基于曲梁理论的褶皱构造中覆岩变形与应力分析
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摘要
为了分析褶皱构造中煤层开采后覆岩的变形与应力,引入了适用于极坐标下曲梁的位移函数,分析了用位移函数表示的曲梁偏微分控制方程和位移分量、应力分量;在此基础上,依据差分原理将微分方程转化为差分方程,并编制了差分方程求解程序。根据某矿区工作面的褶皱构造等地质资料,研究了工作面由向斜槽点推进到背斜脊点过程中覆岩的变形与应力。结果表明:1)覆岩水平位移主要表现为向采空区方向的变形,且最大水平位移主要表现在转折端附近;覆岩垂直位移主要表现为向采空区方向的下沉变形,且在工作面的推进过程中最大垂直位移稳定在距离开切眼70 m附近。2)覆岩中水平方向的应力在切眼前方为拉应力,而工作面后方为压应力;竖直方向的应力主要表现为拉应力,且主要位于工作面后方。
In order to analyze the deformation and stress of overlying rock after coal mining in fold structure, a displacement function suitable for curved beam in polar coordinates was introduced. The partial differential governing equation of curved beam was obtained by theoretical analysis. Then the displacement components and stress components were expressed by using above function. On above basis, these partial differential equations were translated into difference equations. Meanwhile, the program for solving difference equations was compiled according to finite difference method. Based on geological data in a working face(such as fold structure), the deformation and stress of overlying rock in the working face advancing process from syncline bottom to anticline top were studied. Results are as follows: 1) The horizontal displacement of overlying rock is mainly in goaf direction, and its maximum value is close to the hinge zone of fold. In contrast, the vertical displacement of overlying rock is mainly subsidence deformation towards the goaf, and its maximum value is stable at the distance of 70 m away from open-off cut during working face advancing. 2) The horizontal stress in overlying rock is tensile stress in front of the open-off cut, while it is the compressive stress in the rear of working face. The vertical stress in overlying rock is mainly tensile stress, which is mainly in the rear of working face.Meanwhile, the maximum shear stress is mainly in the rear of working face.
In order to analyze the deformation and stress of overlying rock after coal mining in fold structure, a displacement function suitable for curved beam in polar coordinates was introduced. The partial differential governing equation of curved beam was obtained by theoretical analysis. Then the displacement components and stress components were expressed by using above function. On above basis, these partial differential equations were translated into difference equations. Meanwhile, the program for solving difference equations was compiled according to finite difference method. Based on geological data in a working face(such as fold structure), the deformation and stress of overlying rock in the working face advancing process from syncline bottom to anticline top were studied. Results are as follows: 1) The horizontal displacement of overlying rock is mainly in goaf direction, and its maximum value is close to the hinge zone of fold. In contrast, the vertical displacement of overlying rock is mainly subsidence deformation towards the goaf, and its maximum value is stable at the distance of 70 m away from open-off cut during working face advancing. 2) The horizontal stress in overlying rock is tensile stress in front of the open-off cut, while it is the compressive stress in the rear of working face. The vertical stress in overlying rock is mainly tensile stress, which is mainly in the rear of working face.Meanwhile, the maximum shear stress is mainly in the rear of working face.
引文
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[19]SLOBODA A,HONARMANDI P.Generalized elasticity method for curved beam stress analysis:analytical and numerical comparison for a lifting hook[J].Mechanics based Design of Structures and Machines,2007,35(3):319-332.
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[21]徐芝纶.弹性力学简明教程[M].4版.北京:高等教育出版社,2016:59-64.
[22]AHMED S R,HOSSAIN M Z,UDDIN M W.A general mathematical formulation for finite-difference solution of mixed-boundary-value problems of anisotropic materials[J].Computers&Structures,2005,83(1):35-51.
[2]何满朝,钱七虎.深部岩体力学基础[M].北京:科学出版社,2010:1-10.
[3]钱鸣高,缪协兴.采场上覆岩层结构的形态与受力分析[J].岩石力学与工程学报,1995,14(2):97-106.QIAN Minggao,MIAO Xiexing.Theoretical analysis on the structural form and stability of overlying strata in longwall mining[J].Chinese Journal of Rock Mechanics and Engineering,1995,14(2):97-106.
[4]缪协兴,钱鸣高.采场围岩整体结构与砌体梁力学模型[J].矿山压力与顶板管理,1995,12(3/4):3-12.MIAO Xiexing,QIAN Minggao.Solid structure and model of voussoir beam of face surrounding rock[J].Ground Pressure and Strata Control,1995,12(3/4):3-12.
