简支薄壁箱梁自由振动的摄动法解析解
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摘要
为研究简支薄壁箱梁中剪力滞效应对结构动力特性的影响,基于模态摄动法提出了一种求解薄壁箱梁自由振动的新方法.该方法以相同跨度等截面欧拉梁的频率和模态为Ritz基函数,将箱梁自由振动的微分方程组转化为一组非线性代数方程进行求解.对于简支箱梁,可进一步将代数方程组简化为一元二次方程,从而得到精确的特征值和特征向量.在此基础上,利用所得箱梁振动模态的解析表达式,提出了模态剪力滞系数的概念,从而建立了箱梁固有频率和剪力滞效应之间的关系.随后研究了模态剪力滞系数随跨宽比、翼板抗弯刚度和截面抗弯刚度之比的变化规律.计算结果表明:简支梁腹板处剪力滞系数最大且大于1,为正剪力滞效应;随着模态阶数的增加、跨宽比的减小和翼板抗弯刚度和截面抗弯刚度之比的增大,剪力滞效应呈现增大的趋势.
To investigate the shear lag effect of thin-walled box girder on the dynamic characteristics,a new approach was developed to analyze the free vibration of box girders based on the modal perturbation method. The natural modes of vibration of the corresponding prismatic Euler beam with the same length and boundary conditions were used as Ritz base functions. Then,the new method can transform the set of partial differential equations governing the transverse vibration of the box girder into a set of nonlinear algebraic equations. For the simplysupported beams,the algebraic equations were further simplified as quadratic equation with one unknown,so that the exact eigenvalues and eigenvectors could be obtained. The analytical vibration modes of the box girder were used to propose the shear lag coefficients of modes,which illustrates the relationship between the natural frequency and shear lag effect. Numerical examples were used to analyze the shear lag coefficients of modes varying with the ratio between span and width,the second moment of area ratio between flange slab and the full section. The numerical results show that the maximum shear lag coefficients of modes located at the web of the box girder are greater than1,which are positive shear lag effect. As the increase of modes order,the reduction of the ratio between span and width and the increase of the second moment of area ratio between flange slab and the full section,the shear lag coefficients and shear lag effect would be more remarkable.
To investigate the shear lag effect of thin-walled box girder on the dynamic characteristics,a new approach was developed to analyze the free vibration of box girders based on the modal perturbation method. The natural modes of vibration of the corresponding prismatic Euler beam with the same length and boundary conditions were used as Ritz base functions. Then,the new method can transform the set of partial differential equations governing the transverse vibration of the box girder into a set of nonlinear algebraic equations. For the simplysupported beams,the algebraic equations were further simplified as quadratic equation with one unknown,so that the exact eigenvalues and eigenvectors could be obtained. The analytical vibration modes of the box girder were used to propose the shear lag coefficients of modes,which illustrates the relationship between the natural frequency and shear lag effect. Numerical examples were used to analyze the shear lag coefficients of modes varying with the ratio between span and width,the second moment of area ratio between flange slab and the full section. The numerical results show that the maximum shear lag coefficients of modes located at the web of the box girder are greater than1,which are positive shear lag effect. As the increase of modes order,the reduction of the ratio between span and width and the increase of the second moment of area ratio between flange slab and the full section,the shear lag coefficients and shear lag effect would be more remarkable.
引文
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[9]甘亚南,周广春,赫中营.大悬臂板矩形截面箱梁动力反应的分析[J].振动与冲击,2010,29(11):61-65.GAN Y N,ZHOU G C,HE Zhongying.Dynamic response analysis for a large cantilever rectangular box girder in consideration of shear lag effect[J].Journal of Vibration&Shock,2010,29(11):61-65.
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[12]楼梦麟.变参数土层的动力特性和地震反应分析[J].同济大学学报,1997,25(2):155-160.LOU Menglin.Dynamic analysis for modal characteristics and seismic response of soil layer with variable properties[J].Journal of Tongji University,1997,25(2):155-160.
[13]楼梦麟,吴京宁.复杂梁动力问题的近似分析方法[J].上海力学,1997,18(3):234-240.LOU Menglin,WU Jingning.An approach to approximate analysis of dynamic problems of complicated beams[J].Shanghai Journal of Mechanics,1997,18(3):234~240.
[14]张怀静,潘旦光.变截面连续梁动力特性的半解析解法[J].北京科技大学学报,2008,30(6):590-593.ZHANG Huaijing,PAN Danguang.Semi-analytic solution to dynamic characteristics of non-uniform continuous beams[J].Journal of University of Science and Technology Beijing,2008,30(6):590-593.
[15]潘旦光,楼梦麟.变截面Timoshenko简支梁动力特性的半解析解[J].工程力学,2009,26(8):6-9.PAN D G,LOU M L.Semi-analytic solution of dynamic characteristics of non-prismatic timoshenko simply supported beams[J].Engineering Mechanics,2009,26(8):6-9.
