterizes the continuous spatial morphology of the loosened zone in front of the tunnel face, this invention constructs a complete theoretical system for tunnel face stability analysis. By establishing the system's mechanical equilibrium equations through limit analysis, the precise solution to the limit constraint force of the tunnel face is achieved, significantly improving the reliability and accuracy of the calculation results. On this basis, this invention uses the obtained limit constraint force as a key design input, providing a theoretical basis for the scientific determination of anchor bolt support parameters.

[0039] (4) Based on the analytical solution of the ultimate constraint force at the tunnel face derived by the limit equilibrium method, this invention proposes theoretical calculation formulas for the density and length of anchor bolt support, providing a quantitative basis for tunnel face support design. On this basis, the concept of "tunnel face core soil fracture envelope" is innovatively introduced. By using anchor bolts with gradually varying lengths in the lower half of the cross-section, the spatial layout of the tunnel face anchor bolts is optimized, significantly improving the matching degree between the anchor bolt length and the fracture depth of the core soil at the tunnel face. This method not only improves the theoretical system of tunnel face anchor bolt design but also provides systematic theoretical support for the stability analysis of tunnel faces and the optimized arrangement of anchor bolts, effectively enhancing the scientific nature, reliability, and engineering applicability of anchor bolt reinforcement design. Attached Figure Description

[0040] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0041] Figure 1 This is a flowchart of the tunnel face anchor bolt design method based on the generalized logarithmic spiral failure model provided in this embodiment of the invention; Figure 2 This is a three-dimensional geometric model structure diagram of the potential destructive body in front of the tunnel face provided in this embodiment of the invention; Figure 3 This is a simplified diagram of the calculation of the overall moment balance equation for the sliding body in front of the tunnel face provided in this embodiment of the invention; Figure 4 This is a schematic diagram of the core soil fracture envelope provided in an embodiment of the present invention. Detailed Implementation

[0042] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0043] This invention discloses a method for designing anchor bolts at the tunnel face based on a generalized logarithmic spiral failure model, such as... Figure 1 As shown, it includes: S1: Construct a generalized logarithmic spiral failure model. The generalized logarithmic spiral failure model introduces a friction weight coefficient to characterize the contribution weight of the friction angle in the overall strength of the soil and rock mass, and establishes a functional relationship between the friction weight coefficient and the cohesion and internal friction angle of the soil and rock mass, so that the morphology of the fracture surface can smoothly transition between circular arc and standard logarithmic spiral curve according to the soil and rock type. S2: Construct a three-dimensional geometric model of the potential failure body in front of the tunnel face, optimize the upper loosening zone into a combined structure consisting of a hemispherical collapse body and a dome-shaped rectangular bottom silo, and describe the lower slip body using a generalized logarithmic spiral failure model to achieve a continuous and smooth connection between the upper and lower failure bodies in terms of geometric shape. S3: Based on the three-dimensional geometric model and the generalized logarithmic spiral failure model, with the pole of the generalized logarithmic spiral slip surface as the moment center, the overall moment balance equation of the slip body in front of the tunnel face is established, and the functional expression of the horizontal constraint force of the tunnel face with respect to the spiral angle is derived. S4: Iterate through the spiral angle interval to solve the horizontal constraint force function of the working face, take its maximum value as the ultimate constraint force of the working face, and simultaneously obtain the fracture depth under the corresponding spiral angle. S5: Calculate the anchor reinforcement density based on the ultimate constraint force and determine the anchor spacing in combination with the anchor arrangement; at the same time, utilize the spatial distribution characteristics of the fracture depth as the tunnel face elevation changes to construct the fracture envelope of the core soil at the tunnel face, in which the area above the intersection of the envelope uses anchors of equal length, and the area below uses anchors of gradually varying length.

