薄壁曲线箱梁空间振动分析方法
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摘要
通过在结点中引入高阶位移插值函数构造形函数矩阵,基于虚功原理和动力有限元理论,推导每个结点11自由度的薄壁曲线箱梁空间单元刚度矩阵和质量矩阵。在推导刚度矩阵时综合考虑薄壁曲线箱梁的拉压、弯曲、扭转、翘曲、畸变和剪力滞效应;在推导质量矩阵时,考虑扭心、畸心与质心不重合的影响。通过MATLAB编制相应的计算程序,求解薄壁曲线箱梁的前5阶自振频率和相应的振型,经与ANSYS板壳单元数值结果比较,吻合良好。本文沿梁长方向划分为数个单元便取得满意效果,所提出的方法可为桥梁结构抗震、抗风和车致振动计算提供依据。
The form function matrix was created by introducing high order displacement interpolation function in the node. Based on virtual work principle and dynamic finite element theory, the spatial element stiffness matrix and mass matrix of thin-walled box girder with 11 degrees of freedom in each node were deduced. The effects of tension-compression, bending, torsion, warping, distortion and shear lag were comprehensively considered in deducing stiffness matrix. The effects of non-coincidence of mass center with torsion center and distortion center were also considered in deducing mass matrix. The first 5 natural vibration frequencies and corresponding modes of thin-walled curved box girder were solved on MATLAB. A comparison of the results of present work with numerical results of ANSYS shell element shows that they agree well with each other. Satisfactory results can be obtained by dividing the beam into several elements along longitudinal direction. The proposed method can provide a basis for the calculation of seismic, wind and vehicle-induced vibration of bridge structures.
The form function matrix was created by introducing high order displacement interpolation function in the node. Based on virtual work principle and dynamic finite element theory, the spatial element stiffness matrix and mass matrix of thin-walled box girder with 11 degrees of freedom in each node were deduced. The effects of tension-compression, bending, torsion, warping, distortion and shear lag were comprehensively considered in deducing stiffness matrix. The effects of non-coincidence of mass center with torsion center and distortion center were also considered in deducing mass matrix. The first 5 natural vibration frequencies and corresponding modes of thin-walled curved box girder were solved on MATLAB. A comparison of the results of present work with numerical results of ANSYS shell element shows that they agree well with each other. Satisfactory results can be obtained by dividing the beam into several elements along longitudinal direction. The proposed method can provide a basis for the calculation of seismic, wind and vehicle-induced vibration of bridge structures.
引文
[1]张士铎,邓小华,王文州.箱形薄壁梁剪力滞效应[M].北京:人民交通出版社,1998.
[2]张元海,刘泽翔,林丽霞,等.基于势能驻值原理的薄壁箱梁畸变效应分析[J].中南大学学报(自然科学版),2016,47(10):3461-3468.ZHANG Yuanhai,LIU Zexiang,LIN Lixia,et al.Analysis on Distortion Effect of Thin-walled Box Girders Based on Principle of Stationary Potential Energy[J].Journal of Central South University(Science and Technology),2016,47(10):3461-3468.
[3]徐勋,强士中.薄壁箱梁畸变分析理论的研究[J].工程力学,2013,30(11):192-201.XU Xun,QIANG Shizhong.Research on Distortion Analysis Theory of Thin-walled Box Girder[J].Engineering Mechanics,2013,30(11):192-201.
[4]安建,谢凌志,陈宜言,等.波形钢腹板组合箱梁约束扭转频率理论解[J].地震工程与工程振动,2017,37(2):142-150.AN Jian,XIE Lingzhi,CHEN Yiyan,et al.The Restrained Torsion Frequencies Theoretical Solution of Composite Box Girder with Corrugated Steel Webs[J].Earthquake Engineering and Engineering Dynamics,2017,37(2):142-150.
