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改进共旋坐标法的Timoshenko梁单元非线性分析

李东升 高严培 郭鑫

李东升, 高严培, 郭鑫. 改进共旋坐标法的Timoshenko梁单元非线性分析[J]. 工程力学, 2022, 39(11): 22-30, 108. doi: 10.6052/j.issn.1000-4750.2021.07.0508
引用本文: 李东升, 高严培, 郭鑫. 改进共旋坐标法的Timoshenko梁单元非线性分析[J]. 工程力学, 2022, 39(11): 22-30, 108. doi: 10.6052/j.issn.1000-4750.2021.07.0508
LI Dong-sheng, GAO Yan-pei, GUO Xin. NONLINEAR ANALYSIS OF TIMOSHENKO BEAM ELEMENT BASED ON IMPROVED COROTATIONAL FORMULATION[J]. Engineering Mechanics, 2022, 39(11): 22-30, 108. doi: 10.6052/j.issn.1000-4750.2021.07.0508
Citation: LI Dong-sheng, GAO Yan-pei, GUO Xin. NONLINEAR ANALYSIS OF TIMOSHENKO BEAM ELEMENT BASED ON IMPROVED COROTATIONAL FORMULATION[J]. Engineering Mechanics, 2022, 39(11): 22-30, 108. doi: 10.6052/j.issn.1000-4750.2021.07.0508

改进共旋坐标法的Timoshenko梁单元非线性分析

doi: 10.6052/j.issn.1000-4750.2021.07.0508
基金项目: 国家自然科学基金项目(51778103,52078284);广东省自然科学基金项目(2021A1515011770);汕头大学科研启动基金项目(NTF18012)
详细信息
    作者简介:

    高严培(1996−),男,安徽人,硕士生,主要从事结构健康监测研究(E-mail: 19ypgao@stu.edu.cn)

    郭 鑫(1991−),男,内蒙古人,博士生,主要从事结构健康监测研究(E-mail: guo152601@163.com)

    通讯作者:

    李东升(1972−),男,黑龙江人,教授,博士,博导,主要从事结构健康监测研究(E-mail: lids@stu.edu.cn)

  • 中图分类号: TU311

NONLINEAR ANALYSIS OF TIMOSHENKO BEAM ELEMENT BASED ON IMPROVED COROTATIONAL FORMULATION

  • 摘要: 为提高空间Timoshenko梁单元非线性问题的计算精度,在共旋坐标法的基础上,提出了一种改进的Timoshenko梁单元几何非线性分析方法。利用虚功原理得到改进空间梁单元的刚度矩阵;使用有限质点法中的逆向运动思路计算单元局部坐标系下的刚体旋转矩阵;根据整体坐标系与局部坐标系之间旋转角度的转化以及微分关系,求得空间梁单元的切线刚度矩阵;编制了相应的有限元程序,对多个经典的大变形结构进行几何非线性分析。计算结果印证了该文所提出改进方法的正确性,同时与传统共旋坐标法相比,具有更高的精度。
  • 图  1  梁的运动及坐标系

    Figure  1.  Beam kinematics and coordinate systems

    图  2  逆向运动图示

    Figure  2.  Diagram of reverse motion

    图  3  利用改进共旋法进行静位移计算流程图

    Figure  3.  Flow chart of static displacement calculation using improved co-rotational formulation

