脉动风和车辆荷载联合作用下中央扣对悬索桥吊索疲劳损伤的影响

脉动风和车辆荷载联合作用下中央扣对悬索桥吊索疲劳损伤的影响 Effect of central clamps on fatigue damage of suspenders of suspension bridge under joint action of fluctuating wind and vehicle loads 1 2 3 first-author 1 1 2 3 corresp 何旭辉，博士，教授，从事桥梁风工程研究；E-mail：xuhuihe@csu.edu.cn 何旭辉，博士，教授，从事桥梁风工程研究；E-mail：xuhuihe@csu.edu.cn xuhuihe@csu.edu.cn 中南大学 土木工程学院，湖南 长沙，410075 中南大学 土木工程学院，湖南 长沙，410075 School of Civil Engineering, Central South University, Changsha 410075, China School of Civil Engineering, Central South University, Changsha 410075, China 中南大学 高速铁路建造技术国家工程研究中心，湖南 长沙，410075 中南大学 高速铁路建造技术国家工程研究中心，湖南 长沙，410075 National Engineering Research Center for High Speed Railway Construction, Central South University, Changsha 410075, China National Engineering Research Center for High Speed Railway Construction, Central South University, Changsha 410075, China 中南大学 轨道交通工程结构防灾减灾湖南省重点实验室，湖南 长沙，410075 中南大学 轨道交通工程结构防灾减灾湖南省重点实验室，湖南 长沙，410075 Hunan Provincial Key Laboratory for Disaster Prevention and Mitigation of Rail Transit Engineering Structures, Central South University, Changsha 410075, China Hunan Provincial Key Laboratory for Disaster Prevention and Mitigation of Rail Transit Engineering Structures, Central South University, Changsha 410075, China 陈灿华 以杭瑞洞庭大桥为研究对象，建立设置中央扣和无中央扣的大跨度悬索桥有限元模型，研究中央扣对悬索桥跨中短吊索疲劳损伤的影响。首先，采用谐波合成法生成大桥桥址处的脉动风场，采用Monte Carlo法模拟随机车流样本，基于ANSYS软件建立风-车-桥耦合振动分析模型，分析桥梁结构在脉动风和车辆荷载单独与联合作用下的动力响应；其次，采用雨流计数法统计跨中短吊索的应力时程，得到吊索的应力幅值、应力均值和循环次数；最后，基于Miner损伤线性累计理论分析中央扣对跨中短吊索的等效总应力幅值和疲劳损伤度的影响。研究结果表明：中央扣对悬索桥竖弯刚度影响较小，但会提高悬索桥的纵飘刚度和反对称扭转刚度，显著减小荷载作用下缆梁相对位移和跨中短吊索的弯曲应力；中央扣不会改变脉动风荷载作用下跨中短吊索的缆梁相对运动特性，但会改变车辆荷载作用下跨中短吊索的缆梁相对运动特性，并显著降低缆梁相对位移对车速的敏感性；在脉动风和车辆荷载联合作用下，跨中短吊索的等效总应力幅值小于脉动风和车辆荷载单独作用下等效应力幅值的叠加值；在脉动风和车辆荷载联合作用下，中央扣会显著减小跨中短吊索尤其是靠近中央扣位置处吊索的等效总应力幅值和疲劳损伤度。因此，中央扣可以有效提高悬索桥跨中短吊索的疲劳寿命。 The Hangrui Dongting Bridge was taken as study object to investigate the effect of the central clamp on the fatigue damage of short suspenders at the midspan of a suspension bridge via two finite element models with and without central clamps. Firstly, the fluctuating wind speed at the bridge site was simulated using the weighted amplitude wave superposition method. The sample of random traffic flow was simulated using the Monte Carlo approach. The analysis model of wind-vehicle-bridge coupling vibration was established based on ANSYS software to obtain the dynamic responses of the bridge under the action of fluctuating wind and vehicle loads which were alone and combined. Secondly, the program of the rain-flow counting procedure was used to get the equivalent total stress amplitude, the mean stress and the cycle times of short suspenders. Finally, the effect of the central clamp on the equivalent total stress amplitude and fatigue damage of short suspenders was finally quantitatively studied based on the Miner linear cumulative damage theory. The results show that the central clamp has little effect on the vertical stiffness of the suspension bridge, but improves its longitudinal and anti-symmetrical torsional stiffness. Meanwhile, the central clamp significantly reduces the cable-girder relative movement and the bending stress of short suspenders at the midspan under loads. The central clamp does not change the cable-girder relative movement characteristic under fluctuating wind load, but it changes this characteristic under vehicle load and significantly reduces the sensitivity of cable-girder relative movement to the vehicle speed. The equivalent total stress amplitude of short suspenders at the midspan under the action of fluctuating wind and vehicle loads is smaller than the superposition value of the equivalent stress amplitudes caused by fluctuating wind load and vehicle load alone. The central clamp dramatically reduces the equivalent total stress amplitude and fatigue damage of short suspenders at the midspan under fluctuating wind and vehicle loads, particularly for short suspenders near the central clamp. Therefore, the central clamp can effectively improve the fatigue life of short suspenders at the midspan of the suspension bridge. 悬索桥 风-车-桥耦合振动 中央扣 动力特性 缆梁相对位移 疲劳损伤 suspension bridge wind-vehicle-bridge coupling vibration central clamp dynamic characteristics cable-girder relative movement fatigue damage 在悬索桥运营过程中，加劲梁直接受到恒载、风荷载和车辆荷载等的作用，作用力通过吊索传至主缆及锚碇，因此，吊索是悬索桥结构安全的重要保证之一。然而，在风荷载和车辆荷载的作用下，吊索不仅会受到波动的轴向应力，而且会受到由于加劲梁和主缆之间的相对位移而产生的弯曲变形及应力，容易发生疲劳失效 [ 1 1 ] 。 为了减小加劲梁与主缆之间的相对位移，学者们采取了中央扣等结构措施。中央扣的设计和使用可以追溯到塔科马海峡大桥的建设 [ 2 2 ] 。基于有限元分析和现场实测，WANG等 [ 3 3 ] 研究了车辆荷载对悬索桥中央扣屈曲受力的影响；王浩等 [ 4 4 - 9 9 4-9 ] 采用有限元方法研究了中央扣对大跨度悬索桥动力特性的影响，发现中央扣会加强主缆与加劲梁之间的纵向约束，从而提高结构的纵飘刚度，同时增大结构的反对称扭转刚度；彭汪虎等 [ 10 10 ] 基于连续模型推导了悬索桥的扭转基频公式，发现中央扣刚度和跨中缆梁间距是影响悬索桥反对称扭转刚度的主要因素；张寒等 [ 9 9 , 11 11 ] 研究了中央扣对脉动风荷载作用下悬索桥加劲梁位移响应的影响，认为中央扣会减小加劲梁的竖向抖振位移，但会增大加劲梁的扭转抖振位移；徐勋等 [ 5 5 ] 研究了中央扣对等间距车辆荷载作用下四渡河大桥加劲梁应力响应的影响，发现中央扣会增大加劲梁的动态应力响应；王连华等 [ 7 7 ] 研究了中央扣对车辆荷载作用下杭瑞洞庭大桥梁端纵向位移的影响，发现中央扣会减小梁端纵向位移；李光玲等 [ 8 8 ] 研究了中央扣对车辆荷载作用下悬索桥跨中短吊索疲劳损伤的影响，发现中央扣会改变缆梁间的相对运动特性，减小缆梁相对错动产生的弯曲应力，从而显著降低跨中短吊索的疲劳损伤度。 