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考虑弹性压缩的弹性支承抛物线拱内力解析解

李双蓓 梁睿 梅国雄

李双蓓, 梁睿, 梅国雄. 考虑弹性压缩的弹性支承抛物线拱内力解析解[J]. 工程力学, 2023, 40(11): 1-10. doi: 10.6052/j.issn.1000-4750.2022.02.0108
引用本文: 李双蓓, 梁睿, 梅国雄. 考虑弹性压缩的弹性支承抛物线拱内力解析解[J]. 工程力学, 2023, 40(11): 1-10. doi: 10.6052/j.issn.1000-4750.2022.02.0108
LI Shuang-bei, LIANG Rui, MEI Guo-xiong. ANALYTICAL SOLUTION OF INTERNAL FORCE OF PARABOLIC ARCH WITH ELASTIC SUPPORTS CONSIDERING ELASTIC COMPRESSION[J]. Engineering Mechanics, 2023, 40(11): 1-10. doi: 10.6052/j.issn.1000-4750.2022.02.0108
Citation: LI Shuang-bei, LIANG Rui, MEI Guo-xiong. ANALYTICAL SOLUTION OF INTERNAL FORCE OF PARABOLIC ARCH WITH ELASTIC SUPPORTS CONSIDERING ELASTIC COMPRESSION[J]. Engineering Mechanics, 2023, 40(11): 1-10. doi: 10.6052/j.issn.1000-4750.2022.02.0108

考虑弹性压缩的弹性支承抛物线拱内力解析解

doi: 10.6052/j.issn.1000-4750.2022.02.0108
基金项目: 国家自然科学基金项目(51878186,52178321,51738004);广西科技计划项目(桂科AB22036007);广西创新驱动发展专项资金项目(桂科AA18118055)
详细信息
    作者简介:

    李双蓓(1963−),女,广西人,教授,博士,硕导,主要从事结构稳定性与优化研究(E-mail: lsbwh90@163.com)

    梁 睿(1995−),男,云南人,硕士生,主要从事结构稳定性与优化研究(E-mail: 273956341@qq.com)

    通讯作者:

    梅国雄(1975−),男,湖北人,教授,博士,博导,主要从事土力学与基础工程研究 (E-mail: meiguox@163.com)

  • 中图分类号: U441;TU31

ANALYTICAL SOLUTION OF INTERNAL FORCE OF PARABOLIC ARCH WITH ELASTIC SUPPORTS CONSIDERING ELASTIC COMPRESSION

  • 摘要: 为解决非理想边界约束的拱结构内力理论计算问题,将非理想边界约束简化为弹性支承,基于弹性中心法对其力法方程进行简化,考虑弹性压缩的影响,采用精确曲线积分,推导了竖向移动荷载作用下计算弹性支承抛物线拱刚臂长度、常变位、载变位和内力的解析解,研究了弯压刚度比、矢跨比和水平弹性约束对支承处水平推力的影响规律,以及水平弹性约束对拱轴内力分布的影响。研究表明:该文提出的解析解物理概念清晰、正确可靠,可以显式地明确呈现弹性支承相关参数对内力计算的影响过程;不考虑拱肋弹性压缩影响导致的水平推力计算误差随弯压刚度的增大而增大,拱趾支承为刚性约束时误差最大,可以达到27.8%;水平弹性支承对拱轴内力分布和水平推力具有显著的调控作用;水平推力影响系数随矢跨比的增大呈非线性增大,当弯压刚度比为1.93、水平弹性支承柔度系数为0.02时,常见矢跨比对应的水平推力影响系数在0.15左右。
  • 图  1  抛物线拱计算简图

    Figure  1.  Calculation sketch of parabolic arch

    图  2  弹性中心法基本体系

    Figure  2.  Basic system of elastic center method

    图  3  坐标转换后的弹性中心法基本体系

    Figure  3.  Basic system of elastic center method after coordinate transformation

    图  4  不同荷载作用位置的内力分布图(f/l=1/4.8)

    Figure  4.  Distribution of internal forces at different loading positions (f/l=1/4.8)

