## 1 Introduction  
Under the background of the construction of ecological civilization, green development and the realization of \"double carbon\" target in China, filling mining has been gradually developed and applied [1, 2]. Backfilling mining is one of the key techniques of the green mining system, and its development has evolved into various forms, including waste filling, water-gravel-mixed filling, cemented filling, solid filling, paste backfilling, and high-water (super high-water) filling [3, 4]. Among them, the solid filling mining mainly uses the gangue of particle size below 50 mm, through the porous bottom unloading scraper conveyor, transported to the goal for filling; the filling body partially or completely replaces the caving gangue to support the overlying strata [5, 6]. Compared with the traditional caving method, the filling body plays a supporting role in the roof. When the supporting capacity meets the requirements of roof stability, it can weaken the mine pressure behavior of the working face, control the damage of key rock strata, reduce the surface subsidence, and protect the ecological environment and water resources. Backfilling mining is widely used in coal areas under buildings, water bodies, and railways [7, 8]. Massive studies have been conducted by relevant scholars on roof stability control in backfilling mining. The supporting force provided by the backfilling body is not enough to bear the weight of overlying rocks at the initial filling stage, which causes synchronous subsidence of the backfilling body and roof. Strength increases with the continuous compression of the backfilling body and reaches equilibrium with the roof pressure, which stabilizes the roof [9; 10]. [PERSON] et al. established a theoretical model of continuous curved beams of overlying strata in backfilling mining [11; 12]. The quantitative relationship between continuous curved beams and key strata to study the spatiotemporal characteristics of continuous curved beams. [PERSON] et al. established a curvature model of backfilling mining strata to analyze the relationship between the backfill ratio and strata curvature [13]. [PERSON] et al. defined the critical backfilling ratio to solve the critical backfilling ratio for layered fracture of four-layer overlying strata based on the strata movement and control characteristics of backfilling mining [14]. [PERSON] et al. established a coupled roof-controlled mechanical model to analyze the effect law of different factors on the support working resistance under high and low backfilling ratios, as well as the viscoelastic and time effects of the backfilling body [15; 16]. [PERSON] et al. analyzed the self-stability effect of roof arches in goafs with different roof-contacted filling ratios [17]. [PERSON] et al. established a continuous curved mechanical model of backfilling roof strata under non-uniform load to analyze the movement relationship of the overlying strata with different backfilling ratios [18]. The above research positively promotes the development of backfilling mining technology and surrounding rock control theory. The roof-stability control theory is studied from different angles. Roof stability is affected by multiple factors. Therefore, the roof control degrees along strike and dip are different based on different factors, and the quantitative effect research of roof control degrees is not perfect. In this paper, the connotation and characteristics of roof self-stability in solid filling mining are expounded, and the mechanical model of roof along strike and dip direction is constructed. The influence of elastic foundation coefficient of filling body, peak load of support, empty roof distance and footage of working face on roof self-stability along strike direction and the influence of working face length and coal pillar length on roof self-stability along dip direction are studied. Finally, the reliability of the model is verified by the engineering case of Xinjulong Coal Mine.