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超静定梁−柱的解析解研究

李银山 丁千 李子瑞 郭春霞 孙永涛 柳占立

李银山, 丁千, 李子瑞, 郭春霞, 孙永涛, 柳占立. 超静定梁−柱的解析解研究. 力学学报, 2022, 54(11): 1-12 doi: 10.6052/0459-1879-22-337
引用本文: 李银山, 丁千, 李子瑞, 郭春霞, 孙永涛, 柳占立. 超静定梁−柱的解析解研究. 力学学报, 2022, 54(11): 1-12 doi: 10.6052/0459-1879-22-337
Li Yinshan, Ding Qian, Li Zirui, Guo Chunxia, Sun Yongtao, Liu ZhanLi. Analytical solution of statically indeterminate beam-column. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(11): 1-12 doi: 10.6052/0459-1879-22-337
Citation: Li Yinshan, Ding Qian, Li Zirui, Guo Chunxia, Sun Yongtao, Liu ZhanLi. Analytical solution of statically indeterminate beam-column. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(11): 1-12 doi: 10.6052/0459-1879-22-337

超静定梁−柱的解析解研究

doi: 10.6052/0459-1879-22-337
基金项目: 国家自然科学基金项目(12072222, 12072100)资助
详细信息
    作者简介:

    孙永涛, 副教授, 主要研究方向: 非线性振动及控制. E-mail: sunyongtao100@163.com

    柳占立, 教授, 主要研究方向: 计算固体力学. E-mail: liuzhanli@tsinghua.edu.cn

  • 中图分类号: O341

ANALYTICAL SOLUTION OF STATICALLY INDETERMINATE BEAM-COLUMN

  • 摘要: 本文采用渐进积分法研究了超静定梁−柱的弯曲问题. 首先建立超静定梁−柱的四阶挠度微分方程, 考虑到边界条件和连续光滑条件, 采用连续分段独立一体化积分法求解得到了挠度的精确解析解. 为了满足工程设计需要, 构造了超静定梁−柱的四阶挠度微分迭代方程, 选取无轴向力作用时超静定梁的挠曲线作为梁的初函数, 将初函数代入梁的四阶挠度微分迭代方程进行积分, 利用边界条件和连续光滑条件确定积分常数, 得到下一次迭代挠度函数, 依次进行迭代积分运算. 计算出了最大挠度、最大转角和最大弯矩等用轴向力放大系数表示的多项式解析函数解. 本文选取了两种边界条件下受分布力作用的超静定梁−柱进行分析, 计算结果表明, 当超静定梁−柱所受的轴向力小于欧拉临界力的1/2时, 迭代六次误差就可以控制在1%以内; 不仅梁−柱最大位移和最大剪力的大小随轴向力的增大而增大, 而且其位置也随轴向力的增大而发生迁移. 本文的研究对揭示轴向力对超静定梁−柱变形和内力的影响有重要意义, 为超静定梁−柱的实际设计提供了一定的理论基础.

     

  • 图  1  轴向压力与横向分布力共同作用的铰支−固支梁−柱

    Figure  1.  Hinge-fixed beam-column acted by axial force and transverse load

    图  2  不同轴向力下, 横向分布力作用的铰支−固支梁−柱的变形图和内力图

    Figure  2.  Deformation and internal diagram of hinge-fixed beam-column under different axial forces

    图  3  最大挠度、最大剪力位置随轴向力系数κ的迁移

    Figure  3.  Migration of maximum deflection and maximum shear force position with axial force coefficient κ

    图  4  横向分布力作用下铰支−固支梁−柱前六次迭代最大变形和最大弯矩与精确解的对比

    Figure  4.  Comparison between the maximum deflection, maximum angle and maximum bending moment in the first six iterations and the exact solution

    图  5  轴向力与横向分布力作用的固支−固支梁−柱

    Figure  5.  Fixed-fixed beam-column acted by axial force and transverse load

    图  6  不同轴向力下,横向分布力作用的固支−固支梁的变形图和内力图

    Figure  6.  Deformation and internal diagram of fixed-fixed beam-column under different axial forces

    图  7  最大转角和最大剪力位置随轴向力系数$ \kappa $的迁移

    Figure  7.  Migration of maximum angle and maximum shear force position with axial force coefficient κ

    图  8  梁−柱前六次迭代时最大变形和最大弯矩与精确解的对比

    Figure  8.  Comparison between the maximum deflection, maximum angle and maximum bending moment in the first six iterations and the exact solution

    表  1  最大挠度和最大剪力放大系数的位置迁移对照表

    Table  1.   Location transfer comparison table of maximum deflection and maximum shear force amplification factor

