A method for calculating the internal force of the anchor structure of an internal anchor type anchor rod

Abstract

The present invention belongs to the field of engineering protection and governance technology, and discloses a method for calculating the internal force of an anchor structure of an internally anchored anchor rod. The method constructs a first shear stress coupling relationship between the axial displacement and the anchor rod mortar interface based on the anchor rod's force balance condition, axial stress, and axial displacement. The method presets a load transfer mode from the anchor rod to the mortar body to the surrounding rock body in the anchor structure based on a shear lag model, and constructs a second shear stress coupling relationship between the axial displacement and the shear stress of the anchor rod mortar interface and the shear stress of the mortar surrounding rock interface based on the load transfer mode. Based on the first shear stress coupling relationship, the second shear stress coupling relationship, and the force boundary conditions of the anchor structure, an analytical equation for the anchor rod axial force and a group of analytical equations for the internal shear stress are constructed. The method calculates the anchor rod axial force and internal shear stress of the anchor structure based on the physical parameters and construction parameters of the anchor structure. The present invention realizes the transfer analysis of the pullout load in the anchor structure and can calculate the displacement axial force and interface shear stress distribution of the internally anchored anchor rod.

Description

Technical Field

[0001] The invention discloses a method for calculating the internal force of an anchoring structure of an internal anchoring type anchor rod, and belongs to the technical field of engineering protection and treatment. Background Art

[0002] With the rapid development of large-scale water conservancy and hydropower projects in my country, geotechnical anchoring technology is increasingly being used in permanent projects such as steep slopes, underground caverns, and concrete dams. As a core component for project safety, accurate analysis of the internal force distribution of anchoring structures is directly related to the stability and durability of the anchoring system. However, traditional analysis methods have significant limitations in both theoretical models and practical applications.

[0003] Currently, research on the internal forces of anchored structures is primarily based on classical mechanical models such as Mindlin, Kelvin, and Boussinesq. These methods simplify the surrounding rock mass and mortar into homogeneous continua and derive the shear stress distribution at the anchor-mortar interface through global force analysis. While these models provide preliminary calculation methods, they lack sufficient consideration of the independent mechanical properties and interaction mechanisms of the three elements of the anchor structure: the anchor, the mortar, and the surrounding rock. For example, traditional models can only describe the shear stress distribution at the anchor-mortar interface, lacking detailed characterization of key issues such as stress transfer between the mortar and surrounding rock interface and the variation of the anchor axial force along its length. This limitation makes it difficult for traditional methods to fully assess the true response of anchored structures under complex loads, especially leading to biased safety predictions under high stress or long-term service conditions.

[0004] In recent years, the shear lag model, a tool for interfacial stress analysis, has garnered attention due to its successful application in composite materials, such as fiber-reinforced matrices. By assigning fibers, binders, and matrices to anchors, mortar, and surrounding rock, respectively, this model reveals the similarities between anchor structures and composite materials in terms of unidirectional axial load transfer, based on their load-bearing mechanisms. However, existing research on the shear lag model's applicability to geotechnical anchoring still has significant shortcomings: First, the model's analysis of the dual shear stress coupling at the anchor-mortar and mortar-rock interfaces is relatively crude, and the synergistic load transfer relationship between the two interfaces has not yet been established. Second, there is a lack of systematic theoretical derivation of the anchor axial force distribution, the radial deformation of the mortar, and the evolution of the stress field in the surrounding rock mass. Third, existing methods lack a comprehensive internal force calculation system for anchor structures, making it impossible to quantitatively describe the displacement and internal force distribution at any location within the structure. These issues severely limit the practical application of the shear lag model in anchoring engineering. Summary of the Invention

[0005] The purpose of the present invention is to provide a solution to the technical problem that the existing technology is difficult to meet the needs of large-scale engineering for the refined design and safety assessment of anchor structures, and is limited by the homogenization assumption. To achieve the above purpose, the present invention proposes a method for calculating the internal force of the anchor structure of an internal anchoring type anchor rod. The specific scheme is as follows:

[0006] A method for calculating the internal force of an anchor structure of an internal anchor type anchor rod comprises the following steps:

[0007] Step 1: constructing a first shear stress coupling relationship between the axial displacement and the anchor mortar interface according to the force balance condition, axial stress and axial displacement of the anchor;

[0008] Step 2: Preset a load transfer mode from the anchor rod to the mortar body to the surrounding rock mass in the anchor structure based on the shear lag model, and construct a second shear stress coupling relationship between the axial displacement and the shear stress at the anchor rod mortar interface and the shear stress at the mortar surrounding rock interface according to the load transfer mode;

[0009] Step 3: Based on the first shear stress coupling relationship, the second shear stress coupling relationship and the force boundary conditions of the anchor structure, construct an anchor rod axial force analytical equation and an internal shear stress analytical equation group of the anchor structure;

[0010] Step 4: Calculate the anchor rod axial force and internal shear stress of the anchor structure respectively using the anchor rod axial force analytical equation and the internal shear stress analytical equation group according to the physical parameters and construction parameters of the anchor structure.

