A method for calculating the ultimate uplift bearing capacity of a hollow secondary grouting anchor rod

Abstract

This invention belongs to the technical field of ultimate bearing capacity assessment for highway slope anchorage engineering. It provides a method for calculating the ultimate pull-out bearing capacity of hollow secondary grouting anchors, including the following steps: S100, obtaining the cross-sectional area lost at the anchor's opening location and the complete cross-sectional area before the opening; defining the damage factor; S200, determining the material density of the anchor and treating it as an equivalent cylinder of uniform wall thickness; S300, establishing the actual stress expression and the total damage factor expression for the ultimate tensile bearing capacity; S400, converting the actual stress expression for the ultimate tensile bearing capacity; S500, deriving the formula for the ratio of the tensile force of the complete anchor to the tensile force of the anchor with the opening. This invention can systematically evaluate the influence of different perforation diameters on the mechanical properties of the anchor, thus facilitating the scientific determination of the anchoring force of hollow secondary grouting anchors and providing support for the formation of standardized and scientific design methods and systems.

Description

Technical Field

[0001] This invention belongs to the technical field of ultimate bearing capacity assessment for highway slope anchoring engineering, specifically relating to a method for calculating the ultimate pull-out bearing capacity of a hollow secondary grouting anchor. Background Technology

[0002] Grouting steel anchor pipe technology is mainly used for the reinforcement of highway embankments, fractured rock strata, and coal-bearing strata slopes. This technology involves creating grouting holes in seamless steel pipes to achieve grouting reinforcement, relying on interfacial shear force to provide anchoring force and stabilize the soil above the potential slip surface. However, due to the smooth surface of the seamless steel pipe, its initial load transfer capacity and interfacial shear strength are lower than those of traditional threaded steel bar anchors. Furthermore, the currently used seamless steel pipes have relatively thin walls (typically 4–6 mm), limiting the threading depth at the borehole opening, resulting in insufficient anchoring locking force, and overall pull-out resistance still has room for improvement. On the other hand, traditional anchor bolt technology is difficult to control and manage in terms of grouting quality, and existing conventional processes struggle to achieve secondary splitting grouting.

[0003] To improve the applicability and reliability of slope reinforcement systems under extreme rainfall conditions, hollow grouting anchors were optimized, and a new type of hollow grouting threaded anchor was developed, which achieves secondary grouting through external grouting perforations. Compared with ordinary hollow grouting anchors, the hollow secondary grouting anchor has circular perforations at certain intervals on the rod body to facilitate subsequent secondary grouting. However, the circular perforations cause local stress concentration in the rod body, weakening its tensile strength, resulting in a lower tensile bearing capacity of the rod body after perforation compared to hollow threaded anchors without perforations. Currently, the design of hollow secondary grouting anchors is still mainly based on engineering analogies and experience. The selection of rod body model and perforation diameter largely depends on empirical judgment, and its tensile bearing capacity is often referenced from the data of intact hollow threaded anchors, failing to fully consider the impact of perforation. Therefore, it is difficult to accurately assess the actual tensile bearing capacity of the rod body after perforation and to scientifically determine the anchoring force value, which restricts the establishment of a systematic and scientific hollow secondary grouting anchor design system. Summary of the Invention

[0004] To overcome the aforementioned shortcomings of the prior art, the purpose of this invention is to provide a method for calculating the ultimate pull-out bearing capacity of hollow secondary grouting anchors. This method can systematically evaluate the influence of different hole diameters on the mechanical properties of the anchor, thereby facilitating the scientific determination of the anchoring force of hollow secondary grouting anchors and providing support for the formation of standardized and scientific design methods and systems.

[0005] The technical solution adopted by this invention to solve its technical problem is: A method for calculating the ultimate pull-out bearing capacity of a hollow secondary grouting anchor bolt includes the following steps: S100, Obtain the cross-sectional area of ​​the anchor bolt's opening position loss. and the complete cross-sectional area of ​​the opening location before the opening. ;in accordance with and The damage factor is defined as: (1) In the formula, The material damage caused by the opening is a dimensionless parameter. The cross-sectional area lost due to the opening location is called the actual cross-sectional area; The total cross-sectional area of ​​the opening location before the opening is called the nominal cross-sectional area; S200. Determine the material density of the anchor bolt, and treat the anchor bolt as an equivalent cylinder with uniform wall thickness. By Lemaître's equivalent assumption, the effective stress after section damage satisfies the following expression: (2) In the formula, The effective stress after cross-sectional damage. This is the nominal stress after cross-sectional damage; as cross-sectional defects appear and develop, the effective stress of the cross-section will continuously increase with the development of damage; S300, assuming the actual stress at the ultimate tensile bearing capacity is... Combining equation (2), we get: (3) In the formula, This represents the total damage factor of the anchor bolt after it reaches peak stress due to deformation. For hollow, secondary grouting anchor bolts, this damage includes the total damage from the pre-loading opening and the damage caused by deformation of the anchor bolt in the strain-hardening plastic zone. Therefore, the total damage factor... Written as: (4) S400, the total damage factor at peak stress. Approximately equal to : (5) Combining equations (1), (3), and (5), we can obtain: (6) S500, the tensile force of the complete anchor bolt section can be obtained by multiplying the nominal stress by the nominal cross-sectional area, as follows: (7) Combining formulas (6) and (7), we can obtain the formula for the ratio of the tensile strength of a complete anchor bolt to the tensile strength of an open anchor bolt: (8) In the formula, For the complete anchor bolt tensile strength; This refers to the tensile strength of the perforated anchor rod.