上述研究主要分析了中央扣对悬索桥动力特性、脉动风荷载或车辆荷载单独作用下结构动态响应的影响，但关于中央扣对吊索疲劳性能影响的研究非常有限，更少涉及到脉动风和车辆荷载联合作用下的吊索疲劳损伤分析。主要原因如下： 1) 风-车-桥耦合振动分析较难在商业有限元软件中直接实现，以往研究一般把车辆模型简化成集中力荷载模型，通过移动集中力荷载来近似考虑车辆对桥梁的作用，较难与风荷载联合作用； 2) 基于Fortran和C++等平台自编程求解风-车-桥系统动力学方程，该方法通用性较差。 为此，本文作者基于ANSYS平台提出了风-车-桥系统耦合振动分析的实用方法。以杭瑞洞庭大桥为工程背景，研究有中央扣和无中央扣这2种有限元模型在脉动风和车辆荷载联合作用下的吊索疲劳损伤。首先，利用谐波合成法生成所需的脉动风场和Monte-Carlo法模拟随机车流样本；然后，基于ANSYS软件建立桥梁有限元模型和车辆多刚体有限元模型 [ 12 12 ] ，得到桥梁在脉动风和车辆荷载联合作用下的动力响应；最后，基于雨流计数法和Miner损伤线形累计理论分析吊索的疲劳损伤，定量研究中央扣对吊索疲劳损伤的影响。 1 工程背景 1 杭瑞洞庭大桥为双塔双跨钢桁梁悬索桥，双向六车道，跨径布置为1 480.0+453.6 m，如 图1 图1 所示。加劲梁结构形式为板桁结合梁，桁高为 9.0 m，桁宽为35.4 m。中跨主缆矢跨比为1/10，2根主缆中心距为35.4 m，主跨和岳阳跨主缆由175束索股组成，君山跨主缆由181束索股组成。加劲梁与主缆在中跨通过87对吊索连接，吊索编号即MS1~MS87，在君山岸边跨通过25对吊索连接，吊索编号即MS88~MS112；吊索顺桥向的标准间距为16.8 m，跨中吊索间距为17.6 m，在中跨跨中设置5对斜吊索作为中央扣。索塔采用门式框架塔，共设上、下两道横梁。 数据单位：m。 2 悬索桥有限元模型和吊索疲劳损伤模型 1 2.1 　桥梁有限元模型 2 本文利用ANSYS软件建立杭瑞洞庭大桥的三维有限元模型，研究中央扣对吊索疲劳性能的影响。由于本文主要关注吊索而非加劲梁的应力响应，因此，可以提取加劲梁截面特性 [ 13 13 ] 来建立“鱼骨梁”有限元模型，减少计算量。索塔采用空间梁单元Beam188模拟，加劲梁和刚臂采用空间梁单元Beam4模拟，主缆、吊索和中央扣采用空间杆单元Link10模拟，桥面铺装和栏杆等二期恒载采用质量单元Mass21模拟。考虑设置与未设置中央扣2种缆梁连接体系的有限元模型，即无中央扣模型采用“跨中短吊索”的方式模拟，定义为FEM-A，有中央扣模型由“跨中竖向短吊索加5对斜拉索”进行模拟，定义为FEM-B，如 图2 图2 所示。吊索MS41和MS47为跨中处最靠近中央扣外侧的吊索，在荷载作用下，吊索MS41的应力响应更大。因此，选择吊索MS41作为目标吊索。 采用Block Lanczos法分别分析这2种缆梁连接体系有限元模型的结构动力特性，并与文献[ 7 7 ]中设置中央扣的有限元模型(FEM-C)进行对比。有无中央扣悬索桥的主要振型和固有频率如 表1 表1 所示。由 表1 表1 可知： 2 模态振型 FEM-A FEM-B FEM-C [ 7 7 ] 2 纵飘+竖弯 一阶 0.082 1 0.092 1 0.092 7 二阶 0.111 6 0.294 2 — 2 一阶侧弯 正对称 0.055 4 0.055 5 0.056 2 反对称 0.136 9 0.141 2 0.141 8 2 一阶竖弯 正对称 0.117 2 0.115 7 0.114 0 反对称 0.082 1 0.092 1 0.092 7 2 二阶竖弯 正对称 0.152 1 0.151 4 0.150 8 反对称 0.209 5 0.207 9 0.204 8 2 一阶扭转 正对称 0.228 2 0.228 2 0.227 4 反对称 0.270 3 0.300 9 — Hz 1) 设置中央扣后结构一阶纵飘加竖弯(即一阶反对称竖弯)的固有频率提高了12.18%，说明中央扣提高了悬索桥的纵飘刚度； 2) 对比结构的一阶和二阶竖弯振型的固有频率，发现一阶正对称竖弯、二阶正对称竖弯和二阶反对称竖弯的固有频率变化幅度较小，相对误差皆在2%以下，说明中央扣不能有效提高悬索桥的竖向刚度； 3) 设置中央扣后结构的一阶反对称扭转固有频率提高了11.32%，有利于提高桥梁结构的抗风稳定性 [ 14 14 ] ； 4) 本文FEM-B的主要固有频率与文献[ 7 7 ]FEM-C中主要固有频率的相对误差均小于2%，可用于后续的动力响应分析。 2.2 　吊索疲劳损伤模型 2 为得到脉动风荷载和车辆荷载单独或联合作用下悬索桥的应力时程响应，需要先进行静力分析，以得到结构的初始状态；然后，进行时域分析，并通过施加抖振力荷载和改变移动车辆的位置，得到吊索在脉动风和车辆荷载单独或联合作用下的应力响应。 以往研究一般仅考虑吊索的轴向位移引起的轴向应力，并以此作为吊索的总应力来进行疲劳分析。实际上，吊索在运营阶段不仅承受着波动的轴向拉伸变形的影响，还承受缆梁错动产生的弯曲变形的影响。LEE等 [ 15 15 ] 根据理论公式推导，证实了对于悬索桥的吊索，缆梁错动引起的附加轴向应力不可忽略；LIU等 [ 16 16 ] 研究了弯曲应力对跨中短吊索轴向应力的影响，发现跨中短吊索的应力主要由弯曲应力贡献，其吊索的总应力 <math id="M1"><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681495&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681492&type= 1.94733346 2.62466669 可用 式(1) 式(1) 表示： <math display="block" id="M2"><mi>σ</mi><mo>=</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">M</mi></mrow></msub><mo>=</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub><mo>+</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681552&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681551&type= 20.06599998 3.72533321 式中： <math id="M11"><msub><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">T</mi><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681569&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681555&type= 4.48733330 3.72533321 为吊索初始轴向应力。 利用雨流计数法处理吊索的应力，获得荷载作用下吊索的疲劳损伤。采用基于Miner准则的线性疲劳损伤累计理论 [ 17 17 ] ，可得变幅应力循环下结构的总疲劳损伤度 D ，即 <math display="block" id="M12"><mi>D</mi><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi></mrow></munderover></mstyle><mrow><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681574&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681570&type= 14.90133286 8.72066593 式中： <math id="M13"><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681577&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681572&type= 2.70933342 3.72533321 和 <math id="M14"><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681595&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681592&type= 3.38666677 3.72533321 分别为结构在变幅应力幅值 <math id="M15"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>σ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681582&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681581&type= 4.91066647 3.72533321 ( i =1，…， k )作用下循环次数和疲劳寿命； <math id="M16"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681586&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681584&type= 1.86266661 2.62466669 为应力幅值的总数；当总疲劳损伤度 <math id="M17"><mi>D</mi><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681605&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681603&type= 7.95866632 2.70933342 时，吊索发生疲劳破坏。 