A method for determining the optimal pull-out strength of anchor bolts in reinforced slopes

Abstract

A method for determining the optimal pull-out strength of anchor bolts in reinforced slopes, belonging to the field of slope reinforcement technology, includes: establishing a mathematical model of the anchor bolt-reinforced slope; obtaining curves showing the relationship between the slope safety factor and the pull-out strength of anchor bolts under different numbers and bond lengths; obtaining curves showing the variation of slope failure probability and anchor bolt pull-out strength under different numbers and bond lengths; and obtaining the optimal design value of the pull-out strength of anchor bolts in the reinforced slope. This method introduces the failure probability into anchor bolt support engineering, reflecting the reliability of the slope anchor bolt support engineering through the failure probability; and effectively obtains the relationship between pull-out strength and the number of anchor bolts, bond strength and slope stability coefficient. In slope reinforcement engineering, the pull-out characteristics of anchor bolts are fully utilized. The optimal pull-out strength corresponds to the number of anchor bolts and the bond strength, avoiding waste of engineering materials and improving the economy of the support engineering while ensuring slope safety.

Description

Technical Field

[0001] This invention belongs to the field of stability evaluation and disaster prevention technology for reinforced slopes, and specifically relates to a method for determining the optimal pull-out strength of anchor bolts in reinforced slopes. Background Technology

[0002] Currently, anchor bolt reinforcement primarily relies on calculations to determine the required anchoring force, then uses that force to calculate the necessary number of anchor bolts and bond length, and finally verifies the pull-out strength to adjust the bond length. This method yields only a single set of anchor bolt numbers and bond lengths, and it doesn't directly consider pull-out strength during the calculation process. Therefore, the calculated pull-out strength is not optimal, and it fails to effectively determine the relationship between pull-out strength, anchor bolt quantity, and bond strength. In slope reinforcement projects, the optimal pull-out strength of the anchor bolts is not fully utilized, resulting in a certain degree of waste of engineering materials. Summary of the Invention

[0003] To address the above shortcomings, the purpose of this invention is to provide a method for determining the optimal pull-out strength of anchor bolts in reinforced slopes.

[0004] The technical solution adopted in this invention is: a method for determining the optimal pull-out strength of anchor bolts in reinforced slopes, the key technical points of which include the following steps:

[0005] A mathematical model for anchor-reinforced slopes was established. Different numbers of anchors were selected, and different anchor bonding lengths were designed for each number. Under the corresponding number of anchors and bonding lengths, different anchor pull-out strengths were obtained. The slope safety factor corresponding to different anchor pull-out strengths was calculated using the limit equilibrium method. This calculation process was repeated by changing the number of anchors and bonding length until a curve showing the relationship between the slope safety factor and anchor pull-out strength for different numbers of anchors and bonding lengths was obtained. Design values ​​for the anchor pull-out strength of all slopes with a safety factor greater than 1 under different numbers of anchors and bonding lengths were selected. Failure samples of the slopes were obtained using Monte Carlo sampling combined with response surface methodology. The slope failure probability corresponding to the selected design values ​​of the anchor pull-out strength was calculated, and the variation curves of slope failure probability and anchor pull-out strength under different numbers of anchors and bonding lengths were obtained. Based on these variation curves, the optimal design value for the anchor pull-out strength of the reinforced slope was determined.

[0006] In the above scheme, the process of determining the relationship curve between the slope safety factor and the anchor pull-out strength includes the following steps:

[0007] (1) Establish a mathematical model for anchor bolt-reinforced slopes and design them respectively. X and Y The required number of anchor bolts and the anchor bolt bonding length are represented as A1, A2, A3, ... A XB1, B2, B3, ... B Y , give Z The design values ​​of the pull-out strength of each anchor bolt are denoted as C1, C2, C3, ..., C. Z ;

[0008] (2) For a certain required number of anchor bolts A i Bonding length B of anchor bolt j Next, change the pull-out strength of the anchor bolt to obtain C1, C2, C3, ... C Z The corresponding slope safety factors FS 1, FS 2, FS 3...... FS Z , using (C i , FS i ) i =1,2,…Z, plot the slope safety factor as a function of different anchor pull-out strengths. Keep the number of anchors constant, and repeat the above calculation by changing the required anchor bonding length until the slope safety factor as a function of different anchor pull-out strengths under different anchor bonding lengths is obtained. Then change the number of anchors and continue to repeat the above calculation process to finally obtain the slope safety factor as a function of different anchor pull-out strengths under different anchor numbers and anchor bonding lengths.