    κ00.10.20.30.40.50.60.70.80.9
    X00.421540.419830.418030.416120.414080.411900.409570.407060.404370.40146
    βv(κ)11.10971.24671.42281.65761.98612.47883.29974.94149.8666
    X111110.919450.858700.813070.776860.746940.72144
    $ {\beta _{{F_{\text{S}}}}}\left( \kappa \right) $11.01491.03321.06081.11641.25411.49931.94132.86115.6754
    下载: 导出CSV

    表  2  横向分布力作用的铰支−固支梁−柱放大系数对比

    Table  2.   Comparison of hinged-fixed beam-column amplification coefficients under continuous transverse load

    κexact solutionpresent solutionrelative errorexact solutionpresent solutionrelative errorexact solutpresent solutionrelative error
    βv$\; \beta _v^{\left[ 6 \right]}$Δvβθ$\; \beta _\theta ^{\left[ 6 \right]}$θβM$ \;\beta _M^{\left[ 6 \right]} $ΔM
    0110110110
    0.11.10971.10970.00%1.11231.11230.00%1.07431.07430.00%
    0.21.24671.24670.00%1.25291.25290.00%1.16601.16600.00%
    0.31.42281.42250.02%1.43411.43380.02%1.28261.28240.02 %
    0.41.65761.65490.16%1.67611.67330.17%1.43621.43450.12 %
    0.51.98611.97070.78%2.01571.99960.80%1.64901.63930.59 %
    0.62.47632.40762.77%2.52602.45412.85%1.96491.92172.19%
    0.73.29503.02568.18%3.37803.09638.34%2.48682.31756.81%
    0.84.93133.902420.9%5.08484.008721.2%3.52282.876518.3 %
    1
    下载: 导出CSV

    表  3  最大转角和最大剪力放大系数的位置迁移对照表

    Table  3.   Comparison of position migration of maximum angle and maximum shear amplification coefficient

    $ \kappa $00.10.20.30.40.50.60.70.80.9
    $ {X_0} $0.211320.214560.217930.221440.225100.228900.232840.236930.241160.24552
    $ {\beta _\theta }\left( \kappa \right) $11.10981.24731.42451.66141.99382.49353.32805.000110.023
    $ {X_1} $0000.0435620.104710.146440.177250.201190.220490.23648
    $ {\beta _{{F_{\text{S}}}}}\left( \kappa \right) $1111.01131.09331.25681.53752.03703.07106.2292
    下载: 导出CSV

    表  4  分布力作用的固支−固支梁−柱放大系数对比

    Table  4.   Comparison of the fixed-fixed beam-column amplification coefficient under continuous transverse loading

    κexact solutionpresent solutionrelative errorexact solutionpresent solutionrelative errorexact solutionpresent solutionrelative error
    βv$\; \beta _v^{\left[ 6 \right]} $Δvβθ$ \;\beta _\theta ^{\left[ 6 \right]} $ΔθβM$\; \beta _M^{\left[ 6 \right]} $ΔM
    0110110110
    0.11.10961.10960.00%1.10981.10980.00%1.07271.07270.00%
    0.21.24671.24670.00%1.24731.24730.00%1.16241.16240.00%
    0.31.42281.42250.02 %1.42451.42420.02 %1.27661.27640.02%
    0.41.65761.65490.16%1.66141.65870.16 %1.42721.42550.12 %
    0.51.98631.97090.77 %1.99381.97820.78 %1.63591.62640.58%
    0.62.47922.41022.78 %2.47192.40362.76%1.94631.90382.18 %
    0.73.30063.03018.19%3.28683.01908.15 %2.45972.29286.78 %
    0.84.94323.910020.9 %4.91513.891920.8 %3.47992.842618.3 %
    1
    下载: 导出CSV
  • [1] Timoshenko SP, Gere J. Theory of Elastic Stability. New York: McGraw-Hill, 1961
    [2] 邢静忠, 柳春图. 轴向力作用下埋设于线弹性土壤中的悬跨管道振动分析. 工程力学, 2010, 27(3): 193-197

    Xing Jingzhong, Liu Chuntu. Vibration analysis of spanning pipeline buried in linear elastic soil with axial force. Engineering mechanics, 2010, 27(3): 193-197(in Chinese))
    [3] 高德利, 徐秉业. 石油钻井底部钻具组合平面纵横弯曲大挠度分析. 工程力学, 1992(04): 42-49