[0044] In this embodiment, in S1, the friction weighting coefficient is introduced. Based on the Mohr-Coulomb strength criterion, the cohesion c between the soil and rock mass and the internal friction angle of the soil and rock mass were established. Functional relationship: (1) in, This is the friction weighting coefficient. For the cohesion of the soil and rock mass, The internal friction angle of the rock and soil. Where D is the weight of the surrounding rock and D is the tunnel height; Constructing a generalized logarithmic spiral curve using the friction weighting coefficient: (2) in, Polar angle, The initial radius of the generalized logarithmic spiral curve. The corresponding polar angle on the generalized logarithmic spiral curve The radius at that location; When the friction angle within the rock and soil When =0, =0, the generalized logarithmic spiral curve degenerates into a circular arc; When the cohesion of the soil and rock mass is c=0 =1, the generalized logarithmic spiral curve is represented by the standard logarithmic spiral curve; When 0 < When <1, the generalized logarithmic spiral curve transitions smoothly between the circular arc and the standard logarithmic spiral curve.

[0045] This invention proposes an analytical formula for a generalized logarithmic spiral failure model. This invention aims to address the problems of overly idealized assumptions about the fracture surface morphology in traditional methods, resulting in computational models with limited morphology and general applicability. The invention innovatively introduces a friction weighting coefficient. μ is used to characterize the internal friction angle of soil and rock. The weight of the overall strength is determined, and its relationship with cohesion c and internal friction angle is established based on the Mohr-Coulomb strength criterion. The functional expression is derived by using the friction weighting coefficient μ to correlate the fracture surface morphology with the cohesion c and the internal friction angle. When the friction angle When cohesion c = 0 (corresponding to pure cohesive soil), μ = 0, and the generalized logarithmic spiral curve degenerates into a circular arc; when cohesion c = 0 (corresponding to pure sandy soil), μ = 1, and the generalized logarithmic spiral curve exhibits a standard logarithmic spiral curve; when 0 < μ < 1, the generalized logarithmic spiral curve smoothly transitions between the two, exhibiting a transitional form between a circular arc and a standard logarithmic spiral curve. This model is based on the cohesion c and internal friction angle of the soil and rock mass. The corresponding friction weight coefficient μ is calculated to characterize the contribution weight of the internal friction angle in the strength composition of soil and rock masses, thereby adjusting the geometric morphology of the fracture surface. This generalized logarithmic spiral failure model overcomes the limitations of traditional calculation methods in terms of model morphology uniformity and applicability, and has broad engineering applicability. This model highly unifies the mathematical description methods of the spatial morphology of fracture surfaces of various soil and rock masses, and can accurately simulate the continuous failure process and fracture surface morphology characteristics of various soil and rock materials from pure cohesive soil to pure sandy soil, and in between.

[0046] In this embodiment, the present invention optimizes the traditional "prism-wedge" model, transforming the upper loosening zone of the failure body in front of the tunnel face into a combined structure of a domed rectangular bottom silo and a hemispherical collapse body. This achieves a smooth and continuous morphological transition between the upper loosening zone and the lower generalized logarithmic spiral sliding body. A simplified diagram of the specific calculation model is shown below. Figure 2 ,exist Figure 2 In the figure, (a) is an axial view, (b) is a side view, (c) is a front view, and (d) is a top view. In the figure, part I is a hemispherical collapse body, part II is a dome-shaped rectangular bottom silo, and part III is a generalized logarithmic spiral sliding body. The diameter of the hemispherical collapse body is the same as the diameter of the upper bottom of the dome-shaped rectangular bottom silo, which is equal to the width of the lower bottom of the dome-shaped rectangular bottom silo. The line connecting the geometric centers of the upper and lower bottom surfaces of the dome-shaped rectangular bottom silo is perpendicular to the horizontal plane. The height of the generalized logarithmic spiral sliding body is consistent with the tunnel excavation height.

[0047] This invention effectively solves the problem of morphological discontinuity at the junction of the upper and lower parts of the original model. By eliminating cusps and avoiding non-physical features such as geometric protrusions, it significantly improves the geometric rationality and mechanical continuity of the model. This model can more realistically reflect the continuous deformation and progressive failure mechanism of the damaged body in three-dimensional space, thereby accurately characterizing the spatial distribution characteristics of the loosened zone in front of the tunnel face and restoring the actual physical process of its instability and failure.

