考虑弯扭耦合的解析型薄壁梁单元
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摘要
为提高分析薄壁梁杆件弯扭问题的精度和效率,以Vlasov薄壁杆件理论为基础,考虑大位移和截面翘曲的影响,建立弯扭杆件位移控制方程,构造了薄壁梁杆单元的解析位移形函数,采用势能原理建立了薄壁梁杆势能泛函。利用势能驻值变分原理构造了用于内力分析的解析型薄壁梁杆单元列式,给出了解析型单元刚度矩阵,将其与理论解、插值多项式弯扭杆单元进行对比分析。结果表明,采用构造的解析型单元计算的薄壁梁扭转角及翘曲率的精度远高于插值多项式单元,且不需划分单元,即可保证计算结果与理论解的相对误差在0. 065%之内,可用于开口薄壁杆件结构的弯扭耦合受力分析及稳定计算。
To improve the accuracy and efficiency of analysis of thin-walled beams under combined bending and torsion,the displacement controlled equation was established based on the Vlasov's thin-walled member principle,considering the influence of large deformation and warping. The analytical shape function for thin-walled beams was then determined. The potential energy function of thin-walled beams was established by applying the potential energy principle. Analytical element formulations and the stiffness matrix of analytical element for thin-walled beams were obtained via the variational principle of potential energy stationary value. The calculation results of the proposed element,the theoretical solution,and the interpolation polynomial element were compared and analyzed. The results show that the accuracy of the proposed element on calculating the torsional angle and warp rate is significantly higher than that of the interpolation polynomial method. The relative error between the solution of the analytical element and the theoretical solution is less than 0. 065% when only one single element is used. The analytical element can be applied to solve the problems of force analysis and stable calculation for open thin-walled frame structures considering the coupling of bending and torsion.
To improve the accuracy and efficiency of analysis of thin-walled beams under combined bending and torsion,the displacement controlled equation was established based on the Vlasov's thin-walled member principle,considering the influence of large deformation and warping. The analytical shape function for thin-walled beams was then determined. The potential energy function of thin-walled beams was established by applying the potential energy principle. Analytical element formulations and the stiffness matrix of analytical element for thin-walled beams were obtained via the variational principle of potential energy stationary value. The calculation results of the proposed element,the theoretical solution,and the interpolation polynomial element were compared and analyzed. The results show that the accuracy of the proposed element on calculating the torsional angle and warp rate is significantly higher than that of the interpolation polynomial method. The relative error between the solution of the analytical element and the theoretical solution is less than 0. 065% when only one single element is used. The analytical element can be applied to solve the problems of force analysis and stable calculation for open thin-walled frame structures considering the coupling of bending and torsion.
引文
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[10]童根树,张磊.薄壁构件弯扭失稳的一般理论[J].建筑结构学报,2003,24(3):16-24.(TONGGenshu,ZHANG Lei.A general theory about flexuraltorsional buckling of thin-walled members[J].Journal of Building Structures,2003,24(3):16-24.(in Chinese))
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[12]MOHRI F,AZRAR L,POTIER-FERRY M.Flexuraltorsional post-buckling analysis of thin-walled elements with open sections[J].Thin-walled Structures,2001,39(11):907-938.
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[14]SHAKOURZADEH H,GUO Y Q,BATOZ J L.Atorsion bending element for thin-walled beams with open and closed cross sections[J].Computers&Structures,1995,55(6):1045-1054.
[15]李华煜,辛克贵.采用分段样条插值的半离散方法分析薄壁杆件[J].工程力学,1992,9(3):8-22.(LI Huayu,XIN Kegui.Analysis of thin-walled members by a semi-discrete method using sectional spline function[J].Engineering Mechanics,1992,9(3):8-22.(in Chinese))
[16]CORVEC V L.Nonlinear 3D frame element with multiaxial coupling under consideration of local effects[D].Berkeley:University of California,2012:95-128.
[17]CAO Q Q,XU J,CHENG X K,et al.Element for beam analysis based on parabolic distribution stiffness[C]//International Conference on Architectural Engineering and New Materials.Hangzhou:Architectural Engineering and New Materials,2015:131-137.
