基于板模型的顶板力学分析及应用
|
摘要
本文通过理论分析、物理模拟和数值模拟等综合研究的手段和方法,系统地研究了顶板的变形破坏特征,提出了求解悬臂顶板弯曲下沉的边界积分方法。建立了复杂顶板与下伏复杂岩体共同作用的力学模型,并通过变分原理及有限差分原理得到了各种复杂力学模型的解。同时编制了求解非均匀地基上复杂板模型的通用差分程序,可以很好地得到均匀地基及局部塌陷的非均匀地基上岩层板结构的数值解。引入岩石流变理论,根据弹性一粘弹性关系相应原理,对顶板蠕变特征进行了理论分析,得到了黏弹性地基上四边简支和四边固支矩形岩层板结构的应力、弯矩和位移的粘弹性解析解。对于其它黏弹性地基上复杂边界条件的岩层板模型可通过本文编制的数值逆变换程序来得到理论解。
其次,以某煤矿矸石置换宽厚煤柱充填式巷采为工程背景,采用FLAC3D软件,建立了数值计算的模型。通过数值仿真计算,分别得到了顶板、隔离煤柱等的应力及变形规律。理论分析结果和数值计算结论相一致,表明了理论分析和数值计算所得到的结论是实用的。本文的研究成果丰富了顶板理论,为解决我国煤矿顶板问题提供了理论依据和研究方法。同时本文的研究成果已在某些煤矿得到应用和推广,并取得明显的社会经济效益。
In this paper,deformation and destruction characters of structure in rock plate were studied systemically by means of theoretical analysis, physical simulation and numerical simulation, Aiming at the bending problems of cantilever rock plate with different boundary conditions, based on the boundary integral formula for the boundary value problems of biharmonic equation,the bending solutions to the cantilever rock plate under arbitrary lateral loads are gained by this new method. The mechanical model of rock plate in the coalface of the mining is set up, and based on rock rheological theory, viscous-elastic solutions which offer the theoretical gist to rock plate support design in coal mine were obtained. Some FDM calculate programs were worked out, and using these programs we can compute some examples of non-homogeneity foundation and the foundation plate on non-uniform ground. The solutions of the plate laid on elastic foundation are transferred into Laplace space by Laplace transform according to elastic-viscoelastic theory, then the analytic solutions of the plate on viscoelastic foundation were got by inverse Laplace transform, some inverse Laplace transform cannot be transformed easily with the complicated boundary conditions, then they can gained by numerical inverse Laplace transform method.
Second, based on the engineering background of rock plate Structure in Bao Dian coal mine, the numerical calculation models were established by using FLAC3D. The discipline of rock plate structure and supporting pillar was obtained by numerical calculation of models. And important conclusions for the stress and displacement of rock plate structure were got by analyzing the effects of different models. The results of theoretical analysis were in according with the results of numerical calculation. It proved that the conclusions of both theoretical analysis and numerical calculation were credible and applied. The methods in this paper perfected the theory of rock plate and offered the theoretical gist and study methods for resolving the hard question of rock plate in coal mine of our country. The methods have been applied and generalized, and obvious economic and social benefits have been gained.
其次,以某煤矿矸石置换宽厚煤柱充填式巷采为工程背景,采用FLAC3D软件,建立了数值计算的模型。通过数值仿真计算,分别得到了顶板、隔离煤柱等的应力及变形规律。理论分析结果和数值计算结论相一致,表明了理论分析和数值计算所得到的结论是实用的。本文的研究成果丰富了顶板理论,为解决我国煤矿顶板问题提供了理论依据和研究方法。同时本文的研究成果已在某些煤矿得到应用和推广,并取得明显的社会经济效益。
In this paper,deformation and destruction characters of structure in rock plate were studied systemically by means of theoretical analysis, physical simulation and numerical simulation, Aiming at the bending problems of cantilever rock plate with different boundary conditions, based on the boundary integral formula for the boundary value problems of biharmonic equation,the bending solutions to the cantilever rock plate under arbitrary lateral loads are gained by this new method. The mechanical model of rock plate in the coalface of the mining is set up, and based on rock rheological theory, viscous-elastic solutions which offer the theoretical gist to rock plate support design in coal mine were obtained. Some FDM calculate programs were worked out, and using these programs we can compute some examples of non-homogeneity foundation and the foundation plate on non-uniform ground. The solutions of the plate laid on elastic foundation are transferred into Laplace space by Laplace transform according to elastic-viscoelastic theory, then the analytic solutions of the plate on viscoelastic foundation were got by inverse Laplace transform, some inverse Laplace transform cannot be transformed easily with the complicated boundary conditions, then they can gained by numerical inverse Laplace transform method.
