The randomness of rock joint development is an important factor in the uncertainty of geotechnical engineering stability. In this study, a method is proposed to evaluate the reliability of intermittent jointed rock slope. The least squares support vector machine (LSSVM) evolved by a bacterial foraging optimization algorithm (BFOA) is used to establish a response surface model to express the mapping relationship between the intermittent joint parameters and the slope safety factor. The training samples are obtained from the numerical calculation based on the joint finite element method during this process. Considering the randomness of the intermittent joint parameters in the actual project, each parameter is evaluated at different locations on the site, and its distribution characteristics are counted. According to these statistical results, a large number of parameter combinations are obtained through Monte Carlo sampling. The trained machine learning mapping model is used to obtain the slope safety factor corresponding to each group, and these results are then used to obtain the slope reliability. When the research results were applied to slope disaster treatment along the Yalu River in China’s Jilin Province, it was found that the joint length and joint inclination angle both play key roles in rock slope stability, which should receive more attention in the slope treatment. In summary, this study establishes a method for evaluating the reliability of intermittent jointed rock slope based on an evolutionary SVM model, and its feasibility is verified by engineering application.

Evaluation of slope stability should always be considered in geotechnical engineering such as tunnel portal sections, foundation pit slope releases and highway subgrade. Due to the influence of weathering, geological movement and other natural factors, the slope often contains intermittent joints. Variations in joint length, joint inclination angle and other parameters result in uncertainty of the slope stability [

Researchers often try to obtain rock mass parameters through sampling tests to judge the stability of geotechnical engineering [

Machine learning algorithms are continually being developed and applied to express complex mapping relations in geotechnical engineering, such as support vector machine (SVM) [

Most current slope reliability studies assume that the joints are continuous [

Compared with other machine learning models, least squares support vector machine (LSSVM) has faster calculation speed and a more concise theoretical model, which can be applied to data learning under the condition of small samples and is suited for reliability analysis of jointed slopes. To resolve the problem of kernel parameter dependence, which frequently occurs in LSSVM, this study uses the BFOA strategy with good global optimization ability. The joint finite element method is used to simulate the intermittent jointed slope model and to form the learning samples by parameter orthogonal design. Next, the BFOA-LSSVM is trained to express the mapping relationship between the slope joint parameters and the safety factor and is used to replace the finite element in order to speed up the calculation. A large number of parameter groups are established by Monte Carlo sampling, the slope safety factor of each group is obtained by BFOA-LSSVM, and the results were statistically analyzed to obtain the slope reliability.

Due to the randomness of joint parameters, the reliability evaluation of jointed rock slopes is very complicated, and the efficiency of numerical analysis is low. In this study, the BFOA-LSSVM was trained based on the data samples obtained from the joint finite element method, in order to perform the rapid evaluation of the intermittent joint slope reliability.

Reliability is a probabilistic measure of a structure’s ability to perform a predetermined function at a specified time and under specified conditions [

where

where

The relationship between joint parameters and the slope safety factors expressed by

Regression calculation of LSSVM is the process of fitting known data through a hyperplane [

where kernel function

where

LSSVM offers the advantages of simple structure and fast calculation speed, but at the same time, it is also faced with the problem of parameter dependence, which is frequently found in intelligent algorithms. The BFOA strategy is used to find the optimal solution for the parameters of LSSVM in order to obtain a more accurate mapping model.

The BFOA [

(1) Set the BFOA initialization parameters, include the size of the bacterial community _{ed}

(2) Generate initial population randomly in the parameter optimization interval. Each bacterium represents a set of LSSVM parameters (

(3) Divide the learning samples into training samples and testing samples. The current position of each bacterium is taken as the individual value, and then the LSSVM model is substituted for regression prediction to obtain the corresponding safety factor.

(4) Evaluate the adaptive value of the current population through

where

For the expected adaptive value

(5) Adjust the current population with a single step _{c}

(6) Record the position with the minimum fitness value in Step (5), sort the population at that position according to individual fitness value, delete half of the individuals with the larger fitness value and copy the other half to keep the population quantity unchanged. Return to Step (4) until Step (6) is repeated

(7) Delete the current population and return to Step (2). If Step (7) has been performed

(8) The optimal parameter (

The above process obtains a complete BFOA-LSSVM model, but before using it to build the response surface function, it is necessary to prepare the corresponding learning samples through joint finite element calculation.

