Buckling Analysis of Stepped Columns Based on the Improved Fourier Series Method
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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。
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关键词:
- 改进Fourier级数法 /
- 临界荷载 /
- 带约束的势能变分原理 /
- 阶梯柱
Abstract:The elastic buckling of stepped columns with variable cross sections was studied. Firstly, based on the improved Fourier series method, the displacement function of the column was established in the local coordinate system, then the linear equations for buckling loads were obtained with the constrained variational principle of potential energy. The problem was transformed into a matrix eigenvalue problem and the buckling load was obtained from solution of the matrix eigenvalues. Finally, the parameter values in the method were discussed through numerical examples, and the obtained results were compared with the finite element results and previous literature results so as to verify the accuracy of the method. In the presented model the translational and rotational springs were arranged at the 2 ends and the setback cross sections. The method can determine the buckling loads of stepped columns with various elastic boundary conditions accurately in engineering design.
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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
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} $ 10 102 103 104 105 106 107 108 109 1010 P/N 0.8364 3.0870 4.0082 4.1216 4.1332 4.1344 4.1345 4.1345 4.1345 4.1345 
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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
M 4 6 8 10 12 14 16 18 20 P/N 4.2213 4.1393 4.1351 4.1346 4.1345 4.1345 4.1345 4.1345 4.1345
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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
μ2 I1/I2 l1/L 0.2 0.3 0.4 0.5 0.6 0.7 0.8 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
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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/I2 l1/L $ {\tilde K_2} = {\tilde k_2} $ 0 10 103 105 107 108 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
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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/I2 l2/L 0.2 0.3 0.4 0.5 0.6 0.7 0.8 FEM 0.2 2.7955 3.4062 4.2218 5.3094 6.6942 8.1970 9.3302 this paper 2.7955 3.4062 4.2218 5.3094 6.6942 8.1971 9.3303 error δ/% 0.0000 0.0000 0.0000 0.0000 0.0000 0.0012 0.0011 FEM
0.45.0885 5.8256 6.6774 7.6060 8.5098 9.2360 9.6742 this paper 5.0884 5.8256 6.6774 7.6060 8.5099 9.2361 9.6744 error δ/% 0.0020 0.0000 0.0000 0.0000 0.0012 0.0011 0.0021 FEM 0.6 6.9794 7.5759 8.1850 8.7602 9.2438 9.5888 9.7839 this paper 6.9794 7.5759 8.1851 8.7603 9.2439 9.5890 9.7841 error δ/% 0.0000 0.0000 0.0012 0.0011 0.0011 0.0021 0.0020 FEM 0.8 8.5512 8.8764 9.1767 9.4334 9.6315 9.7646 9.8377 this paper 8.5513 8.8764 9.1768 9.4335 9.6316 9.7648 9.8380 error δ/% 0.0012 0.0000 0.0011 0.0011 0.0010 0.0020 0.0030
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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/I2 l2/L 0.2 0.3 0.4 0.5 0.6 0.7 0.8 FEM 0.2 11.1574 13.4813 16.2613 18.9643 20.4594 20.6714 21.0548 this paper 11.1574 13.4813 16.2613 18.9643 20.4594 20.6714 21.0548 error δ/% 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 FEM
0.420.2377 22.7428 24.8856 26.0625 26.3025 26.4150 27.4675 this paper 20.2377 22.7428 24.8856 26.0627 26.3013 26.4170 27.4671 error δ/% 0.0000 0.0000 0.0000 0.0008 0.0046 0.0076 0.0015 FEM 0.6 27.7150 29.4950 30.6450 31.0550 31.0875 31.3475 32.4550 this paper 27.7154 29.4960 30.6447 31.0541 31.0865 31.3493 32.4559 error δ/% 0.0014 0.0034 0.0010 0.0029 0.0032 0.0057 0.0028 FEM 0.8 34.0200 34.8750 35.3150 35.4200 35.4400 35.6755 36.3725 this paper 34.0194 34.8758 35.3153 35.4195 35.4401 35.6754 36.3713 error δ/% 0.0018 0.0023 0.0010 0.0014 0.0003 0.0003 0.0033
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