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1、Finite element method analysis of a concrete bridge repaired with fiber reinforced plastic laminatesJoseph W.Tedesco, J.Michael Stallings.Mahmoud El-Mihilmy, Department of Civil Engineering,Auburn University,Auburn,AL 36849,USAMartin Peltyn and Early,Reno,NV 89502,USAAbstractMany reinforced concrete
2、 bridges throughout the United States on county and state highway systems are deteriorated and/or distressed to such a degree that structural strengthening of the bridge or reducing the allowable truck loading on the bridge by load posting is necessary to extend the service life of the bridge .The s
3、tructural performance of many of these bridges can be improved through external bonding of fiber reinforced plastic(FRP) laminates or plates.This paper summarizes the results of a comprehensive finite element method(FEM)analysis of a deteriorated reinforced concrete bridge repaired with externally b
4、onded FRP laminates.Static and dynamic analyses of the bridge were conducted for conditions both before and after the FRP repairs,with loading by two identical test trucks of known weight and configuration.The results of the FEM analyses were corroborated with field test data.The results of a parame
5、tric study to assess the effects of altering the cross sectional dimensions and material properties of the FRP laminates upon bridge girder deflections and reinforcing steel stresses is also presented.在美国，许多通过各地县，洲际的公路系统的钢筋混凝土桥梁被恶化和/或困扰到这种程度，桥梁结构加固或通过荷载记录减少卡车容许载荷对延长大桥服务年限是必需的。许多这种桥梁的结构性能，可以通过提高外部FRP
6、碾压与金属的连接。本文概述一个广泛运用于恶化的钢筋混凝土桥梁外部FRP碾压维修的有限元分析方法所取得的成果。在利用玻璃钢维修前后桥梁静态和动态分析的桥梁被条件所引导，应该利用两个相同重量和外形的汽车装载。有限元分析的结果可以通过现场测试的数据来保证。变量研究的结果是为了评估对改变交叉组合尺度的影响，玻璃钢碾压材料设备对桥梁的影响与加强钢筋压力同样出现。2019 Elsevier Science Ltd.All rights reserved. 2019年科学Elsevier公司版权所有。1. IntroductionA significant percentage of the bridges
7、 in North America were built after the Second World War.Most of them were originally designed for smaller vehicles, lighter loads and a lower traffic volume than commonly experienced today.Furthermore,over 50% of all bridges in the United States were built before 1940,and approximately 42%of these b
8、ridges are considered to be structurally deficient1.This alarming statistic underscores the importance of developing reliable and cost-effective repair and strengthening techniques for existing bridge structures.第二次世界大战后，在北美建成的大桥中占有重大的比例。他们中的大多数是根据比现在普通经历的更小的交通工具，驳船的装载量以及更低的交通量来设计的，此外，美国超过50%的桥梁中建于1
9、940年以前， 这些桥中大约又42%被认为结构不完善，这些令人担忧的统计数据强调了发展可靠的，具有综合经济效益的维修以及加固现有桥梁结构技术的重要性。In the past,various methods have been used to strengthen bridges and other types of structures.Traditionally,structural rehabilitation has been accomplished by methods such as introducing additional beams to the structure or
10、by strengthening existing beams with externally post-tensioned cables.In recent years,mechanical attachment of steel plates to the tension face of the deficient member has proven to be a successful repair technique for many structures.However,the use of steel plates has many disadvantages,such as co
11、rrosion,difficulty in handling the plates,deterioration of the bond at the steel-concrete interface and the requirement of massive scaffolding during installation. 在过去，各种方法已被用来加固桥梁和其他类型的结构，传统上，诸如增设传统梁结构或者通过布置张拉索来加固现有梁等方法来改善结构已经完成，近年来，对多数结构来说，利用钢金属板机械装置张拉外部不完善的部分已经被证实是一种成功的修复技术，然而，钢金属板的使用有很多的缺点，如腐蚀，金
12、属板处理发杂，钢筋混凝土接触面容易退化以及在厚重的脚手架安置过程中要求必备的条件。Unidirectional fiber reinforced plastic(FRP)sheets made of carbon(CFRP),glass(GFRP)or aramid (AFRP) fibers bonded together with a polymer matrix (e.g.epoxy,polyester,vinylester)are being used as a substitute for steel plates in many repair applications. FRP p