Second, based on the engineering background of rock plate Structure in Bao Dian coal mine, the numerical calculation models were established by using FLAC3D. The discipline of rock plate structure and supporting pillar was obtained by numerical calculation of models. And important conclusions for the stress and displacement of rock plate structure were got by analyzing the effects of different models. The results of theoretical analysis were in according with the results of numerical calculation. It proved that the conclusions of both theoretical analysis and numerical calculation were credible and applied. The methods in this paper perfected the theory of rock plate and offered the theoretical gist and study methods for resolving the hard question of rock plate in coal mine of our country. The methods have been applied and generalized, and obvious economic and social benefits have been gained.
引文
[1] 钱鸣高,刘听成.矿山压力及其控制[M].北京:煤炭工业出版社,1984.
[2] 宋振骐.实用矿山压力[M].徐州:中国矿业大学出版社,1988.
[3] 史红,姜福兴. 采场上覆大厚度坚硬岩层破断规律的力学分析[J]. 岩石力学与工程学报,2004,23(18):3066～3069.
[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,(3,4):3～12.
[5] 陈忠辉,谢和平,王家臣.综放开采顶煤三维变形、破坏的数值分析[J].岩石力学与工程学报,21(3):309～313,2002.
[6] 钱鸣高,缪协兴.采场上覆岩层结构形态与力学分析[J].岩石力学与工程学报,1995,Vol.14,No.3,225～230.
[7] 贾喜荣,翟英达. 采场薄板矿压理论与实践综述[J]. 矿山压力与顶板管理,1999.3(4):25～29.
[8] 钱鸣高,赵国景.老顶断裂前后的矿山压力变化[J].中国矿业学院学报,1986,15(4).11～19.
[9] 姜福兴.薄板力学解在坚硬顶板采场的应用范围[J].西安矿业学院学报,1991,11(2).12～19.
[10] 钱鸣高,石平五. 矿山压力与岩层控制[M]. 徐州:中国矿业大学出版社,2003.
[11] 闫少宏,贾光胜,刘贤龙.放顶煤开采上覆岩层结构向高位转移机理分析[J].矿山压力与顶板管理,1996,(3):3～5.
[12] 张项立,王悦汉.综采放顶煤工作面岩层结构分析[J].中国矿业大学学报,1998,(4):340～343.
[13] 贾喜 荣 , 翟英 达 , 杨 双锁 . 放 顶煤工 作 面 顶 板 岩 层 结 构 及 顶 板 来 压 计 算 [J]. 煤 炭 学报,1998,(4):366～370.
[14] 钱鸣高,缪协兴,许家林等. 岩层控制的关键层理论[M].徐州:中国矿业大学出版社,2003.
[15] 侯忠杰.老顶断裂岩块端角接触尺寸[J].矿山压力与顶板管理,1999,(3—4):29～32.
[16] 钟新谷等. 长壁工作面顶板变形失稳的突变模式[J].湘潭矿业学院学报,1994,9(2):1～6.
[17] 钟新谷等. 顶板岩梁结构的稳定性与支护系统刚度[J].煤炭学报,1995,20(6):601～606.
[18] 谷栓成.层状顶板岩层中巷道稳定性研究[J].建井技术,1991,4:17～19.
