A method for calculating the pullout load of a full-length anchoring anchor rod considering multi-anchoring defects

Abstract

The present application belongs to the technical field of rock mass engineering support, and proposes a calculation method of full-length anchoring anchor rod pulling load considering multiple anchoring defects. The specific steps are: obtaining anchoring interface bonding strength parameters, anchor rod mechanical parameters, anchoring defect parameters and anchoring anchor rod length and diameter; establishing a full-length anchoring anchor rod load transmission mechanical model considering multiple anchoring defects, anchoring interface nonlinear slip and anchor rod plastic hardening, solving the axial stress, shear stress and displacement of each node of the anchor rod anchoring section and the defect section based on the model; obtaining the load-displacement curve of the full-length anchoring anchor rod containing multiple defects, the ultimate pulling load and the peak displacement, which are directly used for support design optimization. The present application considers the nonlinear shear slip behavior of the anchoring interface, the plastic hardening of the anchor rod and the multiple anchoring defects, can more accurately predict the load displacement curve, the ultimate pulling load and the peak displacement of the anchor rod, and has important significance for the design of anchor rods and the evaluation of anchoring quality in rock mass support construction.

Description

Technical Field

[0001] This invention relates to the field of rock mass engineering support technology, specifically to a method for calculating the pull-out load of a full-length anchor bolt considering multiple anchoring defects. Background Technology

[0002] Rock bolts are widely used in the support of mines, tunnels, slopes, and foundation pits. Rock bolt support has gained popularity in rock engineering due to its strong practicality and advantages. Studying the bearing capacity of rock bolts under different influences and the load transfer law of the anchorage section is a primary means of quantitatively evaluating anchorage quality. This helps analyze the failure modes of the anchor body, understand the conditions and catastrophic processes of anchorage failure, and improve and optimize support design to extend the service life of the anchored rock mass. However, due to the concealed environment of the anchored rock mass and the complexity of the stress on the rock bolt, the anchor bolt and surrounding rock are affected by various factors such as groundwater intrusion and corrosion, uneven mixing of the anchoring agent, and rock layer delamination. Defects in the anchoring agent often lead to localized anchor voids, resulting in a loss or reduction of the bond strength between the anchoring agent and the surrounding rock and the anchor bolt body. This continuously deteriorates the anchorage performance and weakens its reinforcement and support effect on the surrounding rock. The occurrence of anchorage defects inevitably affects the anchorage quality and the effective and continuous transfer of the axial load of the anchor bolt. Many scholars have not paid enough attention to the pull-out load transfer law of anchor bolts with anchorage defects, resulting in a lack of understanding of its influence mechanism in existing research. Therefore, with the development of non-destructive testing technology for anchor bolts, it is now possible to locate and determine the size of anchorage defects using certain equipment and techniques. This technological development provides a foundation for research on the pull-out load transfer law and bearing capacity of anchor bolts considering anchorage defects, enabling its application in engineering practice and providing theoretical support for the evaluation of anchorage quality and repair decisions in engineering projects.

[0003] Research on anchor bolt anchoring performance is the foundation and key to anchoring design and analysis. The analysis of the pull-out load transfer mechanism and bearing capacity of anchor bolts is the most important aspect of anchor bolt anchoring performance evaluation. Most existing studies focus on well-anchored anchor bolts, with few studies considering anchor bolts with multiple anchoring defects. Furthermore, existing analyses lack mechanical models that simultaneously consider the nonlinear mechanical properties of the anchoring interface, plastic hardening and fracture of anchor bolts, as well as multiple anchoring defects, making them difficult for engineers to use in practice. Summary of the Invention

[0004] To address the aforementioned problems, this invention provides a method for calculating the pull-out load of full-length anchor bolts considering multiple anchoring defects. This method can be applied to anchoring projects with multiple anchoring defects, nonlinear slippage at the anchoring interface, and plastic hardening or breakage of the anchor bolt, providing a solution for evaluating the anchoring quality and making repair decisions. This method overcomes the theoretical calculation difficulties under various conditions such as anchor bolt yielding, nonlinear interfaces, and numerous anchoring defects, thereby improving the rationality and accuracy of the predicted load-displacement curve.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

[0006] A method for calculating the pull-out load of a full-length anchor bolt considering multiple anchoring defects includes the following steps:

[0007] Step 1: Obtain anchorage interface bond strength parameters, anchorage mechanical parameters, and anchorage defect parameters through short anchorage pull-out tests, anchorage tensile tests, and non-destructive testing of the anchorage. Obtain the anchorage length L and anchorage diameter. Anchorage interface bond strength parameters include peak shear strength, peak shear displacement, and residual shear strength at the anchorage interface. Anchorage mechanical parameters include elastic modulus, yield strength, yield strain, hardening modulus, and ultimate tensile strain. Anchorage defect parameters include the number, location, and size of anchorage defects.

[0008] Step 2: Establish a full-length anchorage load transfer mechanical model that considers multiple anchorage defects, nonlinear slippage at the anchorage interface, and plastic hardening of the anchor rod. Based on this full-length anchorage load transfer mechanical model, solve for the axial stress, shear stress, and displacement of each node in the anchorage section and defect section of the anchor rod.

[0009] Step 3: Based on the solution results of Step 2, obtain the load-displacement curve, ultimate pull-out load, and peak displacement of the anchor rod with multiple anchoring defects along its entire length, which can be directly used for support design optimization.