[0009] In the above scheme, the steps to obtain the curve of slope failure probability versus anchor pull-out strength are as follows:

[0010] (1) Given a certain required number of anchor bolts A i Bonding length B of anchor bolt j Below, in Z The slope safety factor was selected one by one from the design values ​​of the pull-out strength of each anchor bolt. FS Values ​​greater than 1, and retrieve the corresponding values. S The pull-out strength values ​​of the anchor bolts are C1, C2, C3, ..., C. S By changing the required number of anchor bolts and the anchor bolt bonding length, the above operation is repeated until different numbers of anchor bolts and anchor bolt bonding lengths are obtained. Z The slope safety factor corresponding to the design value of the pull-out strength of each anchor bolt FS Values ​​greater than 1, the number of times each is obtained is denoted as follows: S 1, S 2, S 3,...... S r , r = X × YFor different numbers and bond lengths of anchor bolts, the slope safety factor corresponding to the selected anchor bolt pull-out strength design value is established one by one. FS The response surface function between the soil strength parameters;

[0011] (2) In a certain number A of anchor bolts i Anchor bolt bonding length B j Next, selected S The pull-out strength values ​​of the anchor bolts are C1, C2, C3, ..., C. S For each selected anchor pull-out strength value, the soil parameters within the slope, including cohesion c and internal friction angle, are... φ Treat as random variables ， Generate using Matlab software N Soil layer parameter samples T1, T2, ..., T N The established response surface function is used to calculate T1, T2, ..., T N Corresponding safety factor FS 1, FS 2,……, FS N ;like F S i If <1, then T is called T i For invalid samples, repeat this process to obtain a total of [number] samples. W Calculate the failure probability of the slope from an unstable sample. P fi = W / N A total of corresponding S Slope failure probability P f1 , P f2 , P f3 ,...... P fS ,use( C i , P fi ) i =1,2,… S Plot the curve showing the relationship between the slope failure probability and the pull-out strength of the selected anchor bolts;

[0012] (3) Maintain the number of anchor bolts A i Keeping the anchor length constant, change the bonding length of the anchor bolt and repeat step (2) to calculate and obtain different anchor bolt bonding lengths B1, B2, B3, ... B Y Below is the curve showing the relationship between the slope failure probability and the pull-out strength of the selected anchor bolts;

[0013] (4) By changing the number of anchor bolts, repeat steps (2) and (3) until different numbers of anchor bolts A1, A2, A3, ... A X Bonding lengths B1, B2, B3, ... B of different anchor bolts Y Below is the curve showing the relationship between the slope failure probability and the pull-out strength of the selected anchor bolt.

[0014] In the above scheme, the process of obtaining the optimal design value of the pull-out strength of the reinforced slope anchor rod based on the change curve is as follows:

[0015] The optimal value of the anchor pull-out strength in the reinforced slope is determined by consulting the anchor pull-out strength relationship curve.