[19] 朱浮声,王泳嘉,层状岩体等效模型数值分析[J].东北工学院学报,1992,13(6):533～539.
[20] 林崇德.层状岩石顶板破坏机理数值模拟过程分析[J].岩石力学与工程学报,1999,18(.4):392～396.
[21] 邢士军,张连勇,陈龙高. 综放面顶板运移规律实测研究[J]. 矿山压力与顶板管理,2002.2(4):8～10.
[22] Fan Jiarang and Ye Jianqiao,An Extract Solution for the statics and Dynamics of laminated thickplates with orthotropic layers[J]. Int.J.Solids Structures,26(5/6),655～662,1990.
[23] Vlasov,V.Z.,The Method of Initial Function in Problems of the Theory of Thick Plates and Shells[M],9th Cong.Appl.Mech.,Brussels,Belgium,6,321～330,1957.
[24] Fan Jiarang and Ye Jianqiao,Extract Solutions of Bulkling for Simply Supported Thick Laminates[J]. Composite Structures ,24, 23～28,1993.
[25] L.Lehmann,H.Antes Dynamic Structure–Soil-Structure Interaction applying the Symmetric Galerkin Boundary Element Method (SGBEM) [J]. Mechanics Research Communications 2001,28(3) 297～304.
[26] 顾铁凤,黄景.岩裂隙岩体巷道顶板失稳的块体力学分析与支护强度设计[J].湖南科技大学学报,2005,20(4):21～25 .
[27] 王卫军, 侯朝炯, 柏建彪, 等. 综放沿空掘巷顶板受力变形分析[J] . 岩土工程学报, 2001 , 23 (2) : 178～181.
[28] 高 峰,钱鸣高,缪协兴. 老顶给定变形下直接顶受力变形分析[J].岩石力学与工程学报,2000 ,19(2) :145 ～148.
[29] 邓广哲.放顶煤采场上覆岩层运动和破坏规律研究[J].矿山压力与顶板管理,1994,(2):23～26.
[30] 陈俊国, 孟进军, 董正筑. 带状无限长板弹性问题的求解及应用[J]. 实验力学, 1997, 12: 394～399.
[31] S.S. 彭.煤矿地层控制.高博彦,韩持译[M].北京:煤炭工业出版社,1984.
[32] 樊克恭,翟德元,刘锋珍.岩性弱结构巷道顶底板弱结构体破坏失稳分析[J].山东科技大学学报,2004,23(2):11～14 .
[33] 勾攀峰,侯朝炯.回采巷道锚杆支护顶板稳定性分析[J].煤炭学报,1999,24(5):466～470.
[34] 张 伟, 茜平一, 陈晓平, 向慧敏. 流变理论在深基坑开挖中的应用探讨[J]. 武汉大学学报, 2003 (36) 2, 92～96.
[35] 杨挺青,罗文波,徐平等. 黏弹性理论与应用[M]. 北京:科学出版社,2004.
[36] 杨挺青,罗文波,徐平.黏弹性理论及应用[M]. 北京:科学出版社,2004.
[37] 郭大智,任瑞波等. 层状黏弹性体系力学[M]. 哈尔滨:哈尔滨工业大学出版社,2001.
[38] 郭大智,任瑞波.层状粘弹性体系力学[M]. 哈尔滨:哈尔滨工业大学出版社,2001.
[39] 杨挺青, 张晓春.黏弹性薄板蠕变屈曲的载荷-时间特性研究[J].力学学报,2000,32(3):319～325.
[40] 袁镒吾. 悬臂矩形板的对称弯曲[J].力学与实践,1993: 15(4):37～40.
[41] 戈新生、刘松. 悬臂矩形板的对称弯曲与稳定性[J].计算力学学报,1999,16(3):361～365.
[42] 王克林、黄义. 悬臂矩形板的一般解[J].固体力学学报,1986,2(2):147～151.
[43] 张福范. 悬臂矩形板的不对称弯曲[J].固体力学学报,1980,2:171～182.