[0010] The full-length anchorage load transfer mechanical model, considering multiple anchorage defects, nonlinear slippage at the anchorage interface, and plastic hardening of the anchor rod, is specifically as follows: The anchor rod is divided into a free end and a pull-out end; the full-length anchorage load transfer mechanical model is divided into regions, with the area surrounding the anchor rod encased in anchoring agent designated as the anchorage segment and the area without anchoring agent encased as the defect segment; two transition surfaces are defined: the surface transitioning from the anchorage segment to the defect segment along the direction from the free end to the pull-out end is designated as the anchorage-defect transition surface, and the surface transitioning from the defect segment to the anchorage segment is designated as the defect-anchorage transition surface; an analysis unit is established. The analysis unit includes the anchorage-defect transition surface and the adjacent anchorage and defect sections; the analysis units are connected by the defect-anchorage transition surface; the anchor rod is divided into n nodes, with each pair of nodes forming an anchor rod unit with a length of Δx = L / (n-1); the integer i represents the i-th node, and the first node at the free end is defined as i = 1, then the last node at the pull-out end is i = n; in the full-length anchorage load transfer mechanical model, the load-displacement curve at the pull-out end node i = n is obtained by continuously increasing the relative displacement at the free end node i = 1.

[0011] The specific process for solving the load-displacement curve at the pull-out end node i=n is as follows:

[0012] Based on the full-length anchorage load transfer mechanical model considering multiple anchorage defects, nonlinear slippage at the anchorage interface, and plastic hardening of the anchor rod, the following equations are satisfied: s(x) = u(x), where s(x) is the relative displacement of the anchor node at position x (in meters), and u(x) is the axial displacement of the anchor node at position x (in meters). When the deformation of the surrounding rock mass is ignored, s(x) = u(x). According to the regions divided in the full-length anchorage load transfer mechanical model, the basic equation for load transfer in the anchorage section is:

[0013]

[0014] Where, τ r >0, a≥0, b≥0; τ r The residual shear strength at the anchor bolt-anchoring agent interface is expressed in Pa. τ r =γτ u a and b are determined by the peak interfacial shear strength τ. u and its corresponding peak interface shear displacement s u The commonly determined parameters are: γ, which is the ratio of residual shear strength to peak shear strength; the perimeter of the anchor bolt cross-section U = 2πR; and the cross-sectional area of ​​the anchor bolt A = πR. 2 ; x is the position of the node in the anchor bolt, in meters; τ(x) is the shear stress at the anchor bolt-anchoring agent interface at x, in Pa; R is the anchor bolt radius, in meters; E b The elastic modulus of the anchor bolt is expressed in Pa.

[0015] The expression for the nonlinear bi-exponential curve shear slip relationship of the anchorage interface involved in equation (1) is as follows:

[0016] τ=τ r +ae -bs -(a+τ r )e -2bs (2)

[0017] In the formula: τ is the shear stress at the anchor bolt-anchoring agent interface, in Pa; s is the relative displacement of the anchor bolt; the calculation methods for anchor bolt nodes located in the anchoring section and the defect section are different. As the relative displacement at node i=1 increases, the axial stress at each node in the anchor bolt increases accordingly; when the relative displacement at node i=1 continues to increase, the axial stress at a certain node exceeds the anchor bolt yield load σ. y When the relative displacement at node i=1 is increased, the anchor bolt yields. As the relative displacement at node i=1 is increased, the anchor bolt yield point gradually shifts toward the free end. Based on the state of the anchor bolt node, the axial stress, shear stress, and displacement of the anchor bolt node are calculated. The calculation is performed step by step for the analysis unit, and this cycle is repeated until the last analysis unit is calculated. When node i=n, ​​it indicates that the current state calculation is completed. Then, the displacement u(1) of the free end node i=1 is continuously increased to simulate the anchor bolt pull-out process. The axial stress, shear stress, and displacement of the anchor bolt node under the corresponding process are calculated until all anchor bolt nodes are in the slip state. The calculation is stopped when u(1) reaches the set threshold. The program automatically stops the calculation.

[0018] When the anchor node is located in the anchoring section, the axial stress, shear stress and displacement are calculated according to its state.

[0019] 1.1) When the anchor node is located in the anchorage section and is in an elastic state, solve for the axial stress, shear stress and displacement of each node in the anchorage section under this state;

[0020] Using numerical analysis, the differential equation (1) is transformed into a difference equation to obtain the displacement relationship between adjacent node elements; the displacement estimation formula for the next node is then obtained:

[0021]

[0022] In the formula: i-1, i, i+1 represent three adjacent nodes, x(i-1), x(i), x(i+1) represent the coordinates of nodes i-1, i, i+1 respectively, and u(i-1), u(i), u(i+1) represent the displacements of nodes i-1, i, i+1 respectively;

[0023] For node i = 1, u(1) is the input value; node i = 1 is located at the free end, so the axial stress σ(1) = 0 and the shear stress τ(1) = 0 here, so the strain is also 0;

[0024] For node i = 2, the displacement relationship between node i = 1 and node i = 2 can be obtained from equation (1):

[0025]

[0026] The displacements of other nodes are obtained sequentially based on equation (3);

[0027] The shear stress at node i is obtained from the nonlinear bi-exponential curve shear slip relation (2) of the anchorage interface:

[0028] τ(i)=τ r +ae -bu(i) -(a+τ r )e -2bu(i) (5)

[0029] Based on the stress balance analysis of the nodes, the axial stress at node i is:

[0030]

[0031] According to Hooke's law, the strain at node i is:

[0032]

[0033] In the formula: σ(i) is the axial stress at node i, in Pa; ε(i) is the total strain at node i, ε e (i) represents the elastic strain at node i;

[0034] 1.2) When the anchor node is located in the anchorage section and is under plastic loading, solve for the axial stress, shear stress and displacement of each node in the anchorage section under this state;

[0035] If the anchor bolt yields, assume that the calculated axial stress at the node k preceding the yield point is less than the anchor bolt's yield strength σ. y The axial stress at node k+1 is greater than the yield limit of the anchor rod. The yield point k0 is between nodes k and k+1. Nodes 1 to k are the elastic zone, and k+1 to n are the plastic zone. Node k0 is an additional yield point inserted between adjacent nodes k and k+1 and is not included in the original node number.

[0036] In the elastic zone of the anchor bolt, the nodal displacement, shear stress, axial stress, and strain are obtained by equations (3), (5), (6), and (7), respectively.

