Derivation and Solution of Vibration Differential Equation of the
Curved Beam with Eccentric Reinforce

doi:10.3969/j.issn.1674-0696.2017.07.03

Journal of Chongqing Jiaotong University(Natural Science) ›› 2017, Vol. 36 ›› Issue (7): 15-20.DOI: 10.3969/j.issn.1674-0696.2017.07.03

• Bridge & Tunnel Engineering • Previous Articles Next Articles

Derivation and Solution of Vibration Differential Equation of the Curved Beam with Eccentric Reinforce

HE Wenzheng1, CHEN Chen2, LI Pengcheng3, WEN Jingzhou4

（1. School of Urban Construction Engineering, Chongqing Radio & TV University, Chongqing 400052, P. R. China; 2. Department of Architecture Engineering, Yibin Vocational and Technical College, Yibin 644000, Sichuan, P. R. China; 3. Road Research Institute, Chongqing Communications Research and Design Institute, Chongqing 400067, P. R. China; 4. Rock and Soil Engineering Institute, Yunnan Science ＆ Technology Research Institute of Highway, Kunming 650051, Yunnan, P. R. China)

Received:2015-12-02 Revised:2016-07-16 Online:2017-07-31 Published:2017-07-31

偏心布筋曲梁振动微分方程推导和求解

何文正1, 陈晨2, 李鹏程3, 文竞舟4

（1. 重庆广播电视大学城市建设工程学院，重庆 400052；2. 宜宾职业技术学院建筑工程系，四川宜宾 644000； 3. 重庆交通科研设计院道路所，重庆 400067；4. 云南省公路科学技术研究院岩土所，云南昆明 650051）

作者简介:何文正(1982—)，男，重庆人，讲师，硕士，主要从事桥梁检测加固方面的研究。
基金资助:
交通运输部建设科技项目(2013318791440)；重庆广播电视大学重点基金项目(ZD2015-02)

Abstract

Abstract: In order to study the influence of tendon eccentricity and radius on the vibration characteristics of the curved beam, the differential equation of the vibration of the prestressed curved beam considering the influence of initial curvature on vibration was derived by analyzing the micro section of the curved beam under the condition of free vibration. Then the theoretic calculation formula of the first order natural frequency vibration of the eccentric reinforcement simply supported curved beam was deduced by solving the equation. Finally, the correctness of the proposed theoretic formula was verified by the case study.

Key words: bridge engineering, curved beam, eccentric reinforce, initial curvature, natural frequency of vibration

摘要： 为研究预应力筋偏心距和半径等参数对曲梁振动特性的影响，以自由振动状态下的曲梁微段为研究对象，考虑偏心布筋产生的初始曲率对振动的影响，推导了偏心布筋曲梁振动偏微分方程，并求解方程得到了简支曲梁面内振动1阶频率理论计算公式；最后通过实例分析验证了所提理论公式的正确性。

关键词: 桥梁工程, 曲梁, 偏心布筋, 初始曲率, 自振频率

CLC Number:

U441.2

HE Wenzheng1, CHEN Chen2, LI Pengcheng3, WEN Jingzhou4. Derivation and Solution of Vibration Differential Equation of the Curved Beam with Eccentric Reinforce[J]. Journal of Chongqing Jiaotong University(Natural Science), 2017, 36(7): 15-20.

何文正1, 陈晨2, 李鹏程3, 文竞舟4. 偏心布筋曲梁振动微分方程推导和求解[J]. 重庆交通大学学报（自然科学版）, 2017, 36(7): 15-20.