本研究中吊索材料为镀锌高强钢丝，材料的 S - N 曲线采用对数形式表达 [ 17 17 ] ，即 <math display="block" id="M18"><mi mathvariant="normal">l</mi><mi mathvariant="normal">g</mi><mi>N</mi><mo>=</mo><mi>C</mi><mo>-</mo><mi>m</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">Δ</mi><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681620&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681619&type= 24.80733490 3.38666677 式中： <math id="M19"><mi>C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681623&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681622&type= 2.62466669 2.62466669 和 <math id="M20"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681616&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681614&type= 2.70933342 2.62466669 为吊索的材料常数，取 <math id="M21"><mi>C</mi><mo>=</mo><mn mathvariant="normal">13.84</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681627&type= 14.13933277 2.62466669 ， <math id="M22"><mi>m</mi><mo>=</mo><mn mathvariant="normal">3.5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681635&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681633&type= 10.58333302 2.62466669 ； <math id="M23"><mi mathvariant="normal">Δ</mi><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681641&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681659&type= 4.23333359 2.62466669 为材料受到的应力幅值； N 为应力幅值 <math id="M24"><mi mathvariant="normal">Δ</mi><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681641&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681659&type= 4.23333359 2.62466669 作用时疲劳破坏所需的循环次数，即当应力幅值 <math id="M25"><mi mathvariant="normal">Δ</mi><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681641&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681659&type= 4.23333359 2.62466669 循环 N 次后吊索会发生疲劳破坏。 值得注意的是，为了便于比较不同风速和车速下中央扣对吊索疲劳性能的影响，本文基于疲劳损伤等效原则把吊索变幅应力循环转换为等幅应力循环，对应的等效轴向应力幅值为 <math id="M26"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681646&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681643&type= 7.78933382 3.72533321 和等效总应力幅值为 <math id="M27"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681664&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681663&type= 6.43466663 3.72533321 分别按 式(5) 式(5) 和 式(6) 式(6) 计算： <math display="block" id="M28"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi><mi mathvariant="normal">T</mi></mrow></msub><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><mo>∑</mo><mfenced separators="|"><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>⋅</mo><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">T</mi><mi>i</mi></mrow><mrow><mi>m</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mo>∑</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mi>m</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681669&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681666&type= 37.59199905 12.86933327 <math display="block" id="M29"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">q</mi></mrow></msub><mo>=</mo><msup><mrow><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><mo>∑</mo><mfenced separators="|"><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>⋅</mo><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>m</mi></mrow></msubsup></mrow></mfenced></mrow><mrow><mo>∑</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn><mo>/</mo><mi>m</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681661&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681673&type= 36.15266418 12.86933327 3 脉动风作用下中央扣对悬索桥响应的影响 1 3.1 　脉动风场的模拟 2 假设不同方向脉动风速满足一维平稳正态高斯随机过程，采用谐波合成法模拟脉动风场。桥梁的计算风速、湍流度、功率谱密度函数和脉动风的相干函数均按JTG/T 3360-01—2018《公路桥梁抗风设计规范》 [ 18 18 ] 取值，具体参数如 表2 表2 所示。 参数 数值 地面粗糙度指数 <math id="M30"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681679&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681678&type= 1.77800000 2.37066650 0.12 地面粗糙高度 <math id="M31"><msub><mrow><mi>z</mi></mrow><mrow><mn>0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681683&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681681&type= 2.70933342 3.38666677 /m 0.01 加劲梁处平均风 U/ (m∙s -1 ) 5~20 模拟时间步长 <math id="M32"><mi mathvariant="normal">Δ</mi><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681696&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681684&type= 3.13266683 2.37066650 /s 0.1 初始圆频率 <math id="M33"><msub><mrow><mi>ω</mi></mrow><mrow><mn>0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681690&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681688&type= 3.64066648 3.38666677 /(rad∙s -1 ) 0 截止圆频率 <math id="M34"><msub><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">u</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681701&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681699&type= 3.64066648 3.38666677 /(rad∙s -1 ) <math id="M35"><mn>10</mn><mi mathvariant="normal">π</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681711&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681710&type= 5.07999992 2.37066650 频率等分数 N /个 2 048 模拟总持续时间 T /s 600 在风场模拟过程中，顺风向和竖向脉动风功率谱函数分别采用Kaimal谱和Panofsky谱，采用Davenport相干函数表示空间不同位置2个点的相干性。平均风速为15 m/s时，加劲梁跨中处模拟顺风向和竖向脉动风速时程如 图3 图3 所示。为验证模拟风场的可靠性，采用功率谱进行检验，结果如 图4 图4 所示(其中， f = nZ / U ， n 为脉动风的频率； Z 为主梁离水面以上的高度； U 为高度 Z 处的平均风速)。由 图4 图4 可见：模拟风速时程样本的功率谱与目标谱吻合较好，模拟的脉动风场可用于后续的计算。 (a) 顺风向脉动风速；(b) 竖向脉动风速 (a) 顺风向脉动风速；(b) 竖向脉动风速 3.2 　风-桥系统动力学方程 2 在桥梁抖振分析中，脉动风荷载作用下桥梁结构的动力学方程可表达为 <math display="block" id="M36"><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mover accent="true"><mrow><mi mathvariant="bold-italic">u</mi></mrow><mi mathvariant="bold-italic">¨</mi></mover></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></mrow></mfenced><mo>+</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">u</mi><mo>˙</mo></mover></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></mrow></mfenced><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681718&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681733&type= 36.32199860 4.74133301 <math display="block" id="M37"><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></mrow></mfenced><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">g</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">b</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">b</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">b</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681722&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681720&type= 53.25533295 5.67266655 式中： <math id="M38"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681735&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681739&type= 4.74133301 3.72533321 、 <math id="M39"><msub><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681744&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681742&type= 3.80999994 3.72533321 和 <math id="M40"><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681757&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681755&type= 3.89466691 3.72533321 分别为桥梁的质量、阻尼和刚度矩阵； <math id="M41"><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681761&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681759&type= 3.38666677 3.72533321 、 <math id="M42"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">u</mi><mo>˙</mo></mover></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681764&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681763&type= 3.38666677 3.80999994 和 <math id="M43"><msub><mrow><mover accent="true"><mrow><mi mathvariant="bold-italic">u</mi></mrow><mi mathvariant="bold-italic">¨</mi></mover></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681766&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681765&type= 3.38666677 3.80999994 分别为桥梁的位移、速度和加速度向量； <math id="M44"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">g</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681769&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681750&type= 4.91066647 3.72533321 为桥梁的自重荷载； <math id="M45"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681785&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681783&type= 5.24933338 3.72533321 、 <math id="M46"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681772&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681771&type= 6.01133299 3.72533321 和 <math id="M47"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681789&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681788&type= 5.67266655 3.72533321 分别为作用于桥梁上的静风荷载、抖振力荷载和自激力荷载。 作用于桥梁结构上的风荷载主要分为平均风产生的静风荷载、脉动风分量产生的抖振力荷载和风桥相互作用产生的自激力荷载。根据Scanlan的准定常气动力理论计算抖振力(即气动导纳函数取1)，气动自激力根据Scanlan的自激力表达式计算并采用Matrix27单元进行模拟 [ 19 19 ] ，其中，气动导数取自陈政清的试验结果 [ 20 20 ] 。 3.3 　中央扣对悬索桥响应的影响 2 基于建立的悬索桥有限元模型和模拟的脉动风场，采用ANSYS的瞬态动力学分析模块进行有、无中央扣下桥梁抖振的时域分析，获得了加劲梁跨中竖向抖振位移响应和吊索MS41的缆梁相对位移与应力响应。以平均风速为15 m/s时的抖振响应为例，跨中加劲梁竖向抖振位移和缆梁相对位移曲线如 图5 图5 所示。由 图5 图5 可知：FEM-A的跨中竖向抖振位移RMS(即均方根)为0.042 m，FEM-B的竖向抖振位移RMS为0.051 m，较FEM-A增大21.43%，设置中央扣会增大加劲梁跨中竖向抖振位移响应；设置中央扣会大幅减小脉动风作用下吊索MS41处缆梁相对位移，同时由于脉动风的随机性，缆梁相对位移表现出零均值波动状态；中央扣并未改变脉动风作用下缆梁相对运动特性，有、无中央扣悬索桥皆主要以一阶纵飘振型波动。 (a) 跨中竖向位移；(b) 吊索MS41处缆梁相对位移 基于获得的桥梁结构抖振时域分析数据，进一步研究中央扣对悬索桥跨中短吊索应力的影响。当平均风速为15 m/s时，在脉动风作用下，吊索MS41的应力时程如 图6 图6 所示。由 图6 图6 可知：FEM-A的吊索轴向应力RMS为0.824 MPa，FEM-B的轴向应力RMS为1.015 MPa，较FEM-A增大23.18%，设置中央扣会增大脉动风作用下跨中短吊索的轴向应力响应；设置中央扣可以显著减小脉动风作用下跨中短吊索的弯曲应力幅值；另外，FEM-A的吊索弯曲应力远大于轴向应力，因此，弯曲应力所叠加的附加轴向应力不可忽略。同时，吊索MS41的弯曲应力曲线与缆梁相对位移曲线形状类似，主要原因如下：由 式(1) 式(1) 可知吊索的弯曲应力主要与缆梁相对位移及轴向应力相关，而吊索初始轴向应力较大，脉动风引起的轴向应力变化幅值 <math id="M48"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681795&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681793&type= 5.67266655 3.72533321 相对较小，吊索轴向应力变化小，所以，吊索弯曲应力几乎仅与缆梁相对位移相关。 (a) 轴向应力幅值；(b) 弯曲应力 为研究不同风速下中央扣对跨中短吊索等效应力幅值的影响，对平均风速分别为5、10、15和20 m/s时悬索桥吊索的抖振应力响应进行分析。以吊索MS37~MS41为研究对象，不同风速下FEM-A和FEM-B的吊索等效应力幅值如 图7 图7 所示。由 图7 图7 可知：设置中央扣会增大脉动风作用下跨中短吊索的等效轴向应力幅值，且风速越高，应力幅值增大越显著；设置中央扣可以显著减小脉动风作用下跨中短吊索的等效总应力幅值，当风速为 15 m/s时，FEM-A的吊索MS41的等效总应力幅 值为73.07 MPa，FEM-B的等效总应力幅值为 1.40 MPa，较FEM-A减小98.08%。因此，若不考虑吊索的弯曲应力引起的应力幅值，则会过低估计脉动风作用下跨中短吊索的等效总应力幅值。 (a) FEM-A和FEM-B吊索等效轴向应力幅值； (b) FEM-A吊索等效总应力幅值； (c) FEM-B吊索等效总应力幅值 4 车辆荷载作用下中央扣对悬索桥响应的影响 1 4.1 　车辆模型 2 以两轴箱式货车 [ 21 21 ] 作为研究对象，两轴四轮车辆模型由1个车体、4个车轮、一系列弹簧和阻尼组成。假定车体在路面行驶过程中车轮与桥面始终不分离且不发生侧倾，悬挂系统以及车轮的质量均集中在车轮质量块上，不考虑弹簧和阻尼装置的质量。