薄壁箱梁随机特性分析的新型随机有限元法
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摘要
随机性是工程结构的固有特性,因而考虑实际工程中的随机性,对工程进行随机分析具有十分重要的现实意义。本论文研究了基于随机场Karhunen-Loève级数蒙特卡罗有限元法(KLSMCFEM)以及简缩基随机有限元法(RBSFEM);并将上述方法、理论应用于薄壁箱型梁的剪力滞效应和弯曲问题中。
首先研究了随机场离散的局部平均法和Karhunen-Loève级数展开法。并针对特殊的随机场协方差函数形式,给出了特征值与特征函数的解析表达式;对于普通的随机场协方差函数,进一步讨论了特征值问题的Galerkin数值解法。
基于随机场的Karhunen-Loève正交级数展开理论,结合蒙特卡罗有限元法(MCFEM)研究建立了基于随机场Karhunen-Loève级数的蒙特卡罗有限元法(KLSMCFEM)。在随机场Karhunen-Loève级数正交展开理论的基础上,利用预处理的随机Krylov子空间基向量,将结构位移随机响应过程展开为随机简缩基向量展开式,研究建立了简缩基随机有限元法(RBSFEM),导出了结构响应量的统计特征值的计算公式。并用于分析随机参数大变异问题。
鉴于薄壁箱梁是桥梁工程通常采用的一种结构形式,在弯曲荷载下箱梁的剪力滞效应是桥梁发生损伤甚至破坏的重要原因之一,本文研究了箱梁在剪力滞效应下的随机挠曲,结合一维离散有限元方法,提出了考虑剪力滞效应的薄壁箱梁随机分析的新型随机有限元法,包括薄壁箱梁随机特性分析的KLSMCFEM法和RBSFEM法。
Randomness is the natural characteristics of engineering structure, in the normal traditional determinate finite element method these stochastic factors are neglected originally, which cannot reflect the practical situation of engineering structure accurately, sometimes we will underestimate response severely. Therefore, it is especially important and has reality significance to develop stochastic analysis of engineering structure. This dissertation focuses on the new methods of stochastic finite element based on the Karhunen-Loève expansion for the random field. For example, stochastic finite element method based on Monte Carlo and stochastic finite element method based on stochastic reduced basis (RBSFEM).
First, this dissertation focuses on the discretization of random fields which are based on the local average theory and Karhunen-Loève expansion. According to special form of convariance function, we provide the analytic expression of eigenvalues and eigenfunctions. According to normal form of convariance function, we further discuss the numerical solution of eigenvalues and eigenfunctions, such as Galerkin method.
Second, the Karhunen-Loeve series based Monte-Carlo finite element method (KLSMCFEM) is proposed for stochastic analysis of structures for the first time. The Karhunen-Loeve series based the solution process is approximated using basis vectors spanning the preconditioned stochastic Krylov subspace(RBSFEM) is proposed for stochastic analysis of structures. Therefore, the stochastic response in the form of stochastic reduce basis can be used to generate statistical moments and probability distributions. The new methods of stochastic finite element can provide results of comparable or better accuracy particularly for large-scale problems with many random variables.
Third, thin-walled box girder is a structure form that is used in bridge engineering frequently, the shear lag effect of the box girder under Bending Load is one of the important reasons of that bridge damage and even destroy. Combing the one-dimensional stochastic finite element method with the shear lag effect of the box girder theory under Bending Load, The new methods of stochastic finite element based on the shear lag effect of the box girder is proposed, including KLSMCFEM and RBSFEM. This confirms the applicability and superiority of the new methods of stochastic finite element based on the Karhunen-Loève expansion for the random field.
首先研究了随机场离散的局部平均法和Karhunen-Loève级数展开法。并针对特殊的随机场协方差函数形式,给出了特征值与特征函数的解析表达式;对于普通的随机场协方差函数,进一步讨论了特征值问题的Galerkin数值解法。
基于随机场的Karhunen-Loève正交级数展开理论,结合蒙特卡罗有限元法(MCFEM)研究建立了基于随机场Karhunen-Loève级数的蒙特卡罗有限元法(KLSMCFEM)。在随机场Karhunen-Loève级数正交展开理论的基础上,利用预处理的随机Krylov子空间基向量,将结构位移随机响应过程展开为随机简缩基向量展开式,研究建立了简缩基随机有限元法(RBSFEM),导出了结构响应量的统计特征值的计算公式。并用于分析随机参数大变异问题。
鉴于薄壁箱梁是桥梁工程通常采用的一种结构形式,在弯曲荷载下箱梁的剪力滞效应是桥梁发生损伤甚至破坏的重要原因之一,本文研究了箱梁在剪力滞效应下的随机挠曲,结合一维离散有限元方法,提出了考虑剪力滞效应的薄壁箱梁随机分析的新型随机有限元法,包括薄壁箱梁随机特性分析的KLSMCFEM法和RBSFEM法。
Randomness is the natural characteristics of engineering structure, in the normal traditional determinate finite element method these stochastic factors are neglected originally, which cannot reflect the practical situation of engineering structure accurately, sometimes we will underestimate response severely. Therefore, it is especially important and has reality significance to develop stochastic analysis of engineering structure. This dissertation focuses on the new methods of stochastic finite element based on the Karhunen-Loève expansion for the random field. For example, stochastic finite element method based on Monte Carlo and stochastic finite element method based on stochastic reduced basis (RBSFEM).
