## 1 Introduction  
Gob-side entry retaining by roof cutting (GERRC) and pressure relieving is one of the effective technologies for the stable control of the roof in deep-mine gob-side entry retaining. The technical means is by implementing the directional energy accumulation pre-splitting of the roadway roof in advance of the working face, cutting or weakening the mechanical connection between the roadway roof and the working face roof. During the process of cutting the roof and retaining the roadway, large deformation anchor cables and high pre-tension anchor rods (cables) are used as basic support, and stacking support is generally used as temporary support. Gangue blocking mass is used to replace the filling mass next to the roadway, so as to automatically form the roadway after mining the working face [1] and eliminate the isolation coal pillars between the two working faces. The schematic diagram of GERRC is shown in Figure 1. The evolution of the roof structure in different mining stages of GERRC is different, and the stress and deformation of the retaining roof change accordingly [2]. Domestic and foreign scholars have conducted extensive research on the evolution of the roof structure in GERRC. [PERSON] et al. [3, 4] believe that GERRC transforms the \"long cantilever beam structure\" of the gob-side roadway roof into a \"short cantilever beam structure\" through advanced directional energy accumulation blasting. They simplify the stress of the short cantilever beam as uniformly distributed loads to calculate the roof deformation. [PERSON] et al. [5, 6] analyze the roof structure and stress characteristics of different stages of GERRC, simplifying the roof structure of different stages into fixed-end beams, simply supported beams, and cantilever beams. They analyze the deformation characteristics of the gob-side roadway roof and propose roof stability control technology for roadway formation. [PERSON] et al. [7,8] focus on conditions where the rock strata are thin and the loose layer is thick. They simplify the thick loose layer as uniformly distributed loads and assume the thin rock strata as a good elastomer, establishing beam models for fixed-end, simply supported, and cantilever roofs. They analyze the roof deformation during roadway formation. [PERSON] et al. [9,10] compare and analyze the roof structure and stress analysis of conventional gob-side entry retaining and GERRC, believing that advanced pre-splitting roof cutting can effectively reduce the cantilever length of the roadway side roof and reduce the additional load on the side support body. [PERSON] et al. [11] investigate the roof deformation mechanism of self-formed roadways with cross-fault roof cutting and pressure relieving and verify that the \"roof cutting and pressure relieving constant resistance anchor cable\" support method has a good control effect on the roof deformation of cross-fault roadways. [PERSON] et al. [12] believe that during GERRC, the immediate roof and main roof form a combined cantilever structure, and the roof subsidence deformation has a synergistic effect. The shear failure of the anchor bolt is used as the critical index to obtain the interlayer displacement criterion. As the mining depth increases, the deformation of the roadway-surrounding rock also increases and becomes more intense, manifesting as roadway instability and even destructive rock burst, making roof management difficult and support costs high [13]. In terms of roof control in deep-mine roadways, [PERSON], [PERSON], and others [14,15] believe that when the mining depth generally exceeds the first critical depth, the mining rock mass medium is in a stage of large plastic deformation. They propose using roof-cutting and pressure-relieving technology to optimize the stress environment of the roadway-surrounding rock. In terms of support technology, it is necessary to consider the coupling of the support body and the surrounding rock in terms of strength, stiffness, and structure.