Analytical Solutions for the Deflection and Internal Force of a Hard Roof Subjected to a Piecewise Linear Supercharged Load

Abstract

Coal mining generally induces stress concentration around the goaf range, and the hard roof undergoes fracture and instability when the stress exceeds the tensile strength. It is crucial to investigate the mechanical behavior of the hard roof and determine the corresponding influencing factors of fracture. However, the roof load is treated as the normal distribution function or simple piecewise linear function in the existing research, which makes the model applicable only to specific load forms. For this purpose, the eight-segment linear function is used in this study to subdivide the supercharged load acting on the roof based on different foundation conditions, which can help one to avoid complex calculations of mechanical behavior and ensure proximity to the actual load. The coal seam in front of the coal wall is regarded as an elastic foundation, and the deflection curve equation is established by analyzing the force on the roof. The internal force and deformation of the hard roof are obtained before the initial and periodic fractures. The accuracy of this model is verified by comparing it with other models in the literature. The results of parameter sensitivity analysis indicate that with the increase of foundation stiffness, roof thickness, and support resistance, the deflection and bending moment significantly decrease. The deflection, bending moment, and shear force are generally positively correlated with goaf length. When the roof is thin and the coal seam is soft, the initial fracture position occurs at the midspan, while the periodic fracture always occurs inside the coal wall. The fracture position gradually moves toward the coal wall with the increase of coal seam stiffness for both the initial and the periodic fractures. The research results are helpful to understand the internal force and deformation mechanism of the hard roof and provide a valuable theoretical basis for predicting the fracture mode of the roof.