First, this dissertation focuses on the discretization of random fields which are based on the local average theory and Karhunen-Loève expansion. According to special form of convariance function, we provide the analytic expression of eigenvalues and eigenfunctions. According to normal form of convariance function, we further discuss the numerical solution of eigenvalues and eigenfunctions, such as Galerkin method.
Second, the Karhunen-Loeve series based Monte-Carlo finite element method (KLSMCFEM) is proposed for stochastic analysis of structures for the first time. The Karhunen-Loeve series based the solution process is approximated using basis vectors spanning the preconditioned stochastic Krylov subspace(RBSFEM) is proposed for stochastic analysis of structures. Therefore, the stochastic response in the form of stochastic reduce basis can be used to generate statistical moments and probability distributions. The new methods of stochastic finite element can provide results of comparable or better accuracy particularly for large-scale problems with many random variables.
Third, thin-walled box girder is a structure form that is used in bridge engineering frequently, the shear lag effect of the box girder under Bending Load is one of the important reasons of that bridge damage and even destroy. Combing the one-dimensional stochastic finite element method with the shear lag effect of the box girder theory under Bending Load, The new methods of stochastic finite element based on the shear lag effect of the box girder is proposed, including KLSMCFEM and RBSFEM. This confirms the applicability and superiority of the new methods of stochastic finite element based on the Karhunen-Loève expansion for the random field.
引文
[1]安利强,王璋奇,彭震中.二维随机场离散的局部平均法[J].华北电力大学学报,2004,31(3):108-111.
[2]陈虬,刘先斌.随机有限元法及其工程应用[M].成都:西南交通大学出版社 1993.
[3]Yamazaki F,Shinozuka M.Statistical preconditioning in simulation of stochastic vector[J].Fifth International Conference on Structural Safety and Reliability,ICOSSAR,1989,129-136
[4]Shinozuka.M,A.Lence.E.A probabilistic model for spatial distribution of material properties[J].Engng.Fracture Mech.,8(1976):217-227
[5]Handa.K,Anderson.K.Application of finite element methods in statistical analysis of structures[J].Proc.3~(rd)Int.Conf.on Struct.Safety andReliability,Trondheim,Norway,1981:409-417.
[6]Cambou.B.Application of first order uncertainty analysis in the finite element method in linear elasticity[M].Proc.2~(nd)Int.Conf.on Application of Statistics and Probability in Soil and Struct.Engng.,England,1971
[7]Hisada.T,Nakagiri.S.Role of the stochastic finite element method in structural safety and reliability [J].Proc.4th Int Conf.on Struct.Safety and Reliability Kobe,Japan(May,1985):213-219.
[8]Vanmarcke.E.H.Random fields:Analysis and synthesis[M].The MITPress,Cambrige,1983.
[9]朱位秋,任永坚.随机场的局部平均与随机有限元[J].航空学报.1986,7(6):604-611.
[10]朱位秋,任永坚.基于随机场局部平均的随机有限元法[J].固体力学学报.1988,12(4):285-293
[11]Shinozuka M.Deodatis G.Response variability of stochastic finite element systems[J].J.Engng.Mech.,ASCE.114.3.(1988):499-519.
[12]Yamazaki.F,Shinozuka.M,Dasgupta.G..Neumann expansions for stochastic finite element analysis[J].Journal of Engineering Mechanics,1988,114(8),1335-1354
[13]P.D.Spanos and R.Ghanem.Stochastic finite element expansion for random media[J].Journal of Engineering Mechanics 1989,115(5),1035-1053.
[14]P.D.Spanos and R..Ghanem.Stochastic finite element:A spectral approach[M].Springer-Verlag,New York.N.Y.1991.