[18]TANG L F,XU J,WANG H Z,et al.Accurate element of compressive bar considering the effect of displacement[J].Mathematical Problems in Engineering,2015,2015:762539.
[19]李潇,王宏志,李世萍,等.解析型Winkler弹性地基梁单元构造[J].工程力学,2015,32(3):66-72.(LI Xiao,WANG Hongzhi,LI Shiping,et al.Element structure of analytical winkler elastic foundation beam[J].Engineering Mechanics,2015,32(3):66-72.(in Chinese))
[20]李静,蒋秀根,王宏志,等.解析型弹性地基Timoshenko梁单元[J].工程力学,2018,35(2):221-229.(LI Jing,JIANG Xiugen,WANG Hongzhi,et al.Analytical element for Timoshenko beam on elastic foundation[J].Engineering Mechanics,2018,35(2):221-229.(in Chinese))
[21]郭韦佟,丁敏,邓婷,等.圆拱结构静力分析的直接刚度法[J].广西大学学报(自然科学版),2017,42(4):1351-1360.(GUO Weitong,DING Min,DENGTing,et al.Direct stiffness method for static analysis of circular arch structure[J].Journal of Guangxi University(Natural Science Edition),2017,42(4):1351-1360.(in Chinese))
[22]童根树.钢结构的平面外稳定[M].北京:中国建筑工业出版社,2013:108.(TONG Genshu.The outplane stability of steel[M].Beijng:China Architecture&Building Press,2013:108.(in Chinese))
[23]夏文忠.基于Vlasov约束扭转理论的扭杆分析单元构造[D].北京:中国农业大学,2017:66-78.(XIA Wenzhong.Development of constraint torsion elements on torsion bar based on Vlasov theory[D].Beijing:China Agricultural University,2017:66-78.(in Chinese))
[2]李文雄.框架和薄壁杆件的非线性有限元分析模型研究[D].广州:华南理工大学,2015:1.(LIWenxiong.A study on nonlinear finite element analysis models for frame and thin-walled bar[D].Guangzhou:South China University of Technology,2015:1.(in Chinese))
[3]李俊,沈荣瀛,华宏星.考虑翘曲影响的BernoulliEuler薄壁梁的弯扭耦合振动[J].机械强度,2003,25(5):486-489.(LI Jun,SHEN Rongying,HUAHongxing.Coupling bending-torsional vibration of Bernoulli-Euler thin-walled beam including warping effect[J].Journal of Mechanical Strength,2003,25(5):486-489.(in Chinese))
[4]VLASOV V Z.Thin-walled elastic beams[M].2nd ed.Jerusalem:Islael Program for Scientific Translation,1961:1-66.
[5]胡启平,涂佳黄,梁经群.组合断面薄壁杆件弯扭耦合分析[J].工程力学,2010,27(7):52-55.(HU Qiping,TU Jiahuang,LIANG Jingqun.Bendingtorsion coupling analysis of the composite section thinwalled bar structure[J].Engineering Mechanics,2010,27(7):52-55.(in Chinese))
[6]HU Y R,JIN X D,CHEN B Z.A finite element model for static and dynamic analysis of thin-walled beams with asymmetric cross-sections[J].Computers&Structures,1996,61(5):897-908.
[7]KIM N I,KIM M Y.Exact dynamic/static stiffness matrices of non-symmetric thin-walled beams considering coupled shear deformation effects[J].Thin-Walled Structures,2005,43(5):701-734.