[0048] In this embodiment, based on the optimized three-dimensional analysis model of the loosened zone and according to the limit equilibrium theory, the overall moment equilibrium equation of the sliding body in front of the tunnel face is constructed as shown in equation (3) below. A simplified calculation diagram is shown below. Figure 3Taking the pole O of the generalized logarithmic spiral surface (generalized logarithmic spiral curve surface) as the moment equilibrium point, and based on the principle that the resultant moment of all loads on the sliding body about the pole O is zero, the calculation formula for the ultimate support force of the tunnel face is derived. The moment acting on the sliding body in front of the tunnel face includes: the moment generated by the self-weight of the hemispherical collapse body. The moment generated by the self-weight of the dome-shaped rectangular bottom silo The torque generated by the weight of a generalized logarithmic spiral sliding body Generalized logarithmic spiral tangential friction torque Generalized logarithmic spiral sliding body lateral friction torque and the torque corresponding to the support force at the working face .

[0049] 0 (3) Assume the starting point of the slip surface in front of the tunnel face is located at =0, and set the initial radius of the generalized logarithmic spiral curve. According to relevant flow rules, The angle with the horizontal ground is Based on geometric characteristics, the included angle can be derived. , radius of hemispherical collapse body Horizontal offset from the poles of the generalized logarithmic spiral curve to the tunnel face The calculation formulas are as follows (4) to (8): (4) (5) (6) (7) (8) The following details the calculation process for the moment M1 caused by the self-weight of the collapsing hemispherical object. The lever arm of the hemispherical collapse body from the pole O of the generalized logarithmic spiral sliding surface is: Therefore, the torque generated by the self-weight of the collapsing hemispherical body can be obtained. The calculation formula is as follows (9), where The weight of the surrounding rock is [not specified].

[0050] (9) The following details the calculation process for the torque M2 generated by the self-weight of the domed rectangular bottom silo. Assume the length of the bottom rectangle of the domed rectangular silo is L, and the width is... The radius of the upper circular base is The bottom silo height is , The vertical height measured from the bottom surface ( 0≤ ≤ The model has a rounded rectangle cross-section at any height z. The shape of the bottom silo cross-section is composed of half a length. Half width b( ) and corner radius r( ) By definition, these parameters all vary with height. Linear change: When =0 (bottom surface), = L / 2, b(0) = / 2，r(0)=0； = (Top surface) ， = R, b( )=R，r( )=R。 Through linear interpolation, the function expression can be obtained as follows: (10) (11) (12) The area of ​​the rounded rectangle is the total area of ​​the rectangle minus the area of ​​the squares cut off at the four corners, plus the area of ​​the newly added sectors at the four corners. Therefore, the expression for the cross-sectional area of ​​the bottom silo is... This can be derived as equation (13), with the cross section along... ( 0， By integrating, we can obtain the expression for the volume V of the dome-shaped rectangular bottom silo, as shown in equation (14).

[0051] (13) (14) By order ,but: ， 。 By further simplifying equations (13) to (14) into equations (15) to (16): (15) + + (16) By integrating each term and simplifying equation (14), the volume of the domed rectangular bottom silo can be obtained. The calculation formula is shown in equation (17) below: (17) Based on a comprehensive survey of relevant research findings both domestically and internationally, the bottom silo height was statistically determined. Depth of fracture in the core soil at the working face The numerical proportions are detailed in Table 1. A comprehensive analysis of the recommended values ​​shows that the average is 1.95, the median is 2, and the mode is 2 (occurring most frequently, three times). Given that the median and mode are consistent and highly close to the mean, based on the above statistical characteristics, this invention selects... ,in This is the height of the bottom silo. Depth of core soil fracture at the working face Table 1. Statistical Relationship between Silo Height and Core Soil Fracturing Depth at the Working Face

[0052] Length of the bottom rectangle ,width , Substituting into equation 17, the volume formula can be simplified to equation (18). Multiplying the volume by the unit weight of the bottom silo and the lever arm yields the moment of stress caused by the self-weight of the dome-shaped rectangular bottom silo. Calculation formula (19): (18) (19) When the tunnel is shallow, the height of the loosened zone above the arch in front of the tunnel face... When the depth is greater than the tunnel burial depth Q, then Will Substituting Q into equations (17) and (19), we obtain the moment of self-weight of the dome-shaped rectangular-bottomed silo at this time. As shown in equation (20): (20) The following details the calculation process for the torque M3 generated by the self-weight of the generalized logarithmic spiral sliding body. Differentiate the soil mass in the generalized logarithmic spiral slip region vertically, and calculate the self-weight of the differential volume. As shown in equation (21), where: The angle between the tangent direction of the sliding soil mass and the horizontal plane. Given the tunnel width, the unit self-weight moment of the differential soil strip is as follows (24). Integrating the differential soil strip along polar coordinates yields... Expression (25).

