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阶梯柱屈曲的改进Fourier级数分析

陆健炜 鲍四元 沈峰

陆健炜,鲍四元,沈峰. 阶梯柱屈曲的改进Fourier级数分析 [J]. 应用数学和力学,2021,42(12):1229-1237 doi: 10.21656/1000-0887.410373
引用本文: 陆健炜,鲍四元,沈峰. 阶梯柱屈曲的改进Fourier级数分析 [J]. 应用数学和力学,2021,42(12):1229-1237 doi: 10.21656/1000-0887.410373
LU Jianwei, BAO Siyuan, SHEN Feng. Buckling Analysis of Stepped Columns Based on the Improved Fourier Series Method[J]. Applied Mathematics and Mechanics, 2021, 42(12): 1229-1237. doi: 10.21656/1000-0887.410373
Citation: LU Jianwei, BAO Siyuan, SHEN Feng. Buckling Analysis of Stepped Columns Based on the Improved Fourier Series Method[J]. Applied Mathematics and Mechanics, 2021, 42(12): 1229-1237. doi: 10.21656/1000-0887.410373

阶梯柱屈曲的改进Fourier级数分析

doi: 10.21656/1000-0887.410373
基金项目: 国家自然科学基金(11202146);江苏省普通高校研究生科研与实践创新计划项目(2019年)
详细信息
    作者简介:

    陆健炜(1993—),男,硕士生(E-mail:337246553@qq.com)

    鲍四元(1980—),男,副教授,硕士生导师(通讯作者. E-mail:bsiyuan@126.com)

    沈峰(1984—),男,副教授,硕士生导师

  • 中图分类号: O317.3

Buckling Analysis of Stepped Columns Based on the Improved Fourier Series Method

  • 摘要:

    该文对阶梯柱的弹性屈曲问题进行了研究。首先基于改进Fourier级数法采用局部坐标逐段建立阶梯柱的位移函数表达式,然后由带约束的势能变分原理得到含屈曲荷载的线性方程组,利用线性方程组有非零解的条件把问题转化为矩阵特征值问题得到临界载荷,最后讨论方法中的参数取值,并把结果与已有文献和有限元的结果比较,从而验证方法的精度。所提模型在阶梯柱的两端和变截面处引入横向弹簧和旋转弹簧,通过改变弹簧的刚度值模拟不同的边界。所提方法在工程设计中能比较精确地确定各种弹性边界条件下阶梯柱的临界载荷。

  • 图  1  阶梯柱的屈曲模型

    Figure  1.  The buckling model for the stepped column

    图  2  一端弹性一端自由阶梯柱的临界载荷随刚度参数对数的变化图

    Figure  2.  Change of buckling loads of the elastic-free stepped column with non-dimensional spring stiffness on a logarithmic scale

    图  3  三阶对称阶梯柱

    Figure  3.  A three-step symmetric column

    图  4  三阶对称阶梯柱的屈曲模态

    Figure  4.  The buckling modes of a three-stepped symmetric column

    表  1  一端弹性一端自由阶梯柱的屈曲荷载随不同弹簧刚度值的收敛情况

    Table  1.   Convergence of buckling loads of the elastic-free stepped column with different spring stiffness values

    $ {\tilde k_1} = {\tilde K_1} $101021031041051061071081091010
    P/N0.83643.08704.00824.12164.13324.13444.13454.13454.13454.1345
    下载: 导出CSV

    表  2  一端固定一端自由阶梯柱的屈曲荷载随M的收敛情况

    Table  2.   Convergence of buckling loads of the clamped-free stepped column with different M values

    M468101214161820
    P/N4.22134.13934.13514.13464.13454.13454.13454.13454.1345
    下载: 导出CSV

    表  3  一端固定一端自由阶梯柱的计算长度系数及误差

    Table  3.   The effective length coefficients and errors of clamped-free stepped columns

