移动荷载作用下半无限弹性空间中地铁隧道动力响应的频域—波数域比例边界有限元法分析
|
摘要
基于比例边界有限元法与傅里叶积分变换,分析移动荷载作用下半无限弹性空间中地铁隧道动力响应。考虑到移动荷载运动方向与地铁隧道轴线方向的一致性,将半无限弹性空间动力控制方程进行频域—波数域的傅里叶积分变换;在变换域内,隧道径向采用比例坐标系,环向采用有限元意义离散,在计算域内利用Galerkin法,建立频域—波数域内的比例边界有限元方程,进而推导出半无限空间动力刚度的一阶微分矩阵方程;再利用高频渐近展开和傅里叶积分逆变换,得到时间—空间域系统的动力响应。该方法极大地减小了计算分析量,同时避免了无限边界的计算误差。利用该方法进行实例计算的结果表明:随着荷载移动速度的增大,地铁隧道振动波传播到土体表面,引起的土体振动有放大增强的现象,将会对地铁隧道上部结构的安全性造成一定影响。
The dynamic response of metro tunnel under moving load in semi-infinite elastic space was analyzed by using scaled boundary finite element method and the Fourier integral transform.Considering the consistency between the direction of moving load and the axis of metro tunnel,the dynamic governing equations in semi-infinite elastic space were deduced with the Fourier integral transform in frequency-wave domain.Proportional coordinate system was adopted for the radial direction of tunnel and FE discretization for the circumferential direction of tunnel in transform domain.The Galerkin's weighted residual method was used in computational domain to establish the scaled boundary finite element equations in frequencywave domain and then the first-order differential matrix equation of dynamic stiffness in semi-infinite elastic space was deduced.The dynamic response in time-space domain was obtained by means of high frequency asymptotic expansion and Fourier integral inverse transform.The proposed method has greatly reduced computational analysis.At the same time,the calculation error of infinite boundary is avoided.Example calculation results show that,with the increase of load moving speed,the vibration wave of metro tunnel propagates to soil surface and results in the amplification and enhancement of soil vibration,which will cause certain impact on the safety of the superstructure of subway tunnel.
The dynamic response of metro tunnel under moving load in semi-infinite elastic space was analyzed by using scaled boundary finite element method and the Fourier integral transform.Considering the consistency between the direction of moving load and the axis of metro tunnel,the dynamic governing equations in semi-infinite elastic space were deduced with the Fourier integral transform in frequency-wave domain.Proportional coordinate system was adopted for the radial direction of tunnel and FE discretization for the circumferential direction of tunnel in transform domain.The Galerkin's weighted residual method was used in computational domain to establish the scaled boundary finite element equations in frequencywave domain and then the first-order differential matrix equation of dynamic stiffness in semi-infinite elastic space was deduced.The dynamic response in time-space domain was obtained by means of high frequency asymptotic expansion and Fourier integral inverse transform.The proposed method has greatly reduced computational analysis.At the same time,the calculation error of infinite boundary is avoided.Example calculation results show that,with the increase of load moving speed,the vibration wave of metro tunnel propagates to soil surface and results in the amplification and enhancement of soil vibration,which will cause certain impact on the safety of the superstructure of subway tunnel.
引文
[1]EASON G.The Stress Produced in a Semi-Infinite Solid by a Moving Surface Force[J].International Journal of Engineering Science,1965(2):581-609.
