A method for determining the elastic limit uplift resistance of a slope anchor
By constructing a calculation method for the elastic limit pull-out force of anchor bolts, the problem of the inability to accurately determine the elastic limit pull-out force of anchor bolts in existing technologies is solved. This enables the reversible design and safety assessment of anchoring systems, and improves the toughness and reversibility of slope anchoring projects.
Patent Information
- Application Number
- CN202611543779.0
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-10-08
- Publication Date
- 2026-11-06
AI Technical Summary
The lack of existing methods for determining the elastic limit pull-out force of anchor bolts leads to a lack of necessary parameter support for toughness anchoring design, making it impossible to accurately distinguish between the recoverable and unrecoverable aspects of the anchoring system, thus affecting the safety and recoverability of slope anchoring projects.
By constructing the final formula for the elastic limit pull-out force of anchor bolts, and combining information such as anchoring agent material, anchoring stratum, anchor body diameter, and anchoring section length, a calculation method for the elastic limit pull-out force of anchor bolts is established. This method includes the combination of multiple formulas and the derivation of the final formula, which is used to determine the elastic limit pull-out force of anchor bolts.
It provides an accurate criterion for the elastic limit pull-out force of anchor bolts, ensuring that anchor bolts operate within their elastic limit, improving the toughness and safety of slope anchoring projects, avoiding irreversible plastic deformation, and providing key parameters for recoverable design.
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Figure CN123049807A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of slope anchoring engineering technology, specifically relating to a method for determining the elastic limit pull-out force of slope anchors. Background Technology
[0002] In slope anchoring engineering, the ultimate pull-out force of anchor bolts is a core indicator for evaluating their anchoring performance and load-bearing safety. For a long time, domestic and international research and design have focused on the ultimate pull-out force, which is the maximum load that the anchoring system can withstand. The ultimate pull-out force corresponds to the state where the bond strength at the anchoring interface is fully mobilized and ultimately overcome, resulting in the overall pull-out failure of the anchor bolt. Its research has solved the problem of "whether the structure will experience strength failure" and is the cornerstone of traditional anchoring design.
[0003] In recent years, the concept of resilient design has been introduced into the field of geotechnical anchoring. Its core requirement is that when a structure encounters a design load, it must not only avoid collapse but also possess recoverability—that is, after the load exceeds the normal service level, it should produce limited and controllable damage, rather than irreversible plastic failure, so as to quickly restore its usability after an earthquake or disaster. This concept has put forward new requirements for anchoring systems: it is necessary to clearly distinguish between "recoverable elastic working state" and "irrecoverable damage initiation point".
[0004] Against this backdrop, the technical system that relies solely on ultimate pull-out strength as the criterion reveals its structural flaws: ultimate pull-out strength only describes the final failure limit of the anchoring system and cannot define the starting point of damage. The true boundary of recoverability is precisely provided by the elastic ultimate pull-out strength. The elastic ultimate pull-out strength is defined as the critical load at which the shear stress at the interface of the loaded end of the anchor section first reaches its peak shear strength, resulting in initial debonding damage. Before this point, the force-displacement relationship of the anchor rod is linearly elastic, deformation can be fully recovered, and the anchoring system is in an undamaged state; once this point is exceeded, irreversible plastic slip occurs at the interface, stiffness begins to degrade, the anchoring system enters the damage accumulation stage, and recoverability is lost.
[0005] Therefore, the elastic limit pull-out force is the mechanical watershed that distinguishes between "undamaged and recoverable" and "irreversible damage" in anchor toughness design, providing the only effective criterion for the following key design objectives: First, the long-term working load of the anchor must ensure that the anchoring system operates within the elastic limit; second, the energy consumption and damage control of the anchoring system must accurately know when the elastic limit is exceeded in order to limit the degree of damage.
[0006] Existing research has long focused on ultimate bearing capacity, lacking systematic attention to the determination methods and mechanical criteria for elastic ultimate pull-out force, resulting in a lack of necessary parameter support for tough anchorage design. Therefore, there is an urgent need for a technical means to accurately determine the elastic ultimate pull-out force of anchor bolts, providing key parameters and quantitative basis for recoverability-based tough anchorage design. Summary of the Invention
[0007] To address the aforementioned problems in the prior art, this invention provides a method for determining the elastic ultimate pull-out force of slope anchors. For road cut slopes in highway engineering projects where weathered cover is predominant, the method constructs the elastic ultimate pull-out force of the anchor based on fundamental design information such as the type of anchoring agent, the anchoring stratum (rock and soil), the diameter of the anchor body, and the length of the anchoring section. The final formula can be used to determine the elastic limit pull-out force of the anchor bolt, providing key parameters and quantitative basis for the design of resilient anchoring based on recoverability, and providing a reference for the prestressed compensation construction of slope anchoring projects during highway operation.
[0008] The present invention adopts the following technical solution: A method for determining the elastic ultimate pull-out force of slope anchors includes: S1. Obtain the elastic modulus of the anchoring agent material. Anchoring agent material cross-sectional area Elastic modulus of anchor bolt Cross-sectional area of anchor bolt body Anchor body diameter Comprehensive shear stiffness of anchorage interface Anchorage section length Establish the elastic limit pull-out resistance of the anchor bolt. The preliminary formula, which is defined as the first formula; The first formula is: ; in, The shear yield strength of the anchorage interface; Pi; S2. Obtain the cohesion of the soil and rock mass. internal friction angle with soil and rock mass Establish the shear yield strength of the anchorage interface. The formula is defined as the second formula; The second formula is: ; in, This refers to the normal stress at the anchorage interface; S3. Obtain the radial stiffness of the anchor body. Radial stiffness of the soil and rock surrounding the anchor hole Grouting pressure Free contraction displacement of anchor grout Establish normal stress on the anchorage interface The formula is defined as the third formula; The third formula is: ; S4. Based on the first formula, the second formula, and the third formula, establish the elastic limit pull-out force of the anchor bolt. The final formula, which is defined as the fourth formula; The fourth formula is: ; The fourth formula is used to determine the elastic limit pull-out force of the anchor bolt.
