Purpose. Substantiation of expediency and admissibility of use of the simplified calculation models of a coal seam roof for an estimation of its stability under the action of external loadings. Methods. To achieve this purpose, the studies have been performed using the basic principles of the theory of elasticity and bending of plates, in which the coal seam roof is represented as a model of a rectangular plate or a beam with a symmetrical cross-section with different support conditions. Results. To substantiate and select methods for studying the bending deformations of the roof in the coal massif containing the maingates, the three-dimensional base plate model and the beam model are compared, taking into account the kinematic boundary conditions and the influence of external distributed load. Using the theory of plate bending, the equations for determining the deflections of the coal seam roof in three-dimensional basic models under certain assumptions have a large dimension. After the conditional division of the plate into beams of unit width and symmetrical section, when describing the normal deflections of the middle surface of the studied models, the transition from the partial derivative equation to the usual differential equations is carried out. In this case, the studies of bending deformations of roof rocks are reduced to solving a flat problem in the cross-section of the beam. A comparison of solutions obtained by the methods of the three-dimensional theory of elasticity and strength of materials was performed. For a beam with a symmetrical section, the deflection lies in a plane whose angle of inclination coincides with the direction of the applied load. The calculations did not take into account the difference between the intensity of the surface load applied to the beam. Differences in determining the magnitude of the deflections of the roof in the model of the plate concerning the model of the beam reach 5%, which is acceptable for mining problems. Scientific novelty. To study the bending deformations and determine the magnitude of the roof deflection in models under external uniform distributed load, placed within the simulated plate, a strip of unit width was selected, which has a symmetrical cross-section and is a characteristic component of the plate structure and it is considered as a separate load-bearing element with supports, the cross-sections of this element is remained flat when bending. The deflection of such a linear element is described by the differential equations of the bent axis of the beam without taking into account the integral stiffness of the model, and the vector of its complete displacement coincides with the vector of the force line. Practical significance. In the laboratory, to study the bending deformations and their impact on the stability of the coal seam roof under external loads, it is advisable to use a model of a single width beam with a symmetrical section with supports, the type of which is determined by rock pressure control and secondary support of the maingate at the extraction layout of the coal mine.
Total Born cross section of e + e − -pair production in relativistic ion collisions from differential equations
