无限元参与弹性半空间无限域的数值模拟及其特性研究
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摘要
近年来城市轨道交通和高速铁路发展迅速,由此引发的环境振动问题已经引起许多专家和学者的高度重视。现有研究表明：火车运行造成的地面振动幅值很小,可以把半无限土体近似为弹性半空间无限域用以研究运行火车产生的土体振动问题,本文建立了弹性半空间无限域的数值模型并对其相关动力、静力特性进行了分析。
首先,利用映射无限元单元和有限元单元建立了参数可控的弹性半空间无限域静力分析模型；提出了适合求解映射无限元单元刚度矩阵显式表达式的无限元形函数；通过与已知集中荷载作用下半空间无限域的理论解作对比,得到与弹性半空间内部有限元区域相匹配的刚度比值系数取值范围,并分析了利用有限元单元和无限元单元建立的耦合数值模型数值解的准确性,进而利用此数值模型可以求得复杂受力下所关心区域的静力特征；通过与用ansys建立的有限元模型数值解作对比,验证了利用有限元单元和无限元单元建立的耦合数值分析模型的准确性和高效性。
其次,利用衰减无限元单元和有限元单元建立弹性二维无限域动力数值分析模型,基于对火车行驶于铁轨的实际工程背景,本文利用平面应变单元处理弹性半空间无限域的建模问题,即建立了以平面应变单元为基础的二维动力分析数值模型。求得了不计波数的衰减无限元单元刚度矩阵和单元质量矩阵；修改了幅值衰减系数取值；研究了简谐荷载作用下所建二维数值模型的数值结果,并与ansys建立的有限元模型进行比较,验证了利用衰减无限元模拟无限元边界条件的有效性,为三维模型的建立和火车荷载的进一步精确模拟奠定了基础。
The urban rail transit and high speed railway have developed rapidly in recent years, and the environmental vibration caused by them gets many experts and research schloars'attention. According to the current studies, the soil vibration magnitudes induced by moving trains are so small that it is possible to treat the soil as elastic body in the aspect of solving the soil vibration problems. The elastic semi-infinite space's numerical model has been created in the paper with the dynamic and static features researched.
Firstly, the numerical modal with a controllable parameter is created using mapping infinite elements and finite elements. The shape fuction which is suitable for making the explicit expressions of element stiffness matrix is proposed. Compared with the theoretical solutions in point load, the stiffness ratio coefficient suiting for the inside finite elements is got which can be used to get the concerned semi-infinte space's numerical results in complex load and it also can be used to proved the simulations' precision. Compared with the finite element model's numerical solutions, the simulations'high efficiency and precision are proved.
Secondly, using the decay infinite elements and finite elements the elstic2-D infinte area numerical simulation is proposed to research the dynamic characteristics. Base on the engineering back ground of trains moving on tracks, a2-D elstic infinite space dynamic model is proposed with plane starin elements. The stiffness matrix and mass matrix of the decay elements are proposed without considering the wave number. It has changed the amplitude attenuation coefficient. The elastic semi-infinite space's dynamic characteristics are researched under harmonic load, and the effectiveness about using the decay infinite is got throught the compare between the coupling model solution and the finite model solution, which lay a solid foundation for the three-dimensional model and train load's accurate simulation.
首先,利用映射无限元单元和有限元单元建立了参数可控的弹性半空间无限域静力分析模型；提出了适合求解映射无限元单元刚度矩阵显式表达式的无限元形函数；通过与已知集中荷载作用下半空间无限域的理论解作对比,得到与弹性半空间内部有限元区域相匹配的刚度比值系数取值范围,并分析了利用有限元单元和无限元单元建立的耦合数值模型数值解的准确性,进而利用此数值模型可以求得复杂受力下所关心区域的静力特征；通过与用ansys建立的有限元模型数值解作对比,验证了利用有限元单元和无限元单元建立的耦合数值分析模型的准确性和高效性。
其次,利用衰减无限元单元和有限元单元建立弹性二维无限域动力数值分析模型,基于对火车行驶于铁轨的实际工程背景,本文利用平面应变单元处理弹性半空间无限域的建模问题,即建立了以平面应变单元为基础的二维动力分析数值模型。求得了不计波数的衰减无限元单元刚度矩阵和单元质量矩阵；修改了幅值衰减系数取值；研究了简谐荷载作用下所建二维数值模型的数值结果,并与ansys建立的有限元模型进行比较,验证了利用衰减无限元模拟无限元边界条件的有效性,为三维模型的建立和火车荷载的进一步精确模拟奠定了基础。
The urban rail transit and high speed railway have developed rapidly in recent years, and the environmental vibration caused by them gets many experts and research schloars'attention. According to the current studies, the soil vibration magnitudes induced by moving trains are so small that it is possible to treat the soil as elastic body in the aspect of solving the soil vibration problems. The elastic semi-infinite space's numerical model has been created in the paper with the dynamic and static features researched.