[15]李杰.随机结构系统—分析与建模[M].北京:科学出版社,1996.
[16]P.Surya Mohan,P.Nair.A.Keane.Multi-element stochastic reduced basis methods[J].Comput.Methods Appl.Mech.Engrg.2008(197):1495-1506.
[17]P.Nair.Hybridization of stochastic reduced basis methods with polynomial chaos expansions[J].Probabilistic Engineering Mechanics,2006,21,182-192
[18]包世华,周坚.薄壁杆件结构力学[M].北京:中国建筑工业出版社,2006.
[19]Reissner.E.Analysis of shear lag an box beam by principle of minimum potential energy[J]. Quarterly of applied mathematics.1946,5(3):268-278
[20]Kuzmanovic.B.O,Graham.H.J.Shear lag in box griders[J].J Struct Div,ASCE,1981,107(9):1701-1712.
[21]郭金琼,房贞政,罗孝登.箱型梁桥剪滞效应分析[J].土木工程学报,1983,16(1):1-13.
[22]倪元增.槽型宽梁的剪力滞问题[J].土木工程学报,1986,19(4):32-40.
[23]程翔云,罗旗帜.箱梁在压弯荷载共同作用下的剪力滞[J].土木工程学报,1991,24(1):52-64.
[24]Moffatt.K.R,Dowlng.P.J.Shear lag in steel box griders[J].Structral Engineering,1975,53(10):439-448.
[25]魏丽娜,方放,余天庆等.变截面箱形梁桥剪力滞效应的近似计算方法[J].土木工程学报,1997,30(1):64-72.
[26]吴幼明,罗旗帜,岳珠峰.薄壁箱梁剪力滞计算的梁段有限元法[J].中国铁道科学,2003,24(4):64-68.
[27]张士铎.桥梁设计理论[M].北京:人民交通出版社,1984.
[28]杨绿峰,高兑现,李桂青.箱型梁剪力滞效应求解的样条里兹法[J].广西大学学报.1998,3(1):18-23
[29]杨绿峰,秦荣,李桂青.广义参数有限元法分析箱型梁的剪力滞效应[J].贵州工业大学学报.1997,26卷增刊:84-87
[30]高亮,杨绿峰,赵艳林.一维离散有限元法计算箱形梁剪力滞效应[J].广西大学学报.2003,28(3):269-272
[31]Yang L.F.,Leung A.Y.T.,Li Q.S.The stochastic finite segment in the analysis of the shear-lag effect on box-girders.Engineering Structures.2001,23(11)1461-1468.
[32]孙昌,杨绿峰.数值模拟法分析薄壁箱梁的可靠性[J].铁道科学与工程学报.20074(6):25-29.
[33]林道锦,秦权.有限元可靠度分析中随机场离散方法.清华大学学报(自然科学版).2002,42(6):839-842.
[34]Schueller G L.A state-of the art report on computational stochastic mechanics[J].Probabilistic Eng Mach,1997,12(4),197-321.
[35]E.Vanmarcke,M.Shinozuka,S.Nakagiri.Random fields and stochastic finite elements[J].Structural Safety,1986,3:143-166.
[36]M.F.Ngah,A.Young.Application of the spectral stochastic finite element method for performance prediction of composite structures.[J].Composite structyres,2007(78),:447-456
[37]Phoon.K.K,Huang.S.P.Implementation of Karhunen-Loeve expansion for simulation using a wavelet-Galerkin scheme[J].Probabilistic Engineering Mechanics,2002,17:293-303.
[38]Phoon.K.K,Huang.S.P.Comparison between Karhunen-Loeve and wavelet expansions for simulation of Gaussian processes[J].Computers&Structures,2004,82:985-991.
[39]Huang.S.P,Phoon.K.K.Convergence study of the truncated Karhunen-Loeve expansion for simulation of stochastic processes[J].International Journal for Numerical Methods in Engineering,2001,52:1029-1043.
[40]Phoon.K.K.Simulation of second-order processes using Karhunen-Loeve expansion[J].Computers&Structures,2002,80:1049-1060
[41]Phoon.K.K.Simulation of strongly non-Gaussian processes using Karhunen-Loeve expansion.Probabilistic[J].Engineering Mechanics,2005,20:188-198.
[42]R.Ghanem.Efficient solution of stochastic systems:Application to the embankment dam problem,Structural Safety,2007,(29):238-251.