References

［1］赵跃宇，康厚军，冯锐，等．曲线梁研究进展［J］．力学进展，2006，36(2)：170-186．
ZHAO Yueyu, KANG Houjun, FENG Rui, et al. Advances of research on curved beams［J］. Advances in Mechanics, 2006, 36(2): 170-186.
［2］谢秉敏，向中富，王小松，等．基于ANSYS的车桥耦合动力分析［J］．重庆交通大学学报(自然科学版)，2012，31(5)：935-938．
XIE Bingmin, XIANG Zhongfu, WANG Xiaosong, et al. Dynamic analysis method of vehicle-bridge coupling based on ANSYS［J］. Journal of Chongqing Jiaotong University
(Natural Science), 2012, 31(5): 935-938.
［3］ KOU C H, BENZLEY S E, HUANG J Y, et al. Free vibration analysis of curved thin-walled girder bridges［J］. Journal of Structural Engineering, 1992, 118(10):
2890-2910.
［4］单德山．高速铁路曲线梁桥车桥耦合振动分析及大跨度曲线梁桥设计研究［D］．成都：西南交通大学，1999：14-46．
SHAN Deshan. Curved-Girder Bridges and Vehicles Coupled Vibration Analysis and Design Study of the Long-Span Curved-Girder Bridges in High-Speed Railway ［D］.
Chengdu: Southwest Jiaotong University, 1999:14-46.
［5］ YANG Y B, WU C M, YAU J D. Dynamic response of a horizontally curved beam subjected to vertical and horizontal moving loads［J］.Journal of Sound and
Vibration,2001,242(3):519-537.
［6］叶康生，赵雪健．动力刚度法求解平面曲梁面外自由振动问题［J］．工程力学，2012，29(3)：1-8．
YE Kangsheng, ZHAO Xuejian. Dynamic stiffness method for out-of-plane free vibration analysis of planar curved beams［J］. Engineering Mechanics, 2012, 29(3):1-8.
［7］宋郁民，吴定俊，李奇．圆弧曲梁振动微分方程推导及振动特性分析［J］．沈阳建筑大学学报(自然科学版)，2012，28(3)：400-404．
SONG Yumin, WU Dingjun, LI Qi. Derivation of vibration differential equation and analysis of vibration properties about ARC-curved beam ［J］. Journal of Shenyang
Jianzhu University(Natural Science), 2012, 28(3): 400-404.
［8］宋郁民．曲梁振动微分方程组求解［C］//中国力学学会《工程力学》编委会．第23届全国结构工程学术会议论文集：第Ⅰ册．2014：258-264．
SONG Yumin. Solving the differential equations of curved beams ［C］//Editorial Board of Engineering Mechanics, Chinese Society of Mechanics. Proceedings of the
23rd National Conference on Structural Engineering: Volume I. 2014:258-264.
［9］冯东明，李爱群，李建慧，等．独塔斜拉与连续刚构组合桥梁动力特性及参数影响分析［J］．重庆交通大学学报(自然科学版)，2010，29(3)：329-332．
FENG Dongming, LI Aiqun, LI Jianhui, et al. Research on dynamic behavior and parameter influence of single pylon cable-stayed and continuous rigid-frame composite
bridge［J］. Journal of Chongqing Jiaotong University(Natural Science), 2010, 29(3): 329-332.
［10］梁甜甜，向中富，卞明智，等．曲梁预应力摩阻损失试验研究［J］．重庆交通大学学报(自然科学版)，2008，27(增刊)：871-874．
LIANG Tiantian, XIANG Zhongfu, BIAN Mingzhi, et al. Test and study of attrition loss of pre-stressed wire of curved box girder［J］. Journal of Chongqing Jiaotong
University(Natural Science), 2008, 27(Sup): 871-874.
［11］邵容光，夏淦．混凝土弯梁桥［M］．北京：人民交通出版社，1996：212-216．
SHAO Rongguang, XIA Gan. Concrete Curved Girder Bridge［M］. Beijing: China Communication Press, 1996: 212-216.
［12］姚玲森．曲线梁［M］．北京：人民交通出版社，1989：1-14．
YAO Lingsen. Curved Beam［M］. Beijing: China Communications Press, 1989:1-14.
［13］王龙林．简支梁自振频率的预应力效应分析［D］．长春：吉林大学，2013: 53-77．
WANG Longlin. Analysis of Prestressing Effect on Natural Frequency for Simply Supported Beam［D］. Changchun: Jilin University, 2013: 53-77.
［14］刘晶波，杜修力．结构动力学［M］．北京：机械工业出版社，2005：160-169．
LIU Jingbo, DU Xiuli. Structural Dynamics［M］. Beijing: China Machine Press, 2005: 160-169.
［15］李荣．无粘结预应力混凝土梁动力特性研究［D］．包头：内蒙古科技大学，2013：7-33．
LI Rong. Study on Dynamic Peculiarity for Unbonded Pre-stressed Concrete Beam［D］. Baotou: Inner Mongolia University of Science and Technology, 2013:7-33.

