Research on Shear Lag Warping Displacement Modes of Frame-Tube Structures Based on the Hamiltonian Mechanics
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摘要:
以等效连续化方法为基础，在Hamilton力学体系下进行框筒结构剪滞翘曲位移函数精度研究。选用不同类型的函数描述翼缘板的剪滞翘曲位移，考虑等效板的剪切变形以及纵向翘曲，得到不同位移函数下结构的总势能及对应的Lagrange函数。区别于传统变分法，该文在Hamilton力学体系下进行问题研究，导出框筒结构弯曲问题的Hamilton正则方程并利用精细积分法求解，进而计算出柱轴力并进行精度分析。算例验证结果表明：使用该方法分析框筒结构的剪力滞后效应是简单可行的；不同翘曲位移函数的选择对侧移计算结果影响不大，对轴力求解结果影响较大，二次抛物线最能反映等效翼缘板的实际翘曲位移；对比不同形式荷载作用下等效翼缘板中应力分布可知，随着外荷载合力作用点位置的升高，结构顶部负剪力滞后效应逐渐减弱至消失。
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关键词:
- 框筒结构 /
- 剪力滞后效应 /
- 翘曲位移函数 /
- Hamilton对偶求解体系 /
- 精细积分法
Abstract:Based on the equivalent continuity method, the accuracy of the shear lag warping displacement functions for frame-tube structures was studied under the Hamiltonian mechanics. Different types of functions were selected to describe the shear lag warping displacement of the flange plate, and the shear deformation and longitudinal warping of the equivalent plate were considered. The total potential energy of the structure and the corresponding Lagrangian function under different displacement modes were obtained. Not with the traditional variational methods, the problem was studied under the Hamiltonian mechanics system. The Hamiltonian canonical equation for the frame-tube structure was derived and solved with the precise integration method, then the column axial force was calculated and the accuracy was analyzed. The verification results of the calculation examples show that, this method is simple and feasible to analyze the shear lag effects of the frame-tube structures. The choice of different warping displacement functions has little effect on the lateral displacement calculation results, but has great influence on the axial force solution, and the quadratic parabola can best reflect the actual warping displacement distribution of the flange. Comparison of the stress distributions in the equivalent flange under different types of loads indicate that, with the increase of the position of the external load resultant force, the negative shear-lag effect on the top gradually weakens to disappear.
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图 4 不同翘曲函数下结构侧移曲线
Figure 4. Curves of structural lateral displacements with different warping functions
图 5 工况①下不同高度处翼缘应力分布
Figure 5. Flange stress distributions at different heights under working condition ①
图 6 工况②下不同高度处翼缘应力分布
Figure 6. Flange stress distributions at different heights under working condition ②
图 7 工况③下不同高度处翼缘应力分布
Figure 7. Flange stress distributions at different heights under working condition ③
表 1 工况①下结构底层柱轴力计算值及相对误差
Table 1. The axial force values and relative errors of the bottom floor column under working condition ①
column number ref. [1] N/kN function type N/kN relative error δ1/% quadratic function cosine function catenary function exponential function quadratic function cosine function catenary function exponential function 1 0 0 0 0 0 0 0 0 0 2 48.90 51.32 50.08 52.01 47.97 4.95 2.42 6.36 −1.90 3 105.80 108.53 106.07 109.85 102.20 2.58 0.26 3.83 −3.40 4 178.80 177.51 173.87 179.36 168.94 −0.72 −2.76 0.31 −5.52 5 275.90 264.15 259.38 266.35 254.44 −4.26 −6.00 −3.46 −7.78 6 698.00 644.80 636.71 647.67 629.60 −7.62 −8.78 −7.21 −9.80 7 304.50 283.86 291.09 278.81 288.19 −6.78 −4.41 −8.44 −5.36 8 223.00 210.29 220.09 204.51 223.82 −5.70 −1.30 −8.30 0.37 9 159.60 153.06 159.13 150.16 169.32 −4.10 −0.29 −5.92 6.09 10 114.40 112.19 112.36 113.11 123.20 −1.94 −1.79 −1.13 7.69 11 87.20 87.66 82.95 91.57 84.15 0.53 −4.87 5.01 −3.50 12 78.10 79.49 72.92 84.50 58.69 1.77 −6.63 8.20 −24.85 
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表 2 工况①下结构最大侧移值及相对误差
Table 2. Maximum side shift values and relative errors of the structure under working condition ①
ref. [1] u/cm function type u/cm relative error δ2/% quadratic function cosine function catenary function exponential function quadratic function cosine function catenary function exponential function 0.75 0.7685 0.7663 0.7691 0.7638 2.47 2.17 2.55 1.84
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