[8]王世纪.局部封闭和开口薄壁受压构件的弯扭屈曲[J].建筑结构学报,1993,14(5):59-68.(WANGShiji.Torsional-flexural buckling of partially closed and open thin-walled beam-columns[J].Journal of Buildng Structures,1993,14(5):59-68.(in Chinese))
[9]刘占科,周绪红,何子奇,等.复合荷载作用下简支钢梁弹性弯扭屈曲研究[J].建筑结构学报,2014,35(4):78-85.(LIU Zhanke,ZHOU Xuhong,HEZiqi,et al.Study on flexural-torsional buckling of simply supported steel beam under combined loads[J].Journal of Building Structures,2014,35(4):78-85.(in Chinese))
[10]童根树,张磊.薄壁构件弯扭失稳的一般理论[J].建筑结构学报,2003,24(3):16-24.(TONGGenshu,ZHANG Lei.A general theory about flexuraltorsional buckling of thin-walled members[J].Journal of Building Structures,2003,24(3):16-24.(in Chinese))
[11]李彬.薄壁构件稳定性分析[D].大连:大连理工大学,2016:11.(LI Bin.The stability analysis of thinwalled members[D].Dalian:Dalian University of Technology,2016:11.(in Chinese))
[12]MOHRI F,AZRAR L,POTIER-FERRY M.Flexuraltorsional post-buckling analysis of thin-walled elements with open sections[J].Thin-walled Structures,2001,39(11):907-938.
[13]王晓峰,杨庆山.考虑弯扭耦合的空间薄壁截面梁单元刚度矩阵[J].北京交通大学学报,2008,32(4):83-87.(WANG Xiaofeng,YANG Qingshan.Element stiffness matrix of spatial thin-walled beams considering the coupling of flexure and torsion[J].Journal of Beijing Jiaotong University,2008,32(4):83-87.(in Chinese))
[14]SHAKOURZADEH H,GUO Y Q,BATOZ J L.Atorsion bending element for thin-walled beams with open and closed cross sections[J].Computers&Structures,1995,55(6):1045-1054.
[15]李华煜,辛克贵.采用分段样条插值的半离散方法分析薄壁杆件[J].工程力学,1992,9(3):8-22.(LI Huayu,XIN Kegui.Analysis of thin-walled members by a semi-discrete method using sectional spline function[J].Engineering Mechanics,1992,9(3):8-22.(in Chinese))
[16]CORVEC V L.Nonlinear 3D frame element with multiaxial coupling under consideration of local effects[D].Berkeley:University of California,2012:95-128.
[17]CAO Q Q,XU J,CHENG X K,et al.Element for beam analysis based on parabolic distribution stiffness[C]//International Conference on Architectural Engineering and New Materials.Hangzhou:Architectural Engineering and New Materials,2015:131-137.
[18]TANG L F,XU J,WANG H Z,et al.Accurate element of compressive bar considering the effect of displacement[J].Mathematical Problems in Engineering,2015,2015:762539.
[19]李潇,王宏志,李世萍,等.解析型Winkler弹性地基梁单元构造[J].工程力学,2015,32(3):66-72.(LI Xiao,WANG Hongzhi,LI Shiping,et al.Element structure of analytical winkler elastic foundation beam[J].Engineering Mechanics,2015,32(3):66-72.(in Chinese))
[20]李静,蒋秀根,王宏志,等.解析型弹性地基Timoshenko梁单元[J].工程力学,2018,35(2):221-229.(LI Jing,JIANG Xiugen,WANG Hongzhi,et al.Analytical element for Timoshenko beam on elastic foundation[J].Engineering Mechanics,2018,35(2):221-229.(in Chinese))
[21]郭韦佟,丁敏,邓婷,等.圆拱结构静力分析的直接刚度法[J].广西大学学报(自然科学版),2017,42(4):1351-1360.(GUO Weitong,DING Min,DENGTing,et al.Direct stiffness method for static analysis of circular arch structure[J].Journal of Guangxi University(Natural Science Edition),2017,42(4):1351-1360.(in Chinese))
[22]童根树.钢结构的平面外稳定[M].北京:中国建筑工业出版社,2013:108.(TONG Genshu.The outplane stability of steel[M].Beijng:China Architecture&Building Press,2013:108.(in Chinese))
[23]夏文忠.基于Vlasov约束扭转理论的扭杆分析单元构造[D].北京:中国农业大学,2017:66-78.(XIA Wenzhong.Development of constraint torsion elements on torsion bar based on Vlasov theory[D].Beijing:China Agricultural University,2017:66-78.(in Chinese))