[0006] In a preferred embodiment of the present invention, in step S500, a stress concentration factor is introduced into the ratio formula. The following formula is obtained: (9).

[0007] Preferably, the stress concentration factor Determined through experimental testing methods, including: Several rods were selected as test samples. The rods were of the same length and material. Half of the test samples were drilled with holes of a fixed density to form perforated anchor rod samples. The other half of the test samples were left unprocessed and were used as complete anchor rod samples. Tensile testing was conducted on each perforated anchor bolt sample and the intact anchor bolt sample using a tensile testing machine until failure; the ultimate tensile bearing capacity of each sample was recorded. Calculate the average ultimate tensile bearing capacity of the perforated anchor bolt samples and the average ultimate tensile bearing capacity of the intact anchor bolt samples: (10) (11) In the formula, This represents the average value of the ultimate tensile bearing capacity of the perforated anchor bolt samples; , , and This represents the tensile bearing capacity of each numbered anchor bolt sample with perforation, where n≥6; The ultimate tensile bearing capacity of a complete anchor bolt sample; , , and This represents the bearing capacity of each numbered complete anchor sample, where n≥6; The mean estimates for the ultimate tensile bearing capacity of the perforated anchor bolt sample and the mean estimate for the intact anchor bolt sample are obtained by using the t-distribution for small sample estimation. The expressions are as follows: (12) (13) In the formula, This represents the average value of the perforated anchor bolt samples; This represents the mean of the complete anchor bolt sample. Applying formulas (12) and (13) to formula (9) yields the stress concentration factor. : (14).

[0008] Preferably, the stress concentration factor Determined through mechanical interpretation, its calculation formula is derived as follows: (15) In the formula, B The strain-strengthening material constant is obtained by fitting the stress-strain curve of the nonlinear segment before the peak of the steel used. , , , , , , , , , The coefficients in the formula are all calculated according to... and The form is determined, in which and dimensionless coordinates z A finite polynomial, where the coefficients of each term are given by pre-defined rational constants.

[0009] Compared with the prior art, the beneficial effects of the present invention are: This invention uses theoretical calculations to evaluate the impact of different opening diameters on the mechanical properties of anchor bolts, quantifies the changes in tensile bearing capacity of hollow secondary grouting anchor bolts under different opening diameters, thereby helping to determine the anchoring force of hollow secondary grouting anchor bolts and providing support for the formation of standardized and scientific design methods and systems. Attached Figure Description

[0010] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the following description of the embodiments will be briefly introduced. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0011] Figure 1 This is a schematic diagram of a hollow secondary grouting anchor rod.

[0012] Figure 2 This is a schematic diagram of a hollow secondary grouting anchor rod, which is equivalent to a cylinder with equal wall thickness.

[0013] Figure 3 A schematic diagram of a strain-strengthened infinite plate with a circular hole subjected to unidirectional tension.

[0014] Figure 4 This is a schematic diagram showing the force balance conditions around the grouting hole.

[0015] in: 100 - Rod body, 200 - Opening. Detailed Implementation

[0016] To better understand the above-mentioned objectives, features, and advantages of the present invention, the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. Many specific details are set forth in the following description to provide a thorough understanding of the present invention; the described embodiments are merely some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0017] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. The terminology used herein in the description of the invention is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention.

[0018] Example 1 This embodiment discloses a method for calculating the ultimate pull-out bearing capacity of a hollow secondary grouting anchor, including the following steps: S100, Obtain the cross-sectional area of ​​the anchor bolt's opening position loss. and the complete cross-sectional area of ​​the opening location before the opening. ;in accordance with and The damage factor is defined as: (1) In the formula, The material damage caused by the opening is a dimensionless parameter. The cross-sectional area lost due to the opening location is called the actual cross-sectional area; The nominal cross-sectional area is the complete cross-sectional area of ​​the opening location before the opening.

[0019] After holes are drilled on the surface of the hollow threaded anchor rod according to the reasonable density required for the work (the hole spacing is much larger than the hole radius), the drilling will cause a loss effect on the cross section near the area, so that the cross-sectional area of ​​the anchor rod that can withstand external tensile force will be smaller than that of the normal undrilled case, resulting in the actual bearing capacity of the rod being less than that of the intact case.