[0009] Further, step S1 includes: Based on the force balance at the anchorage interface, an equilibrium equation is established, which is defined as the fifth formula. The fifth formula is: ; in, The comprehensive elastic modulus of the anchor body; The cross-sectional area of the anchor body; This refers to the shear displacement of the anchor bolt. The coordinates are along the length of the anchor bolt, with the origin at the bottom of the anchor bolt; for about The second derivative; To ensure the overall shear stiffness of the anchoring interface; Pi; The diameter of the anchor body; Based on the boundary conditions of the anchor bolt at the bottom of the anchor hole, the sixth formula is established; The sixth formula is: ; in, The comprehensive elastic modulus of the anchor body; The cross-sectional area of the anchor body; for For about The first derivative; Based on the boundary conditions of the anchor bolt at the opening of the anchor hole, the seventh formula is established; The seventh formula is: ; in, The comprehensive elastic modulus of the anchor body; The cross-sectional area of the anchor body; for For about The first derivative; The length of the anchorage section; For tension; Based on the anchoring agent material and the anchor rod material, the comprehensive elastic modulus of the anchor body is established. The formula, which is defined as the eighth formula; The eighth formula is: ; in, The elastic modulus of the anchoring agent material; The elastic modulus of the anchor rod; The cross-sectional area of the anchoring agent material; The cross-sectional area of the anchor rod; The cross-sectional area of the anchor body; By combining the fifth, sixth, seventh, and eighth formulas, the shear displacement at different positions along the length of the anchor bolt is established. The formula, which is defined as the ninth formula; The ninth formula is: ; Based on the ninth formula, the interfacial shear stress at a certain position along the length of the anchor bolt is established. The formula, which is defined as the tenth formula; The tenth formula is: ; in, To ensure the overall shear stiffness of the anchoring interface; Based on the tenth formula, establish Shear stress at location The formula is defined as the eleventh formula; The eleventh formula is: ; when Shear stress at location Reaching the shear yield strength of the anchorage interface ,Right now At this point, the anchorage interface reaches the critical point of damage. The load corresponding to this critical point is the elastic ultimate pull-out force, which, according to the equilibrium relationship, is: Therefore, the elastic limit pull-out force of the anchor bolt is established. The preliminary formula, namely the first formula: .
[0010] Further, step S3 includes: Establishing the rebound displacement of the borehole wall in the soil and rock mass With the free shrinkage displacement of the anchor body grout The formula, which is defined as the twelfth formula; The twelfth formula is: ; in, This refers to the radial compressive displacement of the outer surface of the anchor body due to the disappearance of grouting pressure. Establishing the rebound displacement of the borehole wall in the soil and rock mass Difference from the first value The formula is defined as the thirteenth formula; The thirteenth formula is: ; Among them, the first difference Grouting pressure Normal stress at the anchorage interface The difference, i.e. This formula is defined as the fourteenth formula; The radial stiffness of the soil and rock mass surrounding the anchor hole. Its size depends on the type of anchoring strata; some industry standards and data use it as the elastic resistance coefficient or subgrade coefficient (unit: ). ), the parameter values (units) of some typical strata can be obtained by consulting relevant data. This formula is defined as the fifteenth formula; The elastic modulus of the soil and rock mass surrounding the anchor hole wall; The Poisson's ratio of the soil and rock mass surrounding the anchor hole wall; The radius of the anchor body; Establish the radial compressive displacement of the outer surface of the anchor body due to the disappearance of grouting pressure. Normal stress at the anchorage interface The formula is defined as the sixteenth formula; The sixteenth formula is: ; in, The radial stiffness of the anchor body; the anchor body grout is usually cement grout or concrete-like material, which is a standardized industrial material, and its stiffness can be found in relevant literature and industry specifications. Establish the free shrinkage displacement of the anchor body grout With anchor solid radius The formula is defined as the seventeenth formula; The seventeenth formula: ; in, This is the linear shrinkage rate during the solidification of the slurry; Combining the twelfth, thirteenth, sixteenth, and seventeenth formulas, the third formula is established, namely: .
[0011] Furthermore, step S3 also includes: Establish polar coordinates with the center of the anchor body section as the origin of the polar coordinates; Based on polar coordinates, the radial stress of the anchor body is defined. Circumferential stress ; Radial stress based on anchor body and circumferential stress Establish the eighteenth formula; The eighteenth formula is: ; in, The polar radius is in polar coordinates; The differential symbol; The general solution to the stress in the eighteenth formula is: ; in, and These are coefficients to be determined; Determine the undetermined coefficients based on the boundary conditions of the anchor body. Meanwhile, based on the radial stress on the outer surface of the anchor body... The compressive stress generated by formation contraction is continuous and equal. Based on the general solution of the stress in the eighteenth formula, the relationship is obtained as follows: This relation is defined as the nineteenth formula; Establish the physical equation for the anchor body, which is defined as the twentieth formula; The twentieth formula is: ; in, For the circumferential strain of the anchor body; The radial strain of the anchor body; The comprehensive elastic modulus of the anchor body; The Poisson's ratio of the anchor body; Establish the geometric equation of the anchor body, which is defined as the twenty-first formula; The twenty-first formula is: ; in, This refers to the shear displacement of the anchor bolt. The differential symbol; By combining the eighteenth, nineteenth, twentieth, and twenty-first formulas, the radial shear displacement field of the anchor body is established. The formula is defined as the twenty-second formula; The twenty-second formula is: ; in, The Poisson's ratio of the anchor body; The comprehensive elastic modulus of the anchor body; The radial stress is exerted on the outer surface of the anchor body. The polar radius is in polar coordinates; Will By inputting the 22nd formula and disregarding the shear displacement direction represented by the negative sign, the shear displacement at the edge of the anchor body is obtained. The relation is defined as the twenty-third formula. The twenty-third formula is: ; in, The radius of the anchor body; Establish the radial stress on the outer surface of the anchor body Shear displacement at the edge of the anchor body The formula is defined as the twenty-fourth formula; The twenty-fourth formula is: ; in, The radial stiffness of the anchor body; the anchor body grout is usually cement grout or concrete-like material, which is a standardized industrial material, and its stiffness can be found in relevant literature and industry specifications. Combining formulas 23 and 24, the radial stiffness of the anchor body is obtained. The formula is defined as the twenty-fifth formula; The twenty-fifth formula is: .