[44] 曹德欣,曹璎珞著.计算方法[M],中国矿业大学出版社,2001.
[45] K. Chandrashekhara, Theory of plate s[M]. Hyderabad, Universities Press, 2001, 114～121.
[46] Sundara Raja Iyengar,K.T.,Chandrashekhara,K.and Sebastian,V.K.,On the Analysis of Thick Rectangular Plates[J],Ing.Arch.,43(5),317～330,1974.
[47] 杨耀乾. 平板理论[M].北京:中国铁道出版社,1984.
[48] 曲庆璋,章权,季求知等. 弹性板理论[M]. 北京:人民交通出版社,2000.
[49] 徐芝纶. 弹性力学(第三版)[M]. 北京: 高等教育出版社. 1990. 361～371.
[50] 谭学术,鲜学福等.复合岩体力学理论及其应用[M],煤炭工业出版社,1994,北京,1-28,155～199.
[51] 姜福兴. 老顶的基本结构形式[J]. 岩石力学与工程学报,1993,12(4):366～379.
[52] 钱鸣高,茅献彪,缪协兴. 采动覆岩中关键层的破断规律研究[J].中国矿业大学学报,1998,27(1):39～42.
[53] 缪协兴,茅献彪,孙振武,等.采场覆岩中复合关键层的形成条件与判别方法[J].中国矿业大学学报,2005,34(5):547～550.
[54] 王红卫. 薄板理论及其在地下采场中的应用[D].北京:中国矿业大学, 2004. 35 ～57.
[55] 何满潮, 景海河, 孙晓明.软岩工程力学[M],北京:科学出版社,2002.
[56] 陈忠辉, 谢和平, 李全生.长壁工作面采场围岩铰接薄板组力学模型研究[J].煤炭学报,2005,30(2):172～176.
[57] 吴洪词. 长壁工作面基础板结构模型及其来压规律. 煤炭学报[J], 1997, Vol.22, No. 6,259～264.
[58] 杨治林.浅埋煤层长壁开采顶板结构稳定性分析[J].矿山压力与顶板,2005, (2):7～9.
[59] 陈俊国.岩土工程和矿山压力几个问题的研究[D].中国矿业大学,徐州.2005.5.
[60] 闫少宏,贾光胜,刘贤龙.放顶煤开采上覆岩层结构向高位转移机理分析[J].矿山压力与顶板管理,1996,(3):3～5.
[61] 缪协兴, 茅献彪, 胡光伟, 马占国. 岩石(煤)的碎胀与压实特性研究[J]. 实验力学, 1997, 12: 394～399.
[62] 王磊. 斜角板与等腰梯形板弯曲问题的康托络维奇法[J].固体力学学报,1984,3:454～460.
[63] 钱伟长,王刚. 曲线边界薄板弯曲问题的一种新单元-曲边四边形单元[J].应用数学与力学,1990,11(4):295～300.
[64] 黄晋,吕涛. 曲边多角形域上第一类边界积分方程的机械求积算法与分裂外推[J].计算数学,2004,26(1):51～60.
[65] 李俊贤. 杂形板弯曲问题的研究[J].上海力学,1986,4:67～77.
[66] 王志灵,程玉民.拉氏反变换的数值并行算法及其在弹性动力学边界元法中的应用[J].力学季刊,2005,26(2):224～230.
[67] 杨晓霖,徐大.拉普拉斯变换的数值逆在微分方程中的应用[J].湖南师范大学自然科学学报,2004,27(2):21～25.
[68] Barnsley M. F. Fractals Everywhere[M], New York: Academic press, 1988.
[69] 郭广礼. 老采空区上方建筑地基变形机理及其控制[M].徐州:中国矿业大学出版社,2001.
[70] 周振宇等. 建筑物下矸石充填巷采沉陷控制研究[J]. 中国科技论文在线.
[71] 胡炳南.粉煤灰充填对控制岩层移动的理论研究[J].煤矿开采.1991:2:30～32.