车体具有竖向 <math id="M49"><msub><mrow><mi>Z</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681817&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681815&type= 4.14866638 3.72533321 、侧向 <math id="M50"><msub><mrow><mi>Y</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">r</mi></mrow></msub></math> 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stretchy="false">(</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">R</mi><mi mathvariant="normal">i</mi></mrow><mrow><mi mathvariant="normal">y</mi></mrow></msubsup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681843&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681841&type= 14.39333248 4.82600021 分别为第 i 轴左侧(右侧)上悬挂系统和下悬挂系统的竖向及横向弹簧刚度； <math id="M59"><msubsup><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">u</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">i</mi></mrow><mrow><mi mathvariant="normal">v</mi></mrow></msubsup><mo stretchy="false">(</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">u</mi><mi mathvariant="normal">R</mi><mi mathvariant="normal">i</mi></mrow><mrow><mi mathvariant="normal">v</mi></mrow></msubsup><mo stretchy="false">)</mo></math> 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681867&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681865&type= 13.88533401 4.82600021 分别为第 i 轴左侧(右侧)上悬挂系统和下悬挂系统的竖向及横向阻尼； L i ( i =1，2)为第 i 轴到车体质心的距离； b 1 为车轮横向间距的一半； h 1 为车体质心至中央弹簧上平面的垂直高度； h vr 为车体质心到地面的距离。 (a) 侧面图；(b) 断面图 4.2 　随机车流样本的建立 2 桥梁上的车辆荷载具有随机性，杨晓艳 [ 22 22 ] 通过研究发现任意时刻桥上车辆的分布也是服从一定函数的随机分布。为了分析不同车辆荷载类型对跨中短吊索疲劳损伤的影响，采用3种典型车辆荷载类型 [ 23 23 ] 进行分析，并建立随机车流概率分布模型。在模拟车辆过桥时，一般假设车辆的行驶速度恒定，通过调整车间距来考虑车速时刻变化的影响。基于随机车流概率分布模型和车速恒定假定，采用Monte-Carlo法生成的随机车流样本，如 图9 图9 所示。 4.3 　车-桥系统动力学方程 2 假设车轮与桥面始终接触，可知车辆轮胎与桥面接触点具有相同的位移和接触力协调关系，并考虑桥面粗糙度的影响，可得车-桥耦合系统的动力学方程： <math display="block" id="M63"><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><msub><mrow><mi 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681905&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681856&type= 3.80999994 3.72533321 和 <math id="M67"><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi mathvariant="normal">v</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681908&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681892&type= 3.89466691 3.72533321 分别为车辆的质量、阻尼和刚度矩阵； <math id="M68"><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mi mathvariant="normal">v</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681895&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681912&type= 3.38666677 3.72533321 、 <math id="M69"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">u</mi><mo>˙</mo></mover></mrow><mrow><mi mathvariant="normal">v</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681917&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681896&type= 3.38666677 3.80999994 和 <math id="M70"><msub><mrow><mover accent="true"><mrow><mi mathvariant="bold-italic">u</mi></mrow><mi mathvariant="bold-italic">¨</mi></mover></mrow><mrow><mi mathvariant="normal">v</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681901&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681899&type= 3.38666677 3.80999994 分别为车辆的位移、速度和加速度向量； <math id="M71"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">v</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681923&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681920&type= 4.91066647 3.72533321 为车辆作用于桥梁的接触力荷载； <math id="M72"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">g</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681929&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681934&type= 4.91066647 3.72533321 和 <math id="M73"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681948&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681930&type= 4.91066647 3.72533321 分别为车辆的自重荷载和桥梁作用于车辆上的接触力荷载；下标b和v分别代表桥梁和车辆2个子系统。 本文基于大型通用平台ANSYS建立了桥梁车-桥耦合振动分析框架，根据分离迭代法的原理，建立每一时刻的车-桥位移协调与相互作用力的平衡关系，求解车-桥耦合动力时程响应 [ 12 12 ] 。求解过程如下： 1) 采用ANSYS软件建立桥梁结构的三维空间有限元模型。 2) 输入随机车流样本的数据，根据车道位置、数量、车速和车辆参数等，建立车辆多刚体有限元模型，形成车流样本。 3) 输入路面不平顺样本。 4) 根据车桥接触点位移协调原理，结合ANSYS约束方程功能，由梁单元的位移形函数建立任意时刻车轮与桥面接触点的位移约束条件。更新约束方程中车桥接触点的路面不平顺，同时自动满足车桥接触点力的平衡方程。 5) 通过对车辆多刚体有限元模型进行移动，建立每一时刻车轮与所在桥面接触点的位移约束条件，可以较简便地对车流过桥状态下的车-桥耦合动力时程进行分析。 4.4 　单车荷载作用下中央扣对悬索桥位移响应的影响 2 1辆标准的两轴箱式货车以80 km/h匀速过桥，路面粗糙度采用“非常好”等级，单辆箱式货车行驶在该桥上时加劲梁跨中竖向位移曲线如 图10 图10 (a)所示。由 图10 图10 (a)可知：中央扣对单车荷载作用下加劲梁跨中竖向位移响应影响较小，也验证了前述“中央扣对结构竖弯刚度影响较小”的结论。 (a) 跨中竖向位移；(b) 吊索MS41处缆梁相对位移 单车荷载作用下FEM-A和FEM-B的吊索MS41的缆梁相对位移时程曲线如 图10 图10 (b)所示。由 图10 图10 (b)可知：未设置中央扣时，吊索MS41缆梁相对位移时程曲线呈现波动状态，且波动幅值也在不断变化；设置中央扣后，吊索MS41缆梁相对位移以主跨为中心呈现“S”形曲线，即加劲梁以中跨跨中为中心发生整体运动；当车辆运行到边跨时，FEM-A的MS41缆梁相对位移仍出现较大幅度的波动，而FEM-B的MS41缆梁相对位移趋近于零。因此，中央扣主要改变了单车荷载作用下悬索桥缆梁相对运动特性，减小了缆梁错动循环次数和位移幅值，且减小了边跨行车对主跨吊索缆梁相对位移的影响，但对加劲梁竖向位移的影响较小，这与文献[ 8 8 ]的结论一致。另外，文献[ 24 24 ]在ANSYS中通过移动集中力荷载来近似考虑车辆对桥梁的作用，其缆梁相对运动特性与本文以及文献[ 8 8 ]中的结果有所不同，会对后续的计算结果造成误差，这也证明对车-桥耦合振动进行分析的必要性，同时验证了本文计算方法的可靠性。 为研究不同车速下中央扣对缆梁相对位移的影响，选取1辆标准的两轴箱式货车分别以40、60、80和100 km/h匀速过桥，FEM-A和FEM-B吊索MS41的缆梁相对位移时程曲线见 图11 图11 。由 图 11 图 11 (a)可知：FEM-A吊索MS41处的缆梁相对位移时程曲线在不同车速下均呈现不同幅度的波动状态，位移幅值随车速增加而增大，且2个波峰之间的距离也与车速呈正比。由 图11 图11 (b)可知：设置中央扣后，FEM-B吊索MS41处的缆梁相对位移随车速增加基本不变，均呈现“S”形，且最小值的位置均出现在中跨跨中附近，说明中央扣可以有效降低缆梁相对位移对车辆速度的敏感性。 (a) FEM-A；(b) FEM-B 4.5 　随机车流荷载作用下中央扣对悬索桥响应的影响 2 基于模拟的随机车流样本加载车辆位置，进行随机车流-桥梁耦合动力时程分析，研究中央扣对随机车流荷载作用下悬索桥位移响应的影响。假定车流样本以80 km/h匀速过桥，加劲梁跨中竖向位移和吊索MS41处缆梁相对位移时程曲线如 图12 图12 所示。由 图12 图12 (a)可知：设置中央扣对随机车流荷载作用下悬索桥加劲梁跨中竖向位移的影响较小；由 图12 图12 (b)可知：设置中央扣可以显著减小随机车流荷载作用下跨中短吊索MS41处缆梁相对位移，对悬索桥缆梁错动起到较好的抑制效果。 (a) 跨中竖向位移；(b) 吊索MS41处缆梁相对位移 图13 图13 所示为随机车流荷载作用下跨中短吊索MS41的应力时程曲线。