First fracturing of thin plate of main roof with elastic foundation boundary on both sides of the long side of goaf (coal pillars)

The mechanical model of plate structure of main roof with elastic foundation boundary on the two sides of the long side of goaf (coal pillars) and the short side is established in this study. According to the fracture criterion of pri- mary moment of thin plate and combined with finite difference algorithm for partial differential equation,the influence factors and weight relationships of the fracture of main roof are studied and calculated. It is found that ① the support coefficients kc1 ,kc2 and the widths Lc1 ,Lc2 of coal pillars (c1 ,c2 ) can not only obviously affect the values and positions of primary moments on both sides of coal pillar areas,but also obviously affect the values and positions of primary mo- ments of short side region and central region;② the larger the values of elastic modulus E and thickness h of main roof are,the larger the distance from the fracture line of the short-side solid coal area under main roof as well as the coal pillar areas to the surface of solid coal will be;when Lc1 ,Lc2 are larger and kc1 ,kc2 are smaller,the fracture line of coal pillars and the short-side solid coal area is farther away from the surface of coal pillar and closer to the center of coal pillar;③ when the thickness h and elastic modulus E of main roof are larger and the values of Lc1 ,kc1 ,Lc2 and kc2 are smaller,the main roof in zone c1 and zone c2 of coal pillar area will not fracture and the fracture form is the type of asymmetric “ | | -X”;④ when the values of h,E,Lc1 and kc1 are smaller and the values of Lc2 and kc2 are larger,the main roof in zone c1 of coal pillar (weaker) area will not fracture,but the main roof in zone c2 of coal pillar ( stron- ger) area will fracture,so the final fracture form is the type of asymmetric “ C-X”;⑤ when the support coefficients and the width of coal pillars on both sides of goaf are larger as well as the values of E and h are smaller,the fracture form is the type of asymmetric transverse “O-X”;and ⑥ when k,E,h,kc1 and kc2 change,and the ratio k / ( Eh3 ) is unchangeable (the ratio k / kc1 and the ratio k / kc2 are also unchangeable),the values and positions of primary moments and the position of first fracture are invariant.