[0053] (twenty one) (twenty two) (twenty three) (twenty four) (25) The following details the calculation process for the tangential frictional torque M4 of the generalized logarithmic spiral sliding surface. Incremental component of tangential frictional torque along a logarithmic spiral slip surface on a unit soil strip It can be determined by the infinitesimal component of the normal force on the slip surface. With shear force differential To indicate:

[0054] According to the Mohr-Coulomb strength criterion, we have equation (27):

[0055] Will The logarithmic spiral tangential friction torque of the sliding surface can be obtained by integrating along polar coordinates. As shown in equation 28:

[0056] The following details the calculation process for the lateral frictional torque M5 of a generalized logarithmic spiral sliding body. Assuming the vertical earth pressure on the lateral vertical slip surface varies linearly with depth, the differential component of the shear force on the differential soil strip... Represented as Multiplying by the lever arm yields... As shown in expression (31), after integrating along polar coordinates, we can obtain... Expression (32).

[0057]

[0058] in:

[0059]

[0060]

[0061] in:

[0062]

[0063] The following details the calculation process for the moment M6 corresponding to the support force at the tunnel face. Based on geometric relationships, the lever arm of the horizontal support force at the working face is... + Then the torque of the support force at the working face can be obtained. As shown in equation (35), where Let A be the horizontal constraint force at the tunnel face, A be the area of ​​the tunnel face, and D be the tunnel height.

[0064]

[0065] In this embodiment, by combining equations (3) and (35), the following formula for calculating the horizontal constraint force at the working face can be obtained:

[0066] Horizontal constraint force at the working face It concerns the spiral angle of the working face. The function, when the tunnel face becomes unstable, the soil follows a generalized logarithmic spiral surface. Rigid rotation occurs only when the helix angle of the logarithmic spiral is... Only then can the sliding surface simultaneously satisfy the Mohr-Coulomb envelope and the rigid body motion criterion. Therefore, exist Iterate through the intervals. The maximum value of is the limit constraint force required to stabilize the working face.

[0067]

[0068] The specific calculation process is as follows: Spiral angle of the working face Let be a variable, and traverse it within its interval (0, π / 2) with a preset step size: The friction weighting coefficient is calculated based on the flow law, the cohesion of the soil and rock mass, the internal friction angle of the soil and rock mass, the weight of the surrounding rock, and the tunnel height. The initial radius of the generalized logarithmic spiral curve and the angle between the radial direction and the horizontal plane are also determined. ; Solving the initial radius of the generalized logarithmic spiral curve based on the generalized logarithmic spiral relation and geometric boundary conditions. Termination radius of generalized logarithmic spiral curve ; The radius of the hemispherical collapse body is calculated using the initial radius of the generalized logarithmic spiral curve, the radial angle between the generalized logarithmic spiral curve and the horizontal plane, the initial radius of the generalized logarithmic spiral curve, and the termination radius of the generalized logarithmic spiral curve. ; Determine the height of the loosened area above the arch in front of the tunnel face. 5 Is it greater than or equal to the tunnel burial depth Q? If so, then let... =0, and will Replace with corrected torque Substitution The calculation is performed in equation (38) below. The horizontal constraint force of the face corresponding to the current face helical angle is calculated based on equation (38); otherwise, the horizontal constraint force of the face is calculated based on the function expression of the face horizontal constraint force with respect to the helical angle, i.e., equation (36).

[0069] After traversing, select all The maximum value in the range is used as the limit constraint force at the working face. Record the corresponding spiral angle and the radius of the hemispherical collapse body. , The fracture depth is twice the depth of the fracture.

[0070] To verify the correctness of the formula for calculating the ultimate support force required for tunnel face stability derived in this invention, the internationally recognized theoretical models of Murayama and Anagnostou were selected for comparative analysis. The calculation parameters were set as follows: tunnel height D = 10m, tunnel face area A = 80m², tunnel width B = 7.85m, cohesion c = 0kPa, and internal friction angle... =15°~40°, soil weight γ=16kN / m³, the calculation results are detailed in Table 2. The calculation results of the derived formula of this invention are between those of the Murayaman and Anagnostou theoretical models, and show high consistency with the calculation results of the above two methods. The average deviation rate under different friction angles is only 5%, indicating that the formula has good accuracy and engineering applicability.

