Review on Advances in Seismic Research of Large-Span Prestressed-Concrete Continuous Rigid-Frame Bridges
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摘要:
我国已建设大量的大跨PC （prestressed concrete）连续刚构桥，其墩高可达百米及以上，存在遭受强震的可能，尤其是在西部高地震风险区，连续刚构桥主墩与主梁是刚性连接，主梁与桥墩共同承担地震力. 为促进刚构桥的抗震研究，首先，梳理了国内外近期经受地震考验的几座刚构桥的震害表现；然后，从抗震理论及模型试验、减隔震（耗能）设计和震后修复等方面，对连续刚构桥桥墩、上部结构、基础等主要构件以及全桥整体抗震性能等热点问题进行了评述，刚构桥具有良好的抗震性能，高阶效应及墩梁固结处纵桥向弯矩对桥墩地震反映影响较大，模型试验及理论分析中主梁开裂及损伤问题易被忽视，低墩或双柱墩刚构桥已展开墩底及基础隔震研究；最后，对未来可开展研究方向进行了探讨，强震下箱梁的开裂机理及损伤控制，基于新型材料及耗能构件组成的高墩，基础隔震及高墩底部隔震的实用技术，箱梁及空心墩的地震损伤识别及震后修复，（近）跨断层地震作用下刚构桥的渐进倒塌机理与防止.
Abstract:A large number of large-span prestressed concrete (PC) continuous rigid-frame bridges (CRFBs) have been built in China with heights of piers up to 100 m or more. They are likely to suffer strong earthquakes, especially in high seismic risks areas of Western China. The main girder and piers are rigidly connected together in CRFBs, and jointly bear the seismic force in earthquakes. In order to promote the seismic research of CRFB, the seismic damage of several CRFBs which have undergone recent earthquakes at home and abroad was reviewed firstly. Then, from the aspects of seismic theory, model tests, seismic isolation (energy consumption) design and post-earthquake repair, hot issues of the main components such as pier, superstructure, foundation, and the seismic performance of the whole bridge are reviewed. The current research shows that the CRFBs have good seismic performance; the high-order effect and the longitudinal bending moment at the pier-beam consolidation have a greater impact on the seismic response of piers. In the model test and theoretical analysis, the cracking and damage of the main girder are easily ignored, and research on pier bottom and foundation isolation has been carried out for low-pier or double-column CRFBs. Finally, future research directions were explored, including the cracking mechanism and seismic control of box girder under strong earthquakes, the high piers composed of new materials and energy-consuming components, the practical isolation technology of the bottom and foundation of high-piers, the identification of earthquake damage and post-earthquake repair of box girders and hollow piers, the mechanism and prevention of the progressive collapse of rigid frame bridges under the action of (near-fault) cross-fault earthquakes.
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Key words:
- continuous rigid-frame bridge (CRFB) /
- review /
- seismic response /
- seismic design /
- ground motion effect
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固定辙叉是道岔重要组成部分，广泛应用于我国重载和提速铁路. 固定辙叉区轮轨动力相互作用异常剧烈，关键钢轨部件磨耗磨损等病害较区间线路更加频发[1-6]. 我国固定辙叉平均使用寿命仅为美国的40%，欧洲的60%，造成其服役寿命相对较低的主要原因之一是轮载过渡区辙叉钢轨总承载面积相对欧美较小[7].
辙叉区不规则的钢轨廓形是影响轮载过渡区钢轨承载面积的直接因素. 为此，国内外学者从钢轨廓形参数优化角度进行了大量研究：王树国等[7]从理论分析、动力仿真和现场试验角度对辙叉心轨顶宽、查照间隔和护轨轮缘槽宽度等参数进行优化；徐井芒等[8-9]提出基于轮轨廓形净差值的心轨廓形优化方法；曹洋等[10]针对固定辙叉轮载过渡段轮轨匹配性能对心轨关键断面廓形进行优化；沈钢等[11]采用粒子群算法求解优化模型对心轨顶面降低值参数进行优化；张鹏飞等[12]研究翼轨加高值对列车过岔动力特性的影响；Wan等[13-14]采用数值模拟和现场试验分析辙叉区各轮轨动力响应的权重大小，并结合多目标优化方法提出固定辙叉心轨廓形设计方法.