[0037] For the yield point k0, the position x(k0) of the yield point is determined by the stress-strain relationship between the preceding node k and the yield point k0:

[0038]

[0039] The displacement of the yield point k0 is obtained from the displacement calculation formula (3) of adjacent nodes:

[0040]

[0041] The axial stress at the yield point is the yield load σ. y ;

[0042] According to the nonlinear bi-exponential curve shear slip relationship (2) of the anchorage interface, the shear stress τ(k0) at the yield point is:

[0043]

[0044] The elastic strain ε(k0) at the yield point is determined by Hooke's law:

[0045]

[0046] In the formula: x(k0) is the yield point position, in meters; u(k0) is the yield point displacement, in meters; ε(k0) is the yield point strain; τ(k0) is the yield point shear stress, in Pa.

[0047] The anchor node is located in the plastic zone of the anchor. First, the axial stress, shear stress and displacement of node k+1, which is adjacent to the anchor yield point k0 in the plastic zone of the anchor, are calculated. Then, the axial stress, shear stress and displacement of each node in the plastic zone are calculated sequentially along the direction from the free end of the anchor to the pull-out end.

[0048] The nodal displacements in the plastic zone are obtained from equation (3):

[0049]

[0050] The axial stress at node k+1 is obtained from the stress balance equations of adjacent nodes:

[0051]

[0052] From the nonlinear bi-exponential curve shear slip relation (2) of the anchorage interface, the shear stress at node k+1 is:

[0053] τ(k+1)=τ r +ae -bu(k+1) -(a+τ r )e -2bu(k+1) (14)

[0054] The total strain at the joint in the plastic zone is the sum of the elastic strain and the plastic strain:

[0055]

[0056] In the formula, ET Secant modulus;

[0057] For nodes k+1 to n, the axial displacement of the anchor at node i+1 is obtained according to the deformation compatibility relationship:

[0058]

[0059] From the stress balance equations of adjacent nodes, the axial force of the anchor at node i+1 is:

[0060]

[0061] From the nonlinear bi-exponential curve shear slip relation (2) of the anchorage interface, the shear stress at node i+1 is:

[0062] τ(i+1)=τ r +ae -bu(i+1) -(a+τ r )e -2bu(i+1) (18)

[0063] The total strain at node i+1 is equal to the strain ε at yield. y =ε(k0) plus the strain after yielding:

[0064]

[0065] 1.3) When the anchor bolt node is located in the anchorage section and is in a plastic unloading state, solve for the axial stress, shear stress and displacement of each node in the anchorage section under this state;

[0066] After the anchor bolt yields, as the displacement at node i=1 continues to increase, the anchor bolt yield point gradually moves towards the free end. On the other hand, as the node displacement increases, the anchor bolt-anchoring agent interface reaches its peak shear strength and then undergoes plastic softening. When a sufficient number of nodes have their anchor bolt-anchoring agent interfaces enter the plastic softening stage, the axial stress of the anchor bolt decreases.

[0067] According to the stress balance equations of adjacent nodes:

[0068]

[0069] The shear stress at node i+1 is obtained from the nonlinear bi-exponential curve shear slip relation (2) of the anchorage interface:

[0070] τ(i+1)=τ r +ae -bu(i+1) -(a+τ r )e -2bu(i+1) (twenty one)

[0071] The total strain at node i+1 is:

[0072] ε(i+1)=ε p +ε e (twenty two)

[0073] In the formula, ε p Plastic strain;

[0074] Where the elastic strain is:

[0075]

[0076] The axial displacement at node i+1 satisfies the displacement expression (16) of the anchor rod in the plastic zone;

[0077] Since the anchor bolt node is in the plastic unloading stage, the plastic strain of the node remains unchanged at this time, and its value is the plastic strain value before unloading.

[0078] When the anchor node is located in the anchorage defect section, the axial stress, shear stress and displacement are calculated according to its state.

[0079] 2.1) When the anchor bolt node is located in the anchorage defect section and is in an elastic state, solve for the axial stress, shear stress and displacement of each node in the defect section under this state;

[0080] Based on the deformation compatibility relationship, the axial displacement of the anchor at node i is obtained as follows:

[0081]

[0082] The shear stress of each node in the anchorage defect section is 0, and the axial stress of each node is the same and equal to the axial stress of the node located in the anchorage section that is adjacent to the defect section.

[0083] 2.2) When the anchor node is located in the anchorage defect section and the anchor is in a plastic loading state, the axial stress of adjacent nodes is equal, the shear stress is 0, and the axial displacement of all nodes in the defect section area satisfies the expression (24) of the axial displacement in the defect section when it is in an elastic state.

[0084] Based on the activity of the yield point k0, its location is determined by the following two points:

[0085] (a) When the yield point k0 extends from the anchorage section of the subsequent analysis unit across the defect-anchorage transition surface to the anchorage defect area of ​​the current analysis unit, all nodes in the defect area of ​​the analysis unit will simultaneously yield, and the yield point k0 will directly extend from the anchorage-defect transition surface to the anchorage section of the current analysis unit.

[0086] (b) When the anchorage defect segment node is already in a plastic loading state, the yield point k0 is located near the free end of the current calculation node i, and the yield point k0 is located in the anchorage segment of the current analysis unit or a previous analysis unit.

[0087] 2.3) When the anchor node is located in the anchorage defect section and the anchor is in the plastic unloading state, the axial stress of all adjacent nodes is equal and the shear stress is 0. Therefore, the axial displacement of all nodes satisfies the axial displacement in the defect section when it is in the elastic state. Since the anchor node is in the plastic unloading stage, the plastic strain of the node remains unchanged.

[0088] Based on the calculation results of step 2, the load-displacement curve of the anchor pull-out end node n is finally obtained, which is the final load-displacement curve of the full-length anchor bolt with multiple anchoring defects. P = σA, where σ is the stress and A is the cross-sectional area of ​​the anchor bolt. The ultimate pull-out load and peak displacement of the full-length anchor bolt with multiple anchoring defects are obtained, which can be directly used for support design optimization or theoretical analysis.