[0012] Furthermore, step S3 also includes: Establish the boundary condition expression for the shrinkage of the rock and soil mass of the anchor hole wall, which is defined as the twenty-sixth formula; ; in, The radial stress of the rock and soil mass at infinity; This refers to the radial stress of the rock and soil mass at the edge of the borehole wall. The first difference is the grouting pressure. Normal stress at the anchorage interface The difference; By combining the stress general solution of the twenty-sixth formula and the eighteenth formula, a formula for the stress field of the rock and soil mass surrounding the anchor hole wall is established, which is defined as the twenty-seventh formula. The twenty-seventh formula is: ; Establish the physical equation for the rock and soil mass surrounding the anchor hole wall, which is defined as the twenty-eighth formula. The twenty-eighth formula is: ; in, The circumferential strain of the soil and rock mass surrounding the anchor hole wall; The radial strain of the rock and soil surrounding the anchor hole wall; The elastic modulus of the soil and rock mass surrounding the anchor hole wall; The Poisson's ratio of the soil and rock mass surrounding the anchor hole wall; Establish the geometric equation of the rock and soil mass surrounding the anchor hole wall, which is defined as the twenty-ninth formula. The twenty-ninth formula is: ; in, This refers to the shear displacement of the soil and rock mass surrounding the anchor hole wall; The differential symbol; By combining formulas 27, 28, and 29, the shear displacement field of the soil and rock mass surrounding the anchor hole wall is established. The formula is defined as the thirtieth formula; The thirtieth formula is: ; Will By inputting the thirtieth formula and disregarding the shear displacement direction represented by the negative sign, the shear displacement of the soil and rock mass surrounding the anchor hole wall can be obtained. The relation is defined as the thirty-first formula. The thirty-first formula is: ; Establish the first difference Shear displacement of the surrounding soil and rock mass of the anchor hole wall The formula is defined as the thirty-second formula; The thirty-second formula is: ; in, This refers to the radial stiffness of the soil and rock mass surrounding the anchoring hole. Its magnitude depends on the type of anchoring strata. Some industry standards and data use it as the elastic resistance coefficient or subgrade coefficient (unit: ). ), the parameter values (units) of some typical strata can be obtained by consulting relevant data. ); By combining the thirty-first and thirty-second formulas, the radial stiffness of the soil and rock mass surrounding the anchoring hole is obtained. The formula, which is the fifteenth formula, is: .
[0013] Furthermore, based on the ninth formula, the slope of the straight line segment of the anchor bolt's "tension force-shear displacement" function graph is established. The formula is defined as the thirty-third formula; The thirty-third formula is: ; The optimal oblique line formed by fitting the measured coordinates of the anchor bolt's "tension force-shear displacement" points using the least squares method is then obtained. Finally, the slope of the straight line segment of the anchor bolt's "tension force-shear displacement" function graph is derived from the slope of this oblique line. ; Then, using the aforementioned formula 33, the comprehensive shear stiffness of the anchorage interface is calculated in reverse. .
[0014] In one embodiment, the slope It is 21.54 .
[0015] In one embodiment, the anchorage length 3 Comprehensive shear stiffness at the anchorage interface It is 18.49 Cohesion of soil and rock mass 30 Internal friction angle of rock and soil mass 30°; radial stiffness of the anchor body For 10000 Radial stiffness of the soil and rock mass surrounding the anchoring hole 100 The grouting pressure is 0.5. ~0.7 .
[0016] Furthermore, a method for determining the elastic ultimate pull-out force of a slope anchor bolt further includes: S5, inputting the formulas involved in steps S1 to S4 into the controller of the processing system, and calculating the elastic ultimate pull-out force of the anchor bolt through the controller.
[0017] Furthermore, in step S5, parameters affecting the calculation results of the elastic limit pull-out force of the anchor bolt are input into the controller through the control panel of the processing system.
[0018] Compared with the prior art, the beneficial effects of the present invention are as follows: This invention provides a method for determining the elastic limit pull-out force of slope anchors. Through steps S1 to S4, a final formula for the elastic limit pull-out force of the anchor is established. This formula facilitates the determination of the elastic limit pull-out force. When the tension force applied to the anchor in the field is less than the elastic limit pull-out force, the anchor is in a recoverable elastic state. Even if the anchor is deformed under tension, it can return to its original state without structural damage. However, when the tension force applied to the anchor in the field exceeds the elastic limit pull-out force, the anchor crosses the critical point of elastic limit deformation and enters the plastic deformation stage. At this point, structural damage will occur, requiring timely damage control to avoid construction risks caused by plastic failure of the anchor during anchor tensioning.
[0019] Traditional methods use "ultimate pull-out force" to determine the actual bearing capacity of anchor bolts, which cannot describe the "starting point" of anchor bolt damage and can easily lead to irreversible plastic deformation during anchor bolt tensioning. This invention uses "elastic ultimate pull-out force" as the criterion, which can more safely control whether the anchor bolt is in a state of recoverable deformation and improve the long-term service toughness of slope anchor bolts.
[0020] This invention provides a method for determining the elastic limit pull-out force of slope anchors. It establishes a formula for the elastic limit pull-out force of anchors, which facilitates the determination of the elastic limit pull-out force of anchors. This allows people to use the elastic limit pull-out force as the critical load for the elastic limit deformation of anchors, ensuring the long-term toughness of anchors.