[5]谢旭,黄剑源.薄壁箱型梁桥约束扭转下翘曲、畸变、和剪滞效应的空间分析[J].土木工程学报,1995,28(4):3-14.XIE Xu,HUANG Jianyuan.Three Dimensional Analysis for Warping Distortion and Shear Lag Effect of Thin-walled Box Girder Bridge under Restrained Torsion[J].China Civil Engineering Journal,1995,28(4):3-14.
[6]谢旭,黄剑源.薄壁箱型梁剪力滞效应分析的刚度法[J].工程力学,1995,12(2):95-102.XIE Xu,HUANG Jianyuan.Stiffness Method for Analysis of Shear Lag Effect on Thin-walled Box Girder[J].Engineering Mechanics,1995,12(2):95-102.
[7]韦成龙,曾庆元.薄壁曲线箱梁考虑翘曲、畸变和剪滞效应的空间分析[J].土木工程学报,2000,33(6):81-87.WEI Chenglong,ZENG Qingyuan.A New Element for Thin-walled Curved Box Girder Analysis Including Warping Distortion and Shear Lag Effects[J].China Civil Engineering Journal,2000,33(6):81-87.
[8]郑振,林友勤.弯箱梁挠曲扭转分析的刚度法[J].中国公路学报,1999,12(4):50-58.ZHENG Zhen,LIN Youqin.Flexure Torsional Analysis of Curved Box Girders by Stiffness Method[J].China Journal of Highway and Transport,1999,12(4):50-58.
[9]蔡恒,卢海林,汤正.基于板壳有限元理论的薄壁箱梁剪力滞效应[J].铁道科学与工程学报,2017,14(4):779-786.CAI Heng,LU Hailin,TANG Zheng.Shear Lag Effect of Thin-walled Box Girder Based on Shell Finite Element Theory[J].Journal of Railway Science and Engineering,2017,14(4):779-786.
[10]单德山,李乔.曲线弯梁桥振动分析方法[J].土木工程学报,2005,38(10):61-65.SHAN Deshan,LI Qiao.An Analytical Method of Vibration Analysis for Curved-girder Bridges[J].China Civil Engineering Journal,2005,38(10):61-65.
[11]李国豪,石洞,HEINS C P.曲梁桥地震分析的有限单元法[J].同济大学学报,1984(1):1-21.LI Guohao,SHI Dong,HEINS C P.Finite Element Analysis of Seismic Response of Curved Girder Bridges[J].Journal of Tongji University,1984(1):1-21.
[12]项海帆.高等桥梁结构理论[M].2版.北京:人民交通出版社,2013.
[13]孙璐,王文雷,吴国琦,等.客运专线箱梁端部裂纹成因分析及施工控制[J].铁道工程学报,2007(10):46-51.SUN Lu,WANG Wenlei,WU Guoqi,et al.Analyses of the Causes for Crack on the End Side of Box Girder for Railway Passenger Dedicated Line and Its Construction Control[J].Journal of Railway Engineering Society,2007(10):46-51.
[14]胡玉茹.薄壁箱梁的约束扭转和畸变效应分析[D].兰州:兰州交通大学,2015.
[15]何福保.薄壁曲杆的有限元法[J].固体力学学报,1981,2(2):141-157.HE Fubao.The Finite Element Method for Thin-walled Elastic Curved Beams[J].Chinese Journal of Solid Mechanics,1981,2(2):141-157.
[2]张元海,刘泽翔,林丽霞,等.基于势能驻值原理的薄壁箱梁畸变效应分析[J].中南大学学报(自然科学版),2016,47(10):3461-3468.ZHANG Yuanhai,LIU Zexiang,LIN Lixia,et al.Analysis on Distortion Effect of Thin-walled Box Girders Based on Principle of Stationary Potential Energy[J].Journal of Central South University(Science and Technology),2016,47(10):3461-3468.
[3]徐勋,强士中.薄壁箱梁畸变分析理论的研究[J].工程力学,2013,30(11):192-201.XU Xun,QIANG Shizhong.Research on Distortion Analysis Theory of Thin-walled Box Girder[J].Engineering Mechanics,2013,30(11):192-201.