[0089] When there are no defects at the pull-out end, the last anchoring section is used as the last analysis unit.

[0090] The beneficial effects of this invention are that it provides a theoretical method for calculating the pull-out load of a full-length anchor bolt considering multiple anchoring defects. The recursive calculation formula for the pull-out load based on this method has rigorous logic, thus the obtained anchor bolt load-displacement curve, ultimate pull-out load, and corresponding peak displacement are reasonable. The proposed calculation method not only considers the localized anchor void phenomenon often found in actual engineering, but also simultaneously considers the nonlinear slippage of the anchorage interface, as well as the plastic hardening and fracture of the anchor bolt, making the calculation of the anchor bolt pull-out force more consistent with actual engineering practices. By inputting the anchorage interface bond strength parameters, anchor bolt mechanical parameters, anchoring defect parameters, and the length and diameter of the anchor bolt, the yield instability and anchorage interface slippage state of anchor bolts with multiple defects can be quickly and conveniently obtained. The method is simple, the results are specific and clear, and it can be directly used in engineering construction sites. This invention can fully combine the mechanical and geometric parameters of anchor bolts, the mechanical parameters of the anchorage interface, and the mechanical behavior of anchor bolts and the anchorage interface to calculate the ultimate pull-out load and peak displacement of full-length anchor bolts with multiple defects, thereby reducing the subjectivity and uncertainty in anchor bolt design and anchorage quality evaluation and improving its scientific rigor. Attached Figure Description

[0091] Figure 1 Flowchart of the calculation method for pull-out load of full-length anchor bolts considering multiple anchoring defects;

[0092] Figure 2 A mechanical model for load transfer of full-length anchored bolts, considering multiple anchoring defects, nonlinear slippage at the anchoring interface, and plastic hardening of the anchor bolt;

[0093] Figure 3 This is a comparison chart of the ultimate pull-out load and peak value of the full-length anchor bolt under different working conditions. Detailed Implementation

[0094] To make the objectives, technical solutions, and advantages of this invention clearer, the following description is provided in conjunction with the appendix. Figure 1 The calculation methods and embodiments of the present invention will be described in further detail below. It should be understood that the specific embodiments described herein are for illustrative purposes only and are not intended to limit the scope of the invention.

[0095] The preferred embodiment of this invention provides a method for calculating the pull-out load of a full-length anchor bolt considering multiple anchoring defects. The implementation process is as follows: Figure 1 As shown, the specific steps include:

[0096] Step 1: Obtain the elastic modulus E of the anchor bolt through pull-out tests of short anchor bolts, tensile tests of anchor bolts, non-destructive testing of anchor bolts, and tests of the length and diameter of anchor bolts. b Tangent modulus E T Bond strength parameters at the anchorage interface (residual shear strength τ) r Peak shear strength τ u and its corresponding peak interface shear displacement s u The yield strength σ of the anchor bolt y The ultimate tensile strain ε of the anchor bolt ext The number of defects, N, was calculated, and the diameter D = 20 mm and length L = 2.5 m of the anchor rod were obtained through basic measurements. This patent embodiment calculates two cases: one without considering anchor rod yield (setting the yield strength to a larger value) and the other considering anchor rod yield. It is compared with an embodiment that considers both yield and non-yield of a single anchoring defect. Specific parameters are shown in Tables 1 and 2.

[0097] Table 1 Model input parameters without considering anchor bolt yielding

[0098]

[0099] Table 2 Model input parameters considering anchor bolt yielding

[0100]

[0101] This patent embodiment employs three defects, each with a unit length of 0.1m. With the free end as x = 0m, the coordinate intervals of the three defect segments are (0.45m, 0.55m), (0.95m, 1.05m), and (1.45m, 1.55m), respectively.

[0102] Step 2: Establish a full-length anchorage load transfer mechanical model considering multiple anchorage defects, nonlinear slippage at the anchorage interface, and plastic hardening of the anchor rod (e.g., Figure 2 As shown in the figure, the model is divided into regions, and the axial stress, shear stress and displacement of each node in the anchorage section and defect section of the anchor rod are solved based on the model. The specific steps include the following sub-steps.

[0103] Step 2.1: Based on the full-length anchorage load transfer mechanical model considering multiple anchorage defects, nonlinear slippage at the anchorage interface, and plastic hardening of the anchor rod, the anchor rod is divided into a free end and a pull-out end. The model is further divided into regions. The area surrounding the anchor rod that is wrapped with anchoring agent is denoted as the anchorage segment, and the area without defects is denoted as the defect segment. Two transition surfaces are defined: the surface transitioning from the anchorage segment to the defect segment is denoted as the anchorage-defect transition surface, and the surface transitioning from the defect segment to the anchorage segment is denoted as the defect-anchorage transition surface. Analysis elements are established, each consisting of adjacent anchorage segments, defect segments, and the anchorage-defect transition surface. The transition between analysis elements is the defect-anchorage transition surface (if there is no defect at the pull-out end, the last anchorage segment is taken as the last analysis element). The anchor rod is divided into n nodes, and the length of each anchor rod element is Δx = L / (n-1). The integer i represents the i-th node; the first node at the free end is defined as i = 1, and the last node at the pull-out end is defined as i = n. In the load transfer model, the load-displacement curve at the pull-out end node i=n is obtained by continuously increasing the displacement at the free end node i=1.

[0104] Step 2.2: Based on the full-length anchorage load transfer mechanical model considering multiple anchorage defects, nonlinear slippage at the anchorage interface, and plastic hardening of the anchor rod, when the deformation of the surrounding rock mass is neglected, we have s(x) = u(x). According to the model region division in Step 2.1, the basic equation for load transfer in the anchorage section of the anchor rod can be obtained as follows:

[0105]

[0106] Where: the perimeter of the anchor bolt cross-section U = 2πR; the cross-sectional area of ​​the anchor bolt A = πR 2 ; x is the position of the node in the anchor bolt, in meters; τ(x) is the shear stress at the anchor bolt-anchoring agent interface at x, in Pa; u(x) is the axial displacement of the anchor bolt at x, in meters; R is the anchor bolt radius, in meters; E b The elastic modulus of the anchor bolt is expressed in Pa.