[0021] This invention determines the elastic limit pull-out force of anchor bolts through the final formula, providing key parameters and quantitative basis for the design of resilient anchoring based on recoverability, and providing construction reference for prestress compensation in highway slope anchoring projects during the operation period. Attached Figure Description
[0022] The technology of the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments: Figure 1 It is a diagram showing the arrangement of anchor bolts, anchoring agents, and soil / rock mass. Figure 2 It is a cross-sectional view of the anchor body; Figure 3 This is a graph showing the "tension force - shear displacement" data of the anchor bolt measured on-site.
[0023] Figure label: 1-Anchor bolt body; 2-Anchoring agent; 3-Soil and rock mass; -Tension; - Anchorage section length; : Axis coordinates; Origin of coordinate system; : Axis coordinates; : Axis coordinates; -Polar radius; -Polar angle; - Anchor solid radius; -Radial stress on the outer surface of the anchor body; -Optimal diagonal line; - Critical point; -Elastic limit pull-out force; - Ultimate pull-out resistance; - Slope. Detailed Implementation
[0024] The following will provide a clear and complete description of the concept, specific structure, and technical effects of the present invention in conjunction with embodiments and accompanying drawings, so as to fully understand the purpose, solution, and effects of the present invention. It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. The same reference numerals used throughout the accompanying drawings indicate the same or similar parts.
[0025] It should be noted that, unless otherwise specified, when a feature is referred to as "fixed" or "connected" to another feature, it can be directly fixed or connected to the other feature, or indirectly fixed or connected to the other feature. Furthermore, the descriptions of "up," "down," "left," and "right" used in this invention are only relative to the relative positional relationships of the various components of the invention in the accompanying drawings.
[0026] Reference Figure 1 Understand the anchor rod body 1, anchoring agent 2, and soil and rock mass 3; A method for determining the elastic ultimate pull-out force of slope anchors includes: S1. Obtain the elastic modulus of the anchoring agent material. Anchoring agent material cross-sectional area Elastic modulus of anchor bolt Cross-sectional area of anchor bolt body Anchor body diameter Comprehensive shear stiffness of anchorage interface Anchorage section length Establish the elastic limit pull-out resistance of the anchor bolt. The preliminary formula, which is defined as the first formula; The first formula is: ; in, The shear yield strength of the anchorage interface; Pi; Preferably, the anchoring interface refers to the "anchoring agent-soil" interface; in one embodiment, the anchoring agent is a grout formed after the grout solidifies, specifically, if grout is injected into the anchoring hole, the grout will eventually solidify to form a grout; the anchoring body includes the anchoring section of the anchor rod and the grout. S2. Under normal pressure grouting conditions in the anchorage hole, the shear yield strength of the anchorage interface. It is usually equal to the shear yield strength of the surrounding soil and rock mass; Obtaining the cohesion of soil and rock mass internal friction angle with soil and rock mass Establish the shear yield strength of the anchorage interface. The formula is defined as the second formula; The second formula is: ; in, This refers to the normal stress at the anchorage interface; S3. Obtain the radial stiffness of the anchor body. Radial stiffness of the soil and rock surrounding the anchor hole Grouting pressure (unit: ), free shrinkage displacement of anchor grout Establish normal stress on the anchorage interface The formula is defined as the third formula; The third formula is: ; S4. Based on the first formula, the second formula, and the third formula, establish the elastic limit pull-out force of the anchor bolt. The final formula, which is defined as the fourth formula; The fourth formula is: ; The fourth formula is used to determine the elastic limit pull-out force of the anchor bolt.
[0027] In one embodiment, when the anchor is under tension, the "anchoring agent-soil" interface (i.e., the anchoring interface) will have a uniform shear stress distributed on its annular interface when subjected to tensile force. Step S1 includes: Based on the force balance at the anchorage interface, an equilibrium equation is established, which is defined as the fifth formula. The fifth formula is: ; in, The comprehensive elastic modulus of the anchor body; The cross-sectional area of the anchor body; For anchor bolt shear displacement; refer to Figure 1 , Let be the coordinates along the length of the anchor bolt, with the origin at . At the bottom of the anchor bolt; for about The second derivative; To ensure the overall shear stiffness of the anchoring interface; Pi; The diameter of the anchor body; Based on the boundary conditions of the anchor bolt at the bottom of the anchor hole, the sixth formula is established; The sixth formula is: ; in, The comprehensive elastic modulus of the anchor body; The cross-sectional area of the anchor body; for For about The first derivative; Based on the boundary conditions of the anchor bolt at the opening of the anchor hole, the seventh formula is established; The seventh formula is: ; in, The comprehensive elastic modulus of the anchor body; The cross-sectional area of the anchor body; for For about The first derivative; The length of the anchorage section; For tension; Based on the anchoring agent material and the anchor rod material, the comprehensive elastic modulus of the anchor body is established. The formula, which is defined as the eighth formula; The eighth formula is: ; in, The elastic modulus of the anchoring agent material; The elastic modulus of the anchor rod; The cross-sectional area of the anchoring agent material; The cross-sectional area of the anchor rod; The cross-sectional area of the anchor body; By combining the fifth, sixth, seventh, and eighth formulas, the shear displacement at different positions along the length of the anchor bolt is established. The formula, which is defined as the ninth formula; The ninth formula is: ; When the anchorage interface has not entered the plastic stage, the constitutive relationship of the anchorage interface is usually linear. Based on the ninth formula, the interfacial shear stress at a certain position along the length of the anchor bolt is established. The formula, which is defined as the tenth formula; The tenth formula is: ; in, To ensure the overall shear stiffness of the anchoring interface; When the anchor rod is under tension Under the influence of, The shear stress is greatest at the anchorage section at the location (critical section); Based on the tenth formula, establish Shear stress at location The formula is defined as the eleventh formula; The eleventh formula is: ; when Shear stress at location Reaching the shear yield strength of the anchorage interface ,Right now At this point, the anchoring interface reaches the critical point of entering damage (in conjunction with...). Figure 3 Understanding the critical point ), the critical point The corresponding load is the elastic ultimate pull-out force, which can be obtained from the equilibrium relationship: Therefore, the elastic limit pull-out force of the anchor bolt is established. The preliminary formula, namely the first formula: .