[16]PAN Danguang,CHEN Genda,LOU Menglin.A modified modal perturbation method for vibration characteristics of non-prismatic Timoshenko beams[J].Structural Engineering and Mechanics An international Journal,2011,40(5):689-703.
[2]GARA F,LEONI G,DEZI L.A beam finite element including shear lag effect for the time-dependent analysis of steel-concrete composite decks[J].Engineering Structures,2009,31(8):1888-1902.
[3]郭健,孙炳楠.型钢-混凝土箱形桥塔的剪力滞后效应分析[J].哈尔滨工业大学学报,2003,35(增刊):269-271,275.GUO Jian,SUN Bingnan.Analysis of shear lag effect of steelconcrete composite box tower in cable-stayed bridge[J].Journal of Harbin Institute of Technology,2003,35(s):269-271,275.
[4]王浩,李爱群,赵大亮,等.润扬悬索桥钢箱梁受力分析及实验研究[J].哈尔滨工业大学学报,2006,38(7):1062-1064.WANG Hao,LI Aiqun,ZHAO Daliang,et al.FE analysis and experimental study on local stress of steel box girder of runyang suspension bridge[J].Journal of Harbin Institute of Technology,2006,38(7):1062-1064.
[5]ZHANG Yuanhai,LIN Lixia.Shear lag analysis of thin-walled box girders based on a new generalized displacement[J].Engineering Structures,2014,61(1):73-83.
[6]甘亚南,吴亚平,王根会,等.剪力滞效应对简支箱梁自振特性的影响研究[J].兰州铁道学院学报,2002,21(3):23-25.GAN Y N,WU Y P,WANG G H,et al.Influence of shear lag on the characteristic of free vibration to box girder under simply supported[J].Journal of Lanzhou Railway Institute,2002,21(3):23-25
[7]吴有俊,罗旗帜,陈玉骥.考虑剪力滞效应简支箱梁自振方程的求解[J].佛山科学技术学院学报(自然科学版),2010,28(2):12-17.WU Youjun,LUO Qizhi,CHEN Yuji.The Solution of free vibration equation to simply-supported box girder considering shear lag effect[J].Journal of Foshan University(Natural Science Edition),2010,28(2):12-17.
[8]张永健,黄平明.一种计入剪力滞及剪切变形效应的箱梁自振频率计算方法[J].郑州大学学报(工学版),2007,28(1):51-55.ZHANG Yongjian,HUANG Pingming.A new method to calculate the natural frequencies of simply supported box girder including the effect of shear lag and shear deformation[J].Journal of Zhengzhou University(Engineering Science),2007,28(1):51-55.
[9]甘亚南,周广春,赫中营.大悬臂板矩形截面箱梁动力反应的分析[J].振动与冲击,2010,29(11):61-65.GAN Y N,ZHOU G C,HE Zhongying.Dynamic response analysis for a large cantilever rectangular box girder in consideration of shear lag effect[J].Journal of Vibration&Shock,2010,29(11):61-65.
[10]甘亚南,周广春.基于能量变分原理的薄壁箱梁自振特性分析[J].中国公路学报,2007,20(1):73-78.GAN Yanan,ZHOU Guangchun.Analysis of free vibration characteristics of thin-walled box girder based on energy variation principle[J].China Journal of Highway&Transport,2007,20(1):73-78.
[11]ZHOU Wangbao,JIANG Lizhong,YU Zhiwu.Analysis of free vibration characteristic of steel-concrete composite box-girder considering shear lag and slip[J].Journal of Central South University,2013,20:2570-2577.
[12]楼梦麟.变参数土层的动力特性和地震反应分析[J].同济大学学报,1997,25(2):155-160.LOU Menglin.Dynamic analysis for modal characteristics and seismic response of soil layer with variable properties[J].Journal of Tongji University,1997,25(2):155-160.
[13]楼梦麟,吴京宁.复杂梁动力问题的近似分析方法[J].上海力学,1997,18(3):234-240.LOU Menglin,WU Jingning.An approach to approximate analysis of dynamic problems of complicated beams[J].Shanghai Journal of Mechanics,1997,18(3):234~240.
[14]张怀静,潘旦光.变截面连续梁动力特性的半解析解法[J].北京科技大学学报,2008,30(6):590-593.ZHANG Huaijing,PAN Danguang.Semi-analytic solution to dynamic characteristics of non-uniform continuous beams[J].Journal of University of Science and Technology Beijing,2008,30(6):590-593.
[15]潘旦光,楼梦麟.变截面Timoshenko简支梁动力特性的半解析解[J].工程力学,2009,26(8):6-9.PAN D G,LOU M L.Semi-analytic solution of dynamic characteristics of non-prismatic timoshenko simply supported beams[J].Engineering Mechanics,2009,26(8):6-9.
[16]PAN Danguang,CHEN Genda,LOU Menglin.A modified modal perturbation method for vibration characteristics of non-prismatic Timoshenko beams[J].Structural Engineering and Mechanics An international Journal,2011,40(5):689-703.