等/变截面铁摩辛柯梁单元一致质量矩阵分析  

Study on The Consistent Mass Matrix of Timoshenko Beam Element with Constant or Tapered Section

作　　者：胡连超 张军锋[2] 杨作杰 耿玉鹏 叶雨山 李杰[2] HU Lianchao;ZHANG Junfeng;YANG Zuojie;GENG Yupeng;YE Yushan;LI Jie(Communications Construction Company of CSCEC 7th Division Co.,Ltd.,Zhengzhou 450003,China;School of Hydraulics and Civil Engineering,Zhengzhou University,Zhengzhou 450001,China;Henan Puze Expressway Co.,LTD.,Puyang 457000,China)

机构地区：[1]中建七局交通建设有限公司,郑州450003 [2]郑州大学土木工程学院,郑州450001 [3]河南濮泽高速公路有限公司,濮阳457000

出　　处：《结构工程师》2025年第1期34-42,共9页Structural Engineers

基　　金：国家自然科学基金(51508523);河南省交通运输厅科技项目(2021J2);中建七局科技研发课题(CSCEC7b-2023-Z-13)。

摘　　要：因既有研究对铁摩辛柯梁单元质量矩阵的分析过程缺乏系统介绍,为明确其一致质量矩阵的推导方法,以其所用拉格朗日形函数为基础,基于虚功原理,区分伸缩、扭转、挠曲变形和转角变形这四种运动状态,分别给出了2节点和3节点等截面单元在单一运动状态下的质量矩阵,并汇总为完整质量矩阵理论表达式,还针对变截面单元给出了有限元计算中的简化计算方法。另外,介绍了铁摩辛柯梁和欧拉梁在形函数层面上的差异,以及铁摩辛柯梁单元在受弯状态下质量矩阵和刚度矩阵分析中的差异。结果表明,拉格朗日形函数作为铁摩辛柯梁各种受力变形的统一表达,尤其是受弯状态下挠曲和转角变形的独立性,使得所有单独受力模式下的质量矩阵形式相同,均可由相应的截面参数配合同一种形函数来表达,并且最终所得单元的完整质量矩阵有更好的解耦性;对于变截面单元,实用计算中采用独立积分并配合高斯分的简化计算方法,由此所得质量矩阵表达式与等截面梁的形式一致,只是采用等效截面参数替换等截面梁的截面参数。Due to the non-sufficient introduction on the derivation process of mass matrix of Timoshenko beam element in existed studies,the present study was initiated for the derivation of consistent mass matrix of Timoshenko beam element.Based on the Lagrange shape function and the virtual work,the theorical expression of mass matrix for each single motion state was given for the 2-node and 3-node elements separately,including the tension,torsion,bending-deflection and bending-rotation motions,and then the complete mass matrix was got by aggregation.Moreover,the simplified procedure in practical finite element calculation was also proposed for the tapered-section element.Another,the difference of the shape functions between the Timoshenko and Euler beam elements was also elaborated,as well as the difference in the derivation of the mass and stiffness matrixes concerning the bending condition.The results show that the Lagrange shape function was employed for all the deform conditions and especially the uncoupling of the deflection and rotation in bending conditions make the mass matrix for each single motion state shares the same form:each one is expressed by the corresponding section parameter and the same shape function.Additionally,the complete mass matrix based on this foundation is featured by the high degree of decoupling.For the tapered-section element,when the theoretical whole integration is simplified to be independent integration calculated by Gauss integration method in practical application,the obtained mass matrix shares the same form with the constant-beam element,just replacing the constant section parameters by the equivalent section parameters.

关 键 词：一致质量矩阵 铁摩辛柯梁单元 形函数 独立插值 变截面 

分 类 号：TU973[建筑科学—结构工程]