[0020] According to damage mechanics theory, the degradation and decline of material and component properties is essentially a process where crack initiation and propagation reduce the material's load-bearing cross-section, until the crack fully penetrates the material, leading to complete cross-section failure and ultimately, complete material failure. (See also...) Figure 1The anchor rod 100 has a hole 200, which can be equated to the material having a natural macroscopic "crack" or "defect" in the cross section. It can be considered that the material has already suffered irreversible damage. Therefore, the damage factor can be defined as formula (1).

[0021] S200 hollow secondary grouting anchor bolts are threaded anchor bolts, which resemble corrugated pipes in their longitudinal section. Therefore, the cross-sectional dimensions (wall thickness) of the anchor bolt are not completely fixed and fluctuate within a certain range, which poses a challenge to establishing an explicit quantitative expression. To avoid the influence of the "corrugation," given a fixed anchor bolt material density, the anchor bolt is equivalent to a cylinder with a uniform wall thickness, such as... Figure 2 As shown.

[0022] That is, the material density of the anchor rod is determined, and the anchor rod is equivalent to a cylinder with a uniform wall thickness. By Lemaître's equivalent assumption, the effective stress after section damage satisfies the following expression: (2) In the formula, The effective stress after cross-sectional damage. This is the nominal stress after the section is damaged; as the section defect appears and develops, the effective stress of the section will continuously increase as the damage develops.

[0023] S300. It can be deduced from formula (2) that the area of ​​the cross section has been reduced in advance before loading, and the original cross section area is still used for measurement during loading, resulting in the actual stress level of the cross section reaching the bearing capacity limit, but the nominal stress level is lower than that of the complete specimen.

[0024] Therefore, let the actual stress at the ultimate tensile bearing capacity be... Combining equation (2), we get: (3) In the formula, This represents the total damage factor of the anchor bolt after it reaches peak stress due to deformation. For hollow, secondary grouting anchor bolts, this damage includes the total damage from the pre-loading opening and the damage caused by deformation of the anchor bolt in the strain-hardening plastic zone. Therefore, the total damage factor... It can be written as: (4) In the S400 strain hardening stage, damage development in metallic materials is relatively limited, and the stress-strain curve does not exhibit significant nonlinear softening characteristics before reaching its peak. Most of the damage generated by material deformation is concentrated in the stress softening (necking) stage. In contrast, the local damage caused by openings (macroscopic geometric discontinuities) contributes far more than the intrinsic damage generated in the deformation stage of strain hardening; therefore, the total damage factor is... Approximately taken as: (5) Combining equations (1), (3), and (5), we can obtain: (6) S500, the tensile force of the complete anchor bolt section can be obtained by multiplying the nominal stress by the nominal cross-sectional area, as follows: (7) Combining formulas (6) and (7), we can obtain the formula for the ratio of the tensile strength of a complete anchor bolt to the tensile strength of an open anchor bolt: (8) In the formula, For the complete anchor bolt tensile strength; This refers to the tensile strength of the perforated anchor rod.

[0025] However, the premise of establishing damage mechanics theory is that the cross-section has relatively uniform defects and crack initiation and development. If there is a relatively concentrated defect area, it will cause stress concentration. The calculation method derived from damage mechanics theory will have certain deviations in the calculated value under this case. In the case of hollow secondary grouting anchor rod with holes in the side wall of the rod body, there are concentrated defects and non-uniform defects in the cross-section. It is only applicable to the expression derived from traditional damage mechanics theory to calculate and evaluate the tensile bearing capacity of hollow secondary grouting anchor rod. The result will be much higher than the actual value, which is not conservative and increases the engineering risk. In order to reflect the concentrated defects, that is, the influence of single side wall openings, it is necessary to introduce a stress concentration factor to improve equation (8).

[0026] Therefore, a stress concentration factor is introduced into the ratio formula. The following formula is obtained: (9) Stress concentration factor This reflects the effect of the concentrated openings in the sidewalls on the local high stress zone, which increases the effective stress of the entire cross section, causing the actual stress of the cross section to reach the tensile limit value when the nominal stress is smaller.

[0027] In summary, the influence of different opening diameters on the mechanical properties of anchor bolts can be evaluated by formula (9), and the change in tensile bearing capacity of hollow secondary grouting anchor bolts under different opening diameters can be quantified.

[0028] Example 2 This embodiment discloses a method for calculating the stress concentration factor in Embodiment 1. The method. Specifically, for strain-hardened circular holes on the sidewalls, since the hole diameter is small (usually on the order of millimeters) and there are openings on both sides, the stress influence directly on the side holes is very small. Moreover, the opening is not on the same order of magnitude as the rod length and the area opened on the rod surface. Therefore, it can be simplified to a uniaxial tension problem of a circular hole in a strain-hardened infinite plate, such as... Figure 3As shown.