[0028] In one embodiment, during grouting of the anchor hole, the grout acts on the hole wall with fluid pressure; after the grouting pump is removed, the grout solidifies into a slurry (solid). Step S3 includes: Establishing the rebound displacement of the borehole wall in the soil and rock mass With the free shrinkage displacement of the anchor body grout The formula, which is defined as the twelfth formula; The twelfth formula is: ; in, This refers to the radial compressive displacement of the outer surface of the anchor body due to the disappearance of grouting pressure. Rebound displacement of the borehole wall in rock and soil Difference from the first value related; Establishing the rebound displacement of the borehole wall in the soil and rock mass Difference from the first value The formula is defined as the thirteenth formula; The thirteenth formula is: ; Among them, the first difference Grouting pressure Normal stress at the anchorage interface The difference, i.e. This formula is defined as the fourteenth formula; The radial stiffness of the soil and rock mass surrounding the anchor hole. This formula is defined as the fifteenth formula; The elastic modulus of the soil and rock mass surrounding the anchor hole wall; The Poisson's ratio of the soil and rock mass surrounding the anchor hole wall; The radius of the anchor body; Establish the radial compressive displacement of the outer surface of the anchor body due to the disappearance of grouting pressure. Normal stress at the anchorage interface The formula is defined as the sixteenth formula; The sixteenth formula is: ; in, The radial stiffness of the anchor body; Establish the free shrinkage displacement of the anchor body grout With anchor solid radius The formula is defined as the seventeenth formula; The seventeenth formula: ; in, The linear shrinkage rate during solidification of the grout; in one embodiment, cement grout is injected into the anchor hole, and the cement grout solidifies to obtain a grout body, the linear shrinkage rate during solidification of the grout body being 0.5%; Combining the twelfth, thirteenth, sixteenth, and seventeenth formulas, the third formula is established, namely: .
[0029] In one embodiment, for the anchor body (which includes the anchor rod body anchorage section and the anchoring agent (such as grout)), the anchor rod body anchorage section and the anchoring agent maintain deformation coordination within the elastic deformation range; The S3 step also includes: Reference Figure 2 Establish polar coordinates with the center of the anchor body section as the origin of the polar coordinates. , ), The polar radius in polar coordinates. The polar angle is in polar coordinates; Based on polar coordinates, the radial stress of the anchor body is defined. Circumferential stress ; Radial stress based on anchor body and circumferential stress Establish the eighteenth formula; The eighteenth formula is: ; in, The polar radius is in polar coordinates; The differential symbol; The stress general solution (also known as the "Lame solution") of the eighteenth formula is: ; in, and These are coefficients to be determined; Determine the undetermined coefficients based on the boundary conditions of the anchor body. Meanwhile, based on the radial stress on the outer surface of the anchor body... The compressive stress generated by formation contraction is continuous and equal. Based on the general solution of the stress in the eighteenth formula, the relationship is obtained as follows: This relation is defined as the nineteenth formula; Establish the physical equation for the anchor body, which is defined as the twentieth formula; The twentieth formula is: ; in, For the circumferential strain of the anchor body; The radial strain of the anchor body; The comprehensive elastic modulus of the anchor body; The Poisson's ratio of the anchor body; Establish the geometric equation of the anchor body, which is defined as the twenty-first formula; The twenty-first formula is: ; in, This refers to the shear displacement of the anchor bolt. The differential symbol; By combining the eighteenth, nineteenth, twentieth, and twenty-first formulas, the radial shear displacement field of the anchor body is established. The formula is defined as the twenty-second formula; The twenty-second formula is: ; in, The Poisson's ratio of the anchor body; The comprehensive elastic modulus of the anchor body; The radial stress is exerted on the outer surface of the anchor body. The polar radius is in polar coordinates; Will By inputting the 22nd formula and disregarding the shear displacement direction represented by the negative sign, the shear displacement at the edge of the anchor body is obtained. The relation is defined as the twenty-third formula. The twenty-third formula is: ; in, The radius of the anchor body; Establish the radial stress on the outer surface of the anchor body Shear displacement at the edge of the anchor body The formula is defined as the twenty-fourth formula; The twenty-fourth formula is: ; in, The radial stiffness of the anchor body; Combining formulas 23 and 24, the radial stiffness of the anchor body is obtained. The formula is defined as the twenty-fifth formula; The twenty-fifth formula is: .
[0030] In one embodiment, step S3 further includes: Establish the boundary condition expression for the shrinkage of the rock and soil mass of the anchor hole wall, which is defined as the twenty-sixth formula; ; in, The radial stress of the rock and soil mass at infinity; This refers to the radial stress of the rock and soil mass at the edge of the borehole wall. The first difference is the grouting pressure. Normal stress at the anchorage interface The difference; By combining the stress general solution of the twenty-sixth formula and the eighteenth formula, a formula for the stress field of the rock and soil mass surrounding the anchor hole wall is established, which is defined as the twenty-seventh formula. The twenty-seventh formula is: ; Establish the physical equation for the rock and soil mass surrounding the anchor hole wall, which is defined as the twenty-eighth formula. The twenty-eighth formula is: ; in, The circumferential strain of the soil and rock mass surrounding the anchor hole wall; The radial strain of the rock and soil surrounding the anchor hole wall; The elastic modulus of the soil and rock mass surrounding the anchor hole wall; The Poisson's ratio of the soil and rock mass surrounding the anchor hole wall; Establish the geometric equation of the rock and soil mass surrounding the anchor hole wall, which is defined as the twenty-ninth formula. The twenty-ninth formula is: ; in, This refers to the shear displacement of the soil and rock mass surrounding the anchor hole wall; The differential symbol; By combining formulas 27, 28, and 29, the shear displacement field of the soil and rock mass surrounding the anchor hole wall is established. The formula is defined as the thirtieth formula; The thirtieth formula is: ; Will By inputting the thirtieth formula and disregarding the shear displacement direction represented by the negative sign, the shear displacement of the soil and rock mass surrounding the anchor hole wall can be obtained. The relation is defined as the thirty-first formula. The thirty-first formula is: ; Establish the first difference Shear displacement of the surrounding soil and rock mass of the anchor hole wall The formula is defined as the thirty-second formula; The thirty-second formula is: ; in, The first difference is the grouting pressure. Normal stress at the anchorage interface The difference; The radial stiffness of the soil and rock mass surrounding the anchoring hole; By combining the thirty-first and thirty-second formulas, the radial stiffness of the soil and rock mass surrounding the anchoring hole is obtained. The formula, which is the fifteenth formula, is: .