[0011] Preferably, the step 1 specifically includes:

[0012] Based on the anchor rod's force balance condition, the force balance equation between the anchor rod's pull-out force and the anchor rod mortar interface is constructed.

[0013] Determine the linear relationship between the axial stress and axial displacement of the anchor rod based on elasticity theory;

[0014] A first shear stress coupling relationship between the axial displacement and the anchor mortar interface is constructed based on the force balance equation and the linear relationship.

[0015] Preferably, in step 2, constructing a second shear stress coupling relationship between the axial displacement and the interface between the anchor mortar and the mortar surrounding rock according to the load transfer mode specifically includes:

[0016] Determining the shear stress and deformation displacement of the mortar body at a preset position, and constructing a mortar coupling relationship between the shear stress and deformation displacement of the mortar body according to elasticity theory;

[0017] Determine the shear stress and deformation displacement of the surrounding rock mass at a preset position, and construct a surrounding rock coupling relationship between the shear stress and deformation displacement of the surrounding rock mass based on elasticity theory;

[0018] According to the load transfer mode, the mortar coupling relationship and the surrounding rock coupling relationship, a second shear stress coupling relationship between the axial displacement and the shear stress at the anchor mortar interface and the shear stress at the mortar surrounding rock interface is constructed.

[0019] Preferably, the step 3 specifically includes:

[0020] Step 3.1, constructing an axial force analytical equation of the anchor rod at a preset position in the anchor structure based on the first shear stress coupling relationship, the second shear stress coupling relationship, and the force boundary conditions of the anchor rod;

[0021] Step 3.2: Construct an internal shear stress analytical equation group of the anchoring structure based on the second shear stress coupling relationship and the anchor rod axial force analytical equation.

[0022] Preferably, the step 3.1 further comprises:

[0023] According to the first shear stress coupling relationship, the second shear stress coupling relationship and the force boundary conditions of the anchor rod, a displacement analytical equation of the anchor rod at a preset position is constructed.

[0024] Preferably, according to the second shear stress coupling relationship and the anchor rod axial force analytical equation, an internal shear stress analytical equation group of the anchoring structure is constructed, specifically:

[0025] Constructing a first shear stress equation for the anchor mortar interface according to the second shear stress coupling relationship, the load transfer mode, and the displacement analytical equation;

[0026] Constructing a second shear stress equation for the mortar surrounding rock interface based on the first shear stress equation and the load transfer method;

[0027] The first shear stress equation and the second shear stress equation are characterized as the internal shear stress analytical equation group.

[0028] Preferably, according to the second shear stress coupling relationship and the anchor rod axial force analytical equation, an internal shear stress analytical equation group of the anchoring structure is constructed, specifically:

[0029] Constructing a third shear stress equation according to the first transfer mode of the anchor mortar interface in the load transfer mode, the first shear stress equation and the second shear stress equation;

[0030] Constructing a fourth shear stress equation according to the second transfer mode of the mortar surrounding rock interface in the load transfer mode, the first shear stress equation and the second shear stress equation;

[0031] The first shear stress equation, the second shear stress equation, the third shear stress equation and the fourth shear stress equation are characterized as the internal shear stress analytical equation group.

[0032] Preferably, the step 3 further includes:

[0033] According to the linear relationship and the internal shear stress analytical equation group of the anchoring structure, a first displacement equation for a preset position in the mortar body and a second displacement equation for a preset position in the surrounding rock body are constructed.

[0034] Preferably, the step 4 further includes:

[0035] According to the solution of Kelvin problem, the axial force distribution function of anchor rod and the shear stress distribution function of anchor rod mortar interface in the anchor structure are constructed;

[0036] Determining the anchor rod dividing point according to the axial force distribution function and the anchor rod axial force analytical equation;

[0037] The anchor rod axial force and the anchor rod mortar interface shear stress are calculated based on the dividing point.