由 图13 图13 (a)可知：设置中央扣对随机车流荷载作用下跨中短吊索轴向应力幅值的影响较小，甚至可以忽略。由 图13 图13 (b)可知：设置中央扣可以显著减小随机车流荷载作用下跨中短吊索的弯曲应力，且弯曲应力时程曲线与缆梁相对位移时程曲线基本一致。 (a) 轴向应力幅值；(b) 弯曲应力 以吊索MS37~MS41为研究对象，研究随机车流荷载作用时不同车速下中央扣对短吊索疲劳损伤的影响。不同车速下FEM-A和FEM-B吊索的等效应力幅值如 图14 图14 所示。由 图14 图14 可知：设置中央扣会增大随机车流荷载作用下跨中短吊索的等效轴向应力幅值，且车速越高，增大越显著，这与文献[ 8 8 ]的结果一致。设置中央扣可以显著减小随机车流荷载作用下跨中短吊索的等效总应力幅值，当车速为80 km/h时，FEM-A吊索MS41的等效总应力幅值为28.52 MPa，FEM-B的等效总应力幅值为5.62 MPa，较FEM-A减小80.29%。因此，若不考虑吊索的弯曲应力引起的应力幅值，会过低估计随机车流荷载作用下跨中短吊索的等效总应力幅值。 (a) FEM-A和FEM-B吊索等效轴向应力幅值； (b) FEM-A吊索等效总应力幅值； (c) FEM-B吊索等效总应力幅值 5 脉动风和随机车辆荷载联合作用下吊索的疲劳损伤 1 5.1 　风-车-桥系统动力学方程 2 风-车-桥耦合振动系统中汽车和桥梁2个子系统间通过位移和接触力协调关系求解，可得风-车-桥耦合系统的动力学方程为 <math display="block" id="M74"><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">b</mi></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi mathvariant="normal">v</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mrow><mi mathvariant="bold-italic">u</mi></mrow><mi mathvariant="bold-italic">¨</mi></mover></mrow><mrow><mi 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mathvariant="normal">b</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mi mathvariant="normal">v</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">g</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">b</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">b</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">b</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">b</mi><mi mathvariant="normal">v</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">g</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">v</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">v</mi><mi mathvariant="normal">b</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681988&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681986&type= 68.91867065 8.21266651 式中： <math id="M76"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">v</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681972&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681970&type= 5.24933338 3.72533321 为作用于车辆上的风荷载，包括静风荷载和抖振力荷载。 5.2 　跨中短吊索的疲劳等效应力幅值和疲劳损伤 2 为了进一步分析脉动风和随机车辆荷载联合作用下中央扣对跨中短吊索疲劳性能的影响，建立风-车-桥耦合分析模型来实现不同风速和车速下风-车-桥耦合系统动力学方程的求解。 图15 图15 所示为不同风速和车速工况下FEM-A和FEM-B中吊索MS41的等效总应力幅值。由 图15 图15 可知：不设置中央扣时吊索MS41的等效总应力幅值随着风速增加而增大，对风速敏感；等效总应力幅值随车速增加而增大，尤其是低风速时。设置中央扣后吊索MS41的等效总应力幅值随着风速增加而缓慢增大，对风速不敏感；等效总应力幅值随车速增加而增大。中央扣会显著降低脉动风和随机车辆荷载联合作用下跨中短吊索等效总应力幅值。另外，脉动风(风速15 m/s)和随机车辆荷载(车速80 km/h)单独作用时FEM-A中吊索MS41的等效总应力幅值分别为73.07 MPa(如 图7 图7 (b)所示)和28.52 MPa(如 图14 图14 (b)所示)，其叠加值为101.59 MPa；当脉动风(风速15 m/s)和随机车辆荷载(车速80 km/h)单独作用时，FEM-B中吊索MS41的等效总应力幅值分别为1.40 MPa(如 图7 图7 (c)所示)和5.62 MPa(如 图14 图14 (c)所示)，其叠加值为7.02 MPa。在相同风速和车速联合作用下，FEM-A中吊索MS41的等效总应力幅值为58.88 MPa，较叠加值(101.59 MPa)减小42.04%；联合作用时FEM-B的等效总应力幅值为5.86 MPa，较叠加值(7.02 MPa)减小16.52%。说明脉动风和车辆荷载联合作用下的等效总应力幅值小于其单独作用下等效应力幅值的叠加值，这也表明进行风-车-桥耦合振动分析的必要性。 (a) FEM-A；(b) FEM-B 基于谷峰法原理和雨流计数法 [ 25 25 ] 处理脉动风和随机车辆荷载联合作用下吊索MS41的应力时程响应，得到其应力幅值、应力均值和应力循环次数。鉴于实际桥梁通车时风速普遍较小，以风速 5 m/s和车速80 km/h为例，FEM-A和FEM-B短吊索MS41总应力幅值分布如 图16 图16 所示。由 图16 图16 可知：设置中央扣会显著降低脉动风和随机车辆荷载联合作用下吊索MS41的应力幅值和应力均值，但值得注意的是，FEM-B吊索的应力循环次数多于FEM-A吊索的应力循环次数。由 式(3) 式(3) 可知，吊索的疲劳损伤度与应力幅值和应力循环次数有关，故需进一步分析中央扣对跨中短吊索疲劳损伤度的影响。 (a) FEM-A；(b) FEM-B 在脉动风和车辆荷载联合作用下，FEM-A和FEM-B跨中短吊索MS37~MS41的等效总应力幅值和疲劳损伤如 表3 表3 所示。由 表3 表3 可知：设置中央扣可以显著降低脉动风和随机车辆荷载联合作用下短吊索的等效总应力幅值和疲劳损伤度，尤其是靠近中央扣位置的短吊索，设置中央扣后吊索MS41的等效总应力幅值降低了83.03%，疲劳损伤度降低了99.38%。因此，在脉动风和车辆荷载联合作用下，中央扣对悬索桥跨中短吊索的疲劳损伤具有较好的抑制效果，可以为悬索桥的设计、养护和运营提供重要参考。 2 吊索编号 2 吊索长度/m 2 等效总应力幅值/MPa 2 疲劳损伤度 <math id="M77"><mi mathvariant="normal">Δ</mi><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681999&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681998&type= 3.80999994 2.37066650 FEM-A <math id="M78"><mi mathvariant="normal">Δ</mi><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681999&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43681998&type= 3.80999994 2.37066650 FEM-B <math id="M79"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43682003&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43682002&type= 2.53999996 2.37066650 FEM-A <math id="M80"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43682003&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43682002&type= 2.53999996 2.37066650 FEM-B MS37 10.43 19.33 5.24 1.58×10 -7 3.50×10 -9 MS38 9.32 22.24 5.35 2.53×10 -7 3.83×10 -9 MS39 8.35 24.77 5.79 3.56×10 -7 5.04×10 -9 MS40 7.54 27.89 5.63 5.03×10 -7 4.58×10 -9 MS41 6.88 32.88 5.58 7.19×10 -7 4.46×10 -9 6 结论 1 1) 中央扣可以提高悬索桥的纵飘刚度和反对称扭转刚度，但对竖弯刚度的影响较小。 2) 在脉动风荷载作用下，中央扣可以大幅度减小悬索桥缆梁相对位移和跨中短吊索弯曲应力，但不会改变缆梁相对运动特性；同时，中央扣会稍微增大跨中短吊索的轴向应力，但显著减小跨中短吊索的等效总应力幅值。 3) 在车辆荷载作用下，中央扣可以大幅度减小悬索桥缆梁相对位移、跨中短吊索弯曲应力和等效总应力幅值；同时，中央扣会改变缆梁相对运动特性，有效降低缆梁相对位移对车速的敏感性。 4) 在脉动风和车辆荷载联合作用下，中央扣会显著减小跨中短吊索等效总应力幅值和疲劳损伤度，有效提高跨中短吊索尤其是靠近中央扣位置处短吊索的疲劳寿命。脉动风和车辆荷载联合作用下的等效总应力幅值小于脉动风和车辆荷载单独作用下等效应力幅值的叠加值。 