[0031] In one embodiment, the slope of the straight line segment of the anchor bolt "tension force-shear displacement" function graph is established according to the ninth formula. The formula is defined as the thirty-third formula; The thirty-third formula is: ; The optimal oblique line formed by fitting the measured coordinates of the anchor bolt's "tension force-shear displacement" points using the least squares method is then obtained. Finally, the slope of the straight line segment of the anchor bolt's "tension force-shear displacement" function graph is derived from the slope of this oblique line. ; Then, using the aforementioned formula 33, the comprehensive shear stiffness of the anchorage interface is calculated in reverse. .
[0032] For example, an in-situ test of the anchor bolts was conducted at the slope of a road cut, and an anchorage section with a length of 3 was constructed on site. The hollow anchor bolts were anchored to a strongly weathered, fragmented welded tuff stratum. During installation, M30 cement grout was injected into the anchor holes using atmospheric pressure grouting. After several months, the cement grout's bond strength reached the design value. Subsequently, the research team conducted on-site pull-out tests on the hollow anchor bolts and obtained the attached... Figure 3 The experimental data were used; then, the optimal oblique line formed by fitting the measured anchor bolt tension-shear displacement point coordinates was obtained using the least squares method. To obtain the slope It is 21.54 Then, using the aforementioned formula 33, the comprehensive shear stiffness of the anchor bolt interface is calculated in reverse. It is 18.49 .
[0033] In one embodiment, the relevant parameters of the anchor bolt are shown in the table below:
[0034] In one embodiment, the physical and mechanical parameters of the anchoring stratum are shown in the table below:
[0035] In one embodiment, a method for determining the elastic ultimate pull-out force of a slope anchor further includes: S5, inputting the formulas involved in steps S1 to S4 into the controller of the processing system, and calculating the elastic ultimate pull-out force of the anchor through the controller. The controller has a built-in program for running the formulas (such as the fourth formula).
[0036] In one embodiment, the first formula to the fourth formula are all input into the controller of the processing system.
[0037] In another embodiment, the first formula to the thirty-third formula are all input into the controller of the processing system.
[0038] In one embodiment, in step S5, parameters affecting the calculation result of the elastic limit pull-out force of the anchor bolt are input into the controller through the control panel of the processing system.
[0039] In one embodiment, the processing system is a computer, the controller is a processor built into the computer, and the control panel is a touch panel (such as a touch screen, which allows people to input parameters into the controller).
[0040] In another embodiment, the processing system is an industrial computer, the controller is a PLC (i.e., a programmable logic controller), and the control panel is a panel electrically connected to the controller (such as a human-machine interface touch screen, which allows people to input parameters into the controller).
[0041] In one embodiment, the controller of the processing system calculates the elastic limit pull-out force of the anchor bolt to be 416.3 using the fourth formula described in this invention. The on-site anchor pull-out test showed that the elastic limit pull-out force of the anchor was 440. The relative error between the two is 5.4%, which falls within the allowable range of engineering errors; therefore, the elastic limit pull-out force of the anchor bolt of this invention is [not specified]. The final formula (i.e., the fourth formula) can be used to help people predict the elastic limit pull-out force of anchor bolts, which reduces the difficulty of determining the elastic limit pull-out force of anchor bolts.
[0042] In one embodiment, the elastic limit pull-out force of the anchor bolt can be calculated using the fourth formula of the present invention. During anchor bolt construction, the tension force on the anchor bolt is controlled within a range less than the elastic limit pull-out force of the anchor bolt, which can avoid tension damage to the anchor bolt and improve the long-term service toughness of the anchor bolt.
[0043] This invention provides a method for determining the elastic limit pull-out force of slope anchors. Through steps S1 to S4, a final formula for the elastic limit pull-out force of the anchor is established. This formula facilitates the determination of the elastic limit pull-out force. When the tension force applied to the anchor in the field is less than the elastic limit pull-out force, the anchor is in a recoverable elastic state; even if the anchor is deformed under tension, it can return to its original state without structural damage. However, when the tension force applied to the anchor in the field exceeds the elastic limit pull-out force, the anchor has exceeded the critical point of elastic limit deformation (in conjunction with...). Figure 3 Understanding the critical point When the anchor bolt enters the plastic deformation stage, structural damage will occur. Timely damage control is necessary to avoid construction risks caused by plastic failure of the anchor bolt during anchor bolt tensioning.
[0044] Combination Figure 3 To understand this, the traditional method utilizes "ultimate pull-out force". The method of judging the load-bearing safety of anchor bolts under tension loads cannot describe the "starting point" of anchor bolt damage, which can easily lead to irreversible plastic deformation of the anchor bolt under tension. This invention uses the "elastic limit pull-out force" to determine the load-bearing safety of anchor bolts under tension loads. "As a criterion, it is possible to more safely control whether the anchor is in a state of recoverable deformation, thereby improving the long-term service toughness of the slope anchor."
[0045] This invention provides a method for determining the elastic limit pull-out force of slope anchors. It establishes a formula for the elastic limit pull-out force of anchors, which facilitates the determination of the elastic limit pull-out force of anchors. This allows people to use the elastic limit pull-out force as the critical load for the elastic limit deformation of anchors, ensuring the long-term toughness of anchors.
