﻿ 重刚比对悬索静态位形影响的建模与分析
 森林与环境学报  2017, Vol. 37 Issue (3): 302-308 PDF
http://dx.doi.org/10.13324/j.cnki.jfcf.2017.03.009
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#### 文章信息

FENG Huirong, ZHOU Chengjun, ZHOU Xinnian, LIM Cheewah, CHEN Liqun

Modeling and analysis of the static configuration of cableway suspension by the ratio of cable weight to stiffness

Journal of Forest and Environment,2017, 37(3): 302-308.
http://dx.doi.org/10.13324/j.cnki.jfcf.2017.03.009

### 文章历史

1. 上海大学上海市应用数学与力学研究所, 上海 200072;
2. 福建农林大学交通与土木工程学院, 福建 福州 350002;
3. 香港城市大学土木与建筑工程系, 香港 999077;
4. 上海大学力学系, 上海 200444

Modeling and analysis of the static configuration of cableway suspension by the ratio of cable weight to stiffness
FENG Huirong1,2, ZHOU Chengjun2, ZHOU Xinnian2, LIM Cheewah3, CHEN Liqun1,4
1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China;
2. College of Transportation and Civil Engineering, Fujian Agriculture and Forestry University, Fuzhou, Fujian 350002, China;
3. Department of Architecture and Civil Engineering, City University of Hongkong, Hongkong 999077, China;
4. Department of Mechanics, Shanghai University, Shanghai 200444, China
Abstract: This paper intends to investigate the influence of self-weight, stiffness and distributed load on the static equilibrium configuration of suspension cables.The static suspension configuration of a level-supported cableway was analyzed by classifying cable states and force equilibrium analysis in a mathematical model of a cable element.Using dimensionless parameters, the basic static performance of a suspension cable in various conditions were analyzed including a natural state, unload self-weight state and distributed load state. A mathematical model of the static tension considering the cable unit length weight-to-stiffness ratio, deflection, and bearing spacing under the action of self-weight and cable distributed load was established. The paper found that, for a small ratio to an extent of β ≤ 10-2, the cable static state coincided in general. The cable stiffness played a leading role in the static analysis and the static suspension cable can be treated as an inextensible cable. For a high weight-to-stiffness ratio, the self-weight and distributed load were significant in the static state analysis. In these conditions, it was concluded that the greater the self-weight and the distributed load, the larger the central deflection of suspension cable. When the cable length with a small weight-to-stiffness ratio approached the original span, the maximum tension and the horizontal tension obviously increased. On the other hand, the increase in tension was not obvious for a high weight-to-stiffness ratio.Therefore, it was concluded in this paper that the suspension cable with a high weight-to-stiffness ratio was governed by mainly elastic deformation and yield failure due to its small elastic modulus. Consequently, it was not advisable to blindly tighten a suspension cable in engineering practice. Lastly, the configuration parameters of static suspension cables were verified by experiments to verify the feasibility and accuracy of the model.The relative error between the experimental results and theoretical results were within 4.95% and less than 5% of the engineering application accuracy requirements.
Key words: engineering cableway     unit length of cable     weight-to-stiffness ratio     stiffness     static configuration

1 悬索基本静态位形分析 1.1 静力学模型基本假设

(1) 假设悬索支座两端固定，悬索在垂直平面内的任一部分的运动相比于悬索长度非常小。(2) 由于受力是沿悬索长度的张力，假设其应力应变遵循虎克定律。(3) 对悬索静态性能的分析时将所有的摩擦力、空气阻力和支座的窜动位移忽略不计。(4) 在悬索建模和静态位形分析时，考虑悬索自重和悬索刚度的影响。(5) 悬索平均直径相对于其长度非常小，在索长参与计算中可忽略不计。

1.2 悬索基本静态位形分类

 图 1 弹性悬索的3种位形状态 Fig. 1 Three configuration of elastic cable

1.3 分布力与重刚比影响下的静态位形建模

 $e={{\left[ {{\left( \frac{\partial x}{\partial S} \right)}^{2}}+{{\left( \frac{\partial y}{\partial S} \right)}^{2}} \right]}^{1/2}}-1$ (1)

 $T=EeA=Ee\frac{{{A}_{0}}}{1+e}$ (2)