    图  5  不同参数影响下水平推力影响系数ψ的变化曲线

    Figure  5.  Variation curve of influence coefficient of horizontal thrust ψ under different parameters

    图  6  不同柔度系数水平弹性支承抛物线拱内力分布图

    Figure  6.  Distribution of internal forces of parabolic arches with different flexibility coefficients of horizontal elastic supports

    图  7  内力特征值随水平弹性支承柔度变化曲线

    Figure  7.  Variation curve of internal force characteristic value with flexibility coefficient of horizontal elastic support

    表  1  赘余力和单位外荷载作用下基本结构的内力

    Table  1.   Superfluous force and internal forces under the effect of redundant forces or unit external load

    内力赘 余力移动单位荷载
    ${x_1}$$ {x_2} $${x_3}$$\xi < {\xi _{\text{P}}}$$\xi \geqslant {\xi _{\text{P}}}$
    弯矩${\overline M_1}$1${\overline M_2}$$f{\xi ^2} - {y_{\text{S}}}$${\overline M_3}$$x$${M_{\text{P}}}$0${M_{\text{P}}}$$ \mp P{l_1}\left( {\xi - {\xi _{\text{P}}}} \right)$
    轴力$ {\overline N_1} $0$ {\overline N_2} $$\cos \varphi $$ {\overline N_3} $$ - \sin \varphi $$ {N_{\text{P}}} $0$ {N_{\text{P}}} $$ \pm P\sin \varphi $
    剪力${\overline Q_1}$0${\overline Q_2}$$\sin \varphi $${\overline Q_3}$$\cos \varphi $${Q_{\text{P}}}$0${Q_{\text{P}}}$$ \mp P\cos \varphi $
    注:正负半轴上拱轴切线与水平线的夹角$\varphi $异号;当单移动荷载位于右边拱时,表中移动单位荷载MPNPQP表达式的正负号分别取−、+、−;当单移动荷载位于左边拱时,表中移动单位荷载MPNPQP表达式的正负号分别取+、−、+”。
    下载: 导出CSV

    表  2  刚臂长度、常变位及载变位显式解

    Table  2.   Explicit solution of rigid arm length, constant displacement and load displacement

    参数分类参数参数表达式
    刚臂长度${y_{\text{S}}}$$\dfrac{{{C_{{\text{S1}}}}\sqrt {1 + {a^2}}+ 2{\zeta _1}}}{{{C_{{\text{S2}}}}\sqrt {1 + {a^2}} + 2{\zeta _1}}}f$
    常变位${\delta _{11}}$$\dfrac{ { {l_1} } }{ {a{{EI} } } } \cdot ( { {C_1}\sqrt {1 + {a^2} } + 2{\zeta _1} } )$
    ${\delta _{22}}$$\dfrac{ { {l_1} } }{ {a{{EI} } } } \cdot \left[ { {C_2}\sqrt {1 + {a^2} } + 2{ {\left( {f - { {y_{\text{S} } } } } \right)}^2}\zeta + 2l_1^2{\zeta _2} } \right]$
    ${\delta _{33}}$$\dfrac{ { {l_1} } }{ {a{{EI} } } } \cdot ( { {C_3}\sqrt {1 + {a^2} } + 2{\zeta _1}l_1^2 + 2{\zeta _2}l_1^2} )$
    载变位${\varDelta _{ {\text{1P} } } }$$- \dfrac{ {Pl_1^2} }{ {a{{EI} } } } \cdot [ { {C_{ {\text{1P} } } }\sqrt {1 + {a^2} } + {D_{ {\text{1P} } } }\sqrt {1 + { {\left( {a{\xi _{\text{P} } } } \right)}^2} } + \left( {1 - {\xi _{\text{P} } } } \right){\zeta _1} } ]$
    ${\varDelta _{ {\text{2P} } } }$$- \dfrac{ {P{l_1} } }{ {a{{EI} } } } \cdot [ { {C_{ {\text{2P} } } }\sqrt {1 + {a^2} } + {D_{ {\text{2P} } } }\sqrt {1 + { {\left( {a{\xi _{\text{P} } } } \right)}^2} } + \left( {f - {y_{\rm{S} } } } \right)\left( {1 - {\xi _{\text{P} } } } \right){l_1}{\zeta _2} }]$
    ${\varDelta _{ {\text{3P} } } }$$- \dfrac{ {P{l_1} } }{ {a{{EI} } } } \cdot [ { {C_{ {\text{3P} } } }\sqrt {1 + {a^2} } + {D_{ {\text{3P} } } }\sqrt {1 + { {\left( {a{\xi _{\text{P} } } } \right)}^2} } + \left( {1 - {\xi _{\text{P} } } } \right){\zeta _1}l_1^2 + {\zeta _3}l_1^2} ]$
    注:当移动荷载作用在左半拱时即x轴负半轴时,将载变位中的ξP替换为−ξP,且Δ3P取正号。
    下载: 导出CSV