[0107] The expression for the nonlinear bi-exponential curve shear slip relationship of the anchorage interface involved in equation (1) is as follows:

[0108] τ=τ r +ae -bs -(a+τ r )e -2bs(2)

[0109] Where τ r >0, a≥0, b≥0;

[0110]

[0111]

[0112] τ r =γτ u

[0113] In the formula: τ is the shear stress at the anchor bolt-anchoring agent interface, in MPa;

[0114] s represents the relative displacement of the anchor bolt, in meters (m).

[0115] τ r The residual shear strength at the anchor bolt-anchoring agent interface is expressed in MPa.

[0116] a and b are determined by the peak interfacial shear strength τ u and its corresponding peak interface shear displacement s u The parameters are determined together, where γ is the ratio of residual shear strength to peak shear strength.

[0117] Step 2.3: When the anchor node is located in the anchorage section and is in an elastic state, solve for the axial stress, shear stress and displacement of each node in the anchorage section under this condition.

[0118] Using numerical analysis, the differential equation (1) is transformed into a difference equation, yielding the displacement relationship between adjacent node elements. After simplification, the displacement estimation formula for the next node is obtained:

[0119]

[0120] In the formula: i-1, i, i+1 represent three adjacent nodes, x(i-1), x(i), x(i+1) represent the coordinates of nodes i-1, i, i+1 respectively, and u(i-1), u(i), u(i+1) represent the displacements of nodes i-1, i, i+1 respectively.

[0121] For node i = 1, u(1) is the input value. Since this node is located at the free end, the axial stress σ(1) = 0 and the shear stress τ(1) = 0, so the strain is also 0 (the solution for node i described later does not include node i = 1).

[0122] For node i = 2, from equation (1), the displacement relationship between node i = 1 and node i = 2 is:

[0123]

[0124] The displacements of other nodes can be obtained sequentially based on equation (3).

[0125] From the bi-exponential curve shear slip relation (2), the shear stress at node i is:

[0126] τ(i)=τ r +ae -bu(i) -(a+τ r )e -2bu(i) (5)

[0127] From the stress balance analysis of the node, the axial stress at node i is:

[0128]

[0129] According to Hooke's law, the strain at node i is:

[0130]

[0131] In the formula: σ(i) is the axial stress at node i, in MPa; ε(i) is the total strain at node i, ε e (i) represents the elastic strain at node i.

[0132] Step 2.4: When the anchor node is located in the anchorage section and is under plastic loading, solve for the axial stress, shear stress and displacement of each node in the anchorage section under this condition.

[0133] As the displacement at node i=1 increases, the axial stress at each node in the anchor bolt gradually increases. When the displacement at node i=1 continues to increase, the axial stress at a certain node exceeds the anchor bolt yield load σ. y When the displacement at node i=1 continues to increase, it indicates that the anchor bolt has yielded. Furthermore, as the displacement at node i=1 continues to increase, the anchor bolt yield point gradually shifts towards the free end.

[0134] If the anchor bolt yields, assume that the calculated axial stress at the node k preceding the yield point is less than the anchor bolt's yield strength σ. y If the axial stress at node k+1 is greater than the yield strength of the anchor rod, then the yield point k0 is between nodes k and k+1. Therefore, nodes 1 to k are in the elastic zone, k+1 to n are in the plastic zone, and node k0 is an additional yield point inserted between adjacent nodes k and k+1, not included in the original node numbering.

[0135] (1) In the elastic zone of the anchor bolt, the nodal displacement, shear stress, axial stress and strain can be obtained by equations (3), (5), (6) and (7), respectively.

[0136] (2) For the yield point k0, the position x(k0) of the yield point can be determined by the stress-strain relationship between the previous node k and the yield point:

[0137]

[0138] The displacement of the yield point k0 can be obtained from the displacement calculation formula (3) of adjacent nodes:

[0139]

[0140]

[0141] The axial stress at the yield point is the yield load σ. y .

[0142] According to the bi-exponential curve shear slip relation (2), the shear stress τ(k0) at the yield point is:

[0143]

[0144] The elastic strain ε(k0) at the yield point can be determined using Hooke's law:

[0145]

[0146] In the formula: x(k0) is the yield point position, in meters; u(k0) is the yield point displacement, in meters; ε(k0) is the yield point strain; τ(k0) is the yield point shear stress, in Pa.

[0147] (3) When the node is located in the plastic zone of the anchor rod, the axial stress, shear stress and displacement of the node k+1 located in the plastic zone of the anchor rod and adjacent to the yield point k0 of the anchor rod must be calculated first. Then, the axial stress, shear stress and displacement of each node in the plastic zone are calculated sequentially along the direction of the free end of the anchor rod.

[0148] From equation (3), the nodal displacements in the plastic zone are:

[0149]

[0150] From the stress balance equations of adjacent nodes, the axial stress at node k+1 is:

[0151]

[0152] From the bi-exponential curve shear slip relation (2), the shear stress at node k+1 is:

[0153] τ(k+1)=τ r +ae -bu(k+1) -(a+τ r )e -2bu(k+1) (14)

[0154] The total strain at the joint in the plastic zone is the sum of the elastic strain and the plastic strain:

[0155]

[0156] For nodes k+1 to n, the axial displacement of the anchor at node i+1 is obtained according to the deformation compatibility relationship:

[0157]

[0158] From the stress balance equations of adjacent nodes, the axial force of the anchor at node i+1 can be obtained as follows:

[0159]

[0160] From the bi-exponential curve shear slip relation (2), the shear stress at node i+1 is:

[0161] τ(i+1)=τ r +ae -bu(i+1) -(a+τ r )e -2bu(i+1) (18)

[0162] The total strain at node i+1 is equal to the strain ε at yield. y =ε(k0) plus the strain after yielding:

[0163]

[0164] Step 2.5: When the anchor node is located in the anchorage section and is in a state of plastic unloading, solve for the axial stress, shear stress and displacement of each node in the anchorage section under this condition.