[0038] Preferably, the calculation of the anchor rod axial force and the anchor rod mortar interface shear stress according to the dividing point specifically includes:

[0039] Calculate the anchor rod axial force on the side of the dividing point close to the anchoring section end according to the axial force distribution function;

[0040] Calculate the anchor rod axial force on the side of the dividing point away from the anchoring section end according to the anchor rod axial force analytical equation;

[0041] Calculating the anchor mortar interface shear stress at the dividing point close to the anchoring section end side according to the anchor mortar shear stress distribution function;

[0042] The anchor rod mortar interface shear stress at the side of the dividing point away from the anchoring section end is calculated according to the first shear stress equation.

[0043] Beneficial effects: The method of the present invention is based on the shear lag model to derive formulas, with clear physical and mechanical concepts and a high degree of formula refinement, and is mainly suitable for internal anchoring anchor rods.

[0044] The present invention realizes the transfer analysis of pull-out load in the three elements of the anchoring structure (anchor rod, mortar body and surrounding rock mass), and can calculate the displacement and axial force distribution of the internal anchoring type anchor rod, the distribution of shear stress at the anchor rod mortar interface, and the shear stress distribution between the mortar body and the surrounding rock mass.

[0045] The method of the present invention can further calculate the displacement and internal force distribution of any position in the internal anchoring type anchor rod anchoring structure.

[0046] The present invention combines the shear lag model and the Kelvin problem solution to construct a segmented composite calculation method for the anchor rod axial force. The method implements segmented calculation of the anchor rod internal force, thereby improving the accuracy and refinement of the anchor rod internal force. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 Schematic diagram of the anchoring structure calculation model in an embodiment of the present invention;

[0048] Figure 2 Schematic diagram of force analysis of the anchor rod in an embodiment of the present invention.

[0049] In the figure: 1. Anchor rod; 2. Mortar body; 3. Surrounding rock mass. DETAILED DESCRIPTION

[0050] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with specific embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and do not limit the scope of protection of the present invention.

[0051] Through the calculation method of the present invention, engineers can more accurately design projects such as high and steep slopes and tunnel support, significantly improving the safety and economy of anchoring structures.

[0052] The present invention discloses a method for calculating the internal force of an anchor structure of an internal anchor type anchor rod, which specifically includes the following steps:

[0053] Step 1: construct the first shear stress coupling relationship between the axial displacement and the anchor mortar interface according to the force balance condition, axial stress and axial displacement of the anchor;

[0054] Furthermore, step 1 specifically includes: constructing a force balance equation between the pull-out force on the anchor rod and the anchor rod mortar interface based on the force balance condition of the anchor rod;

[0055] like Figure 1 The calculation model diagram of the anchor structure is shown in the figure, where 1 represents the anchor rod, 2 represents the mortar body, and 3 represents the surrounding rock mass. Based on the force balance condition of the anchor rod in the anchor structure, it can be seen that the free section of the anchor rod only bears tension, and the load applied to the internal anchor type anchor rod is transmitted to the mortar body through the bonding and friction between the anchor section of the anchor rod and the anchor rod mortar interface, and then transmitted to the surrounding rock mass through the interaction between the mortar body and the surrounding rock mass; it can be seen from the above that the pull-out force on the anchor rod is Frictional resistance of the mortar to the anchor section of the anchor bolt Equal, such as Figure 2 As shown in the figure 、 are the lengths of the free section and the anchoring section of the anchor rod, respectively.

[0056] From the above we can get:

[0057] (1)

[0058] After finishing, we can get:

[0059] (2)

[0060] Where:

[0061] is the pull-out force on the anchor rod;

[0062] is the radius of the anchor rod (m);

[0063] is the axial stress of any section of the anchor rod (Pa);

[0064] is the interfacial shear stress (Pa) of any section of the anchor mortar.

[0065] Determine the linear relationship between the axial stress and axial displacement of the anchor rod based on elasticity theory;

[0066] Specifically, when the anchor rod is in an elastic deformation state, the linear relationship between the axial stress and the axial displacement of any cross section of the anchor rod is as shown below:

[0067] ;

[0068] Where;

[0069] is the axial stress of any section of the anchor rod (Pa);

[0070] is the elastic modulus of the anchor;

[0071] is the axial displacement of the anchor rod.

[0072] The coupling relationship between the axial displacement of the anchor and the first shear stress of the anchor mortar interface is constructed based on the force balance equation and linear relationship.