Firstly, the numerical modal with a controllable parameter is created using mapping infinite elements and finite elements. The shape fuction which is suitable for making the explicit expressions of element stiffness matrix is proposed. Compared with the theoretical solutions in point load, the stiffness ratio coefficient suiting for the inside finite elements is got which can be used to get the concerned semi-infinte space's numerical results in complex load and it also can be used to proved the simulations' precision. Compared with the finite element model's numerical solutions, the simulations'high efficiency and precision are proved.
Secondly, using the decay infinite elements and finite elements the elstic2-D infinte area numerical simulation is proposed to research the dynamic characteristics. Base on the engineering back ground of trains moving on tracks, a2-D elstic infinite space dynamic model is proposed with plane starin elements. The stiffness matrix and mass matrix of the decay elements are proposed without considering the wave number. It has changed the amplitude attenuation coefficient. The elastic semi-infinite space's dynamic characteristics are researched under harmonic load, and the effectiveness about using the decay infinite is got throught the compare between the coupling model solution and the finite model solution, which lay a solid foundation for the three-dimensional model and train load's accurate simulation.
引文
[1]夏禾,张楠.车辆与结构动力相互作用[M].北京：科学出版社,2005
[2]赵崇斌.无限域中波传播问题的数值计算模拟[J].中国科学,2010,40(7)：828-837
[3]ZHAOChongBin. Coupled method of finite and dynamic infinite elements for simulating wave propagation in elastic solids invoing infinite domains[J]. Technological Sciences,2010,53(6): 1678-1687
[4]廖振鹏.工程波动理论导论[M].北京：科学出版社,2002
[5]吴世明.土介质中的波[M].北京：科学出版社,1997
[6]Mindlin. Force at a point in the interior of a semi-infinte solid[J]. Physics,1936,7(5):195-202
[7]徐志英.以明德林公式为依据的地基中垂直应力的计算公式[J].工程学报,1957
[8]袁聚云,赵锡宏.竖向均布荷载作用在地基内部时的土中应力计算公式[J].上海力学,1995,16(3)：213-221
[9]袁聚云,赵锡宏.水平均布荷载作用在地基内部时的上中应力计算公式[J].上海力学,1995,16(4)：339-346
[10]袁聚云,赵锡宏.竖向线荷载和条形均布荷载作用在地基内部时的上中应力计算公式[J].上海力学,1999,20(2)：156-165
[11]徐远杰.半空间Burgers粘弹性体受集中力的理论解[J].武汉水利水电大学学报,1997,30(2)：37-41
[12]祝彦知.Kelvin长无限体内部受集中力作用时的粘弹性解[J].工业建筑,2005,35(12)：55-60
[13]Biot. General solutions of the equation of elasticity and consolidation for a porous material[J]. Journal of Applied Mechanics,1956, (3):91-96
[14]Philippacopoulos. Buried point source in a poroelastic half-space[J]. Journal of Applied Mechanics,1997,123:860-869