[0029] The stress concentration factor was derived. The calculation formula is as follows: (15) In the formula, B The strain-strengthening material constant is obtained by fitting the stress-strain curve of the nonlinear segment before the peak of the steel used. , , , , , , , , , The coefficients in the formula can be calculated using formula (46) below, and are all calculated according to... and The form is determined, in which and dimensionless coordinates z A finite polynomial, where the coefficients of each term are given by pre-defined rational constants.

[0030] The derivation of the above formula (15) is as follows: By perturbing the elastic mechanical solution of a circular hole in a uniaxially tensioned infinite plate, an approximate analytical solution for the stress field of the circular hole in a uniaxially tensioned infinite plate under strain hardening conditions can be obtained. The strain hardening characteristics of the material can be described using the strain hardening material constant, and the constitutive relation expression for strain-hardened metallic materials can be written as: (16) In the formula, Poisson's ratio of the material; The strain-strengthened material constant can be obtained by fitting it through a uniaxial tensile test of a hollow secondary grouting anchor. For material stress; For material strain; And it satisfies the following stress function governing equations, stress component and stress function relationship formulas, plastic strain and stress component formulas, and boundary conditions (dimensionless): (17) (18) (19) (20) In the formula, Φ The stress function; x and y Subscripts indicate the direction of rectangular coordinates; The stress component is in the x-direction; The stress component is in the y-direction; This refers to the shear stress perpendicular to the x-axis plane and along the y-axis direction; for x Plastic strain components in the direction; The plastic strain component is in the y-direction; The plastic shear strain component is perpendicular to the x-axis plane and along the y-axis direction.

[0031] Under ordinary elastic conditions without considering strain hardening, i.e., the solution of basic elasticity, is a relatively mature existing result, as shown in expressions (21) and (22). It is necessary to make perturbations based on these:

[0032] Equation (21) is a polar coordinate system The stress field under the given conditions, Equation (22) is in a rectangular coordinate system x-y The stress field in polar coordinates is shown, illustrating how the stress field is transformed into a Cartesian coordinate system. The subscript 1 indicates the parameters used in the first asymptotic step of the perturbation method. The radial stress is under ordinary elastic conditions; This refers to the tangential stress under ordinary elastic conditions. For ordinary elastic conditions x Directional stress; For ordinary elastic conditions y Directional stress; For ordinary elastic conditions, perpendicular to x On the axial plane and along y Shear stress in the axial direction.

[0033] Take far-field tensile stress p As perturbation parameters, the stress function and the perturbation expressions for each stress component are constructed, as shown in equations (23) and (24):

[0034] (twenty four) In the formula, subscript 1 represents the parameters for the first asymptotic step of the perturbation method; subscript 2 represents the parameters for the second asymptotic step of the perturbation method. For ordinary elastic conditions x Directional stress; For ordinary elastic conditions y Directional stress; For ordinary elastic conditions, perpendicular to x On the axial plane and along y Shear stress in the axial direction; For the second gradual phase x Directional stress; For the second gradual phase y Directional stress; For the second asymptotic process, perpendicular to x On the axial plane and along y Shear stress in the axial direction; To perform the first asymptotic stress function; To perform the first asymptotic stress function. It is a higher-order infinitesimal of the third order.

[0035] The stress components for the first and third asymptotic processes are shown in equation (25): (25) In the formula, subscript 1 represents the parameters for the first asymptotic step of the perturbation method; subscript 2 represents the parameters for the second asymptotic step of the perturbation method. For ordinary elastic conditions x Directional stress; For ordinary elastic conditions y Directional stress; For ordinary elastic conditions, perpendicular to x On the axial plane and along y Shear stress in the axial direction; For the second gradual phase x Directional stress; For the second gradual phase y Directional stress; For the second asymptotic process, perpendicular to x On the axial plane and along y Shear stress in the axial direction; To perform the first asymptotic stress function; To perform the first asymptotic stress function.

[0036] Equations (24) and (25) in polar coordinates can be written as: (26) In the formula, subscript 1 represents the parameters for the first asymptotic step of the perturbation method; subscript 2 represents the parameters for the second asymptotic step of the perturbation method. The radial stress is under ordinary elastic conditions; This refers to the tangential stress under ordinary elastic conditions. For ordinary elastic conditions, perpendicular to r On the axial plane and along t The shear stress in the axial direction; This refers to the radial stress during the second asymptotic phase. This represents the tangential stress during the second asymptotic phase. For the second asymptotic process, perpendicular to rOn the axial plane and along t Shear stress in the axial direction; It is a higher-order infinitesimal of the third order.