[0165] After the anchor bolt yields, further increases in displacement at node i=1 cause the yield point to gradually move towards the free end. On the other hand, as node displacement increases, the anchor bolt-anchoring agent interface reaches its peak shear strength and then undergoes plastic softening. Therefore, when a sufficient number of nodes enter the plastic softening stage, the axial stress of the anchor bolt will decrease.

[0166] According to the stress balance equations of adjacent nodes, we can obtain:

[0167]

[0168] From the bi-exponential curve shear slip relation (2), the shear stress at node i+1 is:

[0169] τ(i+1)=τ r +ae -bu(i+1) -(a+τ r )e -2bu(i+1) (twenty one)

[0170] The total strain at node i+1 is:

[0171] ε(i+1)=ε p +ε e (twenty two)

[0172] Wherein, the elastic strain is:

[0173]

[0174] Its axial displacement satisfies the displacement expression (16) of the anchor rod in the plastic zone.

[0175] Since the anchor bolt node is in the plastic unloading stage, the plastic strain of the node remains unchanged at this time, and its value is the plastic strain value before unloading.

[0176] Step 2.6: When the anchor node is located in the anchorage defect section and is in an elastic state, solve for the axial stress, shear stress and displacement of each node in the defect section under this condition.

[0177] Based on the deformation compatibility relationship, the axial displacement of the anchor at node i is obtained as follows:

[0178]

[0179] The shear stress at each node within the anchorage defect section is 0, and the axial stress at each node is the same and equal to the axial stress of the node immediately adjacent to the defect section within the anchorage section.

[0180] Step 2.7: When the anchor node is located in the anchorage defect section and the anchor is in a plastic loading state, the axial stress of adjacent nodes is equal, the shear stress is 0, and the axial displacement of all nodes in this area satisfies the expression (24) for the axial displacement in the defect section when it is in an elastic state.

[0181] Based on the activity of the yield point k0, its approximate location can be determined by the following two points:

[0182] (1) If the yield point k0 extends from the anchorage section of the following analysis unit across the defect-anchorage transition surface to the anchorage defect area of ​​the current analysis unit, then all nodes in the defect area of ​​the analysis unit will yield simultaneously, and the yield point k0 will directly extend from the anchorage-defect transition surface to the anchorage section of the current analysis unit.

[0183] (2) If the anchorage defect segment node is already in a plastic loading state, then the yield point k0 must be located near the free end of the current calculation node i, and it can be determined that the yield point k0 is in the anchorage segment of the current analysis unit or a previous analysis unit.

[0184] Step 2.8: If the anchor node is located in the anchorage defect section and the anchor is in the plastic unloading state, the axial stress of all adjacent nodes is equal and the shear stress is 0. Therefore, the axial displacement of all nodes satisfies the axial displacement in the defect section when it is in the elastic state. Since the anchor node is in the plastic unloading stage, the plastic strain of the node remains unchanged.

[0185] Step 2.9: After completing the above calculations for one analysis unit, proceed to the next analysis unit. Repeat this process until the last analysis unit is calculated. When node i = n, the current state calculation is complete. Then, continuously increase the displacement u(1) of the free end node i = 1 to simulate the anchor pull-out process. That is, repeat all the steps except step 1 until all anchor nodes are in a slip state and the calculation stops. Here, a large critical value can be set through testing. When u(1) reaches or exceeds its value, the program automatically stops the calculation.

[0186] Step 3: Based on the calculation results of Step 2, the load-displacement curve of the anchor pull-out end node n is finally obtained (e.g., Figure 3 As shown in Table 3, the ultimate pull-out load and peak displacement of a full-length anchor rod with multiple defects can be obtained. The calculated data can be directly used for support design optimization or theoretical analysis. The load is P = σA (σ is the stress, and A is the cross-sectional area of ​​the anchor rod).

[0187] Table 3 Ultimate pull-out load and peak value of full-length anchor bolts under different working conditions

[0188]

[0189] As shown in Table 3, the ultimate pull-out load and peak displacement of anchor bolts are not only related to the plastic hardening of the anchor bolt, but also closely related to the parameters of anchorage defects (including the number, location, and size of defects). It is a consensus in the industry that when the number of anchorage defects increases and the anchor bolt undergoes plastic yielding, its ultimate pull-out load will decrease, but the peak displacement will increase. The anchor bolt pull-out load calculation method of this invention, which considers the nonlinear shear slip behavior of the anchorage interface, the plastic hardening of the anchor bolt, and multiple anchorage defects, differs significantly from other methods and is more consistent with actual engineering and industry consensus. The obtained data is very close to the actual situation and has excellent accuracy.