    表  3  典型截面数值对比表

    Table  3.   Comparison of results of typical sections

    算例内力截面位置荷载作用位置ξP=0.0荷载作用位置ξP=0.5
    本文解有限元解相对误差/(%)本文解有限元解相对误差/(%)
    算例1(f/l=1/4)弯矩M/(N·m)ξ=−0.5−10.858−10.8030.503−11.932−11.9180.116
    ξ=0.028.02928.2020.614−7.714−7.6351.023
    ξ=0.5−10.858−10.8030.50334.94634.9810.102
    轴力N/Nξ=−0.51.0451.0420.2820.5510.5500.237
    ξ=0.00.9180.9150.3580.5350.5330.280
    ξ=0.51.0451.0420.2820.405/0.8530.404/0.8510.342/0.159
    剪力Q/Nξ=−0.50.0370.0383.832−0.093−0.0930.773
    ξ=0.00.500/−0.5000.500/−0.5000.000/0.0000.1630.1630.047
    ξ=0.5−0.037−0.0383.8320.385/−0.5090.384/−0.5100.152/0.112
    算例2(f/l=1/5)弯矩M/(N·m)ξ=−0.5−10.982−10.9060.697−11.609−11.5860.198
    ξ=0.027.73027.9860.920−7.558−7.4371.612
    ξ=0.5−10.982−10.9060.69734.74534.7940.141
    轴力N/Nξ=−0.51.2531.2510.4630.6780.6750.401
    ξ=0.01.1531.1470.5440.6650.6620.445
    ξ=0.51.2571.2510.4630.558/0.9290.555/0.9260.500/0.298
    剪力Q/Nξ=−0.50.0360.0386.214−0.097−0.0961.206
    ξ=0.00.500/−0.5000.500/−0.5000.000/0.0000.1610.1610.056
    ξ=0.5−0.036−0.0386.2140.397/−0.5320.396/−0.5330.253/0.187
    算例3(f/l=1/6)弯矩M/(N·m)ξ=−0.5−11.046−10.9440.922−11.398−11.3630.307
    ξ=0.027.56727.921.278−7.456−7.2832.347
    ξ=0.5−11.046−10.9440.92234.60434.6680.185
    轴力N/Nξ=−0.51.4751.4650.6820.8050.8000.605
    ξ=0.01.3881.3780.7640.7950.7900.650
    ξ=0.51.4751.4650.6820.704/1.0200.699/1.0150.702/0.483
    剪力Q/Nξ=−0.50.0350.0398.329−0.100−0.0981.735
    ξ=0.00.500/−0.5000.500/−0.5000.000/0.0000.1600.1600.063
    ξ=0.5−0.035−0.0398.3290.403/−0.5450.402/−0.5470.379/0.279
    注:“/”左右两侧的数值分别表示左截面和右截面的数据。
    下载: 导出CSV
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出版历程
  • 收稿日期:  2022-02-01
  • 修回日期:  2022-06-07
  • 录用日期:  2022-06-17
  • 网络出版日期:  2022-06-17
  • 刊出日期:  2023-11-06

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