13、lates are an attractive solution over steel plates because of their ease in handling,resistance to corrosion,light weight and high strength.Recent experimental2-5studies have shown that reinforced concrete beams strengthened with externally bonded FRP laminates can exhibit ultimate load capacities a
14、s high as three times their original capacity depending on the steel ratio,concrete strength,FRP ratio,FRP mechanical properties,properties of the bonding agent,and pre-existing level of damage of the beams.由碳（碳纤维复合材料） ，玻璃（玻璃钢）或芳纶（AFRP）有保证的光纤混合一个聚合物基体（如环氧的化合物，聚酯，乙烯基酯）制成的单向纤维增强塑料（玻璃钢）板在很多维修应用中已经被用来替代
15、钢金属板。FRP金属板是一种比钢金属板更能吸引人的解决方案，因为他们处理简便，耐腐蚀，重量轻和强度高。最近实验 2-5 研究表明，梁在外部通过有保证的FRP碾压来加固的钢筋混凝土梁最后可以承受的最大荷载容许值是最初依靠钢比例，混凝土强度，FRP比例，FRP机械设备，代理连接设备，和梁破坏的现有标准的三倍。While laboratory experiments have illustrated the effectiveness of using FRP in repairs,and some field applications of FRP have been reported,there
16、 is a lack of test data illustrating the field performance of FRP repairs.The behavior of a repaired bridge is quantified by measurements of vertical deflections, strains in the primary flexural reinforcement,and strains on the surfaces of the FRP plates.Typically, these data are recorded from stati
17、c and dynamic field tests conducted with test trucks of known weight and configuration.To this end,a research project was conducted for the Alabama Department of Transportation (ALDOT)6.The overall project included:field application of FRP plates to an existing bridge,field load testing,development
18、of a cross sectional analysis procedure including the FRP,and static and dynamic finite element analyses of the structure.而实验室试验的图解表明，使用FRP在修理，和一些FRP修复范围的应用的效力已被报道，有一个缺乏检验数据说明FRP维修性能的范围。桥梁修复通过垂直偏差的度量单位制，初步弯曲加固拉紧，FRP金属板的表面拉紧来量化。在静态和动态野外试验中利用知道重量和外形的卡车中所取得并记录的有代表性的数据，为此，阿拉巴马州运输部进行了一项研究项目（ALDOT)。该项目总体包
19、括：玻璃钢板在现有桥梁中的应用领域，测试荷载范围，包括玻璃钢板的交叉组合分析程序的发展，动静态结构的有限元分析方法。This paper summarizes the results of a comprehensive finite element method(FEM)analysis of a deteriorated reinforced concrete bridge repaired with externally bonded FRP laminates.Static and dynamic analyses of the bridge were conducted both b
20、efore and after the FRP repairs,with loading by two identical test trucks of known weight and configuration.The results of the FEM analyses were corroborated with field test data.The results of a parametric study to assess the effects of altering the cross sectional and material properties of the FR
21、P laminates upon bridge girder deflections and reinforcing steel stresses is also presented.本文概述用一个全面的有限元方法（ FEM ）分析一个恶化的钢筋混凝土桥利用外形有保证的FRP碾压进行维修的结果，在利用FRP维修前后，加载两辆相同已知重量和外形的卡车来进行桥梁的动静态分析。有限元分析的结果可以通过野外测试数据来保证，一项变量研究评估改变交叉组合和玻璃钢碾压材料设备材料对梁桥的影响的结果，和加强钢压力已经呈现。2.Bridge description and repair planThe brid
22、ge selected for repair was built in 1952,is located on Alabama Highway 110,and consists ofseven 10.36 m simple spans with an east-west orientation.Four standard reinforced concrete girders support the deck in each span.The bridge had a 7.32 m clear roadway width with two 3.20 m traffic lanes and 0.4
23、6 m shoulder on each side as shown in Fig.1.It was originally designed for an AASHO H15-44 design loading which corresponds to an applied live load bending moment for the girders equal to 63%of the AASHTO7HS20-44 loading currently used by ALDOT for bridge design.经过选定维修的桥梁，建于1952年，位于阿拉巴马州公路110 ，由7个跨径
24、为10.36m东西方向的桥跨组成，每个桥跨顶部由四个标准的钢筋混凝土梁支撑，如图一所示，该桥路面宽7.32m，有两个宽3.2m的交通通道以及两侧宽为0.46m的侧翼。它原先的设计适合于AASHOG15-14的设计荷载相当于符合生活中梁使用的荷载弯矩等同于63%的AASHTO7HS20-44加载，通常用于奥尔多桥梁设计。The girders were exhibiting significant flexural cracking caused by routine traffic with heavy trucks over a 40-year period.Flexural cracks
25、were typically spaced along the span at approximately 200- 400 mm,and extended over practically the entire span.One span from the bridge was chosen for repair by the application of FRP plates so that the effectiveness of this type repair could be evaluated.CFRP plates were applied to the bottoms and
