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钢结构焊缝疲劳裂纹检测技术及适用性分析摘要：在钢结构桥梁领域，钢结构桥梁焊缝疲劳裂纹不易发觉，疲劳裂纹进一步扩展会对结构受力、桥面板等产生不利影响，甚至可能导致结构关键部件的突然断裂。
文章论述了钢结构疲劳裂纹开裂机理及影响因素，提出了检测焊疲劳裂纹的相关技术原理和方法以及进行其适用性分析。
关键词：钢结构；焊缝疲劳裂纹；检测方法；适用性分析钢结构桥梁是一种以钢材为原料制造的桥梁，和钢筋混凝土桥梁相比，它具有强度高、自重轻、跨越能力大、抗震好、结构牢靠、环保节能、施工便捷等优点。
随着我国综合实力的增强和钢材质量、产量的大幅提升，钢结构桥梁将会迅速发展。
但随着钢桥的服役年限的增加，桥梁的运营维护就会出现较多问题。
钢结构桥梁主要存在腐蚀、涂装劣化和疲劳三种病害。
腐蚀、涂装劣化病害在桥梁运营维护过程中较易发现，也能够及时采取相应维护措施，而疲劳开裂具有产生机理复杂、开裂初期不容易被发现、检测与修复难、且修复成本高等特点，已经成为钢结构桥梁运营养护过程面临的核心难题之一。
随着钢结构桥梁服役时间的增加，我国的一些大跨径钢结构桥梁也出现了疲劳开裂问题，如虎门大桥、海沧大桥[1-3]等。
钢桥初期疲劳裂纹不易发觉，疲劳裂纹进一步扩展会对结构受力、桥面板等产生不利影响，甚至可能导致结构关键部件的突然断裂，产生事故，造成巨大的经济损失。
钢桥常见疲劳破坏可以分为连接部位结点疲劳开裂、正交异性钢桥面板疲劳开裂、缆索及锚箱疲劳开裂，在这三种常见疲劳开裂模式中，疲劳开裂大多数发生在焊缝处，焊缝疲劳损伤是影响钢结构疲劳破坏的主要因素之一，甚至是首要因素。
相关研究表明，钢结构疲劳损伤多数在焊缝位置处产生[4-5]。
目前对钢结构桥梁焊缝疲劳破坏的研究已成为疲劳破坏领域中的一个重点研究方向，在这一方向中，又可分为疲劳裂纹开裂机理与影响因素研究、疲劳寿命评估研究、疲劳裂纹检测/监测研究、疲劳开裂后的防护措施研究等几个方面。
其中疲劳裂纹检测研究是至关重要的，因为它是识别钢结构桥梁是否发生疲劳裂纹，进而判断结构是否安全、以及确定是否需要采取措施提升性能的关键。
2012年6月上第41卷第366期施工技术CONSTRUCTION TECHNOLOGY11粉房湾长江大桥正交异性桥面板单元制作及变形控制沈念龙，李朝兵，丁瑞平，汪雪风（中建钢构江苏有限公司，江苏靖江214532）［摘要］结合重庆粉房湾长江大桥钢结构工程实例，针对本工程特点与难点，详细介绍了桥面板的制作要点以及变形控制，包括钢板校正、U 形肋拼装、焊接变形控制，板块摆放及约束，焊接顺序和方向，焊接校正方法及操作步骤。
通过对正交异性桥面板单元制作及变形控制技术的研究，并应用于粉房湾长江大桥钢结构工程中，取得了良好的效果。
［关键词］桥梁工程；斜拉桥；正交异性桥面板；变形控制；焊接［中图分类号］TU758．11；U443．31［文献标识码］A［文章编号］1002-8498（2012）11-0011-02Unit Making and Deformation Control of Orthotropic Bridge Deckfor Powder Room Bay Yangtze River BridgeShen Nianlong ，Li Chaobing ，Ding Ruiping ，Wang Xuefeng（China Construction Steel Structure Jiangsu Co．，Ltd．，Jingjiang ，Jiangsu214532，China ）Abstract ：Combined with steel structure engineering of Powder Room Bay Yangtze River Bridge in Chongqing ，based on the engineering characteristics and difficulties ，the authors introduce main points of making for bridge deck ，including plate correction ，U-rib assembling ，welding deformation control ，plate displaying and constraining ，welding sequence and direction ，the method of welding correction ，operation steps．Through study on unit making and deformation control of orthotropic bridge deck ，and application in steel structure engineering of Powder Room Bay Yangtze River Bridge ，good effect is obtained．Key words ：bridges ；cable stayed bridges ；orthotropic bridge decks ；deformation control ；welding ［收稿日期］2012-04-12［基金项目］中建三局课题（CSCEC3B-2011-23）［作者简介］沈念龙，中建钢构江苏有限公司助理工程师，江苏省靖江市江阴-靖江工业园区联心路二圩214532，电话：（0523）84693721，E-mail ：nianlong03@163．com1工程概况重庆粉房湾长江大桥为主跨（216.5+464+216.5）m 双塔双索面半漂浮体系斜拉桥。
正交异性钢桥面板构造细节的疲劳性能研究的开题报告一、选题背景及意义正交异性钢桥面板(Element-Fabricated Steel Orthotropic Deck, 简称EOSD)是近年来在大桥建设领域中广泛应用的桥面结构形式。
EOSD桥面板采用了正交异性钢板结构，是由水平方向的钢板和垂直方向的横筋组成的。
EOSD施工速度快，强度高，使用寿命长，同时还具有较好的舒适性和维护性。