[0037] (27) In the formula, subscript 1 represents the parameters for the first asymptotic step of the perturbation method; subscript 2 represents the parameters for the second asymptotic step of the perturbation method. The radial stress is under ordinary elastic conditions; This refers to the tangential stress under ordinary elastic conditions. For ordinary elastic conditions, perpendicular to r On the axial plane and along t The shear stress in the axial direction; This refers to the radial stress during the second asymptotic phase. This represents the tangential stress during the second asymptotic phase. For the second asymptotic process, perpendicular to r On the axial plane and along t Shear stress in the axial direction; To perform the first asymptotic stress function; To perform the first asymptotic stress function.

[0038] By combining the perturbation parameters equations (24) and (18), we can obtain: (28) In the formula, B The constant for strain-strengthening materials can be obtained by fitting stress-strain curves. It is an infinitesimal of the fourth order.

[0039] Compare By taking the coefficients of the terms and ignoring infinitesimal terms, we obtain the following expression: (29) In the formula, For the second gradual phase x directional plastic strain; For the second asymptotic time y directional plastic strain; For the second asymptotic process, perpendicular to x On the axial plane and along y Axial plastic strain.

[0040] By combining the above expressions (28) and (29) with equations (19) and (20), we can compare them. p The coefficients are used to obtain the first asymptotic stress function expression and boundary condition expression as shown below: (30) (31) In the formula, the subscript r=1 represents the dimensionless case, indicating the position at the edge of the opening; the subscript r=∞ represents the position at infinity, that is, the position at a distance from the center point of the opening >> the radius, usually at least 5 times the radius or more.

[0041] Similarly, combining the above expressions (23), (24), (26), (27), (28), and (29) with expressions (19) and (20), we can compare them. By taking the coefficients, we can obtain the second asymptotic stress function and the second asymptotic boundary condition expression: (32) (33) In the formula, the subscript 2 represents the parameters for the second asymptotic step in the perturbation method. For the second gradual phase x directional plastic strain; For the second asymptotic time y directional plastic strain; For the second asymptotic process, perpendicular to x On the axial plane and along y Axial plastic strain; subscript r=1 indicates the dimensionless case, representing the position at the edge of the opening; subscript r=∞ indicates the position at infinity, that is, the position at a distance from the center point of the opening >> the radius, usually at least 5 times the radius or more.

[0042] Substituting equations (21), (22), and (29) into equation (32), we can see that equation (32) is a linear expression. Therefore, let the solution to equation (32) be: (34) In the formula, This is a particular solution of the stress function during the second asymptotic process; This is the homogeneous solution of the stress function during the second asymptotic process.

[0043] (35) In the formula, For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress-specific solution.

[0044] The particular solution of the second asymptotic stress function can be obtained analytically using complex functions. See expression (36): (36) in,

[0045] in,

[0046] In the formula, the coefficients , and This is a combination of strain components; and They are a pair of conjugate complex numbers.

[0047] Using the complex function integral of equation (36) above, the complex combination of stress components of the particular solution can be obtained. The stress components under the particular solution can be obtained from the stress function. The derivative is obtained. In polar coordinates, the complex combination of stress components is: (37) (38) In the formula, For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress-specific solution.

[0048] By combining expressions (36), (37), and (38), we can obtain the expression for the particular stress component: (39) (40) In the formula, For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress-specific solution.

[0049] Because the coefficients of equations (37) and (38) , and The second asymptotic strain is composed of various strain components. In order to obtain the expression of the particular solution of each stress component in the second asymptotic strain, it is necessary to first solve for each plastic strain component in the second asymptotic strain. , and Here, by combining expressions (21), (22), and (29), we can obtain the specific expressions for the strain components (polar coordinates): (41) in,

[0050] In the formula, B The constant for strain-strengthening materials can be obtained by fitting stress-strain curves.

[0051] Substitute expression (41) back into expression (36) , and The coefficients, when expanded, yield equations (42), (43), and (44): (42) (43) (44) In the formula, B The constant for strain-strengthening materials can be obtained by fitting stress-strain curves.

[0052] Substituting expressions (41), (42), and (43) into equations (39) and (40), we obtain a particular solution for the uniaxial tensile stress field of a circular hole in a stress-strengthened plastic infinite plate, which is applicable to the case of opening holes in hollow secondary grouting anchor rods: (45) (46) in,

[0053] In the formula, B The constant for strain-strengthening materials can be obtained by fitting stress-strain curves; where, For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress-specific solution.

[0054] For the homogeneous solution, its stress function satisfies the following condition: (47) Combining expressions (32), (33), (34), and (35), the boundary conditions for the homogeneous solution can be obtained: (48) In the formula, the subscript r=1 represents the dimensionless case, indicating the position at the edge of the opening; the subscript r=∞ represents the position at infinity, that is, the position at a distance from the center point of the opening >> the radius, usually at least 5 times the radius or more. For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress-specific solution.