26、 GFRP plates to the sides of the stems of the T-beams as shown in Figs.2 and 3 Girder 1 had only the CFRP plates applied to the bottom surface,and served as a standard of comparison to investigate the effects of the GFRP plates bonded to the sides of the other girders.该梁所显示出的明显的弯曲裂缝是由超过40年重卡车的日常交通量造
27、成。弯曲裂缝通常间隔沿跨度约为20-40mm，实际上伸长到整个跨度。桥梁的一跨被选定为应用FRP金属板维修，从中可以评价该类型的维修所产生的效应。CFRP金属板将被用于底部，T梁主梁一边的CFRP金属板如图2和图3所示，1号梁只有CFRP金属板应用到底部表面作为标准与调查CFRP金属板联结到另一梁的一边所产生的效应作对比。Both the CFRP and GFRP plates were unidirectional with the fibers oriented parallel to the longitudinal axis of the plate.The GFRP primary
28、 plates had overall dimensions of 3.28 m356 mm1 mm,and the GFRP splice plates had dimensions of 9143561 mm.All the GFRP plates had an ultimate tensile strength of 448 MPa and a modulus of elasticity of 23,700 MPa.The CFRP primary plates had overall dimensions of 3.09 m267 mm1.3 mm,and the CFRP plate
29、s had dimensions of 9142671.3 mm.All the CFRP plates had an ultimate tensile strength and modulus of elasticity of 1,190 MPa and 121,000 MPa respectively.The FRP plates were bonded to the concrete with a readily available structural adhesive. 无论是碳纤维复合材料和玻璃钢板都是单向与纤维面向平行的纵向轴板。玻璃钢板主要板块的尺寸为3.28 m356 mm1
30、 mm,，其结合板的尺寸为9143561 mm。所有玻璃钢板最后拉伸强度有448 MPa ，弹性模量为23700MPa ，碳纤维复合材料主要板块的尺寸为 3.09 m267 mm1.3 mm，其结合板尺寸为9142671.3 mm，所有碳纤维复合材料最终拉伸强度和弹性模量分别为1190 MPa 和121,000 MPa，可以确保现成结构中FRP金属板与混凝土的容易粘合。3.Field load testsLoad tests were performed both before and after application of the FRP plates with load test tru
31、cks of known axle configuration and weight distribution.The two vehicles used for the load tests were identical load test trucks owned and operated by ALDOT.These trucks had a three-axle configuration with a gross vehicle weight of 346 kN distributed as shown in Fig.4.Static and dynamic tests were p
32、erformed on the bridge.For the static tests,the trucks were positioned with the center axle at midspan in four different transverse positions.Load positions 1 and 2 are illustrated in Fig.5.Load position 3 is the mirror image of position 1 about the bridge centerline,and position 4 is the mirror ima
33、ge of position 2.The spacing between the test trucks,and the distance from the curb to the wheel loads are significantly less than required for design by AASHTO7.The positions were chosen to produce the most extreme load conditions possible.Load position 1 creates the most severe loading on girders
34、1 and 2.Load position 3 creates the most severe loading on girders 3 and 4.The dynamic tests were conducted using the same test vehicles traveling at 80 km/h,side-by-side across the bridge and centered in the traffc lanes.Electrical resistance foil strain gages were used to measure the strain respon
35、se to test truck loads in the reinforcing bars,on the FRP plates and on the surfaces of the concrete girders,both before and after the FRP was applied.Vertical deflections were measured at midspan of each girder with linear variable differential transformers(LVDTs).3.现场荷载试验应用FRP金属板前后进行荷载试验，通过加载两辆已知车
36、轴外形和分配重量的卡车。两辆用来荷载试验的交通工具是由奥本拥有和操作的相同荷载的卡车，该三轴卡车毛重为346 kN，其荷载分布情况如图4所示。桥梁进行了动静态试验，静态试验中，卡车的位置被安排在车轴的中央处于中跨四个不同的横向位置，荷载1和2位置如图5所示。荷载3的位置在图所示中桥梁中心线的1位置，荷载4的位置在图所示中的2位置，试验卡车的间隔和路边与车轮加载处的距离明显小于奥本的设计要求，这些位置是根据可能产生最大荷载的情况而选择的。荷载1在梁1和2有可能产生最大荷载，荷载4在梁3和梁4可能产生最大荷载，动态试验中是利用两辆速度为80 km/h的相同卡车进行的，并行越过大桥并且集中于桥梁的中
37、心线。在应用FRP前后，利用箔电阻应变计来测量试验卡车荷载在钢筋板，FRP金属板和混凝土梁表面上的应变反应，利用线性可变差动变化器(LVDTs)每个梁中跨的竖向影响。4.FEM analysesTo accurately model the bridge,a three dimensional FEM model was constructed.The bridge structure was categorically analyzed using a judiciously selected combination of finite elements.The deck and girde