上述文献研究主要针对辙叉廓形单一几何参数，通过调整局部廓形进行优化设计，并未从固定辙叉几何特征控制参数角度探寻廓形参数化设计方法并分析各参数对辙叉钢轨廓形的影响. 因此，本文基于B样条理论，采用DOE （design of experiments）方法，分析各关键控制参数对心轨廓形变化的影响权重，对实现固定辙叉区钢轨廓形参数化设计具有重要意义.
1. 固定辙叉廓形特征参数
在固定辙叉中，有害空间引起轨距线不连续，且翼轨和心轨廓形呈连续不规则变化. 然而，固定辙叉平面布置、心轨各关键断面廓形以及翼轨断面几何特征具有如下规律：对于任意心轨和翼轨断面，轨顶横坡和轨头侧面纵坡不变；沿辙叉走行方向，心轨和翼轨高度分段线性变化，心轨断面宽度线性增大；心轨和翼轨轮缘槽宽度一定；距轨距测量点标准轨头半宽35.5 mm处的翼轨顶面高度以及翼轨宽度可精确确定. 以60 kg/m钢轨12号固定辙叉[15]为例，心轨轨顶横坡为1∶20，轨头侧面纵坡为1∶5；翼轨工作边纵坡为1∶5，轨顶横坡为1∶20，非工作边侧面纵坡1∶20，如图1所示，W为翼轨轨头宽度.
由此可见，虽然辙叉钢轨轨头宽度和高度沿辙叉方向不断变化，但仍存在保持不变的设计参数，分别为各断面处心轨和翼轨轨头宽度和高度、轨顶横坡、轨头侧面坡度. 本文将以上参数统称为固定辙叉钢轨全断面廓形特征参数，为后文中不断重构更新辙叉钢轨廓形提供了设计依据.
2. 辙叉钢轨廓形参数化拟合方法
2.1 B样条曲线理论
辙叉区钢轨各断面廓形是由多段不同曲率圆弧组成的复杂曲线，整个廓形需保持连续平滑. B样条曲线在计算机辅助设计中具有广泛应用[16]，可通过较少的轮廓关键控制顶点实现达到期望逼近精度的廓形拟合，且具有连续平滑性和强凸包性. 因此，通过B样条曲线拟合辙叉钢轨廓形，对利用辙叉廓形特征参数进行廓形参数化设计具有明显优势. 已有相关研究将其应用于钢轨廓形设计：Yang等[17]采用B样条理论，选取实测廓形33个控制点建立磨耗钢轨廓形曲线，对不同打磨方式钢轨廓形进行磨耗预测；温士明[18]运用B样条曲线对钢轨廓形进行参数化描述，提出钢轨打磨目标廓形优化方法；刘建桥[19]结合遗传算法和B样条曲线，以降低钢轨磨耗为目标确定钢轨优化廓形；唐彦玲等[20]通过B样条曲线对钢轨廓形进行参数化描述，结合遗传算法对曲线地段钢轨廓形进行设计；王亮等[21]采用B样条曲线和NSGA-Ⅱ算法对控制点权因子寻优计算，对钢轨打磨区域轨头廓形进行设计；林凤涛等[22]采用B样条理论对高速铁路辙叉钢轨廓形拟合，基于标准辙叉廓形设置16个控制点设计了钢轨经济性打磨廓形；史振帅[23]以道岔转辙器区域直尖轨为研究对象，采用NURBS理论对道岔直尖轨打磨廓形进行设计重构. 已有文献虽实现了钢轨廓形重构，但仍是通过标准或实测钢轨廓形的稠密离散点数据进行廓形拟合，且大多针对区间线路和可动心辙叉，对固定辙叉钢轨廓形研究较少.
在B样条方法中，任意曲线可用式（1）表示.