 ${{L}_{0}}=2\int _{0}^{\frac{{{l}_{0}}}{2}}\sqrt{{{\left( \frac{\partial x}{\partial S} \right)}^{2}}+{{\left( \frac{\partial y}{\partial S} \right)}^{2}}}\text{d}s$ (3)

 图 2 在自重作用下弹性伸长索的位形 Fig. 2 The configuration of elastic extensible cable with self-weight

x0(S)，y0(S)和T0(S)表示静态参量，可以通过公式(1) 和公式(2) 得出，从公式(2) 中可以得到$\left( 1+e \right)={{\left( \frac{1}{1+e} \right)}^{-1}}={{\left( 1-\frac{e}{1+e} \right)}^{-1}}={{\left( 1-\frac{{{T}_{0}}}{E{{A}_{0}}} \right)}^{-1}}$，从公式(1) 或者图 2(c)可以得到(1+e)2=(x0)2+(y0)2，联立上式可得公式(4)~(6)。

 ${{T}_{0}}=\frac{{{T}_{0}}\left( 0 \right)}{\text{cos}\theta }=\frac{{{H}_{0}}\left( 1+e \right)}{x_{0}^{\prime }}={{\left\{ H_{0}^{2}+{{[\rho g+q\left( x,t \right)]}^{2}}{{s}^{2}} \right\}}^{1/2}}$ (4)
 $x_{0}^{\prime }=\left( 1+e \right)\text{cos}\theta =\frac{{{H}_{0}}}{{{T}_{0}}}{{(1-\frac{{{T}_{0}}}{E{{A}_{0}}})}^{-1}}$ (5)
 $y_{0}^{\prime }=\left( 1+e \right)\text{sin}\theta =\frac{\rho gs+q\left( x,t \right)s}{{{T}_{0}}}{{(1-\frac{{{T}_{0}}}{E{{A}_{0}}})}^{-1}}$ (6)

 ${{T}_{0}}=\rho gL{{\left\{ {{\alpha }^{2}}+{{[1+\bar{q}\left( x,t \right)]}^{2}}{{\xi }^{2}} \right\}}^{1/2}}$ (7)
 $x_{0}^{\prime }=\alpha {{\left\{ {{\alpha }^{2}}+{{[1+\bar{q}\left( x,t \right)]}^{2}}{{\xi }^{2}} \right\}}^{-1/2}}\langle 1-\beta {{\left\{ {{\alpha }^{2}}+{{[1+\bar{q}\left( x,t \right)]}^{2}}{{\xi }^{2}} \right\}}^{1/2}}{{\rangle }^{-1}}$ (8)
 $y_{0}^{\prime }=-[1+\bar{q}\left( x,\text{ }t \right)]\xi {{\left\{ {{\alpha }^{2}}+{{[1+\bar{q}\left( x,\text{ }t \right)]}^{2}}{{\xi }^{2}} \right\}}^{-1/2}}\langle 1-\beta {{\left\{ {{\alpha }^{2}}+{{[1+\bar{q}\left( x,t \right)]}^{2}}{{\xi }^{2}} \right\}}^{1/2}}{{\rangle }^{-1}}$ (9)

ξ=-0.5~0.5并利用边界条件y=(y)ξ=±0.5=0积分方程式，可以得到图 3图 3表示了不同的重刚比下，悬索挠度y随着ξ=s/L的变化关系。重刚比β越大，悬索挠度y也相应增大。当重刚比β=10-2~10-8时，也就是重刚比很小，相对刚度而言，悬索的自重很小时，可伸长的弹性悬索的静态位形基本是重合的，也就是接近于不可伸长的悬索状态[6]。当β≤ 10-2时，此时在106倍率的量级内的无量纲悬索最大挠度y=0.06是基本相同的。并且从两支座向跨中逐渐增大，说明此时悬索重量对悬索的静态位形基本没有影响。但是，当重刚比β≥10-1时，如β=0.10, 0.25, 0.30时，重刚比增加2.5，3.0和1.2倍时，无量纲悬索最大挠度y=0.067, 0.130, 0.158，相应的增幅为1.94，2.36与1.22倍，增幅明显。说明此时悬索的静态位形受悬索的重刚比影响明显。