Numerical calculation models of fractures can be divided into discrete element discontinuous models [

The RS2 finite element software package developed by Canadian company Rocscience was selected as the calculation platform. The jointed rock mass is regarded as a binary structure composed of rock blocks and joints, in which the joint part is simulated by the Goodman unit.

The Goodman unit is a thickness-free unit, which is often used to simulate the soft structural plane in rock mass. It proposes four nodes without thickness and eight degrees of freedom units (as shown in

where

The mechanical properties of friction strength and deformation at the contact surface of the element under external force are described by strength and deformation parameters. Therefore, the Goodman unit can simulate the tangential force and deformation characteristics of joints. RS2 finite element software corrected the Goodman element’s defect of embedding both sides of the contact surface into each other under pressure, allowing the joint parts to dislocate each other, resulting in tangential and normal displacement and plastic yield of the joint part.

The orthogonal design of joint parameters is carried out through engineering investigation to ensure that the design samples can cover the range of the actual parameters. The slope safety factor is obtained by the strength reduction method based on joint finite element calculation, with the two critical conditions being displacement mutation and shear zone transfixion, and the Mohr-Coulomb failure criterion used for both rock blocks and joints. The strength reduction formula is shown in

where

Compared with the limit equilibrium method, the strength reduction method is more suited for the intermittent jointed slope, because it can reflect the stress-strain relation of the slope rock mass, and it does not need to manually set the range of slip surface.

According to the orthogonal design samples of the parameters, the learning samples are formed by the finite element method calculation. This sample is used to train BFOA-LSSVM to establish the response surface between the intermittent joint parameters and the slope safety factor. For an engineering structure that needs to be evaluated, each intermittent joint parameter is evaluated at different locations on the site and its distribution characteristics are counted. The Monte Carlo sampling of these parameters is performed by computer programming, forming a large number of parameter groups, using the trained BFOA-LSSVM model to obtain the slope safety factor corresponding to each group, and then using these results to obtain the slope reliability.

Next, the reliability evaluation model for the intermittent joint slope is established. The logical relationship and calculation process are shown in

The highway from Dandong to Altay in Jilin Province of China has many potential risks in the area of the slope along the Yalu River. In order to ensure the safety of highway operations, a comprehensive slope safety assessment is planned in this area. Using section K235 + 800 ~ K235 + 950 as an example, the slopes of these areas are about 0.5 m away from the left edge of the road. The main body of the slope is composed of strongly weathered basalt, with blocky structure and relatively developed joints and fissures. The engineering position and the actual state of the slope body are shown in

Combined with the geological survey and field investigation results, the distribution range of joint parameters in this section is joint inclination angle

In order to obtain the learning samples required for LSSVM training, an orthogonal design of geometric parameters of intermittent joints is carried out based on the survey results, forming a total of 35 orthogonal test pieces, as shown in Appendix A. Orthogonal design can fully consider the combination of different factors and solve the corresponding slope safety factor from the orthogonal sample through finite element calculation, so as to establish a relatively complete learning sample.

Because the interval in this case study was relatively short and the distribution range of joint parameters was relatively concentrated, the orthogonal design sample was determined to be 35 groups. In the actual application process, when the target interval is long or the joint parameter distribution is relatively scattered, the number of orthogonal samples should be correspondingly expanded to meet the training needs of the machine learning model.

An engineering numerical calculation model was built according to the field survey drawings. The bottom and both sides of the model are fully constrained, while the top part is the free boundary. The entire slope is divided by free triangle elements, and the dimensions of each structure are shown in

In order to record the slope deformation, four displacement measure points were uniformly arranged on the slope surface. In addition to the joint geometric parameters, the rock mass and joint mechanical parameters were sampled on-site and obtained through laboratory tests, as shown in

Rock mass parameters | Joint parameters | ||
---|---|---|---|

Parameters | Value | Parameters | Value |

1.3 | Tangential stiffness (MPa) | 10 | |

0.3 | Normal stiffness (MPa) | 100 | |

0.2 | 0.03 | ||

27 | 23 |

Next, we assigned the physical parameters in

We counted the joint development states in the target area through the investigation of the slope exposed surface. Using the slope of K216 + 750 ~ K216 + 900 as an example, the joint angle in this area is close to 45°, the spacing is about 3 m and the length of a single joint is about 5 m. The results of each parameter are shown in