C(t)=n−1∑v=0Bv,d(t)Qv, (1) 式中：Qv为序号v的廓形关键控制点；n为控制点Qv个数，依次连接控制点可以构成一个特征多边形；t为节点，t0≤t≤tm-1，T=（t0，t1，…，tv，…tm-1），T为节点向量，m=n + d + 1，m为节点个数，d为B样条曲线的阶数；Bv,d(t)为由递归方程表示的d−1次B样条基函数，如式（2）所示.
{Bv,d(t)=t−tvtv+d−tvBv,d−1(t)+tv+d+1−ttv+d+1−tv+1Bv+1,d−1(t),Bv,0={1，tv⩽t⩽tv+1,0，t<tv或t>tv+1. (2) 根据B样条曲线理论与相关工程应用情况可知，采用3次样条可保证拟合曲线2阶导数连续，满足工程实际需要，因此，本文采用3次非均匀B样条曲线拟合辙叉钢轨廓形.
2.2 关键控制点确定
依据心轨和翼轨廓形关键几何特征参数可确定B样条曲线拟合廓形所需关键控制点，由各控制点组成心轨和翼轨廓形拟合控制多边形，如图2所示. 由于心轨断面廓形轴对称，因此只选取心轨中心轴一侧进行廓形参数化说明.
在图2（a）中，心轨廓形拟合的关键控制点为{N1,N2…,Ni}，Ni=(xi，yi)，i =1，2，…，8. 其中：N1为心轨轨顶高度控制点；N2、N3用来调节心轨顶面平直段廓形范围；N4、N6、N8用来调节心轨轨头复合圆弧段廓形；N5为轨顶横坡和侧面纵坡交点；N7为心轨宽度控制点；P为△N4N5N6中垂线与N4N6的交点，为确定控制点N8的辅助点，N8位于N5P的连线上.
在图2（b）中，翼轨廓形拟合的关键控制点为{W1,W2,…,Wj}，j=1，2，…，9. 其中：W1和W9为翼轨轨头宽度控制点；W3为轨顶横坡和翼轨工作边纵坡交点；W5为翼轨高度控制点；W7为轨顶横坡和翼轨非工作边纵坡交点；W2、W4、W6、W8用来调节翼轨轨头两侧复合圆弧段廓形；kW1、kW2、kW3分别为翼轨工作边纵坡、轨顶横坡和非工作边纵坡斜率；hW为翼轨高度；wW为标准轨头宽度的一半，即35.5 mm.
由于心轨和翼轨廓形参数化方法一致，且心轨廓形变化较翼轨复杂，因此，以心轨为例详细说明辙叉钢轨廓形参数化过程. 采用向量法确定各控制点间关系，如式（3）.
{N4N5|N4N5|=(1，k1),N5N6|N5N6|=(1，k2),N4N5=λ1N1N5,N1N2=λ2N1N4,N5N6=λ3N5N7,N2N3=λ4N2N4,N5N8=λ5N5P, (3) 式中：λ1、λ2、λ3、λ4、λ5分别为调节控制点N4、N2、N6、N3、N8位置的比例系数，为可变参数，记为λs∈[0，1]，其中，s=1,2,…，5，λ5用来调节心轨轨头复合圆弧段廓形，简称心轨复合圆弧段比例系数；k1、k2分别为心轨轨顶横坡和侧面纵坡斜率.
当其他控制点一定时，心轨轨头复合圆弧段廓形主要由控制点N8位置决定，为了保持心轨轨头廓形凸曲线的特点，N8取值需在N5P范围内. 依据向量法求三角形垂点P，如式（4）所示.
P=N4+N4N5•N4N6|N4N6|•|N4N6|•(N6−N4). (4) 结合式（3）、（4），根据定比分点原理确定各关键控制点坐标，如式（5）所示.
{(x1,y1)=(0,h),(x2,y2)=(1−λ2)(x1,y1)+λ2(x4,y4)+(0，Δ),(x3,y3)=(1−λ4)(x2,y2)+λ4(x4,y4),(x4,y4)=λ1(x1,y1)+(1−λ1)(x5,y5),(x5,y5)=(h+k2wk2−k1,h+k1(h+k2w)k2−k1),(x6,y6)=(1−λ3)(x5,y5)+λ3(x7,y7),(x7,y7)=(w,0),(x8,y8)=(1−λ5)(x5,y5)+λ5(xp,yp), (5) 式中： (xp，yp)为点P坐标；Δ为心轨平直段廓形修正系数，通过调整Δ使控制点N2与N1在同一水平线上；h为x处心轨高度，x为心轨任意断面距辙叉理论尖端的距离；w为x处心轨轨头半宽.