[0055] As shown in equation (48), the values ​​of the boundary conditions can be obtained from the stress components at r=∞ and r=1 in the particular solution. The stress component expressions in the particular solution contain multivalued functions (such as the logarithmic term lnz), which causes the stress to be non-single-valued in the complex plane. In order to facilitate the solution of the homogeneous solution, an additional complex stress function is added to the particular solution to make its stress components cancel out the multivalued terms in the original particular solution. The flexibility of the complex stress function is used to eliminate the multivaluedness, simplifying the solution process of the subsequent homogeneous solution and ensuring the single-valuedness of the stress components. Referring to the forms of equations (45) and (46), the additional complex stress function shown in equation (49) is introduced through trial calculation: (49) In the formula, and It is a complex stress function introduced to eliminate multivaluedness.

[0056] Based on the additional complex stress function in equation (44), the analytical solution of the stress components under the additional complex stress function is calculated: (50) (51) In the formula, the subscript This represents the analytical solution of the stress components under the additional complex stress function, which will be used later to eliminate the multivalued terms in equations (45) and (46).

[0057] Adding the original particular stress component equations (45) and (46) to the additional stress component equations (50) and (51), and eliminating multi-valued terms by adjusting the coefficients, we obtain the single-valued particular stress expressions, namely equations (52) and (53): (52) (53) In the formula, Represents a single value of a particular solution; , See equation (46); where, For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress-specific solution.

[0058] Combining equations (52) and (53), the boundary conditions of equation (48) can be rewritten as:

[0059] (54)

[0060] In the formula, the subscript r=1 represents the dimensionless case, indicating the position at the edge of the opening; the subscript r=∞ represents the position at infinity, that is, the position at a distance from the center point of the opening >> the radius, usually at least 5 times the radius or more. For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress-specific solution.

[0061] The force equilibrium condition is satisfied around the drilled circular hole, and the force on the surface of the circular hole is directed along... x , y The axial direction is decomposed into and (like Figure 4 ), and maintains equilibrium with the stress around the hole. According to the equilibrium condition and equation (54), we can obtain: (55) In the formula, the subscript r=1 represents the dimensionless case, indicating the position at the edge of the opening; the subscript r=∞ represents the position at infinity, that is, the position at a distance from the center point of the opening >> the radius, which is usually at least 5 times the radius of the opening.

[0062] By integrating equation (54), the surface force complex function boundary conditions required for the homogeneous solution are obtained: (56) (57) (58) In equations (56)-(58), the subscript r=1 indicates the dimensionless case, which means the position is at the edge of the opening; the subscript r=∞ indicates the position is at infinity, which means the position is at a distance >> radius from the center point of the opening, usually at least 5 times the radius of the opening. At the edge of the hole at r=1, the surface force generated by the homogeneous solution is equal in magnitude and opposite in direction to the surface force generated by the particular solution. Treating the boundary conditions as functional equations on the unit circle, the complex stress function at the edge of the hole should satisfy: (59) In the formula, and Let be the complex stress function. According to the Kolosov–Muskhelishvili formula, in the plane elastic complex variable method, the stress field usually requires two complex stress functions to represent it. Furthermore, setting two complex stress functions is also beneficial for solving multivalued problems.

[0063] Using conjugate relations and analytic continuation, and through appropriate transformations, we can obtain expressions for two complex stress functions: (60) (61) In the planar elastic complex variable method, the stress components and the complex stress function satisfy the Kolosov–Muskhelishvili formula, which in polar coordinates is: , xy The first stress invariant and the second stress combination in both coordinate and polar coordinate systems can be expressed as: (62) (63) In the formula, For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress particular solution; For the second asymptotic time Stress-specific solution.

[0064] Combining equations (54) to (63), we can obtain the stress component expression for the homogeneous solution, and combining it with the stress component expression for the particular solution, we get: (64) (65) In the formula, This is the second progressive radial stress; This is the second asymptotic tangential stress; For the second asymptotic perpendicular to r On the axial plane and along t Shear stress in the axial direction.

[0065] By combining equations (64) and (65), we can obtain the following: The expression (66): (66) Substituting equations (66) and (21) into equation (26), considering the dimensionless boundary conditions and neglecting infinitesimal terms, we obtain the expression for the tangential stress of an infinite plate with a circular hole under uniaxial tension under strain hardening: (67) In the formula, B The constant for strain-strengthening materials can be obtained by fitting stress-strain curves; The tangential stress in a circular hole of an infinite plate under strain strengthening is the stress under unidirectional tension.

[0066] According to the definition of stress concentration factor, it is equal to the ratio of the maximum stress (i.e., shear stress) around the circular hole to the far-field tensile force. Since the solution process in this invention uses dimensionless calculation, the expression for stress concentration factor is: (68) In the formula, This represents the maximum stress around the circular hole. For far-field stress; This represents the maximum value of the tangential stress.