[15]许苗健,王城,孟楷.Gibson地基在简谐荷载下的竖向振动分析[J].浙江大学学报,2004,39(8)：1015-1019
[16]]三立忠,陈云敏,吴世民.饱和弹性半空间在低频谐和集中力下的积分形式解[J].水利学报,1996,(2)：84-88
[17]王建华,陆建飞,沈为平.半空间饱和土体在内部简谐垂直力作用下的Green函数[J].水利学报,2001,(3)：54-57
[18]陈胜立,张建民,陈龙珠.饱和地基中点源荷载动力Green函数[J].岩土工程学报,2001,23(4)：423-426
[19]周香莲,F建华,陆建飞.饱和土表面在水平集中荷载作用下的瞬态反反应[J].岩土力学,2002,23(3)：387-391
[20]陈刚,徐长节.半空间饱和土在埋置扭转荷载作用下的响应[J].浙江大学学报,2005,39(10)
[21]徐长节,史焱永.非饱和土中波的传播特性[J].岩土力学,2004,25(3)：354-358
[22]张智卿,王奎华,谢康和¨.非饱和土层中桩的扭转振动响应分析[J].计算机力学学报,2008, 28(6)： 729-734
[23]张智卿,王全华,谢康和.非饱和土层中弹性支承桩的扭转振动响应分析[J].计算力学学报,2008,25(4)：552-556
[24]徐明江,魏德敏,和春保.垂直集中表面荷载作用下非饱和土半空间中的Lamb问题[J].华南理工大学学报,2009,37(12)：105-117
[25]Sneddon. Fourier transform[M]. New York:McGraw-Hill,1951
[26]Cole. Stresses produced in a half plane by moving loads[J]. ApplMech,1958,20(5):433-436
[27]Eason. The stresses produced in a semi-infinite solid by a moving surface force[J]. International Journal of Engineering Sciences,1965,2:581-609
[28]Alabi. A parametric study on some aspects of ground-brone vibrations due to rail traffic[J]. Sound Vibration,153:77-87
[29]Barros. Response of a layered visoelastic half-space to a moving point load[J]. Wave Motion, 1994,19:189-210
[30]Hung H H. Elastic waves in visco-elastic half-space generated by various vehicle loads[J]. Soil Dynamics and Earthquake Engineering,2001,21: 1-7
[31]蒋建群,周华飞,张土乔.弹性半空间体在移动集中荷载作用下的稳态响应[J].岩土工程学报,2004,26(4)
[32]廖振鹏,黄孔亮,杨柏坡等.暂态波透射边界[J].中国科学A辑,1984,(6)：556-564
[33]Eegquist B, Majda A. Absorbing boundary conditions for the numerical simulation of wave[J]. Journal of Solid Structure,1981,17(8):687-697
[34]Lysmer J, Waas G. Shear waves in plane infinite structures[J]. Journal of the Engineering Mechanics, ASCE,1972,98(1):859-877
[35]Lysmer J, Kuhlemeyer R L. Finite dynamic model for infinite media[J]. Journal of the Engineering Mechanics, ASCE,1969,95(4):859-877
[36]Underwood P, Geers T L, Doubly asymptotic, boundary-element analysis of dynamic soil-structure interaction[J]. Journal of Solid Structure,1981,17(8):687-697
[37]刘品波,吕彦东.结构-地基动力相互作用问题分析的一种直接方法[J].土木工程学报,1998,31(3)：55-64
[38]张楚汉,赵崇斌.复杂地基中波动问题的数值模[J].土木工程学报,1987,20(4)：83-96
[39]赵崇斌,张楚汉,张光斗.映射动力无穷元及其特性研究[J].工程与工程振动,J987,7(3)： 1-15
[40]赵崇斌.用无穷元研究土层特性对场地地震反应的影响[J].水利水运科学研究院,1988,(4)： 31-44
[41]赵崇斌,张楚汉,张光斗.用映射动力无穷元求解地基中的三维波动问题[J].中国科学,1988,(10)：1109-1120
[42]赵崇斌,张楚汉,张光斗.弹性和粘弹性地基无限域动力模型[J].岩土工程学报,1991,13(1)：11-23
[43]赵崇斌.模拟波在弹性固体无限域中传播的有限元与动力无穷元耦合方法[J].中国科学,2010,40(12)：1468-1477
[44]燕柳斌.用三维映射无限元模拟重力坝地基[J].学报,1991,10：7-12
