The classification of the stability of surrounding rock is an uncertain system with multiple indices. The Multi-dimensional Cloud Model provides an advanced solution through the use of an improved model of One-dimensional Cloud Model. Setting each index as a one-dimensional attribute, the Multi-dimensional Cloud Model can set the digital characteristics of each index according to the cloud theory. The Multi-dimensional cloud generator can calculate the certainty of each grade, and then determine the stability levels of the surrounding rock according to the principle of maximum certainty. Using this model to 5 coal mine roadway surrounding rock examples and comparing the results with those of One-dimensional and Two-dimensional Cloud Models, we find that the Multi-dimensional Cloud Model can provide a more accurate solution. Since the classification results of the Multi-dimensional Cloud Model are difficult to be presented intuitively and visually, we reduce the Multi-dimensional Cloud Model to One-dimensional and Two-dimensional Cloud Models in order to visualize the results achieved by the Multi-dimensional Cloud Model. This approach provides a more accurate and intuitive method for the classification of the surrounding rock stability, and it can also be applied to other types of classification problems.

Over the last several decades, some of the inevitable requirements and trends regarding global economic development have involved the gain of increased underground space, resources and security. Internationally, there is a great deal of consensus regarding the importance of developing high quality underground engineering projects with high levels of safety [

The stability of surrounding rock depends on the physical-mechanical properties of the rock mass, groundwater conditions, discontinuity parameters and other factors [

As research on the stability of surrounding rock have been expanded, it has been recognized that the collected index information is usually vague, random and uncertain. As such, fuzzy mathematics, PP-PSO (Projection Pursuit-Particle Swarm Optimization), Extension Theory, and Neural Network [

The Multi-dimensional Cloud Model [

The Cloud Model [

The Multi-dimensional Cloud Model is derived from the expansion of the Cloud Model in order to express the concept of multi-index uncertainty. According to the theory behind the Multi-dimensional Cloud Model, the _{1}, Ex_{2}…Ex_{n}_{1}, En_{2}…En_{n}_{1}, He_{2}…He_{n}_{i}_{i}_{i}

Input: expected value (_{1}, Ex_{2}…Ex_{n}_{1}, En_{2}…En_{n}_{1}, He_{2}…He_{n}

(1) Generate an n-dimensional normal random number _{1}, x_{2}…x_{n}_{1}, Ex_{2}…Ex_{n}_{1}, En_{2}…En_{n}

(2) Generate an n-dimensional normal random number _{1}, y_{2}…y_{n}_{1}, En_{2}…En_{n}_{1}, He_{2}…He_{n}

(3) Calculate certainty _{i}

(4) (_{i}, μ_{i}

Repeat Steps (1)∼(4) until N cloud drops are generated.

The selection of indices is the premise of the stability classification of the surrounding rock and also the basis of the model. Based on the existing research results, the stability classification index of the surrounding rock is determined by using association ruler mining and cloud transformation [

Factor | Category | Stable | Relatively stable | Basically stable | Unstable | Extremely unstable |
---|---|---|---|---|---|---|

Rock mass, geological factors | RQD/% | 60–100 | 40–60 | 25–40 | 10–25 | 0–10 |

Uniaxial compressive strength/Mpa | 200–300 | 100–200 | 50–100 | 25–50 | 0–25 | |

Rock integrity coefficient | 0.75–1 | 0.55–0.75 | 0.35–0.55 | 0.15–0.35 | 0–0.15 | |

Groundwater development degree | 0–3 | 3–5 | 5–7 | 7–8 | 8–10 | |

Joint condition | 9–10 | 7–9 | 4–7 | 2–4 | 0–2 | |

Maximum earthquake intensity | 0–3 | 3–5 | 5–7 | 7–8 | 8–10 | |

Engineering factors | Adjacent influence coefficient | 0–0.2 | 0.2–0.4 | 0.4–0.6 | 0.6–0.8 | 0.8–1 |

Design high span ratio | 0–1 | 1–1.5 | 1.5–3.5 | 3.5–4.5 | 4.5–6 | |

Support measures | Superior | Good | General | Poor | Very poor | |

Integrated threshold | 0–1.5 | 1.5–2.5 | 2.5–3.5 | 3.5–4.5 | 4.5–5 |

According to the index classification system in _{ij}, En_{ij}, He_{ij}

In formulas _{ij}^{1} and _{ij}^{2} represent the minimum and maximum boundary values of the index grades respectively. The super entropy (

Grade standard | Numerical characteristics | ||
---|---|---|---|

I | 1 | 0.35 | 0.01 |

II | 2 | 0.35 | 0.01 |

III | 3 | 0.35 | 0.01 |

IV | 4 | 0.35 | 0.01 |

V | 5 | 0.35 | 0.01 |

According to the principles of index system construction, the weights are determined through the improved AHP method [

The importance of the evaluation targets is sorted according to the experts’ opinions. Here, _{i}_{j}_{j+1}_{j}_{j+2}_{i} · r_{i+1}, i = 1, 2, …, m

In the formula _{i}

According to the judgment matrix

The index data _{ij}

_{j}

In order to verify the accuracy of the Model, 5 sections of surrounding rock of a coal mine roadway are selected as classification examples. This mine does not have a systematic understanding of the deformation and destruction of the surrounding rock in its complex roadways. In order to supplement the mine’s geological data, we conducted various tests and a systematic study of the surrounding rock in the mining area. The Hydraulic Fracturing Method and Borehole Penetration Method were used to obtain the data from the surrounding rock.