[2]聂志红,刘宝琛,李亮,等.移动荷载作用下轨道路基动力响应分析[J].中国铁道科学,2006,27(2):15-19.(NIE Zhihong,LIU Baochen,LI Liang,et al.Study on the Dynamic Response of the Track/Subgrade under Moving Load[J].China Railway Science,2006,27(2):15-19.in Chinese)
[3]黄晓吉,扶明福,徐斌.移动环形荷载作用下饱和土中圆形衬砌隧洞动力响应研究[J].岩土力学,2012,33(3):892-898.(HUANG Xiaoji,FU Mingfu,XU Bin.Dynamic Response of a Circular Lining Tunnel in a Saturated Soil Due to Moving Ring Load[J].Rock and Soil Mechanics,2012,33(3):892-898.in Chinese)
[4]贾颖绚,刘维宁,刘卫丰,等.北京地下联络线运营列车振动对既有地铁结构的影响[J].中国铁道科学,2008,29(5):72-76.(JIA Yingxuan,LIU Weining,LIU Weifeng,et al.Influence of Vibration on the Existing Metro Structure Induced by Trains Operated on Beijing Underground Connecting Line[J].China Railway Science,2008,29(5):72-76.in Chinese)
[5]刘卫丰,刘维宁,DEGRANDE G.地铁列车运行引起地表振动的预测模型及其试验验证[J].振动工程学报,2010,23(4):373-378.(LIU Weifeng,LIU Weining,DEGRANDE G.Experimental Validation of a Numerical Model for Prediction of Metro Train-Induced Ground-Surface Vibration[J].Journal of Vibration Engineering,2010,23(4):373-378.in Chinese)
[6]袁万,蔡袁强,史吏,等.基于2.5维有限元饱和土地基中空沟隔振性能研究[J].岩土力学,2013,34(7):2111-2118.(YUAN Wan,CAI Yuanqiang,SHI Li,et al.Study of Vibration-Isolation Efficiency of Open Trench in Saturated Ground by 2.5DFinite Element Method[J].Rock and Soil Mechanics,2013,34(7):2111-2118.in Chinese)
[7]刘晶波,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.(LIU Jingbo,LYandong.A Direct Method for Analysis of Dynamic Soil-Structure Interaction[J].China Civil Engineering Journal,1998,31(3):55-64.in Chinese)
[8]SMITH W D.A Non-Reflecting Plain Boundary for Wave Propagation Problems[J].Journal of Computational Physics,1974,15(4):492-503.
[9]ENGQUIST B,MAJDA A.Absorbing Boundary Conditions for the Numerical Simulation Waves[J].Mathematics of Computation,1977,31(139):629-651.
[10]郭维,周宏业.土—结动力相互作用研究[J].中国铁道科学,2001,22(4):77-83.(GUO Wei,ZHOU Hongye.Study on Soil-Structure Interaction[J].China Railway Science,2001,22(4):77-83.in Chinese)
[11]LIAO Z P,ZHOU Z H,ZHANG Y H.Stable Implementation of Transmitting Boundary in Numerical Simulation of Wave Motion[J].Acta Geophysica Sinica,2002,45(4):554-568.
[12]KEYS R G.Local Absorbing Boundary Conditions for Acoustic Media[J].Geophysics,1985,60(5):892-902.
[13]WOLF J P,SONG C H.Doubly Asymptotic Multidirectional Transmitting Boundary for Dynamic Unbounded MediumStructure-Interaction Analysis[J].Earthquake Engineering and Structural Dynamics,1995,24(2):175-188.
[14]ANDERSEN L,NIELSEN SRK.Boundary Element Analysis of the Steady-State Response of an Elastic Half-Space to a Moving Force on Its Surface[J].Engineering Analysis with Boundary Elements,2003,27:23-38.
[15]RASMUSSEN KM,NIELSEN SRK,KIRKEGAARD PH.Boundary Element Method in Time Domain for a Moving Time-Dependent Force[J].Computer Structure,2001,79:691-701.
[16]徐斌,雷晓燕,徐满清,等.饱和土体中空沟对移动荷载被动隔振的2.5D边界元法分析[J].岩土力学,2012,33(4):1079-1086.(XU Bin,LEI Xiaoyan,XU Manqing,et al.Analysis of Effectiveness of Passive Isolation for Vibration Due to a Moving Load on Saturated Soil by Using Open Trench with 2.5D Boundary Element Method[J].Rock and Soil Mechanics,2012,33(4):1079-1086.in Chinese)
[17]SHENG X,JONES C J C,THOMPSON D J.Prediction of Ground Vibration from Trains Using the Wavenumber Finite and Boundary Element Method[J].Journal of Sound and Vibration,2006,293(3):575-586.