[45]燕柳斌.用映射无限元模拟直立式沉箱基础[J].红水河,1997,16(2)：25-27
[46]向前无限元参与地基基础-坝体动力相互作用分析[D].广西：广西人学,2004
[47]解家毕.带桩多高层建筑共同作用分析方法及其在纠偏中的应用[D].武汉：武汉人学,2004
[48]YAN XiShui, YE HuiFei. 3D time-domain regular grid infinite element in elastic foundation[J]. Technological Sciences,2010,53(5):1413-1423
[49]王后裕,朱可善.三维双向及二向映射无限元[J].工程力学学报,2002,19(3)：95-98
[50]Chongbin Zhao. Dynamic and Transient Infinite Elements[M]. London:Springer,2009
[51]Y B Yang, H H Hung. Wave propagation for train-induced vibrations a finite/infinite element approach[M]. World Scientific,2008
[52]Lamb H. On the Propagation of Tremors over the Surface of an Elastic Solid[J]. Philosophical Transactions of the Royal Society of London,1904,203:1-42
[53]严人觉,王贻荪,韩清宇.动力基础半空间理论概论[M].北京：中国建筑工业出版社,1981
[2]赵崇斌.无限域中波传播问题的数值计算模拟[J].中国科学,2010,40(7)：828-837
[3]ZHAOChongBin. Coupled method of finite and dynamic infinite elements for simulating wave propagation in elastic solids invoing infinite domains[J]. Technological Sciences,2010,53(6): 1678-1687
[4]廖振鹏.工程波动理论导论[M].北京：科学出版社,2002
[5]吴世明.土介质中的波[M].北京：科学出版社,1997
[6]Mindlin. Force at a point in the interior of a semi-infinte solid[J]. Physics,1936,7(5):195-202
[7]徐志英.以明德林公式为依据的地基中垂直应力的计算公式[J].工程学报,1957
[8]袁聚云,赵锡宏.竖向均布荷载作用在地基内部时的土中应力计算公式[J].上海力学,1995,16(3)：213-221
[9]袁聚云,赵锡宏.水平均布荷载作用在地基内部时的上中应力计算公式[J].上海力学,1995,16(4)：339-346
[10]袁聚云,赵锡宏.竖向线荷载和条形均布荷载作用在地基内部时的上中应力计算公式[J].上海力学,1999,20(2)：156-165
[11]徐远杰.半空间Burgers粘弹性体受集中力的理论解[J].武汉水利水电大学学报,1997,30(2)：37-41
[12]祝彦知.Kelvin长无限体内部受集中力作用时的粘弹性解[J].工业建筑,2005,35(12)：55-60
[13]Biot. General solutions of the equation of elasticity and consolidation for a porous material[J]. Journal of Applied Mechanics,1956, (3):91-96
[14]Philippacopoulos. Buried point source in a poroelastic half-space[J]. Journal of Applied Mechanics,1997,123:860-869
[15]许苗健,王城,孟楷.Gibson地基在简谐荷载下的竖向振动分析[J].浙江大学学报,2004,39(8)：1015-1019
[16]]三立忠,陈云敏,吴世民.饱和弹性半空间在低频谐和集中力下的积分形式解[J].水利学报,1996,(2)：84-88
[17]王建华,陆建飞,沈为平.半空间饱和土体在内部简谐垂直力作用下的Green函数[J].水利学报,2001,(3)：54-57
[18]陈胜立,张建民,陈龙珠.饱和地基中点源荷载动力Green函数[J].岩土工程学报,2001,23(4)：423-426
[19]周香莲,F建华,陆建飞.饱和土表面在水平集中荷载作用下的瞬态反反应[J].岩土力学,2002,23(3)：387-391
[20]陈刚,徐长节.半空间饱和土在埋置扭转荷载作用下的响应[J].浙江大学学报,2005,39(10)
[21]徐长节,史焱永.非饱和土中波的传播特性[J].岩土力学,2004,25(3)：354-358
[22]张智卿,王奎华,谢康和¨.非饱和土层中桩的扭转振动响应分析[J].计算机力学学报,2008, 28(6)： 729-734
[23]张智卿,王全华,谢康和.非饱和土层中弹性支承桩的扭转振动响应分析[J].计算力学学报,2008,25(4)：552-556
[24]徐明江,魏德敏,和春保.垂直集中表面荷载作用下非饱和土半空间中的Lamb问题[J].华南理工大学学报,2009,37(12)：105-117
[25]Sneddon. Fourier transform[M]. New York:McGraw-Hill,1951