[0071] Table 2 Comparison of the calculation results of this invention with those of other authoritative theoretical models

[0072] This invention, based on the aforementioned generalized logarithmic spiral failure model and combined with a three-dimensional geometric model of the potential failure body in front of the tunnel face, constructs a complete theoretical system for tunnel face stability analysis. It achieves accurate solutions for the ultimate constraint force of the tunnel face, significantly improving the reliability and accuracy of the calculation results. Furthermore, this invention uses the obtained ultimate constraint force of the tunnel face as a key design input, providing a theoretical basis for the scientific formulation of anchor bolt design parameters.

[0073] In this embodiment, current research on the design parameters of anchor bolts for tunnel faces based on theoretical derivation is still insufficient. The design parameters are mainly based on the project category and lack sufficient theoretical support. This invention proposes a method for determining the anchor bolt reinforcement density and installation length based on the ultimate support force required for stabilizing the tunnel face, aiming to provide theoretical support for the design of anchor bolts for tunnel faces in China.

[0074]

[0075] The tunnel face area is The ultimate anchoring force of a single anchor bolt is The number of anchor bolts is n. The safety factor for the face anchor bolts is 1.0, as the face anchor bolt reinforcement is a temporary structure. The anchor bolt reinforcement density can then be derived. The calculation is as follows:

[0076] According to the relevant provisions of the "Technical Specification for Rock and Soil Anchors and Shotcrete Support Engineering" (GB 50086-2015), and combining the theories of elasticity and shear force transfer, under the condition of shear failure at the interface between the surrounding rock and the grouting body in the anchorage zone:

[0077]

[0078] =

[0079] in, The effective anchorage length (m); The elastic modulus of the anchor rod (kPa); The cross-sectional area of ​​the anchor rod (m²) is the anchor rod body. The perimeter of the anchor rod (m²) is the length of the anchor rod. is the elastic resistance coefficient of the surrounding rock (kN / m³). Where is the diameter of the anchor bolt borehole (m); δ is the attenuation threshold, which is usually taken as 5% for temporary projects; The value is the standard value of bond strength (kPa), which is taken with reference to the "Technical Specification for Rock and Soil Anchors and Shotcrete Support Engineering".

[0080] The anchor reinforcement density can be calculated using the above formula. Then, the spacing of the anchor bolts can be calculated.

[0081] Square arrangement spacing:

[0082] Plum blossom pattern spacing:

[0083] Annular arrangement spacing (default ring spacing is equal to the inner ring spacing):

[0084] like Figure 4 As shown, the fracture surface of the core soil at the tunnel face exhibits a generalized logarithmic spiral curve shape, with the fracture depth decreasing sequentially from the arch crown to the arch foot. Current tunnel face anchor bolt designs typically employ a constant-length scheme, resulting in redundant anchor bolt lengths in the lower half of the cross-section. This invention proposes for the first time the concept of a "fracture envelope of the core soil at the tunnel face," which, when used in conjunction with anchor bolts of gradually varying lengths in the lower half of the cross-section, improves the matching degree between anchor bolt length and fracture depth, thus perfecting the theoretical system of tunnel face anchor bolt design.

[0085] The envelope of the core soil fracture surface at the tunnel face consists of two segments, defined as follows: A perpendicular line is drawn from the intersection of the apex of the generalized logarithmic spiral curve and the tunnel excavation outline, extending downwards towards the tunnel axis. Simultaneously, a tangent line is drawn from the intersection of the apex of the generalized logarithmic spiral curve and the tunnel face; this tangent line intersects the aforementioned perpendicular line. The boundary formed by this perpendicular line and the tangent line is the envelope of the fracture surface, and the height of their intersection point is denoted as... 。

[0086] The length of the anchor bolt at the tunnel face can be calculated using the following formula:

[0087]

[0088] in: The length of the anchor bolt. Anchor bolt installation height, For the effective anchorage length, This refers to the planned excavation length.