[18]ANDERSEN L,JONES C J C.Coupled Boundary and Finite Element Analysis of Vibration from Railway Tunnels—a Comparison of Two and Three Dimensional Models[J].Journal of Sound and Vibration,2006,293(3):575-586.
[19]WOLF J P,SONG C M.Dynamic-Stiffness Matrix in Time Domain of Unbounded Medium by Infinitesimal FiniteElement Cell Method[J].Earthquake Engineering and Structural Dynamics,1994,23:1181-1198.
[20]SONG C M,WOLF J P.The Scaled Boundary Finite-Element Method Alias Consistent Infinitesimal Finite-Element Cell Method[J].Computer Methods in Applied Mechanics and Engineering,1997,147(3/4):329-355.
[21]LI M,ZHANG H,GUAN H,et al.Three-Dimensional Investigation of Wave-Pile Group Interaction Using the Scaled Boundary Finite Element Method.Part I:Theoretical Developments[J].Ocean Engineering,2013,63:174-184.
[22]SONG H,TAO L.An Efficient Scaled Boundary FEM Model for Wave Interaction with a Nonuniform Porous Cylinder[J].International Journal for Numerical Methods in Fluids,2010,63:96-118.
[23]WOLF J P.The Scaled Boundary Finite Element Method[M].Hoboken:John Wiley&Sons Ltd,2003.
[24]OPPENHEIM A V,SCHAFER R W.Discrete-Time Signal Processing[M].London:Prentice Hall,1999.
[2]聂志红,刘宝琛,李亮,等.移动荷载作用下轨道路基动力响应分析[J].中国铁道科学,2006,27(2):15-19.(NIE Zhihong,LIU Baochen,LI Liang,et al.Study on the Dynamic Response of the Track/Subgrade under Moving Load[J].China Railway Science,2006,27(2):15-19.in Chinese)
[3]黄晓吉,扶明福,徐斌.移动环形荷载作用下饱和土中圆形衬砌隧洞动力响应研究[J].岩土力学,2012,33(3):892-898.(HUANG Xiaoji,FU Mingfu,XU Bin.Dynamic Response of a Circular Lining Tunnel in a Saturated Soil Due to Moving Ring Load[J].Rock and Soil Mechanics,2012,33(3):892-898.in Chinese)
[4]贾颖绚,刘维宁,刘卫丰,等.北京地下联络线运营列车振动对既有地铁结构的影响[J].中国铁道科学,2008,29(5):72-76.(JIA Yingxuan,LIU Weining,LIU Weifeng,et al.Influence of Vibration on the Existing Metro Structure Induced by Trains Operated on Beijing Underground Connecting Line[J].China Railway Science,2008,29(5):72-76.in Chinese)
[5]刘卫丰,刘维宁,DEGRANDE G.地铁列车运行引起地表振动的预测模型及其试验验证[J].振动工程学报,2010,23(4):373-378.(LIU Weifeng,LIU Weining,DEGRANDE G.Experimental Validation of a Numerical Model for Prediction of Metro Train-Induced Ground-Surface Vibration[J].Journal of Vibration Engineering,2010,23(4):373-378.in Chinese)
[6]袁万,蔡袁强,史吏,等.基于2.5维有限元饱和土地基中空沟隔振性能研究[J].岩土力学,2013,34(7):2111-2118.(YUAN Wan,CAI Yuanqiang,SHI Li,et al.Study of Vibration-Isolation Efficiency of Open Trench in Saturated Ground by 2.5DFinite Element Method[J].Rock and Soil Mechanics,2013,34(7):2111-2118.in Chinese)
[7]刘晶波,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.(LIU Jingbo,LYandong.A Direct Method for Analysis of Dynamic Soil-Structure Interaction[J].China Civil Engineering Journal,1998,31(3):55-64.in Chinese)
[8]SMITH W D.A Non-Reflecting Plain Boundary for Wave Propagation Problems[J].Journal of Computational Physics,1974,15(4):492-503.
