考虑摩阻力影响的悬索桥索鞍精确算法
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摘要
为了在悬索桥分析计算中精确模拟索鞍位置,给悬索桥的设计、施工和监控等提供参考,提出了一种索鞍位置的解析算法。该算法考虑了主缆与索鞍间摩阻力对索鞍位置的影响,根据欧拉公式,推导了摩阻系数的计算公式,通过建立索鞍数学模型和胡克定律,提出了主缆与索鞍间索鞍摩阻力计算方法。从力学和几何关系出发,建立基于弹性悬链线和静力平衡方程的索鞍计算模型,推导了基于初始索鞍位置坐标与平衡状态索鞍位置坐标间偏差的计算公式,引入雅克比矩阵,通过牛顿迭代法求解,使得偏差小于某一允许限值,即可认为索鞍单元位于平衡位置,同时推导了索鞍单元的切向刚度矩阵。结合解析算法,引入温度伸缩杆的概念,给出了一种适用于有限元分析与数值算法相结合的索鞍模拟方法,可用于精确模拟悬索桥索鞍在施工过程中的顶推,提高主缆在施工过程中线形的计算精度,将算法通过MATLAB程序进行迭代计算。最后,通过2组算例对比验证了该算法的可靠性,并通过改变摩阻力的取值,分析了摩阻力对索鞍位置的影响趋势。研究结果表明:提出的解析算法可精确模拟索鞍位置,公式收敛性好;摩阻力对索鞍位置的影响较小,简化计算中可以忽略,但在精确分析尤其是大跨、多跨悬索桥受力分析时,应予以考虑。
To simulate the position of cable saddle in suspension bridge analysis and calculation precisely, and provide references for the design, construction, and monitoring of suspension bridges, an analytical algorithm were proposed. The influence of frictional resistance between the main cable and cable saddle on the cable saddle position was considered in the algorithm. The calculation formula of the friction coefficient was derived using Euler's formula. The mathematical model and Hooke's law of the saddle was established, and the calculation method of the friction between the main cable and saddle was determined. The cable saddle calculation model was determined on the basis of elastic catenary and static equilibrium equation, considering the relation between mechanics and geometry. The calculation formula was derived based on the deviation between the coordinates of the original cable saddle position and the position of cable saddle in balanced state. Jacobian Matrix was introduced to cyclically solve this problem through Newton's method, cyclic iteration significantly decreases the deviation value considering the prescribed limit, and the cable saddle elements were assumed to achieve balanced position. At the same time, the tangent stiffness matrix was deduced. A cable saddle simulation method which applicable to finite elements was proposed,and the concept of temperature retractable pole was used in simulating the cable saddle of suspension bridge during construction, combined with the analytical algorithm and numerical algorithm. The calculating accuracy of alignment of the main cable during construction was improved,and iterative computation of the MATLAB program was used. The reliability of the algorithm was validated, by comparing two examples, and the trend of the influence of frictional resistance in cable saddle position was demonstrated, by changing the value of frictional resistance. The results show that the proposed analytical algorithm can exactly simulate the cable saddle position, and has good convergence. The influence of frictional resistance on cable saddle position is negligible. Thus, it may be ignored in a simplified calculation. However, especially in force analysis for the long span and multi-span suspension bridges, the influence should be taken into account.
To simulate the position of cable saddle in suspension bridge analysis and calculation precisely, and provide references for the design, construction, and monitoring of suspension bridges, an analytical algorithm were proposed. The influence of frictional resistance between the main cable and cable saddle on the cable saddle position was considered in the algorithm. The calculation formula of the friction coefficient was derived using Euler's formula. The mathematical model and Hooke's law of the saddle was established, and the calculation method of the friction between the main cable and saddle was determined. The cable saddle calculation model was determined on the basis of elastic catenary and static equilibrium equation, considering the relation between mechanics and geometry. The calculation formula was derived based on the deviation between the coordinates of the original cable saddle position and the position of cable saddle in balanced state. Jacobian Matrix was introduced to cyclically solve this problem through Newton's method, cyclic iteration significantly decreases the deviation value considering the prescribed limit, and the cable saddle elements were assumed to achieve balanced position. At the same time, the tangent stiffness matrix was deduced. A cable saddle simulation method which applicable to finite elements was proposed,and the concept of temperature retractable pole was used in simulating the cable saddle of suspension bridge during construction, combined with the analytical algorithm and numerical algorithm. The calculating accuracy of alignment of the main cable during construction was improved,and iterative computation of the MATLAB program was used. The reliability of the algorithm was validated, by comparing two examples, and the trend of the influence of frictional resistance in cable saddle position was demonstrated, by changing the value of frictional resistance. The results show that the proposed analytical algorithm can exactly simulate the cable saddle position, and has good convergence. The influence of frictional resistance on cable saddle position is negligible. Thus, it may be ignored in a simplified calculation. However, especially in force analysis for the long span and multi-span suspension bridges, the influence should be taken into account.
