The second invariant of deviatoric stress of the coal and rock mass is closely related to the distortion energy driving the deformation and failure of the surrounding rock. Based on the second invariant of deviatoric stress, this study built a global model of gob-side entry with different widths of the coal pillar through numerical analysis, and compared and analyzed the evolution law of the surrounding rock distortion energy, plastic location state, and roadway deformation with the width of the coal pillar. This study concluded that the peak distortion energy in the virgin coal rib and the roof and floor of the gob-side entry gradually increases with the reduction in the coal pillar width. When the coal pillar width is 5 m, the second invariant peak value of deviatoric stress in the virgin coal rib reaches the maximum of 294.8 MPa2. When the width of the coal pillar is reduced from 30 m to 5 m, the second invariant of the deviatoric stress in the side of the coal pillar and the roof and floor of the side presents the law of first increasing and then decreasing. The greater the damage degree of the coal pillar, the smaller the distortion energy it contains. The distortion energy is the key factor in driving the deformation and failure of the surrounding rock. The greater the distortion energy, the greater the deformation degree of the surrounding rock, the more vulnerable it is to external mining stress disturbance, and the greater the difficulty in controlling the stability of the roadway-surrounding rock.
The laminated beam with uniformly vertical displacement can be divided into contact region and separation region when bending deformation occurs. Friction between the interfaces will cause the stiffness inconsistency between the separation region and contact region. In this paper, calculation formula of section stiffness considering interface friction effect is derived. Assuming that the beam vibrates freely in equal wavelength and equal stiffness forms respectively, the calculating formulas of natural vibration frequencies of simply supported and cantilever beams are derived. Finally, based on a steel-concrete laminated test beam with uniform vertical displacement, the natural vibration frequencies of the beam are calculated. The conclusions are as follows: The derived formulas can calculate the natural vibration frequencies of laminated beams under different interface states effectively; The influence of friction effect on the vibration frequency of laminated beams becomes more and more obvious with the increase of the order.
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