38、rs were modeled independently of each other in order to simulate the effect of the existing cracks in the girders.Although the original bridge structure is symmetric,the loading conditions were unsymmetric and the FRP bonded side plates were not installed on all girders.Therefore a three-dimensional
39、 model for the entire bridge was required.An isometric view of the FEM model is shown in Fig.6.In the actual bridge,the concrete girders exhibited extensive flexural and shear cracking.These cracks extended through the entire stem of each girder,virtually along their entire length.Therefore the elas
40、tic modulus of the girders was modified to simulate their extensive cracked state.The modulus of elasticity of the deck was calculated using theAmerican Concrete Institute8standard of 4730where fc is the concrete compressive strength in MPa.The bonding of the FRP laminates to the damaged concrete su
41、rfaces prevented the cracks from opening when the bridge was subjected to loading,and tended to stiffen the girders.The increased stiffness of the repaired girders was estimated to be 30-40% greater than that of the unrepaired cracked girders.为了精确桥梁的模型，构造出一个三维的有限元模型，该桥梁结构断然使用了经过明智选择的有限元组合来分析，为了模拟现有梁
42、裂缝的效果，保护层和梁双方独立立模，尽管原先桥结果是均衡对称的，荷载的情况却是不对称的以及由FRP联结一边金属板并没有安装到所有梁，因此，要求一个完整桥梁的三维模型。图6所示的是一个等距的有限元模型，在实际桥梁中，中心梁出现大面积的弯曲以及剪切破裂，这些裂缝透过每个梁整个支撑部分，实质上沿着梁的整个长度。然而梁的弹性模量被修正为模拟他们的大面积的裂缝状态。保护层的弹性模量被适当用于美国混凝土协会的标准其中fc为混凝土的压缩强度，单位MPa。FRP薄片与混凝土被损坏的表面连接防止桥梁受荷载的影响并且趋向于使梁硬化致使裂缝开放，修理过的梁增加的强度估计比未修理的破裂梁的强度大30-40%。The
43、FEM analyses were conducted through implementation of the ADINA9 finite element computer programs.The FEM model consists of 1440 three-dimensional,eight-node solid elements for the concrete deck slab,1280 eight-node solid elements(having a reduced modulus of elasticity to simulate the effect of crac
44、ks)to model the concrete girders,and 160 truss elements to model the steel reinforcing bars(located at the center of gravity,CG,of the reinforcement).The bonded FRP laminates were represented with 160 truss elements to model the CFRP plates located at the mid bottom nodes of the girders.In addition,
45、another 720 truss elements were employed on the sides of the girders to represent the GFRP plates.A typical cross section of the FEM model of the bridge is presented in Fig.7.有限元分析的进行，通过执行艾迪那9的有限元电脑程序，有限元模型是由1440个三维空间、混凝土保护层八个结点立体元素、1280个混凝土梁模型8个节点的立体元素（用简化的弹性模量来模拟裂缝的影响）、以及模拟钢筋保护层的160束元素（位于钢筋重心中央）组成
46、，FRP薄片在梁底部的中央节点与模拟FRP薄片的160束元素的联结被提出异议，除此之外，其余720束元素用于梁一边的象征着金属薄片，桥梁有限元模型一个典型的截面如图7所示。A comprehensive series of three-dimensional FEM analyses were conducted on the bridge structure.Both static and dynamic analyses were performed to verify the accuracy of the FEM model and to replicate the field loa
47、d tests previously described.The dynamic analysis studies included frequency analyses to establish the dynamic characteristics of the bridge,and transient analyses to simulate the dynamic field load tests.A parametric study was also conducted to quantify the effects of varying several cross section
48、characteristics and mechanical properties of the FRP laminates upon the bridge girder structural responses.桥梁结构进行了一系列全面的三维空间有限元分析。不管是动态还是静态的缝隙的执行都是为了检验有限元模型和先前现场荷载试验所描述的准确性，动态分析研究包括在桥梁上安置具有动态特征的频率分析、模仿动态现场荷载试验的短暂性分析。为了量化改变不同的几个截面的特点和FRP薄片的机械材料在桥梁结构上所产生的，进行了一项参数研究。4.1.Frequency analysisThe previously
49、 described FEM model was employed to conduct a frequency analysis of the bridge.This analysis was performed to establish the dynamic characteristics of the bridge and to verify the accuracy of the FEM model by comparing the calculated fundamental frequency and free vibration response with those observed in the field tests.Also,the results of the frequency analysis,along with the damping ratio for the fundamental mode determined from the field tests,were used