[0089] Based on the analytical solution of the ultimate constraint force at the tunnel face derived using the aforementioned limit equilibrium method, this invention further proposes theoretical calculation formulas for the density (spacing) and length of anchor bolt support, providing a reliable quantitative basis for tunnel face support design. Specifically, by establishing the static equilibrium equation between the anchoring force of the anchor bolts at the tunnel face and the ultimate support force required to maintain stability, the reinforcement density of the anchor bolts at the tunnel face is solved. Then, combined with commonly used anchor bolt arrangement forms (such as square, quincunx, and ring), the corresponding anchor bolt spacing calculation formula is derived. Furthermore, this invention innovatively introduces the concept of the "envelope of the core soil rupture at the tunnel face." By calculating the horizontal distance from the envelope to the tunnel face and combining it with the effective anchorage section length at the tunnel face and the planned excavation advance in that section, a theoretical calculation formula for the length of the anchor bolts at the tunnel face is derived. The envelope... The area above the point corresponds to the upper half of the tunnel cross-section, where equal-length anchor bolts are used, with an envelope line. The area below the point corresponds to the lower half of the cross-section, where anchor bolts with gradually varying lengths are arranged. This method significantly optimizes the spatial layout of anchor bolts at the tunnel face, effectively improves the matching accuracy between anchor bolt length and the actual fracture depth of the core soil, further refines the theoretical system of anchor bolt design at the tunnel face, provides systematic theoretical support for stability analysis and optimized anchor bolt arrangement in tunnel engineering, and effectively enhances the scientific nature, reliability, and engineering applicability of anchor bolt design at the tunnel face.

[0090] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to the method section.

[0091] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for designing rock bolts for a tunnel face based on a generalized logarithmic spiral failure model, characterized in that, Comprise: S1: construct a generalized logarithmic spiral failure model, the generalized logarithmic spiral failure model introduces the friction weight coefficient to represent the contribution weight of the internal friction angle of rock and soil in the overall strength, and establishes the function relationship between the friction weight coefficient and the cohesion of rock and soil and the internal friction angle of rock and soil; S2: a three-dimensional geometric model of the potential failure body in front of the working face is constructed, the upper loose zone is optimized into a combined structure composed of a hemispherical collapse body and a circular top rectangular silo, and the lower sliding body is described by the generalized logarithmic spiral failure model; S3: based on the three-dimensional geometric model and the generalized logarithmic spiral failure model, the overall torque balance equation of the sliding body in front of the working face is established with the pole of the generalized logarithmic spiral sliding surface as the torque balance point, and the function expression of the horizontal constraint force of the working face with respect to the spiral opening angle is derived; S4: the horizontal constraint force function is solved in the spiral opening angle interval, and the maximum value is taken as the limit constraint force of the working face, and the corresponding rupture depth under the spiral opening angle is obtained; S5: the anchor reinforcement density is calculated based on the limit constraint force of the working face, and the anchor spacing is determined combined with the anchor arrangement form; meanwhile, the spatial distribution characteristics of the rupture depth with the elevation of the working face are utilized to construct the envelope line of the core soil rupture surface of the working face, wherein the equal-length anchor is used in the area above the intersection height of the envelope line, and the length-graduated anchor is used in the area below.

2. The method of claim 1, wherein, In S1, the friction weighting coefficient is introduced. Based on the Mohr-Coulomb strength criterion, the cohesion c between the soil and rock mass and the internal friction angle of the soil and rock mass were established. Functional relationship: wherein is the tunnel height; The generalized logarithmic spiral curve is constructed through the friction weight coefficient: wherein, is the polar angle, is the initial radius of the generalized logarithmic spiral curve, is the radius at the corresponding polar angle on the generalized logarithmic spiral curve; When the internal friction angle of the rock-soil mass = 0, = 0, the generalized logarithmic spiral curve degenerates into a circular arc. When the cohesion of the rock-soil body c = 0, = 1, the generalized logarithmic spiral curve is a standard logarithmic spiral curve. When 0 < When <1, the generalized logarithmic spiral curve transitions smoothly between the circular arc and the standard logarithmic spiral curve.

3. The method of claim 2, wherein, In the S2, the diameter of the hemispherical collapse body is equal to the upper bottom diameter of the circular top rectangular silo, which is equal to the lower bottom width of the circular top rectangular silo; the connecting line between the geometric centers of the upper and lower bottoms of the circular top rectangular silo is perpendicular to the horizontal plane; the lower sliding body is described by the generalized logarithmic spiral failure model to obtain the generalized logarithmic spiral sliding body, and the height of the generalized logarithmic spiral sliding body is the same as the tunnel excavation height.