    Gao Deli, Xu Bingye. Planar analysis of the static behavior of a bottomhole assembly under large eflection. Engineering mechanics, 1992(04): 42-49(in Chinese)
    [4] 肖建清, 杨玉东, 冯夏庭. 基于梁-柱稳定理论的深埋隧洞岩爆破坏及孕育机理研究. 防灾减灾工程学报, 2014, 34(02): 203-210

    Xiao Jian Qing, Yang Yu dong, Feng Xia ting. Rock burst damage and generation mechanism in deep tunnel based on beam-column stability theory. Journal of Disaster Prevention and Reduction, 2014, 34(02): 203-210(in Chinese))
    [5] 高德利, 黄文君. 井下管柱力学与控制方法若干研究进展. 力学进展, 2021, 51(3): 620-647 doi: 10.6052/1000-0992-21-028

    Gao Deli, Huang Wenjun. Some research advances in downhole tubular mechanics and control methods[J]. Advances in Mechanics, 2021, 51(3): 620-647(in Chinese) doi: 10.6052/1000-0992-21-028
    [6] Wang B, Yu C, Jiang X, et al. A numerical investigation of injury mechanisms and tolerance limit of occupant femur in combined compression-bending load. International Journal of Crashworthiness, 2022, 27(2): 456-465
    [7] 蒋小晴, 杨济匡, 王丙雨, 张维刚. 乘员股骨在轴向压力--弯矩下的损伤生物力学机理研究. 力学学报, 2014, 46(03): 465-474( Jiang Xiaoqing, Yang Jikuang, Wang Bingyu, Zhang Weigang. An investigation of biomechanical mechanisms of occupant femur injuries under compression-bending load Chinese Journal of Theoretical and Applied Mechanics, 2014, 46(03): 465-474(in Chinese)) doi: 10.1080/13588265.2020.1807698

    Wang B, Yu C, Jiang X, et al. A numerical investigation of injury mechanisms and tolerance limit of occupant femur in combined compression–bending load. International Journal of Crashworthiness, 2022, 27(2): 456-465 doi: 10.1080/13588265.2020.1807698
    [8] Chen Wai-Fah , Toshio Atsuta. Theory of beam columns(volume 1). New York: J Ross Publishing, 2008
    [9] 刘鸿文. 高等材料力学. 北京: 高等教育出版社, 1985

    Liu Hongwen. Advanced mechanics of materials. Beijing: Higher Education Press, 1985(in Chinese))
    [10] 唐家祥, 王仕统, 裴若娟. 结构稳定理论. 北京: 中国铁道出版社, 1989

    Tang Jiaxiang, Wang Shitong, Pei Ruojuan. Structural stability theory. Beijing: China Railway Publishing House, 1989 (in Chinese))
    [11] 刘光栋, 罗汉泉. 杆系结构稳定. 北京: 人民交通出版社, 1988

    Liu Guangdong, Luo Hanquan. The structure of the rod system’s stability. Beijing: China Communications Press, 1988(in Chinese))
    [12] 陈连. 求解任意梁的普遍化方法. 机械工程学报, 2004(12): 71-74 doi: 10.3321/j.issn:0577-6686.2004.12.015

    Chen Lian. General method for arbitrary beam analysis. Journal of Mechanical Engineering, 2004(12): 71-74(in Chinese)) doi: 10.3321/j.issn:0577-6686.2004.12.015
    [13] Girhammar UA, Gopu VKA. Composite beam-columns with interlayer slip—exact analysis. Journal of Structural Engineering, 1993, 119(4): 1265-1282 doi: 10.1061/(ASCE)0733-9445(1993)119:4(1265)
    [14] Girhammar UA, Pan DH. Exact static analysis of partially composite beams and beam-columns. International Journal of Mechanical Sciences, 2007, 49(2): 239-255 doi: 10.1016/j.ijmecsci.2006.07.005
    [15] Aristizabal-Ochoa JD. Nonlinear large deflection-small strain elastic analysis of beam-column with semirigid connections. Journal of structural Engineering, 2001, 127(1): 92-96 doi: 10.1061/(ASCE)0733-9445(2001)127:1(92)
    [16] Aristizabal-Ochoa JD. Large deflection and postbuckling behavior of Timoshenko beam–columns with semi-rigid connections including shear and axial effects. Engineering Structures, 2007, 29(6): 991-1003 doi: 10.1016/j.engstruct.2006.07.012
    [17] Giraldo-Londoño O, Monsalve-Giraldo JS, Aristizabal-Ochoa JD. Large- deflection and postbuckling of beam-columns with non-linear semi-rigid connections including shear and axial effects. International Journal of Non-Linear Mechanics, 2015, 77: 85-95 doi: 10.1016/j.ijnonlinmec.2015.07.009
    [18] Liew A, Gardner L. Ultimate capacity of structural steel cross-sections under compression, bending and combined loading, Structures. 2015, 1: 2-11
    [19] 蒋纯志. 分布传递函数方法在梁杆结构分析中的应用. [博士论文]. 国防科学技术大学, 2006