[0016] The beneficial effects of this invention are as follows: The method for determining the optimal pull-out strength of anchor bolts in reinforced slopes includes: establishing a mathematical model of the anchor bolt-reinforced slope; obtaining a curve showing the relationship between the slope safety factor and the pull-out strength of anchor bolts under different numbers and bonding lengths; obtaining a curve showing the variation of the slope failure probability and the pull-out strength of anchor bolts under different numbers and bonding lengths; and obtaining the optimal design value of the pull-out strength of anchor bolts for reinforced slopes based on the variation curves. This method introduces the concept of failure probability into anchor bolt support engineering, reflecting the reliability of slope anchor bolt support engineering through failure probability. By taking pull-out strength into account, the relationship between pull-out strength and the number of anchor bolts, bonding strength and slope stability coefficient is effectively obtained. In slope reinforcement engineering, the pull-out characteristics of anchor bolts are fully utilized. At the optimal pull-out strength, the number of anchor bolts and bonding strength correspond, avoiding waste of engineering materials and improving the economy of support engineering while ensuring slope safety. Attached Figure Description

[0017] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0018] Figure 1 This is a flowchart illustrating a method for determining the optimal pull-out strength of anchor bolts in reinforced slopes according to the present invention.

[0019] Figure 2 This is a model diagram of a reinforced slope (two anchor rods) in an embodiment of the present invention.

[0020] Figure 3 This is a model diagram of a reinforced slope (three anchor rods) in an embodiment of the present invention.

[0021] Figure 4 This is a model diagram of a reinforced slope (four anchor rods) in an embodiment of the present invention.

[0022] Figure 5 This is a graph showing the slope safety factor versus anchor pull-out strength curves for the number and bonding length of two anchor rods in an embodiment of the present invention.

[0023] Figure 6 This is a graph showing the slope safety factor versus anchor pull-out strength curves for the number of three anchor rods and their bonding length in an embodiment of the present invention.

[0024] Figure 7 This is a graph showing the slope safety factor versus anchor pull-out strength curves for the number of four anchor rods and their bonding length in an embodiment of the present invention.

[0025] Figure 8 This is a graph showing the slope failure probability versus anchor pull-out strength curves for the number of two anchor rods and their bonding length in an embodiment of the present invention.

[0026] Figure 9 This is a graph showing the slope failure probability versus anchor pull-out strength curves for the number of three anchor rods and their bonding length in an embodiment of the present invention.

[0027] Figure 10 This is a graph showing the slope failure probability and anchor pull-out strength curves under the number of four anchor rods and their bonding length in an embodiment of the present invention. Detailed Implementation

[0028] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the following description is provided in conjunction with the accompanying drawings. Figures 1-10 The present invention will be further described in detail below with reference to specific embodiments.

[0029] In this embodiment, the slope to be reinforced has a height of 10m and a foundation thickness of 5m. The slope is composed of silty clay with an average soil cohesion (c) of 7 kPa, a coefficient of variation of 0.3, and a standard deviation of 2.1. The average internal friction angle (φ) is 22°, with a coefficient of variation of 0.3 and a standard deviation of 6.6. The unit weight (γ) is 20 kN / m³. Figure 2 , 3 Figures 4 and 5 represent slope model diagrams corresponding to reinforcement with two, three, and four anchor rods, respectively.

[0030] The method for determining the optimal pull-out strength of anchor bolts in reinforced slopes according to this embodiment is as follows: Figure 1 As shown, the specific steps are as follows:

[0031] A numerical model of an anchor-reinforced slope was established using Geo-studio software. Three anchor reinforcement quantities were designed: A1=2, A2=3, and A3=4. For each anchor reinforcement quantity, three anchor bonding lengths were selected: B1=1m, B2=2m, and B3=3m. For each anchor bonding length, seven anchor pull-out strengths were selected: C1=40kPa, C2=50kPa, C3=60kPa, C4=70kPa, C5=80kPa, C6=90kPa, and C7=100kPa. Under different anchor reinforcement quantities and anchor bonding lengths, the safety factor corresponding to the pull-out strength of the seven selected anchors was calculated using the limit equilibrium method. (A...) i B j C k ) i =1,2,3 j =1,2,3 k =1,2,3...7, plot the slope safety factor versus anchor pull-out strength curves for different anchor reinforcement quantities and anchor bonding lengths, as shown below. Figure 5 , 6 As shown in Figures 7 and 8.