[9]ENGQUIST B,MAJDA A.Absorbing Boundary Conditions for the Numerical Simulation Waves[J].Mathematics of Computation,1977,31(139):629-651.
[10]郭维,周宏业.土—结动力相互作用研究[J].中国铁道科学,2001,22(4):77-83.(GUO Wei,ZHOU Hongye.Study on Soil-Structure Interaction[J].China Railway Science,2001,22(4):77-83.in Chinese)
[11]LIAO Z P,ZHOU Z H,ZHANG Y H.Stable Implementation of Transmitting Boundary in Numerical Simulation of Wave Motion[J].Acta Geophysica Sinica,2002,45(4):554-568.
[12]KEYS R G.Local Absorbing Boundary Conditions for Acoustic Media[J].Geophysics,1985,60(5):892-902.
[13]WOLF J P,SONG C H.Doubly Asymptotic Multidirectional Transmitting Boundary for Dynamic Unbounded MediumStructure-Interaction Analysis[J].Earthquake Engineering and Structural Dynamics,1995,24(2):175-188.
[14]ANDERSEN L,NIELSEN SRK.Boundary Element Analysis of the Steady-State Response of an Elastic Half-Space to a Moving Force on Its Surface[J].Engineering Analysis with Boundary Elements,2003,27:23-38.
[15]RASMUSSEN KM,NIELSEN SRK,KIRKEGAARD PH.Boundary Element Method in Time Domain for a Moving Time-Dependent Force[J].Computer Structure,2001,79:691-701.
[16]徐斌,雷晓燕,徐满清,等.饱和土体中空沟对移动荷载被动隔振的2.5D边界元法分析[J].岩土力学,2012,33(4):1079-1086.(XU Bin,LEI Xiaoyan,XU Manqing,et al.Analysis of Effectiveness of Passive Isolation for Vibration Due to a Moving Load on Saturated Soil by Using Open Trench with 2.5D Boundary Element Method[J].Rock and Soil Mechanics,2012,33(4):1079-1086.in Chinese)
[17]SHENG X,JONES C J C,THOMPSON D J.Prediction of Ground Vibration from Trains Using the Wavenumber Finite and Boundary Element Method[J].Journal of Sound and Vibration,2006,293(3):575-586.
[18]ANDERSEN L,JONES C J C.Coupled Boundary and Finite Element Analysis of Vibration from Railway Tunnels—a Comparison of Two and Three Dimensional Models[J].Journal of Sound and Vibration,2006,293(3):575-586.
[19]WOLF J P,SONG C M.Dynamic-Stiffness Matrix in Time Domain of Unbounded Medium by Infinitesimal FiniteElement Cell Method[J].Earthquake Engineering and Structural Dynamics,1994,23:1181-1198.
[20]SONG C M,WOLF J P.The Scaled Boundary Finite-Element Method Alias Consistent Infinitesimal Finite-Element Cell Method[J].Computer Methods in Applied Mechanics and Engineering,1997,147(3/4):329-355.
[21]LI M,ZHANG H,GUAN H,et al.Three-Dimensional Investigation of Wave-Pile Group Interaction Using the Scaled Boundary Finite Element Method.Part I:Theoretical Developments[J].Ocean Engineering,2013,63:174-184.
[22]SONG H,TAO L.An Efficient Scaled Boundary FEM Model for Wave Interaction with a Nonuniform Porous Cylinder[J].International Journal for Numerical Methods in Fluids,2010,63:96-118.
[23]WOLF J P.The Scaled Boundary Finite Element Method[M].Hoboken:John Wiley&Sons Ltd,2003.
[24]OPPENHEIM A V,SCHAFER R W.Discrete-Time Signal Processing[M].London:Prentice Hall,1999.