由式（3）～（5）可知，心轨任意断面廓形各控制点坐标可表示为各特征参数和比例系数的函数. 翼轨同理，则固定辙叉全断面廓形控制点可表示为
{Ni=fcp(h,w,k1,k2,Δ,[λs]),Wj=gwp(hW,wW,kW1,kW2,[λt]), (6) 式中：fcp(·)、gwp(·)分别为求解心轨和翼轨任意断面所有控制点的函数；λt为翼轨控制点调节比例系数，记为λt∈[0，1]，其中，t为其对应的比例系数编号.
综上所述，关键控制点确定所涉及的参数具体包括两类，一类是与廓形相关的有实际物理意义的特征参数，另一类是B样条曲线几何控制点调节比例系数. 为方便后续分析，将上述两类参数统称为辙叉钢轨廓形关键控制参数.
2.3 廓形拟合
以60 kg/m钢轨12号固定辙叉钢轨为例，采用B样条曲线进行廓形构建，心轨和翼轨控制点分布及拟合廓形分别如图3所示.
由图3可知：各控制点位置随心轨轨顶高度和宽度的变化而改变，从而实现心轨廓形连续变化；本文不考虑翼轨高度和轨头宽度的变化，因此翼轨拟合廓形唯一.
在此基础上，结合辙叉区心轨和翼轨走行方向变化，可进一步确定辙叉三维廓形，考虑一侧翼轨的固定辙叉三维廓形如图4所示.
由图4可知，虽然固定辙叉钢轨廓形复杂多变，心轨和翼轨沿辙叉走行方向廓形不断变化，但结合辙叉结构设计中各关键断面的关键控制参数，基于B样条方法可快速确定沿辙叉走行方向的任意截面廓形，该方法可极大拓展以固定辙叉廓形优化设计为基础的轮轨动力仿真以及轮轨接触关系等的研究.
3. 心轨廓形参数试验设计分析方法（DOE）分析
3.1 评价指标
心轨廓形关键控制参数调整会改变心轨廓形，廓形面积差可以直观量化地表达心轨整体廓形的变化情况；与此同时，不同参数对心轨轨头廓形和心轨侧面廓形变化影响程度不同，而心轨轨头廓形变化量和侧面廓形变化量可以更细致地反映各参数对心轨轨头和侧面廓形的单独影响. 因此，为分析各关键控制参数对心轨廓形的影响，选取ΔS以及D1、D2评价各方案廓形与标准廓形的贴合度，如式（7）所示. 本文借鉴《铁路线路修理规则》中对钢轨垂磨和侧磨的定义，选取心轨轨头顶面和侧面某一位置处廓形距离偏差量来定义D1和D2.
{ΔS=|Scase−Sstd|,D1=|Ycase−Ystd|,D2=|Xcase−Xstd|, (7) 式中：Scase为各方案廓形曲线与X坐标轴围成的面积；Sstd为采取表1参数确定的标准廓形与X坐标轴围成的面积；ΔS为各方案廓形与标准廓形面积差，反映心轨整体廓形的变化程度；Ycase和Ystd分别为距心轨中心轴2w/3处，各方案廓形曲线与标准廓形曲线纵坐标值；Xcase和Xstd分别为距心轨顶面h/2处各方案廓形曲线与标准廓形曲线横坐标值；D1和D2分别为各方案廓形与标准廓形在垂向和横向上的廓形偏差.
表 1 DOE全阶乘方案设计Table 1. DOE full factorial scheme design方案 k1 k2 λ5 方案 1 1/30 1/6 1/4 方案 2 1/30 1/6 1/2 方案 3 1/30 1/4 1/4 方案 4 1/30 1/4 1/2 方案 5 1/10 1/6 1/4 方案 6 1/10 1/6 1/2 方案 7 1/10 1/4 1/4 方案 8 1/10 1/4 1/2 标准 1/20 1/5 1/3 在DOE分析中，为明确各关键控制参数在所选合理取值范围内对心轨廓形变化的影响权重E，除了对D1、D2及ΔS各自的均值进行分析，还需对E进行分析，如式（8）所示.