    图  4  集中荷载作用下的大挠度悬臂梁

    Figure  4.  Cantilever of large deflection under concentrated load

    图  5  悬臂梁杆端受集中弯矩

    Figure  5.  Cantilever beam with an end moment

    图  6  悬臂梁的大转动变形

    Figure  6.  Large rotation deformation of a cantilever

    图  7  集中荷载作用下的耦合悬臂梁

    Figure  7.  Coupled cantilever under concentrated load

    图  8  沿梁轴向的位移和旋转参数

    Figure  8.  Displacements and rotation parameters along beam axis

    图  9  预弯曲式悬臂梁

    Figure  9.  Prebend cantilever

    表  1  悬臂梁承受集中荷载时的大挠度变形

    Table  1.   Cantilever's large deformation under concentrated load

    $ P{L}^{2}/EI $ $ w/L $ $ u/L $ ${\theta }_{0}/{\rm rad}$
    解析解 本文 误差/(%) 解析解 本文 误差/(%) 解析解 本文 误差/(%)
    0.2 0.0664 0.0665 0.1506 0.0027 0.0026 2.1852 0.0996 0.0996 0.0000
    0.8 0.2495 0.2499 0.1603 0.0382 0.0381 0.2618 0.3791 0.3791 0.0000
    1.6 0.4294 0.4305 0.2562 0.1186 0.1185 0.0843 0.6707 0.6711 0.0596
    3.0 0.6033 0.6054 0.3481 0.2544 0.2544 0.0000 0.9860 0.9871 0.1116
    4.0 0.6700 0.6728 0.4179 0.3289 0.3290 0.0304 1.1212 1.1227 0.1338
    下载: 导出CSV

    表  2  悬臂梁承受弯矩作用的大变形

    Table  2.   Cantilever’s large deformation under bending moment

    $ k $ 水平位移
    $ u $解析
    解/m
    本文
    解/m
    HAWC2-
    30解/m
    HAWC2-
    50解/m
    本文误
    差/(%)
    HAWC2-
    30误差/(%)
    HAWC2-
    50误差/(%)
    竖直位移
    $ v $解析
    解/m
    本文
    解/m
    HAWC2-
    30解/m
    HAWC2-
    50解/m
    本文误
    差/(%)
    HAWC2-
    30误差/(%)
    HAWC2-
    50误差/(%)
    0.4 −2.4317 −2.4268 −2.4439 −2.4366 0.20 0.50 0.20 5.4987 5.5023 5.5042 5.4987 0.07 0.10 0.00
    0.8 −7.6613 −7.6551 −7.7609 −7.6996 0.08 1.30 0.50 7.1978 7.2168 7.2194 7.2050 0.26 0.30 0.10
    1.2 −11.5591 −11.5684 −11.6978 −11.6053 0.08 1.20 0.40 4.7986 4.8271 5.0145 4.8802 0.59 4.50 1.70
    1.6 −11.8921 −11.9121 −12.0467 −11.9516 0.17 1.30 0.50 1.3747 1.3893 1.6868 1.5080 1.06 22.70 9.70
    2.0 −10.0000 −10.0000 −10.5100 −10.2000 0.00 5.10 2.00 0.0000 0.0000 −0.0080 −0.0100 0.0000 −0.0080 −0.0100
    下载: 导出CSV

    表  3  自由端位移和转角参数比较

    Table  3.   Comparison of tip displacements and rotations

    方法对比及误差分析 解析解/m 经典共旋法解/m HAWC2解/m 本文解/m 共旋法误差/(%) 本文误差/(%) HAWC2误差/(%)
    x方向位移 −0.090 64 −0.090 13 −0.091 64 −0.090 09 0.56 0.61 1.10
    y方向位移 −0.064 84 −0.063 20 −0.067 24 −0.064 76 2.53 0.12 3.70
    z方向位移 1.229 98 1.229 50 1.229 98 1.229 18 0.04 0.06 0.00
    x方向扭转角度 0.184 45 0.184 47 0.188 88 0.184 40 0.01 0.03 2.40
    y方向旋转角度 −0.179 85 −0.179 87 −0.179 87 −0.179 84 0.01 0.01 0.01
    z方向旋转角度 0.004 88 0.005 23 0.004 99 0.005 01 7.17 2.66 2.30
    下载: 导出CSV

    表  4  自由端受力下曲梁端部位移比较

    Table  4.   Comparison of the curved beam tip displacements under a force applied at the free end

    方法对比及
    误差分析
    解析
    本文方
    法解
    经典共
    旋法解
    HAWC2
    本文
    误差/(%)
    共旋法
    误差/(%)
    HAWC2/
    (%)
    $x/{\rm m}$ −23.7 −23.7 −23.8 −24.2 0.2 0.4 2.1
    $y/{\rm m}$ −13.4 −13.6 −13.7 −13.8 1.3 2.2 3.1
    ${\textit{z} }/{\rm m}$ 53.4 53.6 53.7 53.4 0.4 0.5 0.0
    下载: 导出CSV
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出版历程
  • 收稿日期:  2021-07-05
  • 录用日期:  2021-12-31
  • 修回日期:  2021-11-08
  • 网络出版日期:  2021-12-31
  • 刊出日期:  2022-11-01

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