[26]Cole. Stresses produced in a half plane by moving loads[J]. ApplMech,1958,20(5):433-436
[27]Eason. The stresses produced in a semi-infinite solid by a moving surface force[J]. International Journal of Engineering Sciences,1965,2:581-609
[28]Alabi. A parametric study on some aspects of ground-brone vibrations due to rail traffic[J]. Sound Vibration,153:77-87
[29]Barros. Response of a layered visoelastic half-space to a moving point load[J]. Wave Motion, 1994,19:189-210
[30]Hung H H. Elastic waves in visco-elastic half-space generated by various vehicle loads[J]. Soil Dynamics and Earthquake Engineering,2001,21: 1-7
[31]蒋建群,周华飞,张土乔.弹性半空间体在移动集中荷载作用下的稳态响应[J].岩土工程学报,2004,26(4)
[32]廖振鹏,黄孔亮,杨柏坡等.暂态波透射边界[J].中国科学A辑,1984,(6)：556-564
[33]Eegquist B, Majda A. Absorbing boundary conditions for the numerical simulation of wave[J]. Journal of Solid Structure,1981,17(8):687-697
[34]Lysmer J, Waas G. Shear waves in plane infinite structures[J]. Journal of the Engineering Mechanics, ASCE,1972,98(1):859-877
[35]Lysmer J, Kuhlemeyer R L. Finite dynamic model for infinite media[J]. Journal of the Engineering Mechanics, ASCE,1969,95(4):859-877
[36]Underwood P, Geers T L, Doubly asymptotic, boundary-element analysis of dynamic soil-structure interaction[J]. Journal of Solid Structure,1981,17(8):687-697
[37]刘品波,吕彦东.结构-地基动力相互作用问题分析的一种直接方法[J].土木工程学报,1998,31(3)：55-64
[38]张楚汉,赵崇斌.复杂地基中波动问题的数值模[J].土木工程学报,1987,20(4)：83-96
[39]赵崇斌,张楚汉,张光斗.映射动力无穷元及其特性研究[J].工程与工程振动,J987,7(3)： 1-15
[40]赵崇斌.用无穷元研究土层特性对场地地震反应的影响[J].水利水运科学研究院,1988,(4)： 31-44
[41]赵崇斌,张楚汉,张光斗.用映射动力无穷元求解地基中的三维波动问题[J].中国科学,1988,(10)：1109-1120
[42]赵崇斌,张楚汉,张光斗.弹性和粘弹性地基无限域动力模型[J].岩土工程学报,1991,13(1)：11-23
[43]赵崇斌.模拟波在弹性固体无限域中传播的有限元与动力无穷元耦合方法[J].中国科学,2010,40(12)：1468-1477
[44]燕柳斌.用三维映射无限元模拟重力坝地基[J].学报,1991,10：7-12
[45]燕柳斌.用映射无限元模拟直立式沉箱基础[J].红水河,1997,16(2)：25-27
[46]向前无限元参与地基基础-坝体动力相互作用分析[D].广西：广西人学,2004
[47]解家毕.带桩多高层建筑共同作用分析方法及其在纠偏中的应用[D].武汉：武汉人学,2004
[48]YAN XiShui, YE HuiFei. 3D time-domain regular grid infinite element in elastic foundation[J]. Technological Sciences,2010,53(5):1413-1423
[49]王后裕,朱可善.三维双向及二向映射无限元[J].工程力学学报,2002,19(3)：95-98
[50]Chongbin Zhao. Dynamic and Transient Infinite Elements[M]. London:Springer,2009
[51]Y B Yang, H H Hung. Wave propagation for train-induced vibrations a finite/infinite element approach[M]. World Scientific,2008
[52]Lamb H. On the Propagation of Tremors over the Surface of an Elastic Solid[J]. Philosophical Transactions of the Royal Society of London,1904,203:1-42
[53]严人觉,王贻荪,韩清宇.动力基础半空间理论概论[M].北京：中国建筑工业出版社,1981