[72] 刘鹏亮.邢东矿充填巷式开采数值模拟与现场实测研究[D].煤科总院,北京.2007.4.
[73] Itasca Consulting Group,Inc.FLAC3D(Fast Lagrangian Ansalysis of Continua in 3 Dimensions) User Manuals[J]. Version 2.1 ,Minn-eapolis Minnesota,2002.6.
[74] Fan Jiarang and Ye Jianqiao,An Extract Solution for the statics and Dynamics of laminated thick plates with orthotropic layers[J]. Int.J.Solids Structures,26(5/6),655～662,1990.
[75] Xie H. Studies on fractal models of the micro-fracture of marble[J], Chinese Sci. Bulletin,34(15),1292～1296,1989.
[2] 宋振骐.实用矿山压力[M].徐州:中国矿业大学出版社,1988.
[3] 史红,姜福兴. 采场上覆大厚度坚硬岩层破断规律的力学分析[J]. 岩石力学与工程学报,2004,23(18):3066～3069.
[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,(3,4):3～12.
[5] 陈忠辉,谢和平,王家臣.综放开采顶煤三维变形、破坏的数值分析[J].岩石力学与工程学报,21(3):309～313,2002.
[6] 钱鸣高,缪协兴.采场上覆岩层结构形态与力学分析[J].岩石力学与工程学报,1995,Vol.14,No.3,225～230.
[7] 贾喜荣,翟英达. 采场薄板矿压理论与实践综述[J]. 矿山压力与顶板管理,1999.3(4):25～29.
[8] 钱鸣高,赵国景.老顶断裂前后的矿山压力变化[J].中国矿业学院学报,1986,15(4).11～19.
[9] 姜福兴.薄板力学解在坚硬顶板采场的应用范围[J].西安矿业学院学报,1991,11(2).12～19.
[10] 钱鸣高,石平五. 矿山压力与岩层控制[M]. 徐州:中国矿业大学出版社,2003.
[11] 闫少宏,贾光胜,刘贤龙.放顶煤开采上覆岩层结构向高位转移机理分析[J].矿山压力与顶板管理,1996,(3):3～5.
[12] 张项立,王悦汉.综采放顶煤工作面岩层结构分析[J].中国矿业大学学报,1998,(4):340～343.
[13] 贾喜 荣 , 翟英 达 , 杨 双锁 . 放 顶煤工 作 面 顶 板 岩 层 结 构 及 顶 板 来 压 计 算 [J]. 煤 炭 学报,1998,(4):366～370.
[14] 钱鸣高,缪协兴,许家林等. 岩层控制的关键层理论[M].徐州:中国矿业大学出版社,2003.
[15] 侯忠杰.老顶断裂岩块端角接触尺寸[J].矿山压力与顶板管理,1999,(3—4):29～32.
[16] 钟新谷等. 长壁工作面顶板变形失稳的突变模式[J].湘潭矿业学院学报,1994,9(2):1～6.
[17] 钟新谷等. 顶板岩梁结构的稳定性与支护系统刚度[J].煤炭学报,1995,20(6):601～606.
[18] 谷栓成.层状顶板岩层中巷道稳定性研究[J].建井技术,1991,4:17～19.
[19] 朱浮声,王泳嘉,层状岩体等效模型数值分析[J].东北工学院学报,1992,13(6):533～539.
[20] 林崇德.层状岩石顶板破坏机理数值模拟过程分析[J].岩石力学与工程学报,1999,18(.4):392～396.
[21] 邢士军,张连勇,陈龙高. 综放面顶板运移规律实测研究[J]. 矿山压力与顶板管理,2002.2(4):8～10.
[22] Fan Jiarang and Ye Jianqiao,An Extract Solution for the statics and Dynamics of laminated thickplates with orthotropic layers[J]. Int.J.Solids Structures,26(5/6),655～662,1990.
