## 6 Conclusions  
The traditional classification methods for rock mass structural planes have many limitations when dealing with complex rock mass structures and large-scale data. To overcome the drawback of the FCM clustering algorithm easily falling into local optima, in this paper, an improved ABC algorithm is proposed and combined with FCM clustering, and a new FCM-ABC method is developed. Using the measured structural plane data, the specific calculation process, the selection of parameters, the effectiveness of grouping, and the dominant orientation of the proposed method for structural plane occurrence classification are analyzed and discussed, and satisfactory clustering results are achieved, verifying the effectiveness of the method. Also, multiple aspects of the excellent performance of this method for the identification of structural plane sets compared to traditional clustering methods are demonstrated. The approach can not only solve the problem of the sensitivity of traditional clustering algorithms to the initial clustering center but also accelerate the convergence speed of this algorithm and improve the clustering accuracy and stability of the algorithm, showing great potential in solving complex optimization problems. In particular,this method can avoid the subjectivity of artificially defining classification boundaries when the grouping boundary of dominant structural planes is not clear and prior knowledge is insufficient. Additionally, it can give objective and reliable results without relying on engineering experience, providing a reliable basis for the identification and classification of rock discontinuity sets, the optimization design and stability analysis of rock engineering, and the accurate prevention and control of engineering disasters. The method theoretically expands the technology of rock mass structural plane identification and has important application value in practical engineering. Note that to further verify the applicability and reliability of this method, the sample data containing at least three groups of dominant structural planes should be used for clustering analysis based on more parameters in the future. Also, corresponding software platforms to achieve the intelligent and refined grouping of structural planes should be developed to guide engineering practice.  
Conceptualization, P.L. and M.C.; methodology, P.L., T.C. and M.C.; software, T.C.; validation, P.L., T.C. and L.S.; formal analysis, P.L., Y.L. and T.C.; investigation, P.L., Y.L. and P.W.; resources, P.L. and X.Z.; data curation, P.L. and T.C.; writing--original draft preparation, P.L. and T.C.; writing--review and editing, P.L. and X.Z.; visualization, Y.L. and Y.W.; supervision, M.C.; project administration, P.L. and M.C.; funding acquisition, P.L. and M.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Science, Technology & Innovation Project of Xionggan New Area (Grant No. 2023 XAGG0061), the Open Research Fund of the State Key Laboratory of Strata Intelligent Control and Green Mining co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology (Grant No. SICGM202303), the National Natural Science Foundation of China (Grant Nos. 52204084 and 52474091), the Interdisciplinary Research Project for Young Teachers of USTB (Fundamental Research Funds for the Central Universities) (Grant No. FRF-IDRY-GD2-002), and the National Key R&D Program of China (Grant Nos. 2022 YFC2905600 and 2022 YFC3004601).  
Not applicable.  
The datasets used during the current study are available from the corresponding author upon reasonable request.  
The authors are deeply indebted to the financial supporters.  
The authors declare no conflicts of interest.