[0046] This invention determines the elastic limit pull-out force of anchor bolts through the final formula, providing key parameters and quantitative basis for the design of resilient anchoring based on recoverability, and providing construction reference for prestress compensation in highway slope anchoring projects during the operation period.
[0047] Other aspects of the method for determining the elastic limit pull-out force of slope anchors described in this invention are available in the prior art and will not be repeated here.
[0048] 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 determining the elastic ultimate pull-out force of slope anchors, characterized in that, include: S1. Obtain the elastic modulus of the anchoring agent material. Anchoring agent material cross-sectional area Elastic modulus of anchor bolt Cross-sectional area of anchor bolt body Anchor body diameter Comprehensive shear stiffness of anchorage interface Anchorage section length Establish the elastic limit pull-out resistance of the anchor bolt. The preliminary formula, which is defined as the first formula; The first formula is: ； in, The shear yield strength of the anchorage interface; Pi; S2. Obtain the cohesion of the soil and rock mass. internal friction angle with soil and rock mass Establish the shear yield strength of the anchorage interface. The formula is defined as the second formula; The second formula is: ； in, This refers to the normal stress at the anchorage interface; S3. Obtain the radial stiffness of the anchor body. Radial stiffness of the soil and rock surrounding the anchor hole Grouting pressure Free contraction displacement of anchor grout Establish normal stress on the anchorage interface The formula is defined as the third formula; The third formula is: ； S4. Based on the first formula, the second formula, and the third formula, establish the elastic limit pull-out force of the anchor bolt. The final formula, which is defined as the fourth formula; The fourth formula is: ； The fourth formula is used to determine the elastic limit pull-out force of the anchor bolt.
2. The method for determining the elastic ultimate pull-out force of slope anchors according to claim 1, characterized in that, Step S1 includes: Based on the force balance at the anchorage interface, an equilibrium equation is established, which is defined as the fifth formula. The fifth formula is: ； in, The comprehensive elastic modulus of the anchor body; The cross-sectional area of the anchor body; This refers to the shear displacement of the anchor bolt. The coordinates are along the length of the anchor bolt, with the origin at the bottom of the anchor bolt; for about The second derivative; To ensure the overall shear stiffness of the anchoring interface; Pi; The diameter of the anchor body; Based on the boundary conditions of the anchor bolt at the bottom of the anchor hole, the sixth formula is established; The sixth formula is: ； in, The comprehensive elastic modulus of the anchor body; The cross-sectional area of the anchor body; for For about The first derivative; Based on the boundary conditions of the anchor bolt at the opening of the anchor hole, the seventh formula is established; The seventh formula is: ； in, The comprehensive elastic modulus of the anchor body; The cross-sectional area of the anchor body; for For about The first derivative; The length of the anchorage section; For tension; Based on the anchoring agent material and the anchor rod material, the comprehensive elastic modulus of the anchor body is established. The formula, which is defined as the eighth formula; The eighth formula is: ； in, The elastic modulus of the anchoring agent material; The elastic modulus of the anchor rod; The cross-sectional area of the anchoring agent material; The cross-sectional area of the anchor rod; The cross-sectional area of the anchor body; By combining the fifth, sixth, seventh, and eighth formulas, the shear displacement at different positions along the length of the anchor bolt is established. The formula, which is defined as the ninth formula; The ninth formula is: ； Based on the ninth formula, the interfacial shear stress at a certain position along the length of the anchor bolt is established. The formula, which is defined as the tenth formula; The tenth formula is: ； in, To ensure the overall shear stiffness of the anchoring interface; Based on the tenth formula, establish Shear stress at location The formula is defined as the eleventh formula; The eleventh formula is: ； when Shear stress at location Reaching the shear yield strength of the anchorage interface ,Right now At this point, the anchorage interface reaches the critical point of damage. The load corresponding to this critical point is the elastic ultimate pull-out force, which, according to the equilibrium relationship, is: Therefore, the elastic limit pull-out force of the anchor bolt is established. The preliminary formula, namely the first formula: 。 3. The method for determining the elastic ultimate pull-out force of slope anchors according to claim 1, characterized in that, Step S3 includes: Establishing the rebound displacement of the borehole wall in the soil and rock mass With the free shrinkage displacement of the anchor body grout The formula, which is defined as the twelfth formula; The twelfth formula is: ； in, This refers to the radial compressive displacement of the outer surface of the anchor body due to the disappearance of grouting pressure. Establishing the rebound displacement of the borehole wall in the soil and rock mass Difference from the first value The formula is defined as the thirteenth formula; The thirteenth formula is: ； Among them, the first difference Grouting pressure Normal stress at the anchorage interface The difference, i.e. This formula is defined as the fourteenth formula; The radial stiffness of the soil and rock mass surrounding the anchor hole. This formula is defined as the fifteenth formula; The elastic modulus of the soil and rock mass surrounding the anchor hole wall; The Poisson's ratio of the soil and rock mass surrounding the anchor hole wall; The radius of the anchor body; Establish the radial compressive displacement of the outer surface of the anchor body due to the disappearance of grouting pressure. Normal stress at the anchorage interface The formula is defined as the sixteenth formula; The sixteenth formula is: ； in, The radial stiffness of the anchor body; Establish the free shrinkage displacement of the anchor body grout With anchor solid radius The formula is defined as the seventeenth formula; The seventeenth formula: ； in, This is the linear shrinkage rate during the solidification of the slurry; Combining the twelfth, thirteenth, sixteenth, and seventeenth formulas, the third formula is established, namely: 。 