[23] Vlasov,V.Z.,The Method of Initial Function in Problems of the Theory of Thick Plates and Shells[M],9th Cong.Appl.Mech.,Brussels,Belgium,6,321～330,1957.
[24] Fan Jiarang and Ye Jianqiao,Extract Solutions of Bulkling for Simply Supported Thick Laminates[J]. Composite Structures ,24, 23～28,1993.
[25] L.Lehmann,H.Antes Dynamic Structure–Soil-Structure Interaction applying the Symmetric Galerkin Boundary Element Method (SGBEM) [J]. Mechanics Research Communications 2001,28(3) 297～304.
[26] 顾铁凤,黄景.岩裂隙岩体巷道顶板失稳的块体力学分析与支护强度设计[J].湖南科技大学学报,2005,20(4):21～25 .
[27] 王卫军, 侯朝炯, 柏建彪, 等. 综放沿空掘巷顶板受力变形分析[J] . 岩土工程学报, 2001 , 23 (2) : 178～181.
[28] 高 峰,钱鸣高,缪协兴. 老顶给定变形下直接顶受力变形分析[J].岩石力学与工程学报,2000 ,19(2) :145 ～148.
[29] 邓广哲.放顶煤采场上覆岩层运动和破坏规律研究[J].矿山压力与顶板管理,1994,(2):23～26.
[30] 陈俊国, 孟进军, 董正筑. 带状无限长板弹性问题的求解及应用[J]. 实验力学, 1997, 12: 394～399.
[31] S.S. 彭.煤矿地层控制.高博彦,韩持译[M].北京:煤炭工业出版社,1984.
[32] 樊克恭,翟德元,刘锋珍.岩性弱结构巷道顶底板弱结构体破坏失稳分析[J].山东科技大学学报,2004,23(2):11～14 .
[33] 勾攀峰,侯朝炯.回采巷道锚杆支护顶板稳定性分析[J].煤炭学报,1999,24(5):466～470.
[34] 张 伟, 茜平一, 陈晓平, 向慧敏. 流变理论在深基坑开挖中的应用探讨[J]. 武汉大学学报, 2003 (36) 2, 92～96.
[35] 杨挺青,罗文波,徐平等. 黏弹性理论与应用[M]. 北京:科学出版社,2004.
[36] 杨挺青,罗文波,徐平.黏弹性理论及应用[M]. 北京:科学出版社,2004.
[37] 郭大智,任瑞波等. 层状黏弹性体系力学[M]. 哈尔滨:哈尔滨工业大学出版社,2001.
[38] 郭大智,任瑞波.层状粘弹性体系力学[M]. 哈尔滨:哈尔滨工业大学出版社,2001.
[39] 杨挺青, 张晓春.黏弹性薄板蠕变屈曲的载荷-时间特性研究[J].力学学报,2000,32(3):319～325.
[40] 袁镒吾. 悬臂矩形板的对称弯曲[J].力学与实践,1993: 15(4):37～40.
[41] 戈新生、刘松. 悬臂矩形板的对称弯曲与稳定性[J].计算力学学报,1999,16(3):361～365.
[42] 王克林、黄义. 悬臂矩形板的一般解[J].固体力学学报,1986,2(2):147～151.
[43] 张福范. 悬臂矩形板的不对称弯曲[J].固体力学学报,1980,2:171～182.
[44] 曹德欣,曹璎珞著.计算方法[M],中国矿业大学出版社,2001.
[45] K. Chandrashekhara, Theory of plate s[M]. Hyderabad, Universities Press, 2001, 114～121.
[46] Sundara Raja Iyengar,K.T.,Chandrashekhara,K.and Sebastian,V.K.,On the Analysis of Thick Rectangular Plates[J],Ing.Arch.,43(5),317～330,1974.
[47] 杨耀乾. 平板理论[M].北京:中国铁道出版社,1984.
[48] 曲庆璋,章权,季求知等. 弹性板理论[M]. 北京:人民交通出版社,2000.
[49] 徐芝纶. 弹性力学(第三版)[M]. 北京: 高等教育出版社. 1990. 361～371.