[43]Prasanth B.Nair and Andrew J.Keane.Stochastic Reduced Basis Methods.2001
[44]P.Nair.Comparative study of projection schemes for stochastic finite element analysis.Computer methods in applied mechanics and engineering,2006,195,2371-2392
[45]徐钟济.蒙特卡罗方法[M].上海:上海科学技术出版社.1985.
[46]冯允成,邹志红,周泓.离散系统仿真[M].北京:机械工业出版社.1998.
[47]MckayM.D.,BeckmanR.J.,ConverW.J.A Comparison of Three Methods for Selecting Values in the Analysis of Output from A computer Code[J].Technometics.1979(2):239-245.
[48]王洁,武清玺.杆系结构蒙特卡罗法计算及敏感性分析[J].山西建筑.2006,32(3):46-47
[49]孙昌.广义随机空间内薄壁结构可靠度分析的数值方法[学位论文].南宁:广西大学,2005
[50]唐冲.基于随机场正交展开理论的随机有限元法[学位论文].南宁:广西大学,2007.
[51]杨绿峰,高兑现,李桂青.随机结构在随机激励下的二阶摄动随机泛函及其应用[J].应用力学学报,1999,16(4),35-39.
[52]祝云琪.薄壁箱型梁的畸变、剪力滞效应及其可靠性分析[学位论文].南宁:广西大学,2006.
[53]任伟.开口薄壁杆件静力及动力分析的一维数值理论和方法[学位论文].南宁:广西大学,2006.
[2]陈虬,刘先斌.随机有限元法及其工程应用[M].成都:西南交通大学出版社 1993.
[3]Yamazaki F,Shinozuka M.Statistical preconditioning in simulation of stochastic vector[J].Fifth International Conference on Structural Safety and Reliability,ICOSSAR,1989,129-136
[4]Shinozuka.M,A.Lence.E.A probabilistic model for spatial distribution of material properties[J].Engng.Fracture Mech.,8(1976):217-227
[5]Handa.K,Anderson.K.Application of finite element methods in statistical analysis of structures[J].Proc.3~(rd)Int.Conf.on Struct.Safety andReliability,Trondheim,Norway,1981:409-417.
[6]Cambou.B.Application of first order uncertainty analysis in the finite element method in linear elasticity[M].Proc.2~(nd)Int.Conf.on Application of Statistics and Probability in Soil and Struct.Engng.,England,1971
[7]Hisada.T,Nakagiri.S.Role of the stochastic finite element method in structural safety and reliability [J].Proc.4th Int Conf.on Struct.Safety and Reliability Kobe,Japan(May,1985):213-219.
[8]Vanmarcke.E.H.Random fields:Analysis and synthesis[M].The MITPress,Cambrige,1983.
[9]朱位秋,任永坚.随机场的局部平均与随机有限元[J].航空学报.1986,7(6):604-611.
[10]朱位秋,任永坚.基于随机场局部平均的随机有限元法[J].固体力学学报.1988,12(4):285-293
[11]Shinozuka M.Deodatis G.Response variability of stochastic finite element systems[J].J.Engng.Mech.,ASCE.114.3.(1988):499-519.
[12]Yamazaki.F,Shinozuka.M,Dasgupta.G..Neumann expansions for stochastic finite element analysis[J].Journal of Engineering Mechanics,1988,114(8),1335-1354
[13]P.D.Spanos and R.Ghanem.Stochastic finite element expansion for random media[J].Journal of Engineering Mechanics 1989,115(5),1035-1053.
[14]P.D.Spanos and R..Ghanem.Stochastic finite element:A spectral approach[M].Springer-Verlag,New York.N.Y.1991.
[15]李杰.随机结构系统—分析与建模[M].北京:科学出版社,1996.
[16]P.Surya Mohan,P.Nair.A.Keane.Multi-element stochastic reduced basis methods[J].Comput.Methods Appl.Mech.Engrg.2008(197):1495-1506.
[17]P.Nair.Hybridization of stochastic reduced basis methods with polynomial chaos expansions[J].Probabilistic Engineering Mechanics,2006,21,182-192
[18]包世华,周坚.薄壁杆件结构力学[M].北京:中国建筑工业出版社,2006.
[19]Reissner.E.Analysis of shear lag an box beam by principle of minimum potential energy[J]. Quarterly of applied mathematics.1946,5(3):268-278
[20]Kuzmanovic.B.O,Graham.H.J.Shear lag in box griders[J].J Struct Div,ASCE,1981,107(9):1701-1712.
[21]郭金琼,房贞政,罗孝登.箱型梁桥剪滞效应分析[J].土木工程学报,1983,16(1):1-13.