4. The method of claim 3, wherein, In the S3, the overall torque balance equation is: 0 wherein, is the moment of the self-weight of the semi-spherical collapse body, is the moment of the self-weight of the circular dome rectangular bottom silo, is the moment of the self-weight of the generalized logarithmic spiral sliding body, is the moment of the tangential friction of the generalized logarithmic spiral sliding surface, is the moment of the side friction of the generalized logarithmic spiral sliding body, is the moment corresponding to the support force of the working face, and the calculation formula is respectively: wherein, is the radius of the semispherical collapse body, is the horizontal offset of the pole of the generalized logarithmic spiral curve from the tunnel face, is the specific weight of the surrounding rock mass; wherein is the tunnel width; wherein is the radius at the corresponding polar angle on the generalized logarithmic spiral curve, is the face spiral angle; wherein, is the internal friction angle of the rock-soil mass, is the initial radius of the generalized logarithmic spiral curve; wherein, is the initial radius of the generalized logarithmic spiral curve, is the active earth pressure coefficient of the rock-soil mass, ; is the vertical earth pressure of the overlying soil, ; wherein, is the face level restraint force, A is the tunnel face area, and D is the tunnel height.

5. The method of claim 4, wherein, In the S3, the function expression of the horizontal constraint force of the working face with respect to the spiral opening angle is as follows: wherein is the face normal horizontal constraint force.

6. The method of claim 5, wherein, The S4 comprises: with the face spiral opening angle as a variable, the interval (0, π / 2) is traversed in preset steps: According to the flow law, the cohesion of rock-soil, the internal friction angle of rock-soil, the specific weight of surrounding rock and the height of tunnel, the friction weight coefficient is calculated, and the angle between the initial radius of the generalized logarithmic spiral curve and the horizontal plane is determined ; Solving initial radius of generalized logarithmic spiral curve based on generalized logarithmic spiral relationship and geometric boundary condition and termination radius of generalized logarithmic spiral curve ; The radius of the hemispherical collapse body is calculated by the initial radius of the generalized logarithmic spiral curve, the angle between the initial radius of the generalized logarithmic spiral curve and the horizontal plane, and the terminal radius of the generalized logarithmic spiral curve ; Judgment 5 Is it greater than or equal to the tunnel burial depth Q? If so, then let... =0, and will Replace with corrected torque Based on Calculate the horizontal constraint force at the working face corresponding to the current helical angle; otherwise, calculate based on the functional expression of the horizontal constraint force at the working face with respect to the helical angle. After the traversal is completed, the maximum value in all is selected as the limit of the face constraint force , and the corresponding spiral angle and fracture depth are recorded.

7. The method of claim 6, wherein, In the S5, the anchor reinforcement density is calculated based on the limit constraint force of the working face, and the anchor spacing is determined combined with the anchor arrangement form, which comprises: The anchor reinforcement density is calculated based on the limit constraint force of the working face: wherein, is the anchoring density, is the ultimate confinement stress, is the anchoring diameter, is the effective anchoring length, is the standard value of the bond strength; The anchors are arranged in the form of a square grid, and the anchor spacing is: The anchors are arranged in the form of a quincunx, and the anchor spacing is: The anchors are arranged in the form of a ring, and the anchor spacing is: wherein denotes the anchor rod spacing.

8. The method of claim 6, wherein, In the S5, the envelope line of the core soil rupture surface of the working face is composed of two sections, which comprises: From the intersection point of the top end of the generalized logarithmic spiral curve and the tunnel excavation contour line, a vertical line is drawn towards the tunnel axis and extended downward; A tangent line is drawn from the intersection point of the bottom end of the generalized logarithmic spiral curve and the working face, and the tangent line intersects with the vertical line; The boundary formed by the vertical line and the tangent line is the envelope line of the core soil rupture surface of the working face.

9. The method of claim 8, wherein, The anchor length calculation formula is: wherein, is the length of the anchor rod, is the depth of the anchor rod, is the length of the effective anchoring section, is the planned excavation length.

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