[0032] Among the three anchor reinforcement quantities Ai and three anchor bonding lengths Bj, the design values ​​of the pull-out strength of the anchors corresponding to slope safety factors FS greater than 1 were selected one by one. These are: four anchor pull-out strength design values ​​corresponding to the number of anchors A1=2 and the anchor bonding length B3=3m: (2, 3m, 70kPa), (2, 3m, 80kPa), (2, B3, 90kPa), (2, 3m, 100kPa); and five anchor pull-out strength design values ​​corresponding to the number of anchors A2=3 and the anchor bonding length B2=2m: (3, 2m, 60kPa), (3, 2m, 70kPa), (3, 2m, 80kPa), (3, 2m, 90kPa). (3, 2m, 100kPa); 7 anchor pull-out strength design values ​​corresponding to the number of anchors A2=3 and the anchor bonding length B3=3m: (3, 3m, 40kPa), (3, 3m, 50kPa), (3, 3m, 60kPa), (3, 3m, 70kPa), (3, 3m, 80kPa), (3, 3m, 90kPa), (3, 3m, 100kPa); 4 anchor pull-out strength design values ​​corresponding to the number of anchors A3=4 and the anchor bonding length B1=1m. The design values ​​for the pull-out strength of the seven anchor bolts are: (4, 1m, 70kPa), (4, 1m, 80kPa), (4, 1m, 90kPa), (4, 1m, 100kPa); The anchor bolt quantity is A3=4, and the anchor bolt bonding length is B2=2m. The design values ​​are: (4, 2m, 40kPa), (4, 2m, 50kPa), (4, B2, 60kPa), (4, 2m, 70kPa), (4, 2m, 50kPa), (4, 2m, 90kPa), (4, 2m, 100kPa). (kPa); Seven anchor pull-out strength design values ​​corresponding to the number of anchors A3=4 and the anchor bond length B3=3m are established (4, 3m, 40kPa), (4, 3m, 50kPa), (4, 3m, 60kPa), (4, 3m, 70kPa), (4, 3m, 80kPa), (4, 3m, 90kPa), (4, 3m, 100kPa); Response surface functions between the safety factor FS corresponding to the selected anchor pull-out strength design value and the slope soil shear strength parameters are established one by one.

[0033] In each of the selected anchor pull-out strength design values ​​mentioned above, considering the uncertainty of the slope soil shear strength parameters, its cohesion is... c and internal friction angle φ Treating it as a log-normal random variable, it is randomly generated using Matlab software. N =10 5 Let there be random samples T1, T2, ..., T 100000 Then, the established response surface function is used to calculate 10. 5 The slope safety factor corresponding to each random sample is denoted as . FS 1, FS 2,……, FS 10000 .exist N =10 5 Among the safety factors, if the safety factor of the slope FS i If <1, then T is called T i For failed samples, a total of [number] samples were obtained. W One failure sample, calculated according to the failure probability formula. P fi = W / 10 5 The failure probabilities corresponding to the design pull-out strength of all selected anchor bolts are obtained, and the calculation results are as follows: Figure 8 , 9 As shown in Figure 10. When the number of anchor bolts is A1=2 and the anchor bolt bonding length is B3=3m, the failure probabilities corresponding to the pull-out strength design values ​​of the four anchor bolts are 42.41%, 23.70%, 21.71%, and 20.43%, respectively; when the number of anchor bolts is A2=3 and the anchor bolt bonding length is B2=2m, the failure probabilities corresponding to the pull-out strength design values ​​of the five anchor bolts are 28.83%, 7.38%, 1.30%, 0.93%, and 0.67%, respectively; when the number of anchor bolts is A2=3 and the anchor bolt bonding length is B3=3m, the failure probabilities corresponding to the pull-out strength design values ​​of the seven anchor bolts are 30.47%, 4.19%, 2.91%, 1.41%, 0.82%, 0.75%, and 0.58%, respectively; when the number of anchor bolts is A1=2 ... four anchor bolts are 42.41%, 23.70%, 21.71%, and 2 When the number of anchors is A3=4 and the anchor bonding length is B1=1m, the failure probabilities corresponding to the four anchor pull-out strength design values ​​are 45.09%, 19.98%, 7.43%, and 1.31%, respectively. When the number of anchors is A3=4 and the anchor bonding length is B2=2m, the failure probabilities corresponding to the seven anchor pull-out strength design values ​​are 20.07%, 1.34%, 0.45%, 0.14%, 0.09%, 0.07%, and 0.06%, respectively. When the number of anchors is A3=4 and the anchor bonding length is B3=3m, the failure probabilities corresponding to the seven anchor pull-out strength design values ​​are 0.39%, 0.25%, 0.20%, 0.13%, 0.11%, 0.08%, and 0.05%, respectively. Under the anchor pull-out strength design values ​​selected with different numbers of anchors and anchor bonding lengths, using (C i , P fi Plot the curves showing the relationship between slope failure probability and anchor pull-out strength. Based on the required number of anchors, anchor bond length, and expected failure probability, query... Figure 8-10The relationship curve is used to determine the optimal pull-out strength of the anchor bolts in the reinforced slope. For example, when the required number of anchor bolts and the anchor bolt bonding length are 4 and 2m respectively, the expected failure probability is 0.45%, and the optimal pull-out strength design value of the anchor bolts in the reinforced slope is 60kPa.