E=I+−I−V+b−V−b3∑b=1I+−I−V+b−V−b×100%, (8) 式中：I−和I+分别为各控制参数分别取2个水平（−和 + ）时的各贴合度指标均值，V+b和V−b分别为各关键参数的2个水平值（−和 + ）.
3.2 方案设置
在工程实践中，心轨轨头顶面和轨头侧面复合圆弧部分更易发生严重磨耗[24]，k1和k2分别对心轨顶面廓形和轨头侧面廓形起控制作用，比例系数λ5能够调整控制点N8的位置，从而实现轨头侧面复合圆弧段廓形的调整. 因此，采用DOE全阶乘设计方法对心轨廓形关键控制参数k1、k2及λ5进行方案设计. 依据《道岔设计手册》60 kg/m钢轨12号道岔[15]中的心轨标准设计参数，在一定范围内对各参数调整取值进行方案设计，以满足心轨整体廓形符合工程实际. 各方案参数取值如表1.
3.3 重构廓形轮轨匹配特征
轮轨接触点分布规律是轮轨廓形匹配的最直观反映[25]. 选取LMA型车轮踏面，以心轨20 mm断面和50 mm断面为例，分析轮对不同横移量下各方案轮轨接触点分布规律，进一步验证该参数化设计方法的合理性. 各方案心轨拟合廓形如图5所示，各方案轮轨接触点分布情况如图6所示.
图 6 心轨关键断面车轮踏面接触点变化规律Figure 6. Contact point variations on wheel tread for key nose rail sections由图5可知：各方案心轨廓形与标准廓形贴合度良好，说明取得了较好的钢轨廓形重构效果. 所建立的钢轨廓形B样条曲线构造方法为后文参数分析中不断重构更新钢轨廓形奠定了基础.
由图6可知：各方案拟合廓形轮轨接触点分布规律均与现有标准廓形一致；在心轨顶宽20 mm断面，各方案轮轨接触点位置基本一致，仅接触点发生突变对应的横移量有极小差别，说明在心轨顶宽较小时，各参数对轮轨接触点分布影响较小；在心轨顶宽50 mm断面，与标准廓形方案相比，方案1～4中轮轨接触点分布偏向于车轮踏面外侧，变化规律差别较小；而方案5～8则偏向于车轮踏面轮缘侧，变化规律也基本相近；随着心轨顶宽增大，参数k1对轮轨接触点分布的影响较k2和λ5大.
3.4 DOE分析
固定辙叉钢轨廓形影响轮轨匹配性能，因此，明确各控制参数对辙叉钢轨廓形影响权重，对辙叉钢轨廓形参数化设计中的参数调整优化具有重要意义. 以心轨20 mm断面和50 mm断面为例，通过DOE设计方法分析各控制参数对心轨廓形变化的影响效果及影响权重. 图7、8所示为心轨各控制参数对D1、D2和ΔS变化范围的影响. 图9为反映各控制参数对心轨廓形绝对影响效应的Pareto图. 图10为各控制参数对心轨廓形变化的影响权重.
由图7、8可知：在心轨20 mm断面，k2和λ5对ΔS、D1和D2变化范围均影响较大，k1对三者影响较小；在心轨50 mm断面，k1对ΔS、D1影响较大，对D2影响较小，k2和λ5对ΔS、D1和D2影响相当. 说明当心轨轨头宽度较小时，心轨整体廓形变化、心轨顶面和侧面廓形均主要受k2和λ5影响，k1影响较小；随着心轨轨头宽度增大，k1对心轨整体廓形和顶面廓形影响增大，对侧面廓形影响较小；k2和λ5对心轨各部位廓形影响效果相当.
