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浙江大学学报(工学版)
Journal of ZheJiang University (Engineering Science)  2020, Vol. 54 Issue (2): 257-263    DOI: 10.3785/j.issn.1008-973X.2020.02.006
Civil and Transportation Engineering     
Tension analysis of hangers with stepped cross-section based on state space method
Guang OUYANG1(),Tian-jun LI2,Jiang-tao ZHANG1,Jing-feng WANG1,Rong-qiao XU1,*()
1. Department of Civil Engineering, Zhejiang University, Hangzhou 310058, China
2. Faculty of Engineering, China University of Geosciences (Wuhan), Wuhan 430074, China
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Abstract  

An accurate analysis method based on the state space method was developed for the dynamic characteristics of hangers with stepped cross-section for arch bridges using the theory of Euler-Bernoulli beam with axial force. The different cross-section characteristics and material parameters of each part of hangers can be considered in detail as well as the complicated boundary conditions at both ends of the hangers in practical engineering. The relationship of the free vibration frequencies and tension of hangers can then be obtained. It provides a theoretical basis for the so-called frequency method to measure the tension of hangers. The method was verified by the in-situ testing data of tension forces and the results of finite element analysis for the hangers of a practical arch bridge. When the length of the cable segment of a hanger is greater than a certain threshold, a strong linear relation between the effective calculation length of the hanger and the realistic length of the cable segment of the hanger exists. As a result, this method can be combined with the classical string theory, and the effective length of the hanger related to the realistic length of the cable segment can be identified. The classical string theory can be used to calculate the tension of hangers conveniently in practical engineering.

Key words： hanger with stepped cross-section      tension      frequency method      state space method      effective length     
Received: 12 January 2019      Published: 10 March 2020
CLC:  TU 311  
Corresponding Authors: Rong-qiao XU     E-mail: ouyangguang@zju.edu.cn;xurongqiao@zju.edu.cn
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Guang OUYANG
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Cite this article:

Guang OUYANG,Tian-jun LI,Jiang-tao ZHANG,Jing-feng WANG,Rong-qiao XU. Tension analysis of hangers with stepped cross-section based on state space method. Journal of ZheJiang University (Engineering Science), 2020, 54(2): 257-263.

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http://www.zjujournals.com/eng/10.3785/j.issn.1008-973X.2020.02.006     OR     http://www.zjujournals.com/eng/Y2020/V54/I2/257


基于状态空间法的分段变截面吊杆张拉力分析

针对分段变截面拱桥吊杆的动力特性，采用受轴向张拉力作用的Euler-Bernoulli梁理论建立分析模型，开发基于状态空间法的精确分析方法. 该方法详细考虑吊杆各部分不同的截面特性和材料参数，以及吊杆两端实际较为复杂的边界条件，给出吊杆自由振动的频率与其张拉力的关系，为基于频率法的吊杆张拉力测试方法提供理论基础. 通过某实际拱桥的吊杆张拉力测试数据以及有限元分析结果，验证所提方法. 当吊杆的索体长度大于阈值时，吊杆的有效计算长度与吊杆的实际索体长度存在着较强的线性关系，因此可将本研究方法与经典弦理论公式相结合，识别与索体长度相关的吊杆有效计算长度，可在实际工程中运用弦理论公式方便地计算吊杆张拉力.


关键词： 分段变截面吊杆,  张拉力,  频率法,  状态空间法,  有效长度 
Fig.1 Schmatic diagram of hanger of arch bridge
Fig.2 Beam model of hanger with stepped cross-section of arch bridge
参数 Ai /（10?2 m2） Ii /（10?6 m4） Li /m
上叉耳（开叉） 2.09 84.30 0.300
上叉耳（实心） 3.74 151.00 0.136
上锚头 3.55 3.89 0.260
下锚头 3.55 3.89 0.260
连接头 1.77 24.90 0.147
调节套筒 1.77 24.90 0.462
下叉耳（实心） 3.74 151.00 0.136
下叉耳（开叉） 2.09 84.30 0.300
Tab.1 Geometric parameters of hangers
吊杆编号 T/kN f0 /Hz fT /Hz fF /Hz
1）注：括号内数据为本研究结果相对实测结果的误差；2）注：括号内数据为本研究结果相对有限元结果的误差
1# 58.8 10.778 11.328（?4.86%）1） 10.708（0.65%）2）
2# 147.0 12.636 13.477（?6.24%） 12.616（0.16%）
3# 333.2 15.069 14.844（1.52%） 15.042（0.18%）
4# 254.8 16.288 15.527（4.90%） 16.287（0.01%）
5# 390.5 21.929 21.582（1.61%） 21.921（0.04%）
6# 174.4 19.490 19.531（?0.21%） 19.503（?0.07%）
Tab.2 Comparisons of fundamental frequencies calculated by proposed method with actual measurements and finite element analysis results
Fig.3 First-order mode shape of hanger 1#~6#
lc /m f1 /Hz l0 /m l0 /lc （0.984lc+0.339）/m T1 /kN [（T1?T0）/T0]/%
0.5 37.118 2.002 4.005 ? ? ?
1.0 33.294 2.232 2.232 ? ? ?
1.5 30.847 2.409 1.606 ? ? ?
2.0 28.236 2.632 1.316 ? ? ?
2.5 25.194 2.950 1.180 ? ? ?
3.0 22.174 3.352 1.117 3.291 289.211 ?3.60
3.5 19.561 3.800 1.086 3.783 297.364 ?0.88
4.0 17.407 4.270 1.067 4.275 300.717 0.24
4.5 15.643 4.751 1.056 4.767 301.982 0.66
5.0 14.187 5.239 1.048 5.259 302.297 0.77
5.5 12.970 5.730 1.042 5.751 302.169 0.72
6.0 11.942 6.224 1.037 6.243 301.834 0.61
6.5 11.061 6.719 1.034 6.735 301.409 0.47
7.0 10.301 7.216 1.031 7.227 300.952 0.32
7.5 9.637 7.713 1.028 7.719 300.494 0.16
8.0 9.053 8.210 1.026 8.211 300.051 0.02
8.5 8.535 8.708 1.025 8.703 299.629 ?0.12
9.0 8.073 9.207 1.023 9.195 299.231 ?0.26
9.5 7.658 9.705 1.022 9.687 298.859 ?0.38
10.0 7.284 10.204 1.020 10.179 298.511 ?0.50
Tab.3 Recognition results of effective calculation length of hanger
Fig.4 Relationship between effective calculation length and length of cable segment
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