[50] 谭学术,鲜学福等.复合岩体力学理论及其应用[M],煤炭工业出版社,1994,北京,1-28,155～199.
[51] 姜福兴. 老顶的基本结构形式[J]. 岩石力学与工程学报,1993,12(4):366～379.
[52] 钱鸣高,茅献彪,缪协兴. 采动覆岩中关键层的破断规律研究[J].中国矿业大学学报,1998,27(1):39～42.
[53] 缪协兴,茅献彪,孙振武,等.采场覆岩中复合关键层的形成条件与判别方法[J].中国矿业大学学报,2005,34(5):547～550.
[54] 王红卫. 薄板理论及其在地下采场中的应用[D].北京:中国矿业大学, 2004. 35 ～57.
[55] 何满潮, 景海河, 孙晓明.软岩工程力学[M],北京:科学出版社,2002.
[56] 陈忠辉, 谢和平, 李全生.长壁工作面采场围岩铰接薄板组力学模型研究[J].煤炭学报,2005,30(2):172～176.
[57] 吴洪词. 长壁工作面基础板结构模型及其来压规律. 煤炭学报[J], 1997, Vol.22, No. 6,259～264.
[58] 杨治林.浅埋煤层长壁开采顶板结构稳定性分析[J].矿山压力与顶板,2005, (2):7～9.
[59] 陈俊国.岩土工程和矿山压力几个问题的研究[D].中国矿业大学,徐州.2005.5.
[60] 闫少宏,贾光胜,刘贤龙.放顶煤开采上覆岩层结构向高位转移机理分析[J].矿山压力与顶板管理,1996,(3):3～5.
[61] 缪协兴, 茅献彪, 胡光伟, 马占国. 岩石(煤)的碎胀与压实特性研究[J]. 实验力学, 1997, 12: 394～399.
[62] 王磊. 斜角板与等腰梯形板弯曲问题的康托络维奇法[J].固体力学学报,1984,3:454～460.
[63] 钱伟长,王刚. 曲线边界薄板弯曲问题的一种新单元-曲边四边形单元[J].应用数学与力学,1990,11(4):295～300.
[64] 黄晋,吕涛. 曲边多角形域上第一类边界积分方程的机械求积算法与分裂外推[J].计算数学,2004,26(1):51～60.
[65] 李俊贤. 杂形板弯曲问题的研究[J].上海力学,1986,4:67～77.
[66] 王志灵,程玉民.拉氏反变换的数值并行算法及其在弹性动力学边界元法中的应用[J].力学季刊,2005,26(2):224～230.
[67] 杨晓霖,徐大.拉普拉斯变换的数值逆在微分方程中的应用[J].湖南师范大学自然科学学报,2004,27(2):21～25.
[68] Barnsley M. F. Fractals Everywhere[M], New York: Academic press, 1988.
[69] 郭广礼. 老采空区上方建筑地基变形机理及其控制[M].徐州:中国矿业大学出版社,2001.
[70] 周振宇等. 建筑物下矸石充填巷采沉陷控制研究[J]. 中国科技论文在线.
[71] 胡炳南.粉煤灰充填对控制岩层移动的理论研究[J].煤矿开采.1991:2:30～32.
[72] 刘鹏亮.邢东矿充填巷式开采数值模拟与现场实测研究[D].煤科总院,北京.2007.4.
[73] Itasca Consulting Group,Inc.FLAC3D(Fast Lagrangian Ansalysis of Continua in 3 Dimensions) User Manuals[J]. Version 2.1 ,Minn-eapolis Minnesota,2002.6.
[74] Fan Jiarang and Ye Jianqiao,An Extract Solution for the statics and Dynamics of laminated thick plates with orthotropic layers[J]. Int.J.Solids Structures,26(5/6),655～662,1990.
[75] Xie H. Studies on fractal models of the micro-fracture of marble[J], Chinese Sci. Bulletin,34(15),1292～1296,1989.