Parameter | Data range | Mean value ( |
Variance ( |
---|---|---|---|

Joint inclination angle (°) | 0~80 | 43.2 | 6.38 |

Joint spacing (m) | 2~4 | 2.8 | 0.21 |

Joint length (m) | 4~6 | 5.1 | 0.39 |

Persistence coefficient | 0.4~0.8 | 0.55 | 0.017 |

We trained LSSVM using the learning samples obtained in Section 3.2, where the first 30 groups are used as training samples, and the remaining five groups are used as test samples. The parameter optimization results of LSSVM through BFOA are ^{5}. The average safety factor of the target slope is 1.48 and the variance is 0.03.

Some safety factors in

To further illustrate the accuracy of the BFOA-LSSVM response surface model and reliability evaluation method established by this research, several typical sections were selected for a group of experts to conduct an on-site evaluation. The results of this evaluation are shown in

Parameter ( |
||||||
---|---|---|---|---|---|---|

Mileage range | Joint inclination angle (°) | Joint spacing (m) | Joint length (m) | Persistence coefficient | Reliability evaluation results | Expert field analysis |

K217 + 300 ~ K217 + 450 | 27.4/3.17 | 1.5/0.13 | 5.5/0.28 | 0.42/0.013 | 1.278/0.038 | 1.3 |

K217 + 800 ~ K217 + 950 | 35.6/4.22 | 3.1/0.08 | 4.3/0.17 | 0.76/0.022 | 1.152/0.023 | 1.2 |

K219 + 130 ~ K219 + 580 | 63.7/4.31 | 2.4/0.14 | 4.8/0.31 | 0.54/0.015 | 1.842/0.017 | 1.7 |

K221 + 460 ~ K221 + 610 | 42.3/2.58 | 1.8/0.07 | 5.7/0.25 | 0.47/0.016 | 1.153/0.014 | 1.1 |

K227 + 315 ~ K227 + 465 | 54.8/5.15 | 3.5/0.24 | 4.1/0.16 | 0.58/0.020 | 1.874/0.005 | 1.8 |

The above research process established a method to calculate the slope reliability through the joint parameters, through which the sensitivity of the joint parameters can be analyzed based on the changes in reliability.

We adjusted the mean value of the parameters in turn, keeping the other parameters unchanged in order to carry out the Monte Carlo process. The slope reliability index under the condition of different mean values of different parameters is shown in

The reliability index in the figure is defined as:

where

It can be seen in

To illustrate this problem, using the actual measured data of the project based on this research as an example (the project situation described in Chapter 3), we adjusted the regular parameter

Machine learning models such as Gaussian process regression (GPR), artificial neural network (ANN), and LSSVM, which are commonly used in geotechnical engineering, were used to perform regression comparisons on the test samples. The relative error statistics of the prediction results calculated by different algorithms are shown in

It can be seen that the LSSVM has a significant advantage over the other two algorithms in prediction accuracy, which is mainly attributed to the small sample learning ability. The regression accuracy of LSSVM optimized by BFOA is obviously improved. Depending on the robustness and global optimization function of the BFOA, the coupling algorithm can search for the optimal parameters of LSSVM, thereby realizing the accurate expression of the mapping relationship between the intermittent joint parameters and the slope safety factor. The comparison shows that the algorithm proposed in this research is effective.

To study the influence of the number of Monte Carlo sampling on the calculation results, the slope of K216 + 750 ~ K216 + 900 in Section 3.3 was again used as an example, and different sampling comparisons were carried out. The variation coefficient

The

This study established a reliability evaluation method for intermittent jointed rock slope, evaluated the safety factor by strength reduction based on joint finite element calculation, and then developed the BFOA-LSSVM hybrid algorithm/machine learning model to establish a response surface model for expressing the mapping relationship between the slope joint parameters and the safety factors. The successful application of this method on the slope of the Yalu River in China reflects its feasibility. This paper offers the following four main findings.

(1) The reliability evaluation method fully takes into account the uncertainty of the slope joint parameters. It can offer a more comprehensive evaluation of the slope safety results, thereby providing a complete data reference for engineering construction.