[0034] Therefore, comparisons reveal that the impact of anchor pull-out strength on slope stability is often overlooked, leading to insufficient consideration of slope safety. This invention, by designing different numbers and bonding lengths of anchors and selecting different design values ​​for anchor pull-out strength, obtains curves showing the slope safety factor as a function of anchor pull-out strength under different numbers and bonding lengths. Then, according to specifications, design values ​​for anchor pull-out strength corresponding to slope safety factors greater than 1 are selected for different numbers and bonding lengths. Considering the uncertainty of soil parameters, response surface methodology is used to calculate the slope failure probability as a function of the selected anchor pull-out strength design values ​​under different numbers and bonding lengths. Finally, the optimal value for anchor pull-out strength in the reinforced slope is obtained based on the required number of anchors, anchor bonding length, and expected failure probability. Engineering examples have verified that when the number of anchor bolts and the bonding length are constant, the slope failure probability decreases as the pull-out strength of the anchor bolts increases. This fully verifies the influence of the pull-out strength of the anchor bolts on the slope stability. Finally, the optimal pull-out strength of the anchor bolts in the reinforced slope can be determined by consulting the curve based on the required number of anchor bolts, the anchor bolt bonding length, and the expected failure probability.

[0035] In practical engineering, after calculation, the optimal number of anchor bolts and bond strength corresponding to the pull-out strength can be obtained, which can make the slope reinforcement effect more economical.

