Abstract
Bending strata is an inevitable consequence of coal mining, which will waste mineral resources and influence their production. In this paper, the displacement function suitable for curved beam in polar coordinates is introduced, and the partial differential governing equation of curved beam is obtained by theoretical analysis. Then, the expressions of displacement components and stress components are expressed by displacement function. On this basis, the program for solving the partial differential equation is compiled using the difference principle. Finally, these theoretical formulas of curved beam are applied to analyze the displacement distribution of bending rock and the influencing factors of deformation in coal mining. These main results show that the value of deformation is increasing with the increase of inner radius, tectonic stress and mining depth, while it is decreasing with the increase of thickness of overburden layer. Moreover, the advancing angle effects significantly not only the value of deformation, but also the position of max value. The value of deformation is more suitable for mining position at about 30°. Obviously, the inner radius and advancing angle have more effect on deformation than other factors. These conclusions provide scientific basis and reference for coal mine engineering.
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References
Ahmed SR, Idris ABM, Uddin MW (1996) Numerical solution of both ends fixed deep beams. Comput Struct 61(1):21–29
Ahmed SR, Khan MR, Islam KMS, Uddin MW (1998) Investigation of stresses at the fixed end of deep cantilever beams. Comput Struct 69(1):329–338
Ahmed SR, Hossain MZ, Uddin MW (2005) A general mathematical formulation for finite-difference solution of mixed-boundary-value problems of anisotropic materials. Comput Struct 83(1):35–51
Bagci C (1992) A new unified strength of materials solution for stresses in curved beams and rings. J Mech Des 114(2):231–237
Bagci C (1993) Exact elasticity solutions for stresses and deflections in curved beams and rings of exponential and T-sections. J Mech Des 115(3):346–358
Bu WK, Xu H (2019) Curved beam elasticity theory based on the displacement function method using a finite difference scheme. Adv Differ Equ 2019:141
Chianese RB, Erdlac RJ (1988) The general solution to the distribution of stresses in a circular ring compressed by two forces acting along a diameter. Q J Mech Appl Math 41(2):239–247
Cook RD (1989) Axisymmetric finite element analysis for pure moment loading of curved beams and pipe bends. Comput Struct 33(2):483–487
Cook RD (1992) Circumferential stress in curved beams. Am Soc Mech Eng 59(1):224–225
Derakhshan D, Komeili M, Milani AS (2015) An analytical approach to the deflection analysis of woven preforms and composites under tensile loading using the Winkler theory of curved beams. Comput Mater Sci 96:403–410
Dow JO, Jones MS, Harwood SA (1990) A new approach to boundary modeling for finite difference applications in solid mechanics. Int J Numer Methods Eng 30(1):99–113
Durelli AJ, Ranganayakamma B (1989) Parametric solution of stresses in beams. J Eng Mech 115(2):401–415
Gangan P (1985) The curved beam/deep arch/finite ring element revisited. Int J Numer Methods Eng 21(3):389–407
Guo JQ, Yang ZQ, Tang H (2011) The effect of lateral pressure coefficient on deformation and stress distribution rules around roadway. J Min Saf Eng 28(4):566–570 (In Chinese)
He MC, Qian QH (2010) Mechanical foundation for deep rock mass. Science Press, Beijing (In Chinese)
Hu WY, He MC (2008) The present situation and development trend for deep coal resources and the development of geological conditions. Coal Industry Publishing House, Beijing (In Chinese)
Huang YG, Zhang TJ (2019) Plastic failure characteristics and grouting support of deep roadway with high ground stress. J Min Saf Eng 36(5):949–958 (In Chinese)
Li SR, Cao DF, Wan ZQ (2013) Bending solutions of FGM Timoshenko beams from those of the homogenous Euler–Bernoulli beams. Appl Math Model 37:7077–7085
Oden JT, Ripperger EA (1981) Mechanics of elastic structures, 2nd edn. McGraw-Hill Book Company, New York
Ranzi G, Gara F, Leoni G, Bradford MA (2006) Analysis of composite beams with partial shear interaction using available modeling techniques: a comparative study. Comput Struct 84(13–14):930–941
Rattanawangcharoen N, Bai H, Shah AH (2004) A 3D cylindrical finite element model for thick curved beam stress analysis. Int J Numer Methods Eng 59(4):511–531
Richards TH, Daniels MJ (1987) Enhancing finite element surface stress predictions: a semi-analytic technique for axisymmetric solids. J Strain Anal Eng Des 22(2):75–86
Sloboda A, Honarmandi P (2007) Generalized elasticity method for curved beam stress analysis: analytical and numerical comparison for a lifting hook. Mech Based Des Struct Mach 35(3):319–332
Smart J (1987) On the determination of boundary stresses in finite elements. J Strain Anal Eng Des 22(2):87–96
Thurnherr C, Groh RMJ, Ermanni P, Weaver PM (2016) Higher-order beam model for stress predictions in curved beams made from anisotropic materials. Int J Solids Struct 97–98:16–28
Timoshenko SP, Goodier JN (1979) Theory of elasticity. McGraw-Hill Book Company, New York
Tutuncu N (1998) Plane stress analysis of end loaded orthotropic curved beams of constant thickness with applications to full rings. J Mech Des 120(2):368–374
Zhong WX, Xu XS, Zhang HW (1996) On a direct method for the problem of elastic curved beams. Eng Mech 13(4):1–8 (In Chinese)
Acknowledgements
The authors would like to thank the referees for careful reading and several constructive comments and making some useful corrections that have improved the presentation of this paper.
Funding
Financial support for this work, provided by the National Fund for Nature projects (No. 51574228), the Research Foundation of Heze University (No. XY17KJ03, No. XY19BS23) and Engagement Fund of Heze University (NO. XYPY02), the General project of science and technology plan of Shandong University (J17KB044) and General Items of Teaching Reform of Heze University (2016064) are gratefully acknowledged.
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Xu, H., Bu, Wk. Deformation Analysis of Bending Stratum Based on Displacement Function Method. Geotech Geol Eng 38, 2859–2872 (2020). https://doi.org/10.1007/s10706-020-01192-x
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DOI: https://doi.org/10.1007/s10706-020-01192-x