[1]	ZHOU Jianting1,2, LI Ya2, ZHANG Hong2, XIA Runchuan2, YANG Wenqi2. Two Point Corrosion Test Study of Steel Strands Based on Spontaneous Magnetic Flux Leakage Detection [J]. Journal of Chongqing Jiaotong University(Natural Science), 2021, 40(07): 74-81.
[2]	ZHANG Hua1, ZHANG Zhengqi2, CHENG Gao3,4, TIAN Boke1. Rational Distance of Pile Foundation in Precipitous Cliff Section of Loess Area [J]. Journal of Chongqing Jiaotong University(Natural Science), 2021, 40(07): 93-98.
[3]	LIU Jinchun, SONG Zixuan, LIANG Dong. Spatial Effect Analysis of Single Box Three Chambers Continuous Girder Bridge under Transverse Temperature Gradient and Eccentric Vehicle Load [J]. Journal of Chongqing Jiaotong University(Natural Science), 2021, 40(06): 80-86.
[4]	ZHENG Zhi1,2, GENG Bo2, YUAN Pei2, LI Songlin2, HU Zhengtao3. Experimental Study on a New Connection Structure of Anti-collision Device for Pier Composite Materials [J]. Journal of Chongqing Jiaotong University(Natural Science), 2021, 40(05): 66-73.
[5]	ZHANG Yang1, QU Shaoqin1, LU Jiuzhang2, HUO Wenbing3. Method of Fiber Aligning in UHPC and Its Influence on the Tensile Performance [J]. Journal of Chongqing Jiaotong University(Natural Science), 2021, 40(05): 74-80.
[6]	FU Lilei. Correlation Function Method for Parameter Identification Based on Random Excitation Response [J]. Journal of Chongqing Jiaotong University(Natural Science), 2021, 40(04): 70-75.
[7]	SU Xiaobo1, XIAO Lin2. Model Test and Numerical Simulation Research on Reasonable Structure of Steel-Concrete Composite Section in Space Rigid Frame [J]. Journal of Chongqing Jiaotong University(Natural Science), 2021, 40(04): 76-82.
[8]	LIANG Zongbao 1, CHAI Jie1, NA Shouyong2, MA Tianli1, TANG Yu1. Validity Analysis of Bridge Health Monitoring Data Based on Deep Learning [J]. Journal of Chongqing Jiaotong University(Natural Science), 2021, 40(03): 78-83.
[9]	TANG Qizhi1, XIN Jingzhou1,2, ZHOU Jianting1, FU Lei1,3, ZHANG Jun4. Damage Identification of Corroded RC Arch Based on Time Series Analysis:Simulation and Verification [J]. Journal of Chongqing Jiaotong University(Natural Science), 2021, 40(02): 69-77.
[10]	RONG Xian1, 2, YANG Hongwei1, ZHANG Jianxin1, 2. Experimental Research on Hysteresis Behavior of Prefabricated Concrete Beam-Column Joint with Steel End Connector [J]. Journal of Chongqing Jiaotong University(Natural Science), 2021, 40(02): 78-82.
[11]	LIU Xiaohui, XU Yang, CHEN Sitian, XU Lueqin. Seismic Response Analysis of Pedestrian Suspension Bridge [J]. Journal of Chongqing Jiaotong University(Natural Science), 2021, 40(01): 65-71.
[12]	JU Liping1, WANG Yinhui2, LU Xiaohong3, SHAN Jidong4. Failure Mode Analysis of Reinforced Concrete Box Girder under Impact of a Falling Cargo from Passing Truck [J]. Journal of Chongqing Jiaotong University(Natural Science), 2021, 40(01): 72-78.
[13]	WANG Zhen1,2, ZHOU Jianting1,3, ZHANG Jinquan1,4, WU Haijun1, WU Yuexing1. Influence Analysis of Cast-in-Situ Segment Length on Mechanical Behavior of Ten-Thousand-Ton Horizontal Swivel Cable-Stayed Bridge [J]. Journal of Chongqing Jiaotong University(Natural Science), 2020, 39(12): 44-52.
[14]	HAO Xianwu, SHU Peng, HAO Jianming. Static Wind Stability of Long-Span Asymmetric Suspension Bridge [J]. Journal of Chongqing Jiaotong University(Natural Science), 2020, 39(12): 53-59.
[15]	CHEN Sitian, PENG Qing, YANG Min. Human-Induced Vibration Response Analysis of Long-Span Pedestrian Suspension Glass Bridge in Kanggu Parkland [J]. Journal of Chongqing Jiaotong University(Natural Science), 2020, 39(12): 60-66.

Derivation and Solution of Vibration Differential Equation of the Curved Beam with Eccentric Reinforce

偏心布筋曲梁振动微分方程推导和求解

PDF

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 15

Recommended Articles

Metrics