Analytical Solutions for the Deflection and Internal Force of a Hard Roof Subjected to a Piecewise Linear Supercharged Load
Publication: International Journal of Geomechanics
Volume 24, Issue 7
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.
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Data Availability Statement
All data, models, and codes that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
This work was financially supported by the National Natural Science Foundation of China (52074269).
Notation
The following symbols are used in this paper:
- A1–A4, B1–B4, C1–C4, D1–D4, E1–E4, F1–F4, G1–G4, H1–H4
- coefficients;
- EI
- flexural rigidity;
- h
- thickness of roof;
- I
- moment of inertia;
- k
- the elastic foundation stiffness coefficient;
- L
- half of the goaf span;
- Lk
- roof-control distance;
- l1, l2, l3, l4, l5, l6, l7
- distance of each segment;
- L2
- distribution length of the supercharged load;
- M
- bending moment;
- Mmax, Mmin
- maximum and minimum bending moment;
- Q
- shear force;
- q1, q2, q3, q4, q5, q6, q7
- overburden pressure of each segment;
- qo, qm, qk
- minimum, midpoint, and maximum supporting resistance;
- v
- deflection;
- y
- value of the y-coordinate;
- σ
- axial stress;
- midspan deflection; and
- xmax, xmin
- maximum and the minimum bending moment of the roof in the x-axis position.
References
Cao, A. Y., and L. M. Dou. 2008. “Analysis of focal mechanism caused by rupture of stope roof.” Chin. J. Rock Mech. Eng. 27 (S2): 3833–3839.
Dou, L. M., Z. T. Liu, S. G. Cao, and X. R. Wu. 2003. “Analysis of the influence on rock burst caused by hard roof.” Coal Min. Technol. 8 (2): 58–60+66.
Elen, L. W. F. 1965. Differential equations. London: Macmillan Education Press.
Evans, W. H. 1941. “The strength of undermined strata.” Trans. Inst. Min. Metall. 50: 475–500.
Feng, Q., W. W. Liu, S. G. Fu, B. S. Jiang, and L. P. Shi. 2017. “Analytical solution for deformation and internal force of hard roof in stope based on elastic foundation beam.” J. Min. Saf. Eng. 34 (2): 342–347.
Froio, D., and E. Rizzi. 2017. “Analytical solution for the elastic bending of beams lying on a linearly variable Winkler support.” Int. J. Mech. Sci. 227: 1157–1179.
Ji, S., H. He, and J. Karlovšek. 2021. “Application of superposition method to study the mechanical behaviour of overlying strata in longwall mining.” Int. J. Rock Mech. Min. Sci. 146: 104874. https://doi.org/10.1016/j.ijrmms.2021.104874.
Jiang, H. J., S. G. Cao, Y. Zhang, and C. Wang. 2016. “Analytical solutions of hard roof's bending moment, deflection and energy under the front abutment pressure before periodic weighting.” Int. J. Min. Sci. Technol. 26 (1): 175–181. https://doi.org/10.1016/j.ijmst.2015.11.027.
Jin, Z. M., J. P. Wei, and W. X. Jin. 2001. “Distributive characteristic of front abutment pressure in top-coal caving face.” J. Taiyuan Univ. Technol. 32 (3): 216–218.
Ju, J. F., and J. L. Xu. 2013. “Structural characteristics of key strata and strata behaviour of a fully mechanized longwall face with 7.0 m height chocks.” Int. J. Rock Mech. Min. Sci. 58: 46–54. https://doi.org/10.1016/j.ijrmms.2012.09.006.
Li, X. Y., N. J. Ma, Y. P. Zhong, and Q. C. Gao. 2007. “Storage and release regular of elastic energy distribution in tight roof fracturing.” Chin. J. Rock Mech. Eng. 26 (S1): 2786–2793.
Miao, X. X., X. B. Miao, and T. Z. Zhou. 1995. “Beam structure analysis and pressure prediction for elastic foundation of old roof in stope.” Mech. Eng. 5: 21–22+41.
Pan, Y., S. T. Gu, and Y. S. Qi. 2012. “Analytic solution of the tight roof’s bending moment and deflection underswelling distributive supporting pressure.” Chin. J. Rock Mech. Eng. 31 (10): 2035–2063.
Pan, Y., S. T. Gu, and Y. S. Qi. 2013. “Analytic solution of tight roof’s bending moment, deflection and shear force under advanced supercharger load and supporting resistance before first weighting.” Chin. J. Rock Mech. Eng. 32: 1544–1553.
Pan, Y., S. T. Gu, and G. L. Yang. 2015. “Variation of internal force and rebound property of hard roof at initial stage of cracking.” Chin. J. Geotech. Eng. 37 (5): 860–869.
Qian, M. G., X. B. Miao, and X. X. Miao. 1998. “Variation of loads on the key layer of the overlying strata above the working surface.” J. China Coal Soc. 23 (2): 25–29.
Qian, M. G., and X. X. Miao. 1995. “Theoretical analysis on the structural form and stability of overlying strata in longwall mining.” Chin. J. Rock Mech. Eng. 14 (2): 97–106.
Shi, Y. W., and F. M. Han. 1979. “The relationship between the selection of hydraulic support and roof and floor conditions.” Coal Sci. Technol. 7 (3): 11–16.
Zhang, G. C., F. L. He, and L. L. Jiang. 2016. “Analytical analysis and field observation of break line in the main roof over the goaf edge of longwall coal mines.” Math. Probl. Eng. 2016: 4720867.
Zhang, Q., C. H. Peng, R. C. Liu, B. S. Jiang, and M. M. Lu. 2019. “Analytical solutions for the mechanical behaviors of a hard roof subjected to any form of front abutment pressures.” Tunnelling Underground Space Technol. 85: 128–139. https://doi.org/10.1016/j.tust.2018.12.004.
Zhao, G. J., and M. G. Qian. 1987. “Behaviour of overlying hard strata above workings and its effect on roof pressure.” J. China Coal Soc. 3 (9): 1–8.
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© 2024 American Society of Civil Engineers.
History
Received: Nov 29, 2022
Accepted: Jan 13, 2024
Published online: Apr 22, 2024
Published in print: Jul 1, 2024
Discussion open until: Sep 22, 2024
ASCE Technical Topics:
- Coal
- Continuum mechanics
- Cracking
- Design (by type)
- Displacement (mechanics)
- Elastic foundations
- Energy engineering
- Energy sources (by type)
- Engineering fundamentals
- Engineering mechanics
- Foundations
- Fracture mechanics
- Fuels
- Geotechnical engineering
- Load factors
- Non-renewable energy
- Roofs
- Solid mechanics
- Structural design
- Structural engineering
- Structural mechanics
- Structural systems
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