[0067] Based on the understanding of elastic solutions, when At this time, the tangential stress of the polar axis reaches its maximum at the hole wall. After strain-strengthening plasticity occurs, the correction term may slightly shift it, but the location of the maximum stress should still be near it. To simplify the calculation, we have Substituting into equation (67) and combining with equation (68), we can obtain equation (15), that is: (15) In the formula, B The strain-strengthening material constant can be obtained by fitting the stress-strain curve of the nonlinear segment before the peak of the steel used. This is true once the steel used for the rod is determined. B The approximate value can be determined using previous tension test data, thus allowing the determination of the corresponding coefficient. , , , , , , , , , and finally completed The calculation.

[0068] Example 3 This embodiment discloses a method for obtaining the stress concentration factor in Embodiment 1. The method described in this embodiment is mainly determined through experimental testing methods, including: Several rods were selected as test samples. The rods were of the same length and material. Half of the test samples were drilled with holes of a fixed density to form perforated anchor rod samples. The other half of the test samples were left unprocessed and were used as complete anchor rod samples. Tensile testing was conducted on each perforated anchor bolt sample and the intact anchor bolt sample using a tensile testing machine until failure; the ultimate tensile bearing capacity of each sample was recorded. Calculate the average ultimate tensile bearing capacity of the perforated anchor bolt samples and the average ultimate tensile bearing capacity of the intact anchor bolt samples: (10) (11) In the formula, This represents the average value of the ultimate tensile bearing capacity of the perforated anchor bolt samples; , , and This represents the tensile bearing capacity of each numbered anchor bolt sample with perforation, where n≥6; The ultimate tensile bearing capacity of a complete anchor bolt sample; , , and This represents the bearing capacity of each numbered complete anchor sample, where n≥6; The mean estimates for the ultimate tensile bearing capacity of the perforated anchor bolt sample and the mean estimate for the intact anchor bolt sample are obtained by using the t-distribution for small sample estimation. The expressions are as follows: (12) (13) In the formula, This represents the average value of the perforated anchor bolt samples; This represents the mean of the complete anchor bolt sample. Applying formulas (12) and (13) to formula (9) yields: (14).

[0069] Example 4 This embodiment provides an application example of the calculation method for the ultimate pull-out bearing capacity of hollow secondary grouting anchor rods based on Embodiment 1. Specifically, tensile strength data is obtained by testing anchor rods of specification SER32 / 19-280-230-W2.95 and compared with the tensile strength predicted by formula (9) in Embodiment 1.

[0070] The anchor bolt parameters for this test are as follows: (1) Material: HRB500E high-strength steel bar, with a standard value of tensile strength not less than 500MPa; (2) Outer diameter (nominal): 32mm; (3) Nominal inner diameter: 19mm; (4) Wall thickness (nominal): 2.95 mm; (5) Length specifications: The length of the control specimen without holes is 1.2m, and the length of the specimen with holes is 1.2m.

[0071] Five drilling schemes were designed: Option 1: 5 holes - φ6mm (hole spacing approximately 0.25m); Option 2: 7 holes - φ6mm (hole spacing approximately 0.17m); Option 3: 9 holes - φ6mm (hole spacing approximately 0.14m); Option 4: 7 holes - φ8mm; Option 5: 7 holes - φ10mm.

[0072] Three specimens were prepared for each scheme, one of which was a control specimen without holes (0.6m in length) and two of which were specimens with holes (1.2m in length). A total of 15 sets of tests were designed. The detailed parameters of each scheme are shown in Table 1 below.

[0073] Table 1: Detailed parameters of Schemes 1 to 5

[0074] The design ideas of the above five schemes are as follows: (1) To study the influence of the number of holes / hole spacing, Scheme 1, Scheme 2, and Scheme 3 are designed, with a fixed hole diameter of 6 mm (the commonly used hole diameter in engineering), and the number of holes are 5, 7, and 9 respectively, with corresponding hole spacings of 250 mm, 171 mm, and 133 mm. (2) To study the influence of hole diameter, Scheme 2, Scheme 4, and Scheme 5 are designed, with a fixed number of holes of 7, and hole diameters of 6 mm, 8 mm, and 10 mm respectively. (3) Each scheme includes one control specimen without holes and two specimens with holes, and the performance loss rate is quantitatively evaluated through comparative analysis. The test results are shown in Table 2 below. The curve of the complete sample without holes is calculated by fitting the calculation method of Example 3. The tensile strength is predicted by combining relevant parameters and using formula (9) in Example 1, as detailed in Table 2 below.

[0075] Table 2: Predicted Tensile Strength (Ultimate Bearing Capacity) Results

[0076] According to the data in Table 2 above, it can be found that the average tensile strength with holes (obtained through pull-out test) is very close to the tensile strength predicted by expression (9), which can be used in engineering.

[0077] To further determine the applicability of expression (9), tension tests were also conducted on anchor bolts with different specifications and sizes with and without holes, according to the same scheme described above. The diameter of the holes was φ6mm and the density was 100cm7 holes (corresponding to a hole spacing of 171mm). Tension tests were conducted on the same model of the sample with holes and the sample without holes. The sample length was the same as the previous test. See Table 3 below for details.