参考文献 KAWAI Y , SIRINGORINGO D , FUJINO Y . Failure analysis of the hanger clamps of the Kutai-Kartanegara bridge from the fracture mechanics viewpoint [J]. Journal of JSCE , 2014 , 2 ( 1 ): 1 - 6 . Failure analysis of the hanger clamps of the Kutai-Kartanegara bridge from the fracture mechanics viewpoint VIOLA J M , SYED S , CLENANCE J . The new Tacoma narrows suspension bridge: construction support and engineering [C]// Structures Congress 2005 . New York, USA . Reston, VA, USA : American Society of Civil Engineers , 2005 : 1 - 12 . The new Tacoma narrows suspension bridge: construction support and engineering WANG Hao , LI Aiqun , GUO Tong , et al . Accurate stress analysis on rigid central buckle of long-span suspension bridges based on submodel method [J]. Science in China Series E: Technological Sciences , 2009 , 52 ( 4 ): 1019 - 1026 . Accurate stress analysis on rigid central buckle of long-span suspension bridges based on submodel method 王浩 , 李爱群 , 杨玉冬 , 等 . 中央扣对大跨悬索桥动力特性的影响 [J]. 中国公路学报 , 2006 , 19 ( 6 ): 49 - 53 . 中央扣对大跨悬索桥动力特性的影响 WANG Hao , LI Aiqun , YANG Yudong , et al . Influence of central buckle on dynamic behavior of long-span suspension bridge [J]. China Journal of Highway and Transport , 2006 , 19 ( 6 ): 49 - 53 . Influence of central buckle on dynamic behavior of long-span suspension bridge 徐勋 , 强士中 , 贺拴海 . 中央扣对大跨悬索桥动力特性和汽车车列激励响应的影响 [J]. 中国公路学报 , 2008 , 21 ( 6 ): 57 - 63 . 中央扣对大跨悬索桥动力特性和汽车车列激励响应的影响 XU Xun , QIANG Shizhong , HE Shuanhai . Influence of central buckle on dynamic behavior and response of long-span suspension bridge under vehicle group excitation [J]. China Journal of Highway and Transport , 2008 , 21 ( 6 ): 57 - 63 . Influence of central buckle on dynamic behavior and response of long-span suspension bridge under vehicle group excitation HU Tengfei , HUA Xugang , ZHANG Wuwei , et al . Influence of central buckles on the modal characteristics of long-span suspension bridge [J]. Journal of Highway and Transportation Research and Development , 2016 , 10 ( 1 ): 72 - 77 . Influence of central buckles on the modal characteristics of long-span suspension bridge 王连华 , 孙璋鸿 , 崔剑锋 , 等 . 车辆激励下中央扣对悬索桥梁端位移的影响 [J]. 湖南大学学报(自然科学版) , 2019 , 46 ( 3 ): 18 - 24 . 车辆激励下中央扣对悬索桥梁端位移的影响 WANG Lianhua , SUN Zhanghong , CUI Jianfeng , et al . Effects of central buckle on end displacement of suspension bridges under vehicle excitation [J]. Journal of Hunan University(Natural Sciences) , 2019 , 46 ( 3 ): 18 - 24 . Effects of central buckle on end displacement of suspension bridges under vehicle excitation 李光玲 , 苏权科 , 高文博 , 等 . 中央扣对悬索桥动力特性及短吊索车载激励响应的影响 [J]. 中国公路学报 , 2021 , 34 ( 4 ): 174 - 186 . 中央扣对悬索桥动力特性及短吊索车载激励响应的影响 LI Guangling , SU Quanke , GAO Wenbo , et al . Influence of rigid central clamps on dynamic characteristics of suspension bridge and vehicle excitation responses of short suspenders [J]. China Journal of Highway and Transport , 2021 , 34 ( 4 ): 174 - 186 . Influence of rigid central clamps on dynamic characteristics of suspension bridge and vehicle excitation responses of short suspenders 张寒 , 王浩 , 姚程渊 , 等 . 中央扣对大跨度三塔悬索桥抖振性能的影响 [J]. 哈尔滨工程大学学报 , 2021 , 42 ( 3 ): 382 - 387 . 中央扣对大跨度三塔悬索桥抖振性能的影响 ZHANG Han , WANG Hao , YAO Chengyuan , et al . Influence of the central buckle on the buffeting performance of a long-span triple-tower suspension bridge [J]. Journal of Harbin Engineering University , 2021 , 42 ( 3 ): 382 - 387 . Influence of the central buckle on the buffeting performance of a long-span triple-tower suspension bridge 彭旺虎 , 邵旭东 . 设置中央扣悬索桥的扭转自振分析 [J]. 中国公路学报 , 2013 , 26 ( 5 ): 76 - 87 . 设置中央扣悬索桥的扭转自振分析 PENG Wanghu , SHAO Xudong . Analysis of free torsional vibration of suspension bridges with center ties [J]. China Journal of Highway and Transport , 2013 , 26 ( 5 ): 76 - 87 . Analysis of free torsional vibration of suspension bridges with center ties 王浩 , 李爱群 . 中央扣对大跨度悬索桥风致抖振响应的影响 [J]. 土木工程学报 , 2009 , 42 ( 7 ): 78 - 84 . 中央扣对大跨度悬索桥风致抖振响应的影响 WANG Hao , LI Aiqun . Influence of central buckle on wind-induced buffeting response of long-span suspension bridges [J]. China Civil Engineering Journal , 2009 , 42 ( 7 ): 78 - 84 . Influence of central buckle on wind-induced buffeting response of long-span suspension bridges 刘世忠 . 双层公路钢桁梁桥车桥耦合振动研究 [D]. 西安 : 长安大学 , 2015 : 36 - 81 . 双层公路钢桁梁桥车桥耦合振动研究 LIU Shizhong . Research on vehicle-bridge coupled vibration of double-deck highway steel truss bridge [D]. Xi'an : Changan University , 2015 : 36 - 81 . Research on vehicle-bridge coupled vibration of double-deck highway steel truss bridge 祝卫亮 , 葛耀君 . 悬索桥桁架加劲梁动力等效成等截面欧拉梁方法 [J]. 哈尔滨工业大学学报 , 2020 , 52 ( 9 ): 17 - 24 . 悬索桥桁架加劲梁动力等效成等截面欧拉梁方法 ZHU Weiliang , GE Yaojun . Simplified method for dynamic equivalent Euler beam of truss girder in suspension bridge [J]. Journal of Harbin Institute of Technology , 2020 , 52 ( 9 ): 17 - 24 . Simplified method for dynamic equivalent Euler beam of truss girder in suspension bridge 曾德礼 . 柔性中央扣对单跨悬索桥的影响分析 [J]. 铁道建筑 , 2019 , 59 ( 11 ): 16 - 18 . 