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基于共旋法与稳定函数的几何非线性平面梁单元

邓继华 谭建平 谭平 田仲初

邓继华, 谭建平, 谭平, 田仲初. 基于共旋法与稳定函数的几何非线性平面梁单元[J]. 工程力学, 2020, 37(11): 28-35. doi: 10.6052/j.issn.1000-4750.2020.01.0012
引用本文: 邓继华, 谭建平, 谭平, 田仲初. 基于共旋法与稳定函数的几何非线性平面梁单元[J]. 工程力学, 2020, 37(11): 28-35. doi: 10.6052/j.issn.1000-4750.2020.01.0012
Ji-hua DENG, Jian-ping TAN, Ping TAN, Zhong-chu TIAN. A GEOMETRIC NONLINEAR PLANE BEAM ELEMENT BASED ON COROTATIONAL FORMULATION AND ON STABILITY FUNCTIONS[J]. Engineering Mechanics, 2020, 37(11): 28-35. doi: 10.6052/j.issn.1000-4750.2020.01.0012
Citation: Ji-hua DENG, Jian-ping TAN, Ping TAN, Zhong-chu TIAN. A GEOMETRIC NONLINEAR PLANE BEAM ELEMENT BASED ON COROTATIONAL FORMULATION AND ON STABILITY FUNCTIONS[J]. Engineering Mechanics, 2020, 37(11): 28-35. doi: 10.6052/j.issn.1000-4750.2020.01.0012

基于共旋法与稳定函数的几何非线性平面梁单元

doi: 10.6052/j.issn.1000-4750.2020.01.0012
基金项目: 广东省自然科学基金项目(2015A030310141);中国博士后基金项目(2014M562154);国家自然科学基金项目(51478049);湖南省科技重大专项(2015GK1001-1);长沙理工大学土木工程优势特色重点学科创新性项目(2015ZDXK03);长沙理工大学研究生“实践创新与创业能力提升计划”项目(SJCX202001);长沙理工大学青年教师成长计划项目(2019QJCZ059)
详细信息
    作者简介:

    谭建平(1996−),男,湖南人,硕士生,主要从事大跨径桥梁施工监控研究(E-mail: 834115536@qq.com)

    谭 平(1973−),男,湖南人,研究员,博士,博导,主要从事结构抗震和防灾减灾研究(E-mail: ptan@gzu.edu.cn)

    田仲初(1963−),男,湖南人,教授,博士,博导,主要从事大跨径桥梁施工监控研究(E-mail: 382525361@qq.com)

    通讯作者:

    邓继华(1975−),男,湖南冷水江人,副教授,博士,硕导,主要从事桥梁与结构非线性研究(E-mail: jihuadeng@sina.com)

  • 中图分类号: TV332.12

A GEOMETRIC NONLINEAR PLANE BEAM ELEMENT BASED ON COROTATIONAL FORMULATION AND ON STABILITY FUNCTIONS

  • 摘要: 建立一个准确、高效的几何非线性梁单元对于描述杆系结构的非线性行为至关重要。该文基于共旋坐标法和稳定函数提出了一种几何非线性平面梁单元。该单元在形成中把变形和刚体位移分开,局部坐标系内采用稳定函数以考虑单元P-δ效应的影响,从局部坐标系到结构坐标系的转换则采用共旋坐标法以及微分以考虑几何非线性,给出了几何非线性平面梁单元在结构坐标系下的全量平衡方程和切线刚度矩阵;在此基础上根据带铰梁端弯矩为零的受力特征,导出了能考虑梁端带铰的单元切线刚度矩阵表达式。通过多个典型算例验证了算法与程序的正确性、计算精度和效率。
  • 图  1  变形前后平面梁单元

    Figure  1.  Plane beam element before and after deformation

    图  2  梁单元及微段

    Figure  2.  beam element and micro-segment

    图  3  投影增量法示意

    Figure  3.  Projection increment method

    图  4  Lee’s框架及荷载 /cm

    Figure  4.  Lee’s frame geometry and loading

    图  5  Lee’s框架

    Figure  5.  Lee’s frame

    图  6  端部承受集中荷载的悬臂梁

    Figure  6.  Cantilever subjected to concentrated load at free end

    图  7  对角点受拉力作用时的铰接方棱形框架

    Figure  7.  Diamond-shaped frame under a pare of opposite concentrated tensions in opposite angles

    表  1  悬臂梁自由端的位移

    Table  1.   Displacement of cantilever at free end

    荷载步数方法1个单元2个单元
    $u$$w$$\theta $$u$$w$$\theta $
    3152.33587.9181.45053.89383.4981.435
    256.76184.7781.478
    4152.33587.9181.45053.89383.4981.435
    2136.0593.2821.803
    5152.33587.9181.45053.89383.4981.435
    256.76184.7781.478
    6152.33587.9181.45053.89383.4981.435
    263.94493.2821.803
    7152.33587.9181.45053.89383.4981.435
    263.94493.2821.803
    解析解:$u$=55.5 $w$=81.06 $\theta $=1.430
    注:短横线表示该单元划分及荷载步条件下非线性计算失败。
    下载: 导出CSV

    表  2  计算误差

    Table  2.   Calculation error /(%)

    方法1个单元2个单元
    $u$$w$$\theta $$u$$w$$\theta $
    15.708.461.382.903.010.35
    215.2115.0826.052.274.593.31
    下载: 导出CSV

    表  3  铰接方棱形框架位移

    Table  3.   Displacement of diamond-shaped frame

    单元数$u/L$$w/L$${\theta _0}$最大误差/(%)
    10.46180.26247.6
    20.46190.25071.39976.9
    30.46520.24761.46382.6
    40.46510.24591.48161.5
    解析解0.46600.24381.5035
    注:在每根杆件划分成1个单元时,${\theta _0}$=0显然失真。
    下载: 导出CSV
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  • 收稿日期:  2020-01-07
  • 修回日期:  2020-07-16
  • 刊出日期:  2020-11-25

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