由图9可知：在心轨20 mm断面，k1和λ5对ΔS影响效应较大，各参数对D1和D2影响效应相当且均较小；在心轨50 mm断面，各参数对ΔS影响效应最大，D1次之，对D2影响很小，其中，k1对ΔS影响效应最大，k2和λ5对ΔS影响效应相当；各参数对D1影响效应相当.
由图10可知：1） 在心轨20 mm断面，k1和λ5对ΔS和D1影响权重较小，而k2影响较大. 其中，ΔS对应的E分别为21.08%、56.89%和22.02%，D1对应的E分别为8.42%、61.95%、29.63%. 说明在心轨20 mm断面，k1和λ5对心轨整体廓形和顶面廓形的影响权重远不及k2，主要原因是在心轨轨头宽度较小时，侧面高度相对较大，心轨侧面廓形对心轨廓形变化起主导作用. 2） 在心轨50 mm断面，k1、k2和λ5对ΔS和D1的影响权重依次减小. 其中，ΔS对应的E分别为55.90%、33.38%、10.72%，D1对应的E分别为64.81%、26.71%、8.48%，说明随着心轨轨头宽度增大，各关键控制参数在方案所选取值范围内时，k1对心轨顶面和心轨整体廓形的影响权重逐渐明显，k2和λ5的影响逐渐减弱. 3） 在心轨20 mm断面，k1、k2和λ5对心轨侧面廓形偏差D2对应的E分别为9.93%、76.82%、13.25%；在心轨50 mm断面，k1、k2和λ5对心轨侧面廓形偏差D2对应的E分别为14.09%、66.04%、19.87%；由此可见，由于不同心轨轨头宽度对应的高度变化不大，因此，无论心轨宽度大小，心轨侧面廓形主要受k2影响较大，受λ5影响次之，受k1影响最小.
4. 结 论
1） 基于B样条理论，提出了一种考虑固定辙叉全断面钢轨廓形几何特征的参数化设计方法，该方法利用B样条曲线定义工业产品几何形状的优势，通过固定辙叉钢轨全断面廓形关键特征参数确定任意断面廓形控制点，进而实现钢轨廓形参数化重构；提出了钢轨重构廓形贴合度评价指标和关键控制参数对廓形贴合度影响权重指标，可直观量化地反映廓形参数变化对心轨廓形的影响.
2） 在心轨20 mm和50 mm断面，各方案心轨重构廓形与标准廓形贴合度良好，不同横移量下轮轨接触点分布规律符合实际. 各参数对轮轨接触点变化规律与心轨顶面宽度有关，在心轨顶宽较小时，各参数对轮轨接触点变化影响较小，在心轨顶宽较大时，k1对轮轨接触点分布的影响较k2和 λ5大.
3） 当心轨轨头宽度较小时，心轨整体廓形、心轨顶面和侧面廓形变化范围主要受k2和λ5影响，k1影响较小；k2对各部分廓形变化影响权重较大，k1和λ5影响较小；随着心轨轨头宽度的增大，k1对心轨整体廓形和顶面廓形变化范围和影响权重均逐渐增大，k2和λ5的影响逐渐减弱. 由此可见，各控制参数对心轨各部分廓形影响权重不同，通过考虑各控制参数对钢轨各部分廓形的影响权重，有针对性地重点考虑相应的控制参数进行廓形方案设计，避免了经验—性能计算—修正等反复试凑的传统设计过程，使辙叉廓形设计更具有针对性和可预测性.
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图 1 刚构桥的震害
Figure 1. Seismic damage of prestressed concrete continuous rigid-frame bridges
图 6 钢桁腹板PC刚构桥
Figure 6. Prestressed concrete continuous rigid-frame bridge with steel truss webs
表 1 发生震害的刚构桥基本信息
Table 1. Basic information of continuous rigid-frame bridges damaged in earthquakes
m 大桥名称 跨径 主墩墩高 能登岛大桥 75.0 + 108.5 + 75.0 24.4/24.4 庙子坪特大桥 125.0 + 220.0 + 125.0 102.5/99.5 阿苏长阳大桥 39.3 + 91.0 + 91.0 + 53.3 37.0/68.0/33.0 注：能登岛大桥跨径及墩高为估算值. 
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