[22]倪元增.槽型宽梁的剪力滞问题[J].土木工程学报,1986,19(4):32-40.
[23]程翔云,罗旗帜.箱梁在压弯荷载共同作用下的剪力滞[J].土木工程学报,1991,24(1):52-64.
[24]Moffatt.K.R,Dowlng.P.J.Shear lag in steel box griders[J].Structral Engineering,1975,53(10):439-448.
[25]魏丽娜,方放,余天庆等.变截面箱形梁桥剪力滞效应的近似计算方法[J].土木工程学报,1997,30(1):64-72.
[26]吴幼明,罗旗帜,岳珠峰.薄壁箱梁剪力滞计算的梁段有限元法[J].中国铁道科学,2003,24(4):64-68.
[27]张士铎.桥梁设计理论[M].北京:人民交通出版社,1984.
[28]杨绿峰,高兑现,李桂青.箱型梁剪力滞效应求解的样条里兹法[J].广西大学学报.1998,3(1):18-23
[29]杨绿峰,秦荣,李桂青.广义参数有限元法分析箱型梁的剪力滞效应[J].贵州工业大学学报.1997,26卷增刊:84-87
[30]高亮,杨绿峰,赵艳林.一维离散有限元法计算箱形梁剪力滞效应[J].广西大学学报.2003,28(3):269-272
[31]Yang L.F.,Leung A.Y.T.,Li Q.S.The stochastic finite segment in the analysis of the shear-lag effect on box-girders.Engineering Structures.2001,23(11)1461-1468.
[32]孙昌,杨绿峰.数值模拟法分析薄壁箱梁的可靠性[J].铁道科学与工程学报.20074(6):25-29.
[33]林道锦,秦权.有限元可靠度分析中随机场离散方法.清华大学学报(自然科学版).2002,42(6):839-842.
[34]Schueller G L.A state-of the art report on computational stochastic mechanics[J].Probabilistic Eng Mach,1997,12(4),197-321.
[35]E.Vanmarcke,M.Shinozuka,S.Nakagiri.Random fields and stochastic finite elements[J].Structural Safety,1986,3:143-166.
[36]M.F.Ngah,A.Young.Application of the spectral stochastic finite element method for performance prediction of composite structures.[J].Composite structyres,2007(78),:447-456
[37]Phoon.K.K,Huang.S.P.Implementation of Karhunen-Loeve expansion for simulation using a wavelet-Galerkin scheme[J].Probabilistic Engineering Mechanics,2002,17:293-303.
[38]Phoon.K.K,Huang.S.P.Comparison between Karhunen-Loeve and wavelet expansions for simulation of Gaussian processes[J].Computers&Structures,2004,82:985-991.
[39]Huang.S.P,Phoon.K.K.Convergence study of the truncated Karhunen-Loeve expansion for simulation of stochastic processes[J].International Journal for Numerical Methods in Engineering,2001,52:1029-1043.
[40]Phoon.K.K.Simulation of second-order processes using Karhunen-Loeve expansion[J].Computers&Structures,2002,80:1049-1060
[41]Phoon.K.K.Simulation of strongly non-Gaussian processes using Karhunen-Loeve expansion.Probabilistic[J].Engineering Mechanics,2005,20:188-198.
[42]R.Ghanem.Efficient solution of stochastic systems:Application to the embankment dam problem,Structural Safety,2007,(29):238-251.
[43]Prasanth B.Nair and Andrew J.Keane.Stochastic Reduced Basis Methods.2001
[44]P.Nair.Comparative study of projection schemes for stochastic finite element analysis.Computer methods in applied mechanics and engineering,2006,195,2371-2392
[45]徐钟济.蒙特卡罗方法[M].上海:上海科学技术出版社.1985.
[46]冯允成,邹志红,周泓.离散系统仿真[M].北京:机械工业出版社.1998.
[47]MckayM.D.,BeckmanR.J.,ConverW.J.A Comparison of Three Methods for Selecting Values in the Analysis of Output from A computer Code[J].Technometics.1979(2):239-245.
[48]王洁,武清玺.杆系结构蒙特卡罗法计算及敏感性分析[J].山西建筑.2006,32(3):46-47
[49]孙昌.广义随机空间内薄壁结构可靠度分析的数值方法[学位论文].南宁:广西大学,2005
[50]唐冲.基于随机场正交展开理论的随机有限元法[学位论文].南宁:广西大学,2007.
[51]杨绿峰,高兑现,李桂青.随机结构在随机激励下的二阶摄动随机泛函及其应用[J].应用力学学报,1999,16(4),35-39.
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