[5]许家林,钱鸣高.覆岩采动裂隙分布特征研究[J].矿山压力与顶板管理,1997,14(3/4):210-212.XU Jialin,QIAN Minggao.Study of features of distribution of overlaying stratum fractures caused by mining[J].Ground Pressure and Strata Control,1997,14(3/4):210-212.
[6]汪锋,许家林,谢建林,等.上覆煤层开采后下伏煤层卸压机理分析[J].采矿与安全工程学报,2016,33(3):398-402.WANG Feng,XU Jialin,XIE Jianlin,et al.Pressure-relief mechanism of lower seam extraction after upper seam extraction[J].Journal of Mining&Safety Engineering,2016,33(3):398-402.
[7]王磊,郭广礼,张鲜妮,等.基于关键层理论的长壁垮落法开采老采空区地基稳定性评价[J].采矿与安全工程学报,2010,27(1):57-61.WANG Lei,GUO Guangli,ZHANG Xianni,et al.Evaluation on stability of old goaf foundation by long-wall caving based on the key stratum theory[J].Journal of Mining&Safety Engineering,2010,27(1):57-61.
[8]涂敏,付宝杰.关键层结构对保护层卸压开采效应影响分析[J].采矿与安全工程学报,2011,28(4):536-541.TU Min,FU Baojie.Analysis of the effect of key strata structure on relief-pressure mining in protective seam[J].Journal of Mining&Safety Engineering,2011,28(4):536-541.
[9]付宝杰,高明中,涂敏,等.关键层的复合效应及其对矿压显现的影响[J].采矿与安全工程学报,2016,33(2):220-225.FU Baojie,GAO Mingzhong,TU Min,et al.Composite effect of key stratum and its influence on strata behaviors[J].Journal of Mining&Safety Engineering,2016,33(2):220-225.
[10]李永明,刘长友,黄炳香.急斜煤层覆岩关键层对防水煤柱尺寸的影响[J].采矿与安全工程学报,2012,29(2):226-231.LI Yongming,LIU Changyou,HUANG Bingxiang.Influence of key stratum on waterproof coal pillar size in steep seam[J].Journal of Mining&Safety Engineering,2012,29(2):226-231.
[11]蒋金泉,张培鹏,秦广鹏,等.一侧采空高位硬厚关键层破断规律与微震能量分布[J].采矿与安全工程学报,2015,32(4):523-529.JIANG Jinquan,ZHANG Peipeng,QIN Guangpeng,et al.Fracture laws of one-side mined high-position hard thick key strata and microseismic energy distribution[J].Journal of Mining&Safety Engineering,2015,32(4):523-529.
[12]ODEN J T,RIPPERGER E A.Mechanics of elastic structures[M].2nd ed.New York:McGraw-Hill Book Company,1981:50-60.
[13]TIMOSHENKO S P,GOODIER J N.Theory of elasticity[M].3rd ed.New York:McGraw-Hill Book Company,1979:30-42.
[14]CHIANESE R B,ERDLAC R J.The general solution to the distribution of stresses in a circular ring compressed by two forces acting along a diameter[J].The Quarterly Journal of Mechanics and Applied Mathematics,1988,41(2):239-247.
[15]BAGCI C.A new unified strength of materials solution for stresses in curved beams and rings[J].Journal of Mechanical Design,1992,114(2):231-237.
[16]BAGCI C.Exact elasticity solutions for stresses and deflections in curved beams and rings of exponential and T-sections[J].Journal of Mechanical Design,1993,115(3):346-358.
[17]COOK R D.Circumferential stress in curved beams[J].The American Society of Mechanical Engineers,1992,59(1):224-225.
[18]TUTUNCU N.Plane stress analysis of end loaded orthotropic curved beams of constant thickness with applications to full rings[J].Journal of Mechanical Design,1998,120(2):368-374.
[19]SLOBODA A,HONARMANDI P.Generalized elasticity method for curved beam stress analysis:analytical and numerical comparison for a lifting hook[J].Mechanics based Design of Structures and Machines,2007,35(3):319-332.
[20]钟万勰,徐新生,张洪武.弹性曲梁问题的直接法[J].工程力学,1996,13(4):1-8.ZHONG Wanxie,XU Xinsheng,ZHANG Hongwu.On a direct method for the problem of elastic curved beams[J].Engineering Mechanics,1996,13(4):1-8.
[21]徐芝纶.弹性力学简明教程[M].4版.北京:高等教育出版社,2016:59-64.
[22]AHMED S R,HOSSAIN M Z,UDDIN M W.A general mathematical formulation for finite-difference solution of mixed-boundary-value problems of anisotropic materials[J].Computers&Structures,2005,83(1):35-51.
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