Study on roof deflection distribution of ultra-long working faces based on matrix displacement method

To investigate the effects of roof conditions, sidewall conditions, and support conditions on roof deflection variations in ultra-long working faces, and to apply beam model theory to hydraulic support design, this study is based on the ultra-long working face 132202 of Xiaobaodang No.2 Mine. A two-dimensional continuous beam model supported on elastic supports was established, reflecting the relationships among hydraulic supports, sidewalls, and roofs in actual production. Using the displacement method, beam model elements were encoded, and the element stiffness matrix for each beam segment was calculated. By employing the element assembly method, the global stiffness matrix and equivalent nodal loads of the ultra-long beam model with elastic supports were computed. Through the matrix displacement method, the global deflection distribution of the beam and the end forces of beam elements were derived, enabling the calculation of support reaction forces. Parametric analyses were conducted on variables such as support width, number of supports, equivalent stiffness, working face length, sidewall stiffness, roof elastic modulus, moment of inertia, and loads induced by adjacent strata. Their impacts on the global deflection distribution were examined. A cubic polynomial was used to precisely fit the initial-to-peak segment of the deflection curve on one side. Validation was performed using results from 3DEC numerical simulations incorporating pile structural elements for support, alongside field monitoring data from electro-hydraulic control systems. The validation confirmed consistency among theoretical calculations, numerical simulations, and field data, demonstrating that the 2D beam model reasonably explains the tri-peak loading characteristics observed in ultra-long working faces. Additionally, the deflection curve from the edge of the working face to the adjacent peak value aligns with a cubic polynomial distribution. This study deepens the application of beam models in mining and provides guidance for hydraulic support design in ultra-long working faces.