[4]安建,谢凌志,陈宜言,等.波形钢腹板组合箱梁约束扭转频率理论解[J].地震工程与工程振动,2017,37(2):142-150.AN Jian,XIE Lingzhi,CHEN Yiyan,et al.The Restrained Torsion Frequencies Theoretical Solution of Composite Box Girder with Corrugated Steel Webs[J].Earthquake Engineering and Engineering Dynamics,2017,37(2):142-150.
[5]谢旭,黄剑源.薄壁箱型梁桥约束扭转下翘曲、畸变、和剪滞效应的空间分析[J].土木工程学报,1995,28(4):3-14.XIE Xu,HUANG Jianyuan.Three Dimensional Analysis for Warping Distortion and Shear Lag Effect of Thin-walled Box Girder Bridge under Restrained Torsion[J].China Civil Engineering Journal,1995,28(4):3-14.
[6]谢旭,黄剑源.薄壁箱型梁剪力滞效应分析的刚度法[J].工程力学,1995,12(2):95-102.XIE Xu,HUANG Jianyuan.Stiffness Method for Analysis of Shear Lag Effect on Thin-walled Box Girder[J].Engineering Mechanics,1995,12(2):95-102.
[7]韦成龙,曾庆元.薄壁曲线箱梁考虑翘曲、畸变和剪滞效应的空间分析[J].土木工程学报,2000,33(6):81-87.WEI Chenglong,ZENG Qingyuan.A New Element for Thin-walled Curved Box Girder Analysis Including Warping Distortion and Shear Lag Effects[J].China Civil Engineering Journal,2000,33(6):81-87.
[8]郑振,林友勤.弯箱梁挠曲扭转分析的刚度法[J].中国公路学报,1999,12(4):50-58.ZHENG Zhen,LIN Youqin.Flexure Torsional Analysis of Curved Box Girders by Stiffness Method[J].China Journal of Highway and Transport,1999,12(4):50-58.
[9]蔡恒,卢海林,汤正.基于板壳有限元理论的薄壁箱梁剪力滞效应[J].铁道科学与工程学报,2017,14(4):779-786.CAI Heng,LU Hailin,TANG Zheng.Shear Lag Effect of Thin-walled Box Girder Based on Shell Finite Element Theory[J].Journal of Railway Science and Engineering,2017,14(4):779-786.
[10]单德山,李乔.曲线弯梁桥振动分析方法[J].土木工程学报,2005,38(10):61-65.SHAN Deshan,LI Qiao.An Analytical Method of Vibration Analysis for Curved-girder Bridges[J].China Civil Engineering Journal,2005,38(10):61-65.
[11]李国豪,石洞,HEINS C P.曲梁桥地震分析的有限单元法[J].同济大学学报,1984(1):1-21.LI Guohao,SHI Dong,HEINS C P.Finite Element Analysis of Seismic Response of Curved Girder Bridges[J].Journal of Tongji University,1984(1):1-21.
[12]项海帆.高等桥梁结构理论[M].2版.北京:人民交通出版社,2013.
[13]孙璐,王文雷,吴国琦,等.客运专线箱梁端部裂纹成因分析及施工控制[J].铁道工程学报,2007(10):46-51.SUN Lu,WANG Wenlei,WU Guoqi,et al.Analyses of the Causes for Crack on the End Side of Box Girder for Railway Passenger Dedicated Line and Its Construction Control[J].Journal of Railway Engineering Society,2007(10):46-51.
[14]胡玉茹.薄壁箱梁的约束扭转和畸变效应分析[D].兰州:兰州交通大学,2015.
[15]何福保.薄壁曲杆的有限元法[J].固体力学学报,1981,2(2):141-157.HE Fubao.The Finite Element Method for Thin-walled Elastic Curved Beams[J].Chinese Journal of Solid Mechanics,1981,2(2):141-157.
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