[0073] Specifically, by substituting the linear relationship into the force balance equation obtained by sorting out, that is, Equation (2), we can obtain the coupling relationship between the axial displacement and the first shear stress of the anchor mortar interface after the anchor is subjected to force.

[0074] The first shear stress coupling relationship is specifically the relationship between the axial displacement of the anchor rod and the shear stress at the anchor rod mortar interface:

[0075] (3)

[0076] Where:

[0077] is the axial displacement of the anchor rod;

[0078] is the shear stress at the anchor-mortar interface (Pa).

[0079] Step 2: Based on the shear lag model, a load transfer mode from the anchor rod to the mortar body to the surrounding rock mass in the anchor structure is preset, and a second shear stress coupling relationship between the axial displacement and the shear stress at the anchor rod mortar interface and the shear stress at the mortar surrounding rock interface is constructed according to the load transfer mode;

[0080] Specifically, the shear lag model was first used for interfacial stress analysis in composite materials. The fibers, binder, and matrix involved in the shear lag model share similar stress characteristics to anchor cables and bolts: the fibers in the shear lag model can be considered anchors, the binder simulates the mortar, and the matrix corresponds to the surrounding rock mass. From a stress analysis perspective, both anchors and fibers experience unidirectional axial loads and therefore share similar stress characteristics. Therefore, the shear lag model is used to predefine the load transfer mechanism for anchor structures.

[0081] Furthermore, in step 2, the second shear stress coupling relationship between the axial displacement and the interface between the anchor mortar and the mortar surrounding rock is constructed according to the load transfer mode, which specifically includes:

[0082] Determine the shear stress and deformation displacement of the mortar body at a preset position, and construct the mortar coupling relationship between the shear stress and deformation displacement of the mortar body based on elastic theory;

[0083] Determine the shear stress and deformation displacement of the surrounding rock mass at the preset position, and construct the surrounding rock coupling relationship between the shear stress and its deformation displacement based on elastic theory;

[0084] According to the load transfer mode, mortar coupling relationship and surrounding rock coupling relationship, the second shear stress coupling relationship between axial displacement and the shear stress of the anchor mortar interface and the shear stress of the mortar surrounding rock interface is constructed.

[0085] Specifically, according to the load transfer mode of the shear lag model: the load is transferred in the mortar body and the surrounding rock mass in the form of shear load, it can be obtained that in the load transfer mode, the shear stress gradually decays along the radial direction of the anchor structure, and the attenuation relationship is shown in the following formula:

[0086] (4)

[0087] Where:

[0088] ;

[0089] is the shear stress at the anchor-mortar interface (Pa);

[0090] is the shear stress at the mortar-rock interface (Pa);

[0091] is the shear stress at the boundary of the surrounding rock mass;

[0092] is the shear stress of any section inside the mortar and surrounding rock mass;

[0093] is the effective radius of any position inside the mortar body and surrounding rock mass;

[0094] is the outer radius of the mortar body;

[0095] The range of action of the tensile force on the surrounding rock mass;

[0096] is the Poisson's ratio of the rock mass.

[0097] According to elasticity theory, the shear stress at any point inside the mortar body and the deformation displacement at that point, and the shear stress at any point in the surrounding rock body and the deformation displacement at that point have the following relationship:

[0098] (5)

[0099] Where:

[0100] is the shear stress at any point in the mortar;

[0101] is the shear stress at any point in the surrounding rock mass;

[0102] is the axial displacement of any point inside the mortar body;

[0103] is the axial displacement of any point inside the surrounding rock mass;

[0104] is the shear modulus of the mortar (Pa);

[0105] is the shear modulus of the surrounding rock mass (Pa).

[0106] The above formula represents the coupling relationship between the shear stress and displacement of the mortar body and the surrounding rock mass. By sorting out formulas (4) and (5), we can obtain:

[0107] (6)

[0108] Where:

[0109] 、 is an arbitrary constant.

[0110] Then, based on the boundary condition that the displacements at the interface between the anchor rod, mortar body, and surrounding rock mass are equal, the relationship between the anchor rod axial displacement and the shear stress at the interface between the anchor rod, mortar body, and surrounding rock mass, i.e., the second shear stress coupling relationship, can be obtained as shown in the following equation:

[0111] (7)

[0112] Step 3: Based on the first shear stress coupling relationship, the second shear stress coupling relationship and the force boundary conditions of the anchor structure, construct an anchor rod axial force analytical equation and an internal shear stress analytical equation group of the anchor structure;

[0113] Furthermore, step 3 specifically includes:

[0114] Step 3.1: Based on the first shear stress coupling relationship, the second shear stress coupling relationship, and the anchor rod's force boundary conditions, an analytical equation for the anchor rod's axial force at a preset position in the anchoring structure is constructed.