[0078] Table 3: Predicted Tensile Strength (Ultimate Bearing Capacity) of Anchor Bolt Openings of Different Specifications and Sizes

[0079] Similarly, the average tensile strength with holes (obtained through pull-out tests) is very close to the tensile strength predicted by expression (9). Although the prediction error varies to some extent between different batches, it can be used in engineering and has great application significance.

[0080] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Therefore, any modifications, equivalent changes, and alterations made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the scope of the present invention.

Claims

1. A method for calculating the ultimate pull-out bearing capacity of a hollow secondary grouting anchor, characterized in that, Includes the following steps: S100, Obtain the cross-sectional area of ​​the anchor bolt's opening position loss. and the complete cross-sectional area of ​​the opening location before the opening. ;in accordance with and The damage factor is defined as: （1） In the formula, The material damage caused by the opening is a dimensionless parameter. The cross-sectional area lost due to the opening location is called the actual cross-sectional area; The total cross-sectional area of ​​the opening location before the opening is called the nominal cross-sectional area; S200. Determine the material density of the anchor bolt, and treat the anchor bolt as an equivalent cylinder with uniform wall thickness. By Lemaître's equivalent assumption, the effective stress after section damage satisfies the following expression: （2） In the formula, The effective stress after cross-sectional damage. This is the nominal stress after cross-sectional damage; as cross-sectional defects appear and develop, the effective stress of the cross-section will continuously increase with the development of damage; S300, assuming the actual stress at the ultimate tensile bearing capacity is... Combining equation (2), we get: （3） In the formula, This represents the total damage factor that occurs when the anchor bolt reaches its peak stress due to deformation. For hollow secondary grouting anchors, this damage includes the total damage from the pre-loading opening and the damage caused by deformation of the anchor in the strain-strengthened plastic zone. Therefore, the total damage factor is... Written as: （4） S400, the total damage factor at peak stress. Approximately equal to : （5） Combining equations (1), (3), and (5), we can obtain: （6） S500, the tensile force of the complete anchor bolt section can be obtained by multiplying the nominal stress by the nominal cross-sectional area, as follows: （7） Combining formulas (6) and (7), we can obtain the formula for the ratio of the tensile strength of a complete anchor bolt to the tensile strength of an open anchor bolt: （8） In the formula, For the complete anchor bolt tensile strength; This refers to the tensile strength of the perforated anchor rod.

2. The method for calculating the ultimate pull-out bearing capacity of a hollow secondary grouting anchor bolt according to claim 1, characterized in that, In step S500, a stress concentration factor is introduced into the ratio formula. The following formula is obtained: （9）。 3. The method for calculating the ultimate pull-out bearing capacity of a hollow secondary grouting anchor bolt according to claim 2, characterized in that, Stress concentration factor Determined through experimental testing methods, including: Several rods were selected as test samples. The rods were of the same length and material. Half of the test samples were drilled with holes of a fixed density to form perforated anchor rod samples. The other half of the test samples were left unprocessed and were used as complete anchor rod samples. Tensile testing was conducted on each perforated anchor bolt sample and the intact anchor bolt sample using a tensile testing machine until failure; the ultimate tensile bearing capacity of each sample was recorded. Calculate the average ultimate tensile bearing capacity of the perforated anchor bolt samples and the average ultimate tensile bearing capacity of the intact anchor bolt samples: （10） （11） In the formula, This represents the average value of the ultimate tensile bearing capacity of the perforated anchor bolt samples; , , and This represents the tensile bearing capacity of each numbered anchor bolt sample with perforation, where n≥6; The ultimate tensile bearing capacity of a complete anchor bolt sample; , , and This represents the bearing capacity of each numbered complete anchor sample, where n≥6; The mean estimates for the ultimate tensile bearing capacity of the perforated anchor bolt sample and the mean estimate for the intact anchor bolt sample are obtained by using the t-distribution for small sample estimation. The expressions are as follows: （12） （13） In the formula, This represents the average value of the perforated anchor bolt samples; This represents the mean of the complete anchor bolt sample. Applying formulas (12) and (13) to formula (9) yields the stress concentration factor. : （14）。 4. The method for calculating the ultimate pull-out bearing capacity of the hollow secondary grouting anchor bolt according to claim 2, characterized in that, Stress concentration factor Determined through mechanical interpretation, its calculation formula is derived as follows: （15） In the formula, B The strain-strengthening material constant is obtained by fitting the stress-strain curve of the nonlinear segment before the peak of the steel used. , , , , , , , , , The coefficients in the formula are all calculated according to... and The form is determined, in which and dimensionless coordinates z A finite polynomial, where the coefficients of each term are given by pre-defined rational constants.

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