 图 3 悬索挠度y随ξ的变化关系 Fig. 3 The relationship of deflection yand ξ
2 不同重刚比的悬索静态性能分析

 图 4 不同重刚比下悬索单元最大拉力与水平支座间距的关系 Fig. 4 Relationship between Tmax/ρgL and l0/L of cable with different ratio of weight to stiffness
 图 5 不同重刚比下悬索水平拉力与水平支座间距的关系 Fig. 5 Relationship between H0 /ρgL and l0/L of cable with different ratio of weight to stiffness
 图 6 不同重刚比下悬索中央挠度与水平支座间距的关系 Fig. 6 Relationship between fD/L and l0/L of cable with different ratio of weight to stiffness

3 小重刚比悬索静态参数测定与结果分析

 型号与直径Type and diameter l0/m l0/L β(×10-6) fD/m 理论对照Theoretical checks fD/m 相对误差Relative error /% H0/N 理论对照Theoretical checks H0/N 相对误差Relative error /% Tm/N 理论对照Theoretical hecks Tm/N 相对误差Relative error /% 6×7-2 0.000 0.00 1.91 1.902 2.000 4.90 0.000 0.000 0.00 0.301 0.290 3.79 0.800 0.20 1.91 1.846 1.891 2.38 0.064 0.061 4.92 0.304 0.290 4.83 1.610 0.40 1.91 1.683 1.711 1.64 0.132 0.127 3.94 0.306 0.294 4.08 2.410 0.60 1.91 1.431 1.445 0.97 0.201 0.197 2.03 0.312 0.305 2.30 3.200 0.80 1.91 1.057 1.062 0.47 0.301 0.292 3.08 0.361 0.350 3.14 3.644 0.91 1.91 0.771 0.782 1.41 0.436 0.420 3.81 0.473 0.456 3.73 4.000 1.00 1.91 0.032 0.031 3.23 9.551 9.334 2.32 9.552 9.336 2.31 4.010 1.002 5 1.91 0.024 0.023 4.35 241.059 231.073 4.32 241.077 231.091 4.32 6×7-3 0.000 0.00 4.18 1.902 2.000 4.90 0.000 0.000 0.00 0.642 0.636 0.94 0.800 0.20 4.18 1.846 1.891 2.38 0.138 0.135 2.22 0.651 0.637 2.20 1.610 0.40 4.18 1.683 1.711 1.64 0.283 0.278 1.80 0.657 0.644 2.02 2.410 0.60 4.18 1.431 1.445 0.97 0.441 0.431 2.32 0.685 0.670 2.24 3.200 0.80 4.18 1.057 1.062 0.47 0.650 0.641 1.40 0.779 0.768 1.43 3.600 0.91 4.18 0.771 0.782 1.41 0.930 0.920 1.09 1.012 1.001 1.10 4.000 1.00 4.18 0.032 0.031 3.23 21.162 20.472 3.37 21.165 20.474 3.38 4.010 1.002 5 4.18 0.023 0.022 4.55 519.757 506.772 2.56 519.792 506.806 2.56 6×7-5 0.000 0.00 3.47 1.901 2.000 4.95 0.000 0.000 0.00 1.212 1.188 2.02 0.800 0.20 3.47 1.850 1.891 2.17 0.256 0.251 1.99 1.212 1.190 1.85 1.610 0.40 3.47 1.635 1.711 4.44 0.553 0.531 4.14 1.251 1.202 4.08 2.410 0.60 3.47 1.428 1.445 1.18 0.812 0.807 0.62 1.259 1.251 0.64 3.200 0.80 3.47 1.028 1.062 3.20 1.220 1.207 1.08 1.450 1.434 1.12 3.600 0.90 3.47 0.698 0.717 2.65 1.790 1.743 2.70 1.920 1.870 2.67 4.000 1.00 3.47 0.033 0.032 3.13 40.120 38.239 4.92 40.126 38.244 4.92 4.005 1.001 3 3.47 0.024 0.023 4.26 989.162 946.605 4.50 989.236 946.676 4.50 1)此表为提高小挠度时的计算精度，不以小数点后的位数一致，而以小数点后的有效数字一致。Note: to improve the deflection caculation accuracy in the table, number after radix pointis calculate dusing significance digit.

4 讨论与结论

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