[0115] Furthermore, step 3.1 also includes constructing an analytical displacement equation of the anchor rod at a preset position based on the first shear stress coupling relationship, the second shear stress coupling relationship, and the force boundary conditions of the anchor rod.

[0116] Specifically, the second shear stress coupling relationship, formula (7), is substituted into the first shear stress coupling relationship, formula (3), and coupled with the force boundary conditions of the internal anchoring anchor rod to obtain the displacement analytical equation of the anchor rod preset position and the anchor rod axial force analytical equation.

[0117] The boundary conditions specifically include:

[0118] when hour, , is the initial tension force of the anchor (N);

[0119] when hour, .

[0120] The analytical equation for the displacement of the anchor at any position is:

[0121] (8)

[0122] Where: ;

[0123] The analytical equation of the axial force at any position of the anchor is:

[0124] (9)

[0125] Step 3.2: Based on the second shear stress coupling relationship and the anchor axial force analytical equation, construct the internal shear stress analytical equation group of the anchor structure.

[0126] Furthermore, based on the second shear stress coupling relationship and the anchor axial force analytical equation, the internal shear stress analytical equation group of the anchor structure is constructed, specifically:

[0127] The first shear stress equation of the anchor mortar interface is constructed based on the second shear stress coupling relationship, load transfer mode and displacement analytical equation;

[0128] Constructing a second shear stress equation of the mortar surrounding rock interface based on the first shear stress equation and the load transfer method;

[0129] The first shear stress equation and the second shear stress equation are characterized as an internal shear stress analytical equation group.

[0130] Specifically, by coupling the second shear stress coupling relationship, load transfer mode and displacement analytical equation, the first shear stress equation of the anchor mortar interface is obtained as shown below:

[0131] (10)

[0132] According to the first shear stress equation and the load transfer method, the second shear stress equation of the mortar surrounding rock interface is obtained as follows:

[0133] (11)

[0134] Furthermore, based on the second shear stress coupling relationship and the anchor axial force analytical equation, the internal shear stress analytical equation group of the anchor structure is constructed, specifically:

[0135] The third shear stress equation is constructed according to the first transfer mode, the first shear stress equation and the second shear stress equation of the anchor mortar interface in the load transfer mode;

[0136] The fourth shear stress equation is constructed according to the second transfer mode of the mortar-rock interface, the first shear stress equation and the second shear stress equation in the load transfer mode;

[0137] The first shear stress equation, the second shear stress equation, the third shear stress equation and the fourth shear stress equation are characterized as a set of internal shear stress analytical equations.

[0138] Specifically, by coupling the first shear stress equation and the second shear stress equation with the first transfer mode representing the anchor mortar interface in the load transfer mode, the third shear stress equation at any position in the mortar body is obtained as follows:

[0139] (12)

[0140] By coupling the first and second shear stress equations with the second load transfer mode representing the mortar-rock interface, the fourth shear stress equation at any position in the surrounding rock mass is obtained:

[0141] (13)

[0142] Furthermore, after step 3, the following steps are also included:

[0143] According to the linear relationship and the analytical equation group of the internal shear stress of the anchoring structure, the first displacement equation of the preset position in the mortar body and the second displacement equation of the preset position in the surrounding rock body are constructed.

[0144] Specifically, by coupling the linear relationship, the first shear stress equation, and the third shear stress equation, the first displacement equation at any position of the mortar body is obtained as follows:

[0145] (14)

[0146] By coupling the linear relationship, the second shear stress equation, and the fourth shear stress equation, the second displacement equation at any position of the surrounding rock mass is obtained as follows:

[0147] (15)

[0148] Furthermore, before step 4, the steps include: obtaining physical parameters and construction parameters of the anchoring structure;

[0149] Physical parameters include the elastic modulus of the anchor rod in the anchoring structure, the shear modulus of the mortar, the shear modulus of the surrounding rock mass and Poisson's ratio;

[0150] The construction parameters include the initial tension force of the anchor rod in the anchoring structure, the free section length of the anchor rod and the anchoring section length.

[0151] Step 4: Based on the physical parameters and construction parameters of the anchor structure, the anchor axial force and internal shear stress of the anchor structure are calculated using the anchor axial force analytical equation and the internal shear stress analytical equation group.