The selected 5 sections of the roadway rock formations consist primarily of sandstone, which is stable and relatively complete. Project 1 and Project 2 are distributed in a low stress area with stable surrounding rock and a weak degree of groundwater development, but there are many weak intercalations that use bolting net combined support; Project 3 is distributed in a moderately high stress area with good rock integrity, a high degree of groundwater development, the weak rock stratum is thick, the joints are well developed, and here, a bolt-mesh-anchor combined support is used; Project 4 is distributed in a medium stress area with relatively good integrity, no obvious cracks, the adjacent projects have a great impact, and use a bolt-net-beam-cable combined support; Project 5 is distributed in a medium stress area, with obvious fissures and weak intercalations, low rock strength, a bolt-mesh-anchor combined support is used, and there are no large deformation and other accidents within the service life of different projects. The index data for the surrounding rock are shown in

Factor | Category | Surrounding rock 1 | Surrounding rock 2 | Surrounding rock 3 | Surrounding rock 4 | Surrounding rock 5 |
---|---|---|---|---|---|---|

Rock mass, geological factors | RQD/% | 38 | 57.9 | 59.4 | 48.7 | 57.5 |

Uniaxial compressive strength/Mpa | 18.22 | 17.26 | 13.96 | 14.29 | 13.83 | |

Rock integrity coefficient | 0.4 | 0.35 | 0.6 | 0.5 | 0.55 | |

Groundwater development degree | 2 | 2 | 6.7 | 3.6 | 5.5 | |

Joint condition | 5 | 5 | 8 | 6 | 7 | |

Maximum earthquake intensity | 6 | 6 | 6 | 6 | 6 | |

Engineering factors | Adjacent influence coefficient | 0.45 | 0.35 | 0.25 | 0.41 | 0.32 |

Design high span ratio | 1.39 | 1.02 | 1.16 | 1.79 | 1.29 | |

Support measures | Superior | Good | General | Poor | Very poor |

The weight of each index is determined by AHP as (0.148, 0.0485, 0.16, 0.074, 0.042, 0.0275, 0.0835, 0.166, 0.2505). Through incorporating the surrounding rock data and index weights into formula

Surroundingrock number | I | II | III | IV | V | Multi-Dcloud model | One-Dcloud model | Two-D |
---|---|---|---|---|---|---|---|---|

1 | 0.0656 | 0.1841 | 0.5657 | 0.1975 | 0.0488 | III | III | III |

2 | 0.1577 | 0.2261 | 0.3291 | 0.2214 | 0.0575 | III | III | III |

3 | 0.108 | 0.6246 | 0.3088 | 0.0295 | 0.0472 | II | III | II |

4 | 0.2034 | 0.4341 | 0.3415 | 0.0011 | 0.0547 | II | II | II |

5 | 0.2386 | 0.6072 | 0.2106 | 0.0004 | 0.0469 | II | III | II |

As in

The visualization chart of the N-dimensional Cloud Model is (N + 1)-dimensional, that is, the visualization chart of the Two-dimensional Cloud Model is three-dimensional, so the Multi-dimensional Cloud Model is difficult to visualize. Therefore, this paper reduces the dimensionality of the Multi-dimensional Cloud Model to a One-dimensional Cloud Model or Two-dimensional Cloud Model to realize the visualization of the Multi-dimensional Cloud Model.

In the Multi-dimensional Cloud Model, each index is constructed into one-dimensional Cloud Models and then coupled into a Multi-dimensional Cloud Model. The structure of the Index Cloud Model is shown in formulas

Index | Maximum certainty | Grade | Numerical characteristics |
---|---|---|---|

RQD | 0.5716 | III | (2.7, 0.2, 0.011) |

Uniaxial compressive strength | 0.5416 | V | (5.0, 0.19, 0.011) |

Rock integrity coefficient | 0.7691 | III | (3.1, 0.27, 0.015) |

Groundwater development degree | 0.7691 | I | (1.0, 0.27, 0.015) |

Joint condition | 0.8928 | III | (3.0, 0.31, 0.017) |

Maximum earthquake intensity | 1.000 | III | (3.0, 0.35, 0.02) |

Adjacent influence coefficient | 0.7691 | III | (2.9, 0.27, 0.015) |

Design high span ratio | 0.9231 | II | (2.2, 0.32, 0.018) |

Support measures | 0.7691 | IV | (4.0, 0.27, 0.015) |

In formulas _{i*}_{i1}, μ_{i2}_{ip}

The stability classification of surrounding rock 1 is represented by the cloud charts shown as

We then coupling each index cloud chart of the surrounding rock 1 and perform dimension reduction to obtain a one-dimensional grade cloud chart. Then, by comparing the cloud chart with the grade standard cloud chart (

The classification of surrounding rock stability is determined by various uncertain factors, resulting in an extremely complex problem. This paper introduces a Multi-dimensional Cloud Model that has obvious advantages in terms of both the qualitative and quantitative transformations of the stability evaluation of the surrounding rock. The multi-dimensional cloud model takes each evaluation index as a one-dimensional attribute, which can effectively improve the processing speed of index data. Compared with one-dimensional cloud model and two-dimensional cloud model, the one-dimensional cloud model and the two-dimensional cloud model lack the consideration of the correlation between multiple indicators, and it is difficult to reflect the evaluation results under the combined influence of multiple factors, which affects the accuracy of the results to a certain extent. The multi-dimensional cloud model can not only consider the impact of the interaction between multiple indicators, but also systematically reflect the impact of the coupling of various indicators on the comprehensive evaluation results.

Through verification by example, the results obtained are found to be similar to those of other models, and are also consistent with the actual situation, which indicates that the Multi-dimensional Cloud Model evaluation method is effective and feasible with respect to the evaluation of surrounding rock stability. Furthermore, the Multi-dimensional Cloud Model reflects the overall situation of surrounding rock stability and the uncertainty of each index belonging to different grades, which also provides a new reference for surrounding rock stability evaluation and other similar projects.