To characterize the deformation and internal force distribution of the hard roof during coal seam mining and to assess the sensitivity of key parameters on its mechanical response

we model the coal seam and the immediate roof as an elastic foundation and select a unit width rock beam at the center of the working face as the research object. Based on the theory of elastic foundation beams and key strata

we establish an elastic foundation beam model for the main roof and obtain analytical solutions for roof deflection

bending moment

and shear force prior to initial fracture. We analyze the influence of five key parameters—the elastic modulus; the stiffness of the coal seam and the immediate roof; the thickness of the overlying soft-rock strata; the influence range of the advanced abutment pressure; and the main-roof thickness—on roof deflection

bending moment

and shear force Finally

using Design-Expert software

we design a five factor

three level response surface scheme and perform a sensitivity analysis of single factors and factor interactions on the hard roof’s mechanical characteristics. Results show that (1) the stiffness of the coal seam and the immediate roof significantly affect the overall roof deformation

while other factors primarily influence the goaf roof. Additionally

increasing the thickness of the overlying soft-rock strata increases roof deflection

whereas other factors have the opposite effect. (2) If the thickness of the overlying soft-rock strata increases linearly

the goaf roof bending moment and shear force also increase linearly. The influence range of the advanced abutment pressure and the main-roof thickness significantly affect the bending moment and shear force in the roof in front of the working face

whereas the elastic modulus and the stiffness of the coal seam and the immediate roof have little to no effect. (3) Factors governing geometric characteristics and load distribution—such as the thickness of the overlying soft-rock strata

the influence range of the advanced abutment pressure

and the main-roof thickness—generally have a greater effect on rock-beam deflection and internal forces than the mechanical properties of the rock mass (e.g.

elastic modulus and the stiffness of the coal seam and the immediate roof). The model accounts for the dynamic variation of the upper load during coal-seam mining

providing a theoretical basis for studying the deformation and internal-force distribution of the main roof throughout mining.