引文
[1] 肖海波,俞亚南,高庆丰.自锚式悬索桥主缆成桥线形分析[J].浙江大学学报:工学版,2004,38(11):1470-1473. XIAO Hai-bo,YU Ya-nan,GAO Qing-feng.Analysis of finished main cable of self-anchored suspension bridges[J].Journal of Zhejiang University:Engineering Science,2004,38(11):1470-1473.
[2] 秦凤江,周绪红,梁博文,等.大跨度自锚式悬索桥主梁钢-混结合段模型试验[J].中国公路学报,2018,31(9):52-64. QIN Feng-jiang,ZHOU Xu-hong,LIANG Bo-wen,et al.Experiment on steel-concrete joint of hybrid girder of a long-span self-anchored suspension bridge[J].China Journal of Highway and Transport,2018,31(9):52-64.
[3] 龙关旭,黄平明,袁婷,等.非线性随机车流-自锚式悬索桥耦合振动分析系统[J].中国公路学报,2018,31(7):147-155. LONG Guan-xu,HUANG Ping-ming,YUAN Ting,et al.Analytic system of nonlinear random traffic flow-self-anchored suspension bridge coupling vibration[J].China Journal of Highway and Transport,2018,31(7):147-155.
[4] PEVROT A H,GOULOIS A M.Analysis of cable structures[J].Computers and Structures,1979,10(5):805-813.
[5] 罗喜恒.悬索桥主缆线形的鞍座影响[J].公路交通科技,2005,22(8):36-39,48. LUO Xi-heng.Effect of saddle on cable shape of suspension bridges[J].Journal of Highway and Transportation Research and Development,2005,22(8):36-39,48.
[6] 李传习,王雷,刘光栋,等.悬索桥索鞍位置的分离计算法[J].中国公路学报,2005,18(1):63-68. LI Chuan-xi,WANG Lei,LIU Guang-dong,et al.Separate calculation method on suspension bridge saddle’s position[J].China Journal of Highway and Transport,2005,18(1):63-68.
[7] 李传习.混合梁悬索桥非线性精细计算理论及其应用[D].长沙:湖南大学,2006. LI Chuan-xi.A refined non-linear analysis method for suspension bridge with hybrid beam and its application[D].Changsha:Hunan University,2006.
[8] 王邵锐,周志祥,高燕梅,等.悬索桥索鞍预偏量的牛顿-拉斐森算法[J].中国公路学报,2016,29(1):82-88. WANG Shao-rui,ZHOU Zhi-xiang,GAO Yan-mei,et al.Newton-Raphson algorithm for pre-offsetting of cable saddle on suspension bridge[J].China Journal of Highway and Transport,2016,29(1):82-88.
[9] 李乔,张清华,周凌远,等.武汉鹦鹉洲长江大桥主缆与中主鞍座间抗滑移试验研究报告[R].成都:西南交通大学,2011. LI Qiao,ZHANG Qing-hua,ZHOU Ling-yuan,et al.The test report of slip behavior of cable against saddle for Yingwuzhou Yangtze River Bridge[R].Chengdu:Southwest Jiaotong University,2011.
[10] 张清华,李乔.悬索桥主缆与鞍座间摩擦特性试验研究[J].土木工程学报,2013,46(4):85-92. ZHANG Qing-hua,LI Qiao.Study on cable-saddle frictional characteristics of long-span suspension bridges[J].China Civil Engineering Journal,2013,46(4):85-92.
[11] 张清华,李乔,周凌远.悬索桥主缆与鞍座摩擦特性理论分析方法[J].中国公路学报,2014,27(1):44-50. ZHANG Qing-hua,LI Qiao,ZHOU Ling-yuan.Theoretical analysis of cable-saddle friction characteristics for suspension bridges[J].China Journal of Highway and Transport,2014,27(1):44-50.