[0152] Specifically, the initial tension force of the anchor rod, the length of the free section and the anchoring section of the anchor rod, the elastic modulus of the anchor rod in the anchoring structure, the shear modulus of the mortar body, the shear modulus of the surrounding rock body and the Poisson's ratio are substituted into formulas (8) to (15). The displacement and axial force distribution of the internal anchoring type anchor rod, the shear stress distribution at the anchor rod mortar interface, the shear stress distribution in the mortar body and the surrounding rock body, and the displacement and internal force distribution at any position in the anchoring structure of the internal anchoring type anchor rod can be obtained.

[0153] Furthermore, after step 4, the following steps are also included:

[0154] Optimizing physical and / or construction parameters of the anchoring structure.

[0155] Based on the calculation results obtained above, the axial force distribution curves of anchor rods under different anchor rod types (including different steel materials and diameter changes), different mortar ratios, different anchor rod pull-out loads, and different anchoring lengths are compared. The effective anchoring length can be determined through the axial force distribution of the internal anchoring anchor rod, and the design can be further optimized and the construction process can be adjusted.

[0156] The shear stress distribution curves of the anchor-mortar-surrounding rock interface under different anchor types (including different steel materials and diameter changes), different mortar ratios, different anchor pull-out loads, and different anchoring lengths are compared. By comparing the bond strength between the selected anchor and mortar, and between the mortar and surrounding rock, and the shear stress distribution at the anchor-mortar-surrounding rock interface, the possible debonding failure is determined, and the design is further optimized and the construction process is adjusted.

[0157] Compare the axial displacement of anchor rods under different anchor rod types (including different steel materials and diameter changes), different mortar ratios, different anchor rod pull-out loads, and different anchoring lengths, as well as the distribution patterns of the displacements of the mortar body and the surrounding rock mass, to determine whether the displacement within the anchoring structure is within the displacement control range, and then optimize the design and adjust the construction process.

[0158] The method of the present invention is based on the shear lag model to derive formulas, has clear physical and mechanical concepts, and has a high degree of formula refinement. It is mainly suitable for internal anchoring type anchor rods.

[0159] The present invention realizes the transfer analysis of pull-out load in the three elements of the anchoring structure (anchor rod, mortar body and surrounding rock mass), and can calculate the displacement and axial force distribution of the internal anchoring type anchor rod, the distribution of shear stress at the anchor rod-mortar body interface, and the shear stress distribution between the mortar body and the surrounding rock mass.

[0160] The method of the present invention can further calculate the displacement and internal force distribution of any position in the internal anchoring type anchor rod anchoring structure.

[0161] This method, based on the characteristics of the rock mass at the construction site, can compare the effects of different anchor types (including different steel materials and diameter variations) and different mortar ratios on the distribution characteristics of the internal forces in the anchor structure, thereby optimizing the anchor material that balances safety and economic efficiency. It can also study the distribution of internal forces in the anchor structure under different anchor pullout loads and anchor lengths, guiding the optimal design of internal anchor bolt construction methods. This method provides a theoretical reference for stability assessment and safety control in geotechnical anchoring projects.

[0162] In a further embodiment, in order to further improve the accuracy and refinement of the anchor rod internal force analysis, it also includes constructing the axial force distribution function of the anchor rod in the anchoring structure and the shear stress distribution function of the anchor rod mortar interface based on the solution of the Kelvin problem.

[0163] Specifically, the solution to the Kelvin problem refers to a point in an infinite space where the volume force is ignored. Under concentrated force The solution to this problem is The expression for axial displacement is:

[0164] (16)

[0165] Where:

[0166] ;

[0167] , are the shear modulus and Poisson's ratio of the semi-infinite space body respectively;

[0168] 、 、 Respectively represent the semi-infinite space body in 、 、 The range of the direction axis;

[0169] Represents any point in a semi-infinite space to The distance of the point.

[0170] If it is assumed that during the tensioning process of the anchor rod, the anchor rod, mortar and surrounding rock mass are in an elastic state and no relative cracks are generated; and if it is assumed that the properties of the rock mass and the mortar mass are similar, or that the mortar mass is relatively thin, the Kelvin problem solution can be used to analyze the forces acting on the anchoring section of the anchor rod.

[0171] At the origin Department, , then (16) can be simplified to:

[0172] (17)

[0173] Where: is the comprehensive elastic modulus of the surrounding rock mass and mortar (Pa).