[0036] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for determining the optimal pull-out strength of anchor bolts in reinforced slopes, characterized in that, Includes the following steps: A mathematical model for anchor-reinforced slopes was established. Different numbers of anchors were selected, and different anchor bonding lengths were designed for different numbers of anchors. Under the corresponding number of anchors and bonding lengths, different anchor pull-out strengths were obtained. The slope safety factor corresponding to different anchor pull-out strengths was calculated using the limit equilibrium method. By changing the number of anchors and bonding length, the above calculation process was repeated until the relationship curve between the slope safety factor and the anchor pull-out strength under different numbers of anchors and bonding lengths was obtained. We selected the design values ​​of the pull-out strength of anchor bolts corresponding to slope safety factors greater than 1 under different anchor bolt numbers and bonding lengths. We used Monte Carlo sampling combined with response surface methodology to obtain slope failure samples and calculated the slope failure probability corresponding to the selected design values ​​of the pull-out strength of the anchor bolts. We obtained the variation curves of slope failure probability and anchor bolt pull-out strength under different anchor bolt numbers and bonding lengths. Based on the variation curves, we obtained the optimal design value of the pull-out strength of the anchor bolts for the reinforced slope. The process of determining the relationship curve between the slope safety factor and the anchor pull-out strength includes the following steps: (1) Establish a mathematical model for anchor bolt-reinforced slopes and design them respectively. X and Y The required number of anchor bolts and the anchor bolt bonding length are represented as A1, A2, A3, ... A X B1, B2, B3, ... B Y , give Z The design values ​​of the pull-out strength of each anchor bolt are denoted as C1, C2, C3, ..., C. Z ; (2) For a certain required number of anchor bolts A i Bonding length B of anchor bolt j Next, change the pull-out strength of the anchor bolt to obtain C1, C2, C3, ... C Z The corresponding slope safety factors FS 1, FS 2, FS 3...... FS Z , using (C i , FS i ) i =1,2,…Z, plot the slope safety factor with different anchor pull-out strengths, keep the number of anchors constant, change the required anchor bonding length, repeat the above calculation until the slope safety factor with different anchor pull-out strengths under different anchor bonding lengths is obtained, then change the number of anchors and continue to repeat the above calculation process to finally obtain the slope safety factor with different anchor pull-out strengths under different anchor numbers and anchor bonding lengths; The steps for obtaining the curves showing the relationship between slope failure probability and anchor pull-out strength are as follows: (1) Given a certain required number of anchor bolts A i Bonding length B of anchor bolt j Below, in Z The slope safety factor was selected one by one from the design values ​​of the pull-out strength of each anchor bolt. FS Values ​​greater than 1, and retrieve the corresponding values. S The pull-out strength values ​​of the anchor bolts are C1, C2, C3, ..., C. S By changing the required number of anchor bolts and the anchor bolt bonding length, the above operation is repeated until different numbers of anchor bolts and anchor bolt bonding lengths are obtained. Z The slope safety factor corresponding to the design value of the pull-out strength of each anchor bolt FS Values ​​greater than 1, the number of times each is obtained is denoted as follows: S 1, S 2, S 3,...... S r , r = X × Y For different numbers and bond lengths of anchor bolts, the slope safety factor corresponding to the selected anchor bolt pull-out strength design value is established one by one. FS The response surface function between the soil strength parameters; (2) In a certain number A of anchor bolts i Anchor bolt bonding length B j Next, selected S The pull-out strength values ​​of the anchor bolts are C1, C2, C3, ..., C. S For each selected anchor pull-out strength value, the soil parameters within the slope, including cohesion c and internal friction angle, are... φ Treat as random variables ， Generate using Matlab software N Soil layer parameter samples T1, T2, ..., T N The established response surface function is used to calculate T1, T2, ..., T N Corresponding safety factor FS 1, FS 2,……, FS N ;like F S i If <1, then T is called T i For invalid samples, repeat this process to obtain a total of [number] samples. W Calculate the failure probability of the slope from an unstable sample. P fi = W / N A total of corresponding S Slope failure probability P f1 , P f2 , P f3 ,...... P fS ,use( C i , P fi ) i =1,2,… S Plot the curve showing the relationship between the slope failure probability and the pull-out strength of the selected anchor bolts; (3) Maintain the number of anchor bolts A i Keeping the anchor length constant, change the bonding length of the anchor bolt and repeat step (2) to calculate and obtain different anchor bolt bonding lengths B1, B2, B3, ... B Y Below is the curve showing the relationship between the slope failure probability and the pull-out strength of the selected anchor bolts; (4) By changing the number of anchor bolts, repeat steps (2) and (3) until different numbers of anchor bolts A1, A2, A3, ... A X Bonding lengths B1, B2, B3, ... B of different anchor bolts Y Below is the curve showing the relationship between the slope failure probability and the pull-out strength of the selected anchor bolt.

2. The method for determining the optimal pull-out strength of anchor bolts in reinforced slopes as described in claim 1, characterized in that, The process of obtaining the optimal design value of the pull-out strength of the reinforced slope anchor based on the variation curve is as follows: The optimal value of the anchor pull-out strength in the reinforced slope is determined by consulting the anchor pull-out strength relationship curve.

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