Claims

1. A method for calculating the pull-out load of a full-length anchor bolt considering multiple anchoring defects, characterized in that, Includes the following steps: Step 1: Obtain anchorage interface bond strength parameters, anchorage mechanical parameters, and anchorage defect parameters through short anchorage pull-out tests, anchorage tensile tests, and non-destructive testing of the anchorage. Obtain the anchorage length L and anchorage diameter. Anchorage interface bond strength parameters include peak shear strength, peak shear displacement, and residual shear strength at the anchorage interface. Anchorage mechanical parameters include elastic modulus, yield strength, yield strain, hardening modulus, and ultimate tensile strain. Anchorage defect parameters include the number, location, and size of anchorage defects. Step 2: Establish a full-length anchorage load transfer mechanical model that considers multiple anchorage defects, nonlinear slippage at the anchorage interface, and plastic hardening of the anchor rod. Based on this full-length anchorage load transfer mechanical model, solve for the axial stress, shear stress, and displacement of each node in the anchorage section and defect section of the anchor rod. The full-length anchorage load transfer mechanical model considering multiple anchorage defects, nonlinear slippage at the anchorage interface, and plastic hardening of the anchor rod is specifically as follows: The anchor rod is divided into a free end and a pull-out end; the full-length anchorage load transfer mechanical model is divided into regions, with the area surrounding the anchor rod wrapped with anchoring agent designated as the anchorage segment and the area without anchoring agent wrapping designated as the defect segment; two transition surfaces are defined: the surface transitioning from the anchorage segment to the defect segment in the direction from the free end to the pull-out end is designated as the anchorage-defect transition surface, and the surface transitioning from the defect segment to the anchorage segment is designated as the defect-anchorage transition surface; an analysis unit is established. The analysis unit includes the anchorage-defect transition surface and the adjacent anchorage and defect sections; the analysis units are connected by the defect-anchorage transition surface; the anchor rod is divided into n nodes, with each pair of nodes forming an anchor rod element with a length of ∆x=L / (n-1); the integer i represents the i-th node, and the first node at the free end is defined as i=1, then the last node at the pull-out end is i=n; in the full-length anchorage load transfer mechanical model, the load-displacement curve at the pull-out end node i=n is obtained by continuously increasing the relative displacement at the free end node i=1; Step 3: Based on the solution results of Step 2, obtain the load-displacement curve, ultimate pull-out load, and peak displacement of the anchor rod with multiple anchoring defects along its entire length, which can be directly used for support design optimization.

2. The method for calculating the pull-out load of a full-length anchor bolt considering multiple anchoring defects according to claim 1, characterized in that, The specific process for solving the load-displacement curve at the pull-out end node i=n is as follows: Based on the full-length anchorage load transfer mechanical model considering multiple anchorage defects, nonlinear slippage at the anchorage interface and plastic hardening of the anchor rod, the load transfer is satisfied by s(x) = u(x), where s(x) is the relative displacement of the anchor node at x, in meters. u(x) represents the axial displacement of the anchor node at position x, in meters (m). Based on the regions defined in the mechanical model of load transfer in the full-length anchor bolt, the fundamental equation for load transfer in the anchor bolt anchorage section is: （1） in, >0, a 0, b 0; The residual shear strength at the anchor bolt-anchoring agent interface is expressed in Pa. ); ; ; a and b are determined by the peak interfacial shear strength. and its corresponding peak interface shear displacement Commonly determined parameters The ratio of residual shear strength to peak shear strength; perimeter of the anchor bolt cross-section. ; Anchor bolt cross-sectional area x represents the position of the node within the anchor bolt, in meters (m). Let x be the shear stress at the anchor bolt-anchoring agent interface, in Pa, and R be the anchor bolt radius, in meters. The elastic modulus of the anchor bolt is expressed in Pa. The expression for the nonlinear bi-exponential curve shear slip relationship of the anchorage interface involved in equation (1) is as follows: （2） In the formula: σ represents the shear stress at the anchor bolt-anchoring agent interface, in Pa; s represents the relative displacement of the anchor bolt. The calculation methods for anchor bolt nodes located in the anchorage section and the defect section are different. As the relative displacement at node i=1 increases, the axial stress at each node in the anchor bolt increases accordingly. When the relative displacement at node i=1 continues to increase, the axial stress at a certain node exceeds the anchor bolt yield load. When the relative displacement at node i=1 is increased, the anchor bolt yields. As the relative displacement at node i=1 is increased, the anchor bolt yield point gradually shifts toward the free end. Based on the state of the anchor bolt node, the axial stress, shear stress, and displacement of the anchor bolt node are calculated. The calculation is performed step by step on the analysis unit, and this cycle is repeated until the last analysis unit is calculated. When node i=n, ​​it indicates that the current state calculation is completed. Then, the displacement u(1) of the free end node i=1 is continuously increased to simulate the anchor bolt pull-out process. The axial stress, shear stress, and displacement of the anchor bolt node under the corresponding process are calculated until all anchor bolt nodes are in the slip state. The calculation is stopped when u(1) reaches the set threshold. The program automatically stops the calculation.