    Jiang Chunzhi. The Application of Distributed Transfer Function Method to Girders and Beams Analysis. [PhD Thesis]. Graduate School of National University of Defense Technology, 2006 (in Chinese)
    [20] Arboleda-Monsalve LG, Zapata-Medina DG, Aristizabal-Ochoa JD. Timoshenko beam-column with generalized end conditions on elastic foundation: Dynamic-stiffness matrix and load vector. Journal of Sound and Vibration, 2008, 310(4-5): 1057-1079 doi: 10.1016/j.jsv.2007.08.014
    [21] Untaroiu CD. A numerical investigation of mid-femoral injury tolerance in axial compression and bending loading. International Journal of Crashworthiness, 2010, 15(1): 83-92 doi: 10.1080/13588260903047671
    [22] Untaroiu CD, Yue N, Shin J. A finite element model of the lower limb for simulating automotive impacts. Annals of biomedical engineering, 2013, 41(3): 513-526 doi: 10.1007/s10439-012-0687-0
    [23] Russell SG. Informal Numerical Methods for Generalized Beam-ColumnProblems//55th AIAA/ASMe/ASCE/AHS/SC Structures, Structural Dynamics, and Materials Conference. 2014: 0160
    [24] Zhang Z, Xu S, Nie B, et al. Experimental and numerical investigation of corroded steel columns subjected to in-plane compression and bending. Thin-Walled Structures, 2020, 151: 106735 doi: 10.1016/j.tws.2020.106735
    [25] Yang L, Wang Y, Richard Liew JY. Compression-Bending Strength Model for Corrugated Steel Tube Confined Reinforced Concrete Section. Journal of Structural Engineering, 2021, 147(11): 04021187 doi: 10.1061/(ASCE)ST.1943-541X.0003148
    [26] 王秋维, 赵航, 史庆轩, 等. 内置非贯通型钢STRC柱压弯承载力计算研究. 工程力学, 2022, 39: 1-12 doi: 10.6052/j.issn.1000-4750.2021.04.0271

    (WANG Qiuwei, ZHAO Hang, SHI Qingxuan, et al. CALCULATION ON COMPRESSION-BENDING CAPACITY OF THE STRC COLUMN WITH BUILT-IN NON-THROUGH SECTION STEEL. Engineering Mechanics, 2022, 39: 1-12 (in Chinese)). doi: 10.6052/j.issn.1000-4750.2021.04.0271
    [27] Huang Z, Uy B, Li D, et al. Behaviour and design of ultra-high-strength CFST members subjected to compression and bending. Journal of Constructional Steel Research, 2020, 175: 106351 doi: 10.1016/j.jcsr.2020.106351
    [28] Fang H, Chan TM, Young B. Experimental and numerical investigations of octagonal high-strength steel tubular stub columns under combined compression and bending. Journal of Structural Engineering, 2021, 147(1): 04020282 doi: 10.1061/(ASCE)ST.1943-541X.0002848
    [29] 吴艳艳, 李银山, 魏剑伟, 等. 求解超静定梁的分段独立一体化积分法. 工程力学, 2013, 30(S1): 11-14 doi: 10.6052/j.issn.1000-4750.2012.04.S006

    WU Yan-yan, LI Yin-shan, WEI Jian-wei, et al. A subsection independently systematic integral method for solving problems of statically indeterminate beam. Engineering Mechanics, 2013, 30(S1): 11-14(in Chinese)) doi: 10.6052/j.issn.1000-4750.2012.04.S006
    [30] 李银山. 材料力学(上册). 北京: 人民交通出版社, 2014

    Li Yinshan. Mechanics materials(Volume one). Beijing: China Communications Press, 2014(in Chinese)
    [31] 李银山. 材料力学(下册). 北京: 人民交通出版社, 2015

    Li Yinshan. Mechanics materials(Volume two). Beijing: China Communications Press, 2015(in Chinese)
    [32] 李银山, 韦炳威, 李彤, 等. 复杂载荷下变刚度超静定梁快速解析求解. 工程力学, 2016, 33(S1): 33-38

    Li Yinshan, WeiBingwei, Li Tong, et al. A fast analytical solution for staticallyindeterminate beam of variable stiffness under complicated load. Engineering Mechanics, 2016, 33(S1): 33-38(in Chinese))
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  • 收稿日期:  2022-07-18
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