Analytical solutions of hard roof’s bending moment, deflection and energy under the front abutment pressure before periodic weighting

The distribution of front abutment pressure is closely related to the force, deformation and energy distribution of hard roof before periodic weighting. So it is necessary to carry out research on the relationship between them. According to front abutment pressure distribution feature, using the location of peak front abutment pressure as the dividing point, coal seam is divided into two parts along the direction of mining: the yield zone that is the area between the faceline and the point where the maximum front abutment pressure occurs, and the elastic zone that is the part before the point of peak front abutment pressure. The proposed mechanical model of unit width hard roof at the panel center before periodic weighting consists of five parts including the yield zone. All parameters of the deflection equations for each of the five parts that satisfy the continuity conditions and natural boundary conditions are obtained by using the Matlab software. The continuous curves of front abutment pressure, deflection, bending moment and bending strain energy density distribution of hard roof are obtained by iterative approximation method, and the relationship between the yield zone width and the above curves are analyzed in detailed.