[12] MCDONALD B M,PEYROT A H.Analysis of cables suspended in sheaves[J].Journal of Structural Engineering,1988,114(3):693-706.
[13] 魏建东,刘忠玉.悬索桥结构分析中索鞍的精确模拟[J].工程力学,2006,23(7):114-118. WEI Jian-dong,LIU Zhong-yu.A saddle model in finite element analysis of suspension bridges[J].Engineering Mechanics,2006,23(7):114-118.
[14] 叶觉明.大跨度三塔悬索桥主缆系统施工技术创新及改进[J].桥梁建设,2015,45(1):6-12. YE Jue-ming.Innovation and improvement of construction techniques for main cable systems of long span three-tower suspension bridges[J].Bridge Construction,2015,45(1):6-12.
[15] 侯苏伟,诸葛萍,强士中,等.悬索桥CFRP主缆与鞍座间摩擦学性能试验研究[J].西南交通大学学报,2011,46(3):391-397. HOU Su-wei,ZHUGE Ping,QIANG Shi-zhong,et al.Experimental investigation of friction properties between CFRP main cable and saddle of suspension bridge[J].Journal of Southwest Jiaotong University,2011,46(3):391-397.
[16] 曹竞.潮河大桥基准索股架设计算与控制[J].沈阳大学学报:自然科学版,2017,29(3):238-243. CAO Jing.Calculation and control of datum strand erection of Chaohe Bridge[J].Journal of Shenyang University:Natural Science,2017,29(3):238-243.
[17] 段茂军,李建慧,索小灿,等.混凝土自锚式悬索桥鞍座的精细化模拟方法[J].南京工业大学学报:自然科学版,2016,38(6):112-115,123. DUAN Mao-jun,LI Jian-hui,SUO Xiao-can,et al.Refined analysis method of saddle for self-anchored concrete suspension bridge[J].Journal of Nanjing Technology University:Natural Science,2016,38(6):112-115,123.
[18] 唐茂林,沈锐利,强士中.大跨度悬索桥丝股架设线形计算的精确方法 [J].西南交通大学学报,2001,36(3):303-307. TANG Mao-lin,SHEN Rui-li,QIANG Shi-zhong.An accurate calculation method for erecting curves of wire strands of long suspension bridges[J].Journal of Southwest Jiaotong University,2001,36(3):303-307.
[19] 潘世建,杨盛福.东航道悬索桥[M].北京:人民交通出版社,2002. PAN Shi-jian,YANG Sheng-fu.Cable suspension bridge over East Channel[M].Beijing:China Communications Press,2002.
[20] TAKENA K,SASAKI M,HATA K,et al.Slip behavior of cable against saddle in suspension bridges[J].Journal of Structural Engineering,1992,118(2):377-391.
[21] HASEGAWA K,KOJIMA H,SASAKI M,et al.Frictional resistance between cable and saddle equipped with friction plate[J].Journal of Structural Engineering,1995,121(1):1-14.
[22] 周凌远,李乔.缆索与鞍座间的摩擦特性[J].兰州理工大学学报,2011,37(2):117-121. ZHOU Ling-yuan,LI Qiao.Characteristics of friction between cable and saddle[J].Journal of Lanzhou University of Technology,2011,37(2):117-121.
[23] 吉林,陈策,冯兆祥.三塔悬索桥中塔主缆与鞍座间抗滑移试验研究[J].公路,2007(6):1-6. JI Lin,CHEN Ce,FENG Zhao-xiang.Experimental study on anti-sliding between main cable and saddle of three tower suspension bridge[J].Highway,2007(6):1-6.
[24] 诸葛萍,强士中,任伟平,等.碳纤维主缆鞍座处抗滑移试验[J].深圳大学学报:理工版,2011,28(5):395-399. ZHUGE Ping,QIANG Shi-zhong,REN Wei-ping,et al.Experiment of slip behavior of CFRP cable against saddle[J].Journal of Shenzhen University:Science and Engineering,2011,28(5):395-399.
[25] DENG T,LUO X H.Study for reasonable value of friction coefficient between main cable and saddle[J].Engineering Sciences,2012,10(3):72-76.
[26] 李喜平.三塔悬索桥结构行为特点研究[J].公路,2013(5):104-108. LI Xi-ping.Study on structural behavior characteristics of three tower suspension bridge[J].Highway,2013(5):104-108.