3. The method for calculating the pull-out load of a full-length anchor bolt considering multiple anchoring defects according to claim 2, characterized in that, When the anchor node is located in the anchoring section, the axial stress, shear stress and displacement are calculated according to its state. 1) When the anchor node is located in the anchorage section and is in an elastic state, solve for the axial stress, shear stress and displacement of each node in the anchorage section under this state; Using numerical analysis, the differential equation (1) is transformed into a difference equation to obtain the displacement relationship between adjacent node elements; the displacement estimation formula for the next node is then obtained: （3） In the formula: i-1, i, i+1 represent three adjacent nodes, x(i-1), x(i), x(i+1) represent the coordinates of nodes i-1, i, i+1 respectively, and u(i-1), u(i), u(i+1) represent the displacements of nodes i-1, i, i+1 respectively; For node i=1, u(1) is the input value; node i=1 is located at the free end, so the axial stress here is... With shear stress If so, then the strain is also 0; For node i=2, the displacement relationship between node i=1 and node i=2 can be obtained from equation (1): （4） The displacements of other nodes are obtained sequentially based on equation (3); The shear stress at node i is obtained from the nonlinear bi-exponential curve shear slip relation (2) of the anchorage interface: （5） Based on the stress balance analysis of the nodes, the axial stress at node i is: （ ） （6） According to Hooke's law, the strain at node i is: （ ） （7） In the formula: Let be the axial stress at node i, in Pa. Let the total strain be at node i. For the elastic strain of node i; 2) When the anchor bolt node is located in the anchorage section and is under plastic loading, solve for the axial stress, shear stress and displacement of each node in the anchorage section under this state; If the anchor bolt yields, assume that the calculated axial stress at the node k preceding the yield point is less than the anchor bolt's yield strength. The axial stress at node k+1 is greater than the yield strength of the anchor bolt, and the yield point is... Between nodes k and k+1, nodes 1 to k are the elastic region, and nodes k+1 to n are the plastic region. These are additional yield points inserted between adjacent nodes k and k+1, and are not included in the original node numbering. In the elastic zone of the anchor bolt, the nodal displacement, shear stress, axial stress, and strain are obtained by equations (3), (5), (6), and (7), respectively. For yield point The node k preceding the yield point and the yield point The stress-strain relationship determines the location of the yield point x ( ): （8） The yield point is obtained from the displacement of adjacent nodes using formula (3). The displacement is: )（9） The axial stress at the yield point is the yield load. ; According to the nonlinear bi-exponential curve shear slip relationship (2) of the anchorage interface, the shear stress at the yield point : （10） Elastic strain at yield point Determined by Hooke's Law: = （11） In the formula: The yield point location is in meters (m). This is the yield point displacement, in meters (m). The yield point strain; The yield point shear stress is expressed in Pa. The anchor bolt node is located in the plastic zone of the anchor bolt. First, determine the location of the node within the plastic zone and the anchor bolt yield point. The axial stress, shear stress, and displacement of adjacent node k+1 are calculated, and then the axial stress, shear stress, and displacement of each node in the plastic zone are calculated sequentially along the direction from the free end of the anchor rod to the pull-out end. The nodal displacements in the plastic zone are obtained from equation (3): （12） The axial stress at node k+1 is obtained from the stress balance equations of adjacent nodes: （13） From the nonlinear bi-exponential curve shear slip relation (2) of the anchorage interface, the shear stress at node k+1 is: （14） The total strain at the joint in the plastic zone is the sum of the elastic strain and the plastic strain: （15） In the formula, E T Secant modulus; For nodes k+1 to n, the axial displacement of the anchor at node i+1 is obtained according to the deformation compatibility relationship: （16） From the stress balance equations of adjacent nodes, the axial force of the anchor at node i+1 is: （17） From the nonlinear bi-exponential curve shear slip relation (2) of the anchorage interface, the shear stress at node i+1 is: （18） The total strain at node i+1 is equal to the strain at yield. Including the strain after yielding: （19） 3) When the anchor bolt node is located in the anchorage section and is in a plastic unloading state, solve for the axial stress, shear stress and displacement of each node in the anchorage section under this state; After the anchor bolt yields, as the displacement at node i=1 continues to increase, the yield point of the anchor bolt gradually moves towards the free end. On the other hand, as the node displacement increases, the anchor bolt-anchoring agent interface reaches its peak shear strength and then undergoes plastic softening. When a sufficient number of nodes have their anchor bolt-anchoring agent interfaces enter the plastic softening stage, the axial stress of the anchor bolt decreases. According to the stress balance equations of adjacent nodes, we get: （20） The shear stress at node i+1 is obtained from the nonlinear bi-exponential curve shear slip relation (2) of the anchorage interface: （21） The total strain at node i+1 is: （22） In the formula, Plastic strain; Where the elastic strain is: （23） The axial displacement at node i+1 satisfies the displacement expression (16) of the anchor rod in the plastic zone; Since the anchor bolt node is in the plastic unloading stage, the plastic strain of the node remains unchanged at this time, and its value is the plastic strain value before unloading.

4. The method for calculating the pull-out load of a full-length anchor bolt considering multiple anchoring defects according to claim 2 or 3, characterized in that, When the anchor node is located in the anchorage defect section, the axial stress, shear stress and displacement are calculated according to its state. 1) When the anchor node is located in the anchorage defect section and is in an elastic state, solve for the axial stress, shear stress and displacement of each node in the defect section under this state; Based on the deformation compatibility relationship, the axial displacement of the anchor at node i is obtained as follows: （24） The shear stress of each node in the anchorage defect section is 0, and the axial stress of each node is the same and equal to the axial stress of the node located in the anchorage section that is adjacent to the defect section. 2) When the anchor node is located in the anchorage defect section and the anchor is in a plastic loading state, the axial stress of adjacent nodes is equal, the shear stress is 0, and the axial displacement of all nodes in the defect section area satisfies the expression (24) of the axial displacement in the defect section when it is in an elastic state. According to yield point The location of an activity can be determined by the following two points: (a) When the yield point If the anchorage segment of a subsequent analysis element extends beyond the defect-anchorage transition surface into the anchorage defect region of the current analysis element, then all nodes in the defect region of that analysis element will simultaneously yield, and the yield point will be... Extend directly across the anchorage-defect transition surface to the anchorage segment of the current analysis element; (b) When the anchorage defect segment node is already in a state of plastic loading, the yield point Located near the free end of the current computation node i, and at the yield point It is located in the anchoring segment of the current analysis unit or a previous analysis unit; 3) When the anchor node is located in the anchorage defect section and the anchor is in the plastic unloading state, the axial stress of all adjacent nodes is equal and the shear stress is 0. Therefore, the axial displacement of all nodes satisfies the axial displacement in the defect section when it is in the elastic state. Since the anchor node is in the plastic unloading stage, the plastic strain of the node remains unchanged.

5. The method for calculating the pull-out load of a full-length anchor bolt considering multiple anchoring defects as described in claim 1, characterized in that, Based on the calculation results of step 2, the load-displacement curve of the anchor pull-out end node n is finally obtained, which is the final load-displacement curve of the full-length anchor bolt with multiple anchoring defects. , Given stress and the cross-sectional area of ​​anchor rod A, the ultimate pull-out load and peak displacement of a full-length anchor rod with multiple anchoring defects are obtained, which can be directly used for support design optimization or theoretical analysis.

6. The method for calculating the pull-out load of a full-length anchor bolt considering multiple anchoring defects according to claim 2, characterized in that, When there are no defects at the pull-out end, the last anchoring section is used as the last analysis unit.

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