柔性中央扣对单跨悬索桥的影响分析 ZENG Deli . Analysis on influence of flexible central buckle on single span suspension bridge [J]. Railway Engineering , 2019 , 59 ( 11 ): 16 - 18 . Analysis on influence of flexible central buckle on single span suspension bridge LEE M , KIM H K . Angular change and secondary stress in main cables of suspension bridges [J]. International Journal of Steel Structures , 2016 , 16 ( 2 ): 573 - 585 . Angular change and secondary stress in main cables of suspension bridges LIU Zhongxiang , GUO Tong , HEBDON M H , et al . Corrosion fatigue analysis and reliability assessment of short suspenders in suspension and arch bridges [J]. Journal of Performance of Constructed Facilities , 2018 , 32 ( 5 ): 04018060 . Corrosion fatigue analysis and reliability assessment of short suspenders in suspension and arch bridges 崔晨星 , 宋力 , 陈桂香 , 等 . 碳化与列车荷载耦合作用下城市轨道交通U型梁腐蚀疲劳寿命评估 [J]. 中南大学学报(自然科学版) , 2022 , 53 ( 2 ): 747 - 755 . 碳化与列车荷载耦合作用下城市轨道交通U型梁腐蚀疲劳寿命评估 CUI Chenxing , SONG Li , CHEN Guixiang , et al . Corrosion-fatigue life assessment of U-typed beam in urban rail transit under coupled action of carbonation and train loads [J]. Journal of Central South University(Science and Technology) , 2022 , 53 ( 2 ): 747 - 755 . Corrosion-fatigue life assessment of U-typed beam in urban rail transit under coupled action of carbonation and train loads JTG/T 3360-01—2018 . 公路桥梁抗风设计规范 [S]. JTG/T 3360-01—2018 . Wind-resistant design specification for highway bridges [S]. 曾宪武 , 韩大建 . 大跨桥梁风致抖振时域分析及在ANSYS中的实现 [J]. 桥梁建设 , 2004 , 34 ( 1 ): 9 - 12 . 大跨桥梁风致抖振时域分析及在ANSYS中的实现 ZENG Xianwu , HAN Dajian . Time-domain analysis of wind-induced buffet on long-span bridges and implementation of analysis in ANSYS [J]. Bridge Construction , 2004 , 34 ( 1 ): 9 - 12 . Time-domain analysis of wind-induced buffet on long-span bridges and implementation of analysis in ANSYS 陈政清 . 大跨径板桁结合型加劲梁悬索桥抗风性能研究 [R]. 长沙 : 湖南大学 , 2018 : 142 - 147 . 大跨径板桁结合型加劲梁悬索桥抗风性能研究 CHEN Zhengqing . Study on wind-resistance performance of large-span suspension bridges with plate-truss combined stiffened deck [R]. Changsha : Hunan University , 2018 : 142 - 147 . Study on wind-resistance performance of large-span suspension bridges with plate-truss combined stiffened deck 韩艳 , 李浩 , 薛繁荣 . 基于车桥耦合的钢管混凝土拱桥车桥冲击系数研究 [J]. 铁道科学与工程学报 , 2016 , 13 ( 3 ): 493 - 499 . 基于车桥耦合的钢管混凝土拱桥车桥冲击系数研究 HAN Yan , LI Hao , XUE Fanrong . Research on the impact coefficient of the concrete filled steel tube arch bridge based on the vehicle-bridge coupled vibration [J]. Journal of Railway Science and Engineering , 2016 , 13 ( 3 ): 493 - 499 . Research on the impact coefficient of the concrete filled steel tube arch bridge based on the vehicle-bridge coupled vibration 杨晓艳 . 公路桥梁车辆荷载及可靠度研究 [D]. 大连 : 大连理工大学 , 2018 : 30 - 46 . 公路桥梁车辆荷载及可靠度研究 YANG Xiaoyan . Studies on vehicle loads of highway bridge and reliability [D]. Dalian : Dalian University of Technology , 2018 : 30 - 46 . Studies on vehicle loads of highway bridge and reliability 韩万水 . 风-汽车-桥梁系统空间耦合振动研究 [D]. 上海 : 同济大学 , 2006 : 40 - 56 . 风-汽车-桥梁系统空间耦合振动研究 HAN Wanshui . Three-dimensional coupling vibration of wind-vehicle-bridge system [D]. Shanghai : Tongji University , 2006 : 40 - 56 . Three-dimensional coupling vibration of wind-vehicle-bridge system LIU Zhongxiang , GUO Tong , HUANG Lingyu , et al . Fatigue life evaluation on short suspenders of long-span suspension bridge with central clamps [J]. Journal of Bridge Engineering , 2017 , 22 ( 10 ): 04017074 . Fatigue life evaluation on short suspenders of long-span suspension bridge with central clamps 鲁乃唯 , 刘扬 , 邓扬 . 随机车流作用下悬索桥钢桥面板疲劳损伤与寿命评估 [J]. 中南大学学报(自然科学版) , 2015 , 46 ( 11 ): 4300 - 4306 . 随机车流作用下悬索桥钢桥面板疲劳损伤与寿命评估 LU Naiwei , LIU Yang , DENG Yang . Fatigue damage and life assessment for steel decks of suspension bridge under stochastic traffic flow [J]. Journal of Central South University(Science and Technology) , 2015 , 46 ( 11 ): 4300 - 4306 . Fatigue damage and life assessment for steel decks of suspension bridge under stochastic traffic flow 10.11817/j.issn.1672-7207.2023.04.017 U448.27 A 1672-7207（2023）04-1393-14 严磊, 李嘉隆, 何旭辉. 脉动风和车辆荷载联合作用下中央扣对悬索桥吊索疲劳损伤的影响[J]. 中南大学学报(自然科学版), 2023, 54(4): 1393-1406. 严磊, 李嘉隆, 何旭辉. 脉动风和车辆荷载联合作用下中央扣对悬索桥吊索疲劳损伤的影响[J]. 中南大学学报(自然科学版), 2023, 54(4): 1393-1406. YAN Lei, LI Jialong, HE Xuhui. Effect of central clamps on fatigue damage of suspenders of suspension bridge under joint action of fluctuating wind and vehicle loads[J]. Journal of Central South University(Science and Technology), 2023, 54(4): 1393-1406. YAN Lei, LI Jialong, HE Xuhui. Effect of central clamps on fatigue damage of suspenders of suspension bridge under joint action of fluctuating wind and vehicle loads[J]. Journal of Central South University(Science and Technology), 2023, 54(4): 1393-1406. 湖南省自然科学基金资助项目(2020JJ5754)；国家自然科学基金资助项目(52178516，51925808，51808563)；桥梁结构抗风技术交通行业重点实验室开放课题(KLWRTBMC18-03) Project(2020JJ5754) supported by the Natural Science Foundation of Hunan Province; Projects(52178516, 51925808, 51808563) supported by the National Natural Science Foundation of China; Project(KLWRTBMC18-03) supported by the Open Research Fund of Key Laboratory of Wind Resistance Technology of Bridges of China 2022-05-02 2022-07-25 2023-04-26 2023-05-22 37692425 中南大学学报(自然科学版) 4 54