一种考虑截面完整变形特征的空间三维梁有限元模型
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摘要
提出了一种基于截面插值的三维空间梁模型。首先引入截面插值函数——拉格朗日函数描述截面形状,以位移向量为未知变量描述截面位移,在此基础上依据插值理论构造梁单元位移场,不同于传统梁单元通过假定的中性轴挠度和转角来确定梁截面各点位移,该梁模型摒弃了中性轴假设与平截面假设,通过截面插值函数得到梁截面面内、面外变形;然后通过有限元理论推导了梁单元刚度矩阵与质量矩阵,并采用MATLAB编制了相应的有限元程序;最后通过几个典型算例展开分析,并将该单元的计算结果与经典梁理论和商用有限元软件数值计算结果进行对比,算例结果表明该单元有着更好的适用性和更高的精度。
A type of space beam model based on cross-section interpolations is proposed.Firstly,the Lagrange functions are introduced as the interpolation function to describe the shape of beam cross-section,and the displacement vectors are used as unknown variables to describe the displacements of the cross-section.On this basis,the displacement field of the beam element is constructed according to the interpolation theory.The displacements of the beam in the conventional beam element are determined by the deflection and rotation of the assumed neutral axis,while the new beam element rejects the neutral axis hypothesis and flat section hypothesis,and the deformation of the beam cross-section is obtained by the interpolation functions.Then the stiffness matrix and the mass matrix of the beam element are derived by the finite element theory,and the finite element program is compiled by MATLAB.Finally,the analysis of several typical examples is carried out in this paper,and the results are compared with those of the classical beam element and commercial finite element software.The numerical results show that the new beam element has better applicability and higher precision.
A type of space beam model based on cross-section interpolations is proposed.Firstly,the Lagrange functions are introduced as the interpolation function to describe the shape of beam cross-section,and the displacement vectors are used as unknown variables to describe the displacements of the cross-section.On this basis,the displacement field of the beam element is constructed according to the interpolation theory.The displacements of the beam in the conventional beam element are determined by the deflection and rotation of the assumed neutral axis,while the new beam element rejects the neutral axis hypothesis and flat section hypothesis,and the deformation of the beam cross-section is obtained by the interpolation functions.Then the stiffness matrix and the mass matrix of the beam element are derived by the finite element theory,and the finite element program is compiled by MATLAB.Finally,the analysis of several typical examples is carried out in this paper,and the results are compared with those of the classical beam element and commercial finite element software.The numerical results show that the new beam element has better applicability and higher precision.
引文
[1]周倩南.基于位移变形理论的空间梁模型分析与研究[D].杭州:浙江大学,2015:1-2.Zhou Qiannan.Spatial beam model analysis and research based on displacement deformation theory[D].Hangzhou:Zhejiang University,2015:1-2.
[2]高阳,王敏中.梁理论的发展历史及其方法论[C].第三届全国力学史与方法论学术研讨会,中国,兰州,2007.Gao Yang,Wang Minzhong.The development history and methodology of beam theory[C].The Third National Symposium on History and Methodology of Mechanics,Lanzhou,China,2007.
[3]Timoshenko S P,LXVI.On the correction for shear of the differential equation for transverse vibrations of prismatic bars[J].Philosophical Magazine Series 6,1921,41(245):744-746.
[4]Hughes T J R,Taylor R L,Kanoknukulchai W.Asimple and efficient finite element for plate bending[J].International Journal for Numerical Methods in Engineering,2010,11(10):1529-1543.
[5]Przemieniecki J S.Theory of Matrix Structural Analysis[M].McGraw-Hill,1968.
[6]胡海昌.弹性力学的变分原理及其应用[M].北京:科学出版社,1981:139-144.Hu Haichang.Variational Principle of Elastic Mechanics and Its Application[M].Beijing:Science Press,1981:139-144.
[7]夏桂云,曾庆元,李传习,等.建立Timoshenko深梁单元的新方法[J].交通运输工程学报,2004,4(2):27-32.Xia Guiyun,Zeng Qingyuan,Li Chuanxi,et al.A new method for establishing Timoshenko deep beam element[J].Journal of Transportation Engineering,2004,4(2):27-32.
[8]夏桂云,曾庆元.深梁理论的研究现状与工程应用[J].力学与实践,2015,37(3):302-316.Xia Guiyun,Zeng Qingyuan.Timoshenko beam theory and its applications[J].Mechanics and Practice,2015,37(3):302-316.
[9]Stephen N G,Levinson M.A second order beam theory[J].Journal of Sound&Vibration,1979,67(3):293-305.
[10]Levinson M.A new rectangular beam theory[J].Journal of Sound&Vibration,1981,74(1):81-87.
[11]Heyliger P R,Reddy J N.A higher order beam finite element for bending and vibration problems[J].Journal of Sound&Vibration,1988,126(2):309-326.