[27] 陈永健,房贞政,张超.三塔自锚式悬索桥主索鞍约束体系研究[J].福州大学学报:自然科学版,2011,39(3):428-433. CHEN Yong-jian,FANG Zhen-zheng,ZHANG Chao.Research on main saddle’s restriction of self-anchored suspension bridge with three towers[J].Journal of Fuzhou University:Natural Science,2011,39(3):428-433.
[2] 秦凤江,周绪红,梁博文,等.大跨度自锚式悬索桥主梁钢-混结合段模型试验[J].中国公路学报,2018,31(9):52-64. QIN Feng-jiang,ZHOU Xu-hong,LIANG Bo-wen,et al.Experiment on steel-concrete joint of hybrid girder of a long-span self-anchored suspension bridge[J].China Journal of Highway and Transport,2018,31(9):52-64.
[3] 龙关旭,黄平明,袁婷,等.非线性随机车流-自锚式悬索桥耦合振动分析系统[J].中国公路学报,2018,31(7):147-155. LONG Guan-xu,HUANG Ping-ming,YUAN Ting,et al.Analytic system of nonlinear random traffic flow-self-anchored suspension bridge coupling vibration[J].China Journal of Highway and Transport,2018,31(7):147-155.
[4] PEVROT A H,GOULOIS A M.Analysis of cable structures[J].Computers and Structures,1979,10(5):805-813.
[5] 罗喜恒.悬索桥主缆线形的鞍座影响[J].公路交通科技,2005,22(8):36-39,48. LUO Xi-heng.Effect of saddle on cable shape of suspension bridges[J].Journal of Highway and Transportation Research and Development,2005,22(8):36-39,48.
[6] 李传习,王雷,刘光栋,等.悬索桥索鞍位置的分离计算法[J].中国公路学报,2005,18(1):63-68. LI Chuan-xi,WANG Lei,LIU Guang-dong,et al.Separate calculation method on suspension bridge saddle’s position[J].China Journal of Highway and Transport,2005,18(1):63-68.
[7] 李传习.混合梁悬索桥非线性精细计算理论及其应用[D].长沙:湖南大学,2006. LI Chuan-xi.A refined non-linear analysis method for suspension bridge with hybrid beam and its application[D].Changsha:Hunan University,2006.
[8] 王邵锐,周志祥,高燕梅,等.悬索桥索鞍预偏量的牛顿-拉斐森算法[J].中国公路学报,2016,29(1):82-88. WANG Shao-rui,ZHOU Zhi-xiang,GAO Yan-mei,et al.Newton-Raphson algorithm for pre-offsetting of cable saddle on suspension bridge[J].China Journal of Highway and Transport,2016,29(1):82-88.
[9] 李乔,张清华,周凌远,等.武汉鹦鹉洲长江大桥主缆与中主鞍座间抗滑移试验研究报告[R].成都:西南交通大学,2011. LI Qiao,ZHANG Qing-hua,ZHOU Ling-yuan,et al.The test report of slip behavior of cable against saddle for Yingwuzhou Yangtze River Bridge[R].Chengdu:Southwest Jiaotong University,2011.
[10] 张清华,李乔.悬索桥主缆与鞍座间摩擦特性试验研究[J].土木工程学报,2013,46(4):85-92. ZHANG Qing-hua,LI Qiao.Study on cable-saddle frictional characteristics of long-span suspension bridges[J].China Civil Engineering Journal,2013,46(4):85-92.
[11] 张清华,李乔,周凌远.悬索桥主缆与鞍座摩擦特性理论分析方法[J].中国公路学报,2014,27(1):44-50. ZHANG Qing-hua,LI Qiao,ZHOU Ling-yuan.Theoretical analysis of cable-saddle friction characteristics for suspension bridges[J].China Journal of Highway and Transport,2014,27(1):44-50.
[12] MCDONALD B M,PEYROT A H.Analysis of cables suspended in sheaves[J].Journal of Structural Engineering,1988,114(3):693-706.
[13] 魏建东,刘忠玉.悬索桥结构分析中索鞍的精确模拟[J].工程力学,2006,23(7):114-118. WEI Jian-dong,LIU Zhong-yu.A saddle model in finite element analysis of suspension bridges[J].Engineering Mechanics,2006,23(7):114-118.