[0174] According to the load-displacement reciprocity theorem, the elongation of the anchor rod is equal to the displacement of the surrounding anchor body, and the equilibrium equation is established:

[0175] (18)

[0176] Calculating the above formula, we can get:

[0177] (19)

[0178] Where:

[0179] is an arbitrary constant;

[0180] ;

[0181] is the radius of the anchor rod (m);

[0182] is the Poisson's ratio of the rock mass;

[0183] is the combined elastic modulus of the surrounding rock mass and mortar mass (Pa), and It can be calculated by the following formula:

[0184] ;

[0185] In this formula: is the elastic modulus of the mortar (Pa); is the elastic modulus of the surrounding rock mass (Pa), is the elastic modulus of the anchor (Pa).

[0186] Substitute the boundary conditions: anchor section end (the junction of the anchor section and the free section in the anchor rod), anchor section bottom ; .

[0187] The distribution function of the shear stress at the anchor mortar interface along the axis of the anchor section can be obtained: And the axial force distribution function , that is, the shear stress distribution function of the anchor mortar interface and the anchor axial force distribution function in the anchor structure:

[0188] (20)

[0189] (twenty one)

[0190] Then, the anchor dividing point is determined according to the axial force distribution function and the anchor axial force analytical equation;

[0191] Specifically, by setting formula (21) and formula (9) equal, we can analytically obtain the anchor rod at a certain position in the anchoring section: When the axial forces are equal, the position Mark the dividing point of the anchor.

[0192] The calculation method of the axial force of the anchor rod is determined based on the dividing point.

[0193] Furthermore, the anchor rod axial force on the side where the dividing point is close to the anchor section end is calculated according to the axial force distribution function; the anchor rod axial force on the side where the dividing point is far from the anchor section end is calculated according to the anchor rod axial force analytical equation.

[0194] Specifically, it has been verified that the axial force distribution function of the anchor rod constructed based on the solution of the Kelvin problem can better reflect the change trend of the anchor rod axial force on the side where the dividing point is close to the end of the anchor section; while the axial force analytical equation of the anchor rod constructed based on the shear lag model can better reflect the change trend of the axial force of the anchor rod on the side where the dividing point is far from the end of the anchor section.

[0195] Based on this, the calculation formula for the anchor axial force determined according to the dividing point is as follows:

[0196] (twenty two)

[0197] Furthermore, an embodiment of the present invention also includes calculating the anchor mortar interface shear stress on the side of the dividing point close to the anchor section end based on the anchor mortar shear stress distribution function; and calculating the anchor mortar interface shear stress on the side of the dividing point away from the anchor section end based on the first shear stress equation.

[0198] Based on this, the calculation formula of the shear stress at the anchor mortar interface determined according to the dividing point is as follows:

[0199] (twenty three)

[0200] The present invention employs the aforementioned segmented composite calculation method to construct a formula for calculating the anchor rod axial force. Specifically, using the intersection of the two calculation models as the boundary, the Kelvin problem solution is used to describe the interfacial bond-slip behavior on the side of the segment near the anchor end, while the shear lag model is used to characterize the shear stress diffusion characteristics on the side of the segment far from the anchor end. The prediction accuracy of this segmented composite method is more consistent with the physical nature of the mechanical response of the anchor interface than a single method.

[0201] The above descriptions are merely several embodiments of the present invention and do not constitute any form of limitation to the present invention. Although the present invention is disclosed as above in terms of preferred embodiments, they are not intended to limit the present invention. Any technician familiar with the present profession who, without departing from the scope of the technical solution of the present invention, makes slight changes or modifications using the technical contents disclosed above are equivalent to equivalent implementation cases and fall within the scope of the technical solution.