    μ2I1/I2l1/L
    0.20.30.40.50.60.70.8
    this paper 2.5 1.404 1.302 1.216 1.139 1.077 1.034 1.010
    ref. [5] 1.39 1.30 1.22 1.14 1.08 1.03 1.01
    error δ/% 1.007 0.154 0.328 0.088 0.278 0.389 0.000
    this paper 2 1.274 1.208 1.146 1.093 1.051 1.022 1.007
    ref. [5] 1.27 1.21 1.14 1.09 1.05 1.02 1.01
    error δ/% 0.315 0.165 0.526 0.275 0.095 0.196 0.297
    this paper 1.75 1.211 1.158 1.110 1.069 1.038 1.021 1.006
    ref. [5] 1.21 1.16 1.11 1.07 1.04 1.02 1.00
    error δ/% 0.083 0.172 0 0.093 0.192 0.098 0.6
    this paper 1.5 1.144 1.107 1.074 1.050 1.025 1.011 1.003
    ref. [5] 1.14 1.11 1.07 1.05 1.02 1.01 1.00
    error δ/% 0.351 0.270 0.374 0.000 0.490 0.099 0.3
    this paper 1 1.000 1.000 1.000 1.000 1.000 1.000 1.000
    ref. [5] 1.00 1.00 1.00 1.00 1.00 1.00 1.00
    error δ/% 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    下载: 导出CSV

    表  4  一端固定一端弹性阶梯柱的屈曲载荷值(单位: N)

    Table  4.   The buckling loads of the clamped-free stepped column (unit: N)

    I1/I2l1/L$ {\tilde K_2} = {\tilde k_2} $
    010103105107108
    2.5 0.3 4.1119 5.5297 10.5149 10.6476 10.6489 10.6489
    0.5 3.8030 24.2575 43.1497 44.0280 44.3069 44.0370
    2 0.3 5.1399 6.5968 13.1087 13.3091 13.3111 13.3111
    0.5 4.1345 27.6652 50.3485 51.6053 51.6181 51.6183
    1.75 0.3 5.8742 7.3510 14.9482 15.2101 15.2127 15.2127
    0.5 4.3168 29.9558 55.0281 56.5625 56.5782 56.5783
    1.5 0.3 6.8532 8.3501 17.3881 17.7446 17.7481 17.7481
    0.5 4.5108 32.8297 60.6880 62.5874 62.6069 62.6071
    1.0 0.3 10.2799 11.8173 25.8119 26.6141 26.6221 26.6222
    0.5 4.9349 41.3393 75.9055 78.9252 78.9565 78.9567
    下载: 导出CSV

    表  5  两端铰支阶梯柱的屈曲系数及误差

    Table  5.   Buckling factors and errors of stepped columns with hinged ends

    I1/I2l2/L
    0.20.30.40.50.60.70.8
    FEM0.22.79553.40624.22185.30946.69428.19709.3302
    this paper2.79553.40624.22185.30946.69428.19719.3303
    error δ/%0.00000.00000.00000.00000.00000.00120.0011
    FEM
    0.4
    5.08855.82566.67747.60608.50989.23609.6742
    this paper5.08845.82566.67747.60608.50999.23619.6744
    error δ/%0.00200.00000.00000.00000.00120.00110.0021
    FEM0.66.97947.57598.18508.76029.24389.58889.7839
    this paper6.97947.57598.18518.76039.24399.58909.7841
    error δ/%0.00000.00000.00120.00110.00110.00210.0020
    FEM0.88.55128.87649.17679.43349.63159.76469.8377
    this paper8.55138.87649.17689.43359.63169.76489.8380
    error δ/%0.00120.00000.00110.00110.00100.00200.0030
    下载: 导出CSV

    表  6  两端固定阶梯柱的屈曲系数及误差

    Table  6.   Buckling factors and errors of stepped columns with clamped ends

    I1/I2l2/L
    0.20.30.40.50.60.70.8
    FEM0.211.157413.481316.261318.964320.459420.671421.0548
    this paper11.157413.481316.261318.964320.459420.671421.0548
    error δ/%0.00000.00000.00000.00000.00000.00000.0000
    FEM
    0.4
    20.237722.742824.885626.062526.302526.415027.4675
    this paper20.237722.742824.885626.062726.301326.417027.4671
    error δ/%0.00000.00000.00000.00080.00460.00760.0015
    FEM0.627.715029.495030.645031.055031.087531.347532.4550
    this paper27.715429.496030.644731.054131.086531.349332.4559
    error δ/%0.00140.00340.00100.00290.00320.00570.0028
    FEM0.834.020034.875035.315035.420035.440035.675536.3725
    this paper34.019434.875835.315335.419535.440135.675436.3713
    error δ/%0.00180.00230.00100.00140.00030.00030.0033
    下载: 导出CSV
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出版历程
  • 收稿日期:  2020-12-07
  • 录用日期:  2021-07-31
  • 修回日期:  2021-07-30
  • 网络出版日期:  2021-11-25
  • 刊出日期:  2021-12-01

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