[12]Kant T,Gupta A.A finite element model for a higherorder shear-deformable beam theory[J].Journal of Sound&Vibration,1988,125(2):193-202.
[13]Soldatos K P,Elishakoff I.A transverse shear and normal deformable orthotropic beam theory[J].Journal of Sound&Vibration,1992,155(3):528-533.
[14]Marur S R,Kant T.On the angle ply higher order beam vibrations[J].Computational Mechanics,2007,40(1):25-33.
[15]Larbi L O,Kaci A,Houari M S A,et al.An efficient shear deformation beam theory based on neutral surface position for bending and free vibration of functionally graded beams[J].Mechanics Based Design of Structures&Machines,2013,41(41):421-433.
[16]Subramanian P.Dynamic analysis of laminated composite beams using higher order theories and finite elements[J].Composite Structures,2006,73(3):342-353.
[17]iMek M,Kocatürk T.Free vibration analysis of beams by using a third-order shear deformation theory[J].Sadhana,2007,32(3):167-179.
[18]王勖成.有限单元法[M].北京:清华大学出版社,2003:111-113.Wang Xucheng.Finite Element Method[M].Beijing:Tsinghua University Press,2003:111-113.
[19]Bathe Klaus-Jürgen.Finite Element Procedures in Engineering Analysis[M].Prentice-Hall,Inc,1982.
[20]O1ate E.Structural Analysis with the Finite Element Method[M].The Netherlands:Springer,2009.
[2]高阳,王敏中.梁理论的发展历史及其方法论[C].第三届全国力学史与方法论学术研讨会,中国,兰州,2007.Gao Yang,Wang Minzhong.The development history and methodology of beam theory[C].The Third National Symposium on History and Methodology of Mechanics,Lanzhou,China,2007.
[3]Timoshenko S P,LXVI.On the correction for shear of the differential equation for transverse vibrations of prismatic bars[J].Philosophical Magazine Series 6,1921,41(245):744-746.
[4]Hughes T J R,Taylor R L,Kanoknukulchai W.Asimple and efficient finite element for plate bending[J].International Journal for Numerical Methods in Engineering,2010,11(10):1529-1543.
[5]Przemieniecki J S.Theory of Matrix Structural Analysis[M].McGraw-Hill,1968.
[6]胡海昌.弹性力学的变分原理及其应用[M].北京:科学出版社,1981:139-144.Hu Haichang.Variational Principle of Elastic Mechanics and Its Application[M].Beijing:Science Press,1981:139-144.
[7]夏桂云,曾庆元,李传习,等.建立Timoshenko深梁单元的新方法[J].交通运输工程学报,2004,4(2):27-32.Xia Guiyun,Zeng Qingyuan,Li Chuanxi,et al.A new method for establishing Timoshenko deep beam element[J].Journal of Transportation Engineering,2004,4(2):27-32.
[8]夏桂云,曾庆元.深梁理论的研究现状与工程应用[J].力学与实践,2015,37(3):302-316.Xia Guiyun,Zeng Qingyuan.Timoshenko beam theory and its applications[J].Mechanics and Practice,2015,37(3):302-316.
[9]Stephen N G,Levinson M.A second order beam theory[J].Journal of Sound&Vibration,1979,67(3):293-305.
[10]Levinson M.A new rectangular beam theory[J].Journal of Sound&Vibration,1981,74(1):81-87.
[11]Heyliger P R,Reddy J N.A higher order beam finite element for bending and vibration problems[J].Journal of Sound&Vibration,1988,126(2):309-326.
[12]Kant T,Gupta A.A finite element model for a higherorder shear-deformable beam theory[J].Journal of Sound&Vibration,1988,125(2):193-202.
[13]Soldatos K P,Elishakoff I.A transverse shear and normal deformable orthotropic beam theory[J].Journal of Sound&Vibration,1992,155(3):528-533.
[14]Marur S R,Kant T.On the angle ply higher order beam vibrations[J].Computational Mechanics,2007,40(1):25-33.
[15]Larbi L O,Kaci A,Houari M S A,et al.An efficient shear deformation beam theory based on neutral surface position for bending and free vibration of functionally graded beams[J].Mechanics Based Design of Structures&Machines,2013,41(41):421-433.
[16]Subramanian P.Dynamic analysis of laminated composite beams using higher order theories and finite elements[J].Composite Structures,2006,73(3):342-353.
[17]iMek M,Kocatürk T.Free vibration analysis of beams by using a third-order shear deformation theory[J].Sadhana,2007,32(3):167-179.
[18]王勖成.有限单元法[M].北京:清华大学出版社,2003:111-113.Wang Xucheng.Finite Element Method[M].Beijing:Tsinghua University Press,2003:111-113.
[19]Bathe Klaus-Jürgen.Finite Element Procedures in Engineering Analysis[M].Prentice-Hall,Inc,1982.
[20]O1ate E.Structural Analysis with the Finite Element Method[M].The Netherlands:Springer,2009.