(2) The BFOA-LSSVM hybrid response surface model, which can accurately express the mapping relationship between joint parameters and slope safety factors, achieved rapid slope evaluation. Comparative calculations show that this model performs better than GPR and ANN in evaluation of intermittent jointed rock slope.

(3) Joint length and joint angle play key roles in the intermittent jointed rock slope stability. Attention should be paid to the development of joint length in the construction process, and anchor injection reinforcement should be carried out according to the joint inclination angle state.

(4) During the application of the method established in this study, it is recommended that the Monte Carlo sampling number be no less than to ensure the accuracy of results.

The joint survey in this study is a statistical analysis of the slope surface, which leads to conservative evaluation results. In future research, it is recommended that geological radar and other technical means be used to detect the joint development inside the slope to obtain more accurate evaluation results. Further studies should also focus on the randomness of the shear strength parameters of the slope rock mass.

The authors thank LetPub (

Joint parameters | Joint inclination angle |
Joint spacing |
Joint length |
Persistence coefficient |
Safety factor |
---|---|---|---|---|---|

Experiment 1 | 0 | 4.0 | 4.0 | 0.4 | 1.95 |

Experiment 2 | 0 | 3.5 | 4.5 | 0.5 | 1.96 |

Experiment 3 | 0 | 3.0 | 5.0 | 0.6 | 1.98 |

Experiment 4 | 0 | 2.5 | 5.5 | 0.7 | 1.98 |

Experiment 5 | 0 | 2.0 | 6.0 | 0.8 | 1.92 |

Experiment 6 | 20 | 4.0 | 4.5 | 0.6 | 1.91 |

Experiment 7 | 20 | 3.5 | 5.0 | 0.7 | 1.72 |

Experiment 8 | 20 | 3.0 | 5.5 | 0.8 | 1.65 |

Experiment 9 | 20 | 2.5 | 6.0 | 0.4 | 1.32 |

Experiment 10 | 20 | 2.0 | 4.0 | 0.5 | 1.82 |

Experiment 11 | 40 | 4.0 | 5.0 | 0.8 | 1.29 |

Experiment 12 | 40 | 3.5 | 5.5 | 0.4 | 1.59 |

Experiment 13 | 40 | 3.0 | 6.0 | 0.5 | 1.35 |

Experiment 14 | 40 | 2.5 | 4.0 | 0.6 | 1.78 |

Experiment 15 | 40 | 2.0 | 4.5 | 0.7 | 1.03 |

Experiment 16 | 60 | 4.0 | 5.5 | 0.5 | 1.77 |

Experiment 17 | 60 | 3.5 | 6.0 | 0.6 | 1.69 |

Experiment 18 | 60 | 3.0 | 4.0 | 0.7 | 1.85 |

Experiment 19 | 60 | 2.5 | 4.5 | 0.8 | 1.75 |

Experiment 20 | 60 | 2.0 | 5.0 | 0.4 | 1.73 |

Experiment 21 | 80 | 4.0 | 6.0 | 0.7 | 1.97 |

Experiment 22 | 80 | 3.5 | 4.0 | 0.8 | 1.97 |

Experiment 23 | 80 | 3.0 | 4.5 | 0.4 | 1.96 |

Experiment 24 | 80 | 2.5 | 5.0 | 0.5 | 1.96 |

Experiment 25 | 80 | 2.0 | 5.5 | 0.6 | 1.87 |

Experiment 26 | 60 | 2.5 | 6.0 | 0.7 | 1.68 |

Experiment 27 | 40 | 3.5 | 4.5 | 0.4 | 1.82 |

Experiment 28 | 80 | 2.5 | 5.0 | 0.5 | 1.98 |

Experiment 29 | 0 | 2.0 | 5.0 | 0.5 | 1.90 |

Experiment 30 | 80 | 4.0 | 5.5 | 0.6 | 1.96 |

Experiment 31 | 20 | 4.0 | 4.0 | 0.6 | 1.68 |

Experiment 32 | 40 | 2.0 | 4.0 | 0.7 | 1.54 |

Experiment 33 | 20 | 3.0 | 6.0 | 0.4 | 1.67 |

Experiment 34 | 0 | 3.5 | 5.5 | 0.8 | 1.92 |

Experiment 35 | 60 | 3.0 | 4.5 | 0.8 | 1.72 |