4. The method for determining the elastic ultimate pull-out force of slope anchors according to claim 3, characterized in that, Step S3 also includes: Establish polar coordinates with the center of the anchor body section as the origin of the polar coordinates; Based on polar coordinates, the radial stress of the anchor body is defined. Circumferential stress ; Radial stress based on anchor body and circumferential stress Establish the eighteenth formula; The eighteenth formula is: ； in, The polar radius is in polar coordinates; The differential symbol; The general solution to the stress in the eighteenth formula is: ； in, and These are coefficients to be determined; Determine the undetermined coefficients based on the boundary conditions of the anchor body. Meanwhile, based on the radial stress on the outer surface of the anchor body... The compressive stress generated by formation contraction is continuous and equal. Based on the general solution of the stress in the eighteenth formula, the relationship is obtained as follows: This relation is defined as the nineteenth formula; Establish the physical equation for the anchor body, which is defined as the twentieth formula; The twentieth formula is: ； in, For the circumferential strain of the anchor body; The radial strain of the anchor body; The comprehensive elastic modulus of the anchor body; The Poisson's ratio of the anchor body; Establish the geometric equation of the anchor body, which is defined as the twenty-first formula; The twenty-first formula is: ； in, This refers to the shear displacement of the anchor bolt. The differential symbol; By combining the eighteenth, nineteenth, twentieth, and twenty-first formulas, the radial shear displacement field of the anchor body is established. The formula is defined as the twenty-second formula; The twenty-second formula is: ； in, The Poisson's ratio of the anchor body; The comprehensive elastic modulus of the anchor body; The radial stress is exerted on the outer surface of the anchor body. The polar radius is in polar coordinates; Will By inputting the 22nd formula and disregarding the shear displacement direction represented by the negative sign, the shear displacement at the edge of the anchor body is obtained. The relation is defined as the twenty-third formula. The twenty-third formula is: ； in, The radius of the anchor body; Establish the radial stress on the outer surface of the anchor body Shear displacement at the edge of the anchor body The formula is defined as the twenty-fourth formula; The twenty-fourth formula is: ； in, The radial stiffness of the anchor body; Combining formulas 23 and 24, the radial stiffness of the anchor body is obtained. The formula is defined as the twenty-fifth formula; The twenty-fifth formula is: 。 5. The method for determining the elastic ultimate pull-out force of slope anchors according to claim 4, characterized in that, Step S3 also includes: Establish the boundary condition expression for the shrinkage of the rock and soil mass of the anchor hole wall, which is defined as the twenty-sixth formula; ； in, The radial stress of the rock and soil mass at infinity; This refers to the radial stress of the rock and soil mass at the edge of the borehole wall. The first difference is the grouting pressure. Normal stress at the anchorage interface The difference; By combining the stress general solution of the twenty-sixth formula and the eighteenth formula, a formula for the stress field of the rock and soil mass surrounding the anchor hole wall is established, which is defined as the twenty-seventh formula. The twenty-seventh formula is: ； Establish the physical equation for the rock and soil mass surrounding the anchor hole wall, which is defined as the twenty-eighth formula. The twenty-eighth formula is: ； in, The circumferential strain of the soil and rock mass surrounding the anchor hole wall; The radial strain of the rock and soil surrounding the anchor hole wall; The elastic modulus of the soil and rock mass surrounding the anchor hole wall; The Poisson's ratio of the soil and rock mass surrounding the anchor hole wall; Establish the geometric equation of the rock and soil mass surrounding the anchor hole wall, which is defined as the twenty-ninth formula. The twenty-ninth formula is: ； in, This refers to the shear displacement of the soil and rock mass surrounding the anchor hole wall; The differential symbol; By combining formulas 27, 28, and 29, the shear displacement field of the soil and rock mass surrounding the anchor hole wall is established. The formula is defined as the thirtieth formula; The thirtieth formula is: ； Will By inputting the thirtieth formula and disregarding the shear displacement direction represented by the negative sign, the shear displacement of the soil and rock mass surrounding the anchor hole wall can be obtained. The relation is defined as the thirty-first formula. The thirty-first formula is: ； Establish the first difference Shear displacement of the surrounding soil and rock mass of the anchor hole wall The formula is defined as the thirty-second formula; The thirty-second formula is: ； in, The radial stiffness of the soil and rock mass surrounding the anchoring hole; By combining the thirty-first and thirty-second formulas, the radial stiffness of the soil and rock mass surrounding the anchoring hole is obtained. The formula, which is the fifteenth formula, is: 。 6. The method for determining the elastic ultimate pull-out force of slope anchors according to claim 5, characterized in that, Based on the ninth formula, the slope of the straight line segment of the anchor bolt tension-shear displacement function graph is established. The formula is defined as the thirty-third formula; The thirty-third formula is: ； The optimal oblique line formed by fitting the measured coordinates of the anchor bolt tension-shear displacement points using the least squares method is then obtained. Finally, the slope of the straight line segment of the anchor bolt tension-shear displacement function graph is derived from the slope of this oblique line. ; Then, using the aforementioned formula 33, the comprehensive shear stiffness of the anchorage interface is calculated in reverse. .
7. The method for determining the elastic ultimate pull-out force of slope anchors according to claim 6, characterized in that, slope It is 21.54 .
8. A method for determining the elastic ultimate pull-out force of a slope anchor bolt according to any one of claims 1 to 6, characterized in that, Anchorage length 3 Comprehensive shear stiffness at the anchorage interface It is 18.49 Cohesion of soil and rock mass 30 Internal friction angle of rock and soil mass 30°; radial stiffness of the anchor body For 10000 Radial stiffness of the soil and rock mass surrounding the anchoring hole 100 The grouting pressure is 0.
5. ~0.7 .
9. A method for determining the elastic ultimate pull-out force of a slope anchor bolt according to any one of claims 1 to 6, characterized in that, Also includes: S5. Input the formulas involved in steps S1 to S4 into the controller of the processing system, and use the controller to calculate the elastic limit pull-out force of the anchor bolt.
10. The method for determining the elastic ultimate pull-out force of a slope anchor bolt according to claim 9, characterized in that, In step S5, parameters affecting the calculation results of the elastic limit pull-out force of the anchor bolt are input into the controller through the control panel of the processing system.