[14] 叶觉明.大跨度三塔悬索桥主缆系统施工技术创新及改进[J].桥梁建设,2015,45(1):6-12. YE Jue-ming.Innovation and improvement of construction techniques for main cable systems of long span three-tower suspension bridges[J].Bridge Construction,2015,45(1):6-12.
[15] 侯苏伟,诸葛萍,强士中,等.悬索桥CFRP主缆与鞍座间摩擦学性能试验研究[J].西南交通大学学报,2011,46(3):391-397. HOU Su-wei,ZHUGE Ping,QIANG Shi-zhong,et al.Experimental investigation of friction properties between CFRP main cable and saddle of suspension bridge[J].Journal of Southwest Jiaotong University,2011,46(3):391-397.
[16] 曹竞.潮河大桥基准索股架设计算与控制[J].沈阳大学学报:自然科学版,2017,29(3):238-243. CAO Jing.Calculation and control of datum strand erection of Chaohe Bridge[J].Journal of Shenyang University:Natural Science,2017,29(3):238-243.
[17] 段茂军,李建慧,索小灿,等.混凝土自锚式悬索桥鞍座的精细化模拟方法[J].南京工业大学学报:自然科学版,2016,38(6):112-115,123. DUAN Mao-jun,LI Jian-hui,SUO Xiao-can,et al.Refined analysis method of saddle for self-anchored concrete suspension bridge[J].Journal of Nanjing Technology University:Natural Science,2016,38(6):112-115,123.
[18] 唐茂林,沈锐利,强士中.大跨度悬索桥丝股架设线形计算的精确方法 [J].西南交通大学学报,2001,36(3):303-307. TANG Mao-lin,SHEN Rui-li,QIANG Shi-zhong.An accurate calculation method for erecting curves of wire strands of long suspension bridges[J].Journal of Southwest Jiaotong University,2001,36(3):303-307.
[19] 潘世建,杨盛福.东航道悬索桥[M].北京:人民交通出版社,2002. PAN Shi-jian,YANG Sheng-fu.Cable suspension bridge over East Channel[M].Beijing:China Communications Press,2002.
[20] TAKENA K,SASAKI M,HATA K,et al.Slip behavior of cable against saddle in suspension bridges[J].Journal of Structural Engineering,1992,118(2):377-391.
[21] HASEGAWA K,KOJIMA H,SASAKI M,et al.Frictional resistance between cable and saddle equipped with friction plate[J].Journal of Structural Engineering,1995,121(1):1-14.
[22] 周凌远,李乔.缆索与鞍座间的摩擦特性[J].兰州理工大学学报,2011,37(2):117-121. ZHOU Ling-yuan,LI Qiao.Characteristics of friction between cable and saddle[J].Journal of Lanzhou University of Technology,2011,37(2):117-121.
[23] 吉林,陈策,冯兆祥.三塔悬索桥中塔主缆与鞍座间抗滑移试验研究[J].公路,2007(6):1-6. JI Lin,CHEN Ce,FENG Zhao-xiang.Experimental study on anti-sliding between main cable and saddle of three tower suspension bridge[J].Highway,2007(6):1-6.
[24] 诸葛萍,强士中,任伟平,等.碳纤维主缆鞍座处抗滑移试验[J].深圳大学学报:理工版,2011,28(5):395-399. ZHUGE Ping,QIANG Shi-zhong,REN Wei-ping,et al.Experiment of slip behavior of CFRP cable against saddle[J].Journal of Shenzhen University:Science and Engineering,2011,28(5):395-399.
[25] DENG T,LUO X H.Study for reasonable value of friction coefficient between main cable and saddle[J].Engineering Sciences,2012,10(3):72-76.
[26] 李喜平.三塔悬索桥结构行为特点研究[J].公路,2013(5):104-108. LI Xi-ping.Study on structural behavior characteristics of three tower suspension bridge[J].Highway,2013(5):104-108.
[27] 陈永健,房贞政,张超.三塔自锚式悬索桥主索鞍约束体系研究[J].福州大学学报:自然科学版,2011,39(3):428-433. CHEN Yong-jian,FANG Zhen-zheng,ZHANG Chao.Research on main saddle’s restriction of self-anchored suspension bridge with three towers[J].Journal of Fuzhou University:Natural Science,2011,39(3):428-433.