Claims

1. A method for calculating the internal force of an internal anchoring type anchor rod, characterized in that: The following steps are involved: Step 1: constructing a first shear stress coupling relationship between the axial displacement and the anchor mortar interface according to the force balance condition, axial stress and axial displacement of the anchor; Step 1 specifically includes: constructing a force balance equation between the pull-out force on the anchor and the anchor mortar interface based on the anchor force balance condition; Determine the linear relationship between the axial stress and axial displacement of the anchor rod based on elasticity theory; Constructing a first shear stress coupling relationship between the axial displacement and the anchor mortar interface according to the force balance equation and the linear relationship; Step 2: Preset a load transfer mode from the anchor rod to the mortar body to the surrounding rock mass in the anchor structure based on the shear lag model, and construct a second shear stress coupling relationship between the axial displacement and the shear stress at the anchor rod mortar interface and the mortar surrounding rock interface according to the load transfer mode; Specifically, the method includes: determining the shear stress and deformation displacement of the mortar body at a preset position, and constructing a mortar coupling relationship between the shear stress and deformation displacement of the mortar body according to elasticity theory; Determine the shear stress and deformation displacement of the surrounding rock mass at a preset position, and construct a surrounding rock coupling relationship between the shear stress and deformation displacement of the surrounding rock mass based on elasticity theory; According to the load transfer mode, the mortar coupling relationship and the surrounding rock coupling relationship, a second shear stress coupling relationship between the axial displacement and the shear stress at the anchor mortar interface and the mortar surrounding rock interface is constructed; Step 3: Based on the first shear stress coupling relationship, the second shear stress coupling relationship and the force boundary conditions of the anchor structure, construct an anchor rod axial force analytical equation and an internal shear stress analytical equation group of the anchor structure; Specifically comprising: step 3.1, constructing an anchor rod axial force analytical equation of the anchor rod at a preset position in the anchoring structure according to the first shear stress coupling relationship, the second shear stress coupling relationship and the anchor rod force boundary conditions; Step 3.2: construct an internal shear stress analytical equation group of the anchoring structure based on the second shear stress coupling relationship and the anchor axial force analytical equation; Step 4: Calculate the anchor rod axial force and internal shear stress of the anchor structure using the anchor rod axial force analytical equation and the internal shear stress analytical equation group according to the physical parameters and construction parameters of the anchor structure; After step 4, the following steps are also included: constructing the axial force distribution function of the anchor rod and the shear stress distribution function of the anchor rod mortar interface in the anchor structure according to the solution of the Kelvin problem; Determining the anchor rod dividing point according to the axial force distribution function and the anchor rod axial force analytical equation; The anchor rod axial force and the anchor rod mortar interface shear stress are calculated based on the dividing point.

2. The method for calculating internal forces of an anchoring structure according to claim 1, characterized in that: The step 3.1 further includes: According to the first shear stress coupling relationship, the second shear stress coupling relationship and the force boundary conditions of the anchor rod, a displacement analytical equation of the anchor rod at a preset position is constructed.

3. The method for calculating internal forces of an anchoring structure according to claim 2, characterized in that: According to the second shear stress coupling relationship and the anchor rod axial force analytical equation, the internal shear stress analytical equation group of the anchor structure is constructed, specifically: Constructing a first shear stress equation for the anchor mortar interface according to the second shear stress coupling relationship, the load transfer mode, and the displacement analytical equation; Constructing a second shear stress equation for the mortar surrounding rock interface based on the first shear stress equation and the load transfer method; The first shear stress equation and the second shear stress equation are characterized as the internal shear stress analytical equation group.

4. The method for calculating internal forces of an anchoring structure according to claim 3, characterized in that: According to the second shear stress coupling relationship and the anchor rod axial force analytical equation, the internal shear stress analytical equation group of the anchor structure is constructed, specifically: Constructing a third shear stress equation according to the first transfer mode of the anchor mortar interface in the load transfer mode, the first shear stress equation and the second shear stress equation; Constructing a fourth shear stress equation according to the second transfer mode of the mortar surrounding rock interface in the load transfer mode, the first shear stress equation and the second shear stress equation; The first shear stress equation, the second shear stress equation, the third shear stress equation, and the fourth shear stress equation are characterized as the internal shear stress analytical equation group.

5. The method for calculating internal forces of an anchoring structure according to claim 1, characterized in that: After step 3, the following steps are also included: According to the linear relationship and the internal shear stress analytical equation group of the anchoring structure, a first displacement equation for a preset position in the mortar body and a second displacement equation for a preset position in the surrounding rock body are constructed.

6. The method for calculating internal forces of an anchoring structure according to claim 3, characterized in that: The calculation of the anchor axial force and the anchor mortar interface shear stress according to the dividing point specifically includes: Calculate the anchor rod axial force on the side of the dividing point close to the anchoring section end according to the axial force distribution function; Calculate the anchor rod axial force on the side of the dividing point away from the anchoring section end according to the anchor rod axial force analytical equation; Calculating the anchor mortar interface shear stress at the dividing point close to the anchoring section end side according to the anchor mortar shear stress distribution function; The anchor rod mortar interface shear stress at the side of the dividing point away from the anchoring section end is calculated according to the first shear stress equation.

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