因此，目前大桥建设中越来越多地采用EOSD桥面板。
然而，由于真实工况下的荷载、温度等因素的影响，EOSD桥面板的疲劳性能需要得到重视，以提高桥梁的安全性和寿命。
因此，进行正交异性钢桥面板构造细节的疲劳性能研究，有着重要的现实意义。
二、研究内容本研究着眼于EOSD桥面板的构造细节，重点是正交异性钢板和横筋之间的连接部位，通过模拟桥面板在真实工况下的荷载、温度等因素下的受力情况，研究EOSD桥面板的疲劳性能。
具体研究内容包括：1. EOSD桥面板构造细节的设计原理和结构组成的介绍以及疲劳破坏的机理分析。
2. 采用ANSYS软件对不同疲劳循环次数下的EOSD桥面板进行有限元分析，确定桥面板的应力分布规律和应力集中部位，挖掘其疲劳破坏的原因。
3. 根据分析结果，探讨EOSD桥面板构造细节改进的方案，提出有效的优化措施，旨在延长桥梁的使用寿命和安全性。
三、研究方法1. 搜集EOSD桥面板的相关资料及标准要求，了解其设计原理、构造特点以及疲劳破坏机理。
2. 建立EOSD桥面板的有限元模型，考虑其受力情况，并在不同载荷工况下进行有限元分析，得到桥面板的应力分布以及应力集中位置等参数。
3. 根据分析结果，制定EOSD桥面板构造细节优化方案，如加强钢板与横筋的连接等。
4. 进行疲劳试验，验证分析结果，并评估优化方案的有效性。
四、预期成果本研究的预期成果包括：1. 对EOSD桥面板构造细节的疲劳性能进行深入研究，提高EOSD 桥面板的疲劳寿命和安全性，具有重要的工程实践意义。
第 55 卷第 2 期2024 年 2 月中南大学学报(自然科学版)Journal of Central South University (Science and Technology)V ol.55 No.2Feb. 2024考虑残余应力的钢桥面板−肋双面焊裂纹应力强度因子计算方法肖新辉1，陈方怀1，张海萍1，刘扬2，肖康海1(1. 湖南工业大学 土木工程学院，湖南 株洲，412007；2. 长沙理工大学 土木工程学院，湖南 长沙，410114)摘要：建立焊接分析有限元模型，对顶板−纵肋双面焊构造的焊接过程进行数值模拟，拟合得到顶板焊趾细节沿板厚方向分布的横向残余应力分布经验公式；建立钢桥面板断裂力学数值模型，结合统一的权函数表达式，推导适用于顶板焊趾处裂纹最深点和表面点应力强度因子的新权函数，并将权函数计算的应力强度因子与有限元计算的应力强度因子进行对比。
研究结果表明：顶板−纵肋双面焊顶板焊趾处残余应力沿板厚方向处于拉—压—拉状态，呈正弦函数分布；在二次应力分布下，权函数法与有限元法计算所得顶板焊趾处裂纹最深点应力强度因子最大相对误差为7.4%，表面点应力强度因子最大相对误差为4.1%；在焊接残余应力场下，权函数法与有限元法计算所得顶板焊趾处裂纹最深点应力强度因子最大相对误差为7.6%，表面点应力强度因子最大相对误差为8.6%；权函数法能有效计算钢桥面板−肋双面焊顶板焊趾处疲劳裂纹应力强度因子。
关键词：正交异性钢桥面板；权函数法；疲劳裂纹；应力强度因子；焊接残余应力中图分类号：U443.32 文献标志码：A 文章编号：1672-7207（2024）02-0810-12Calculation method of stress intensity factor for crack of rib-to-deck double-sided welded joints in steel bridge deck consideringresidual stressXIAO Xinhui 1, CHEN Fanghuai 1, ZHANG Haiping 1, LIU Yang 2, XIAO Kanghai 1(1. School of Civil Engineering, Hunan University of Technology, Zhuzhou 412007, China;2. School of Civil Engineering, Changsha University of Science and Technology, Changsha 410114, China)收稿日期： 2023 −06 −22； 修回日期： 2023 −08 −20基金项目(Foundation item)：国家自然科学基金资助项目(51908068)；湖南省自然科学基金资助项目(2021JJ40171，2023JJ50188)；湖南省教育厅科学研究项目(23A0435, 22B0567, 22C0300)；长沙理工大学交通基础设施安全风险管理行业重点实验室(18KF04) (Project(51908068) supported by the National Natural Science Foundation of China; Projects(2021JJ40171, 2023JJ50188) supported by the Natural Science Foundation of Hunan Province; Projects(23A0435, 22B0567, 22C0300) supported by the Scientific Research Foundation of Education Department of Hunan Province; Project(18KF04) supported by the Open Fund of Industry Key Laboratory of Traffic Infrastructure Security Risk Management, Changsha University of Science & Technology)通信作者：陈方怀，博士，讲师，从事钢结构疲劳研究；E-mail ：********************.cnDOI: 10.11817/j.issn.1672-7207.2024.02.030引用格式： 肖新辉, 陈方怀, 张海萍, 等. 考虑残余应力的钢桥面板−肋双面焊裂纹应力强度因子计算方法[J]. 中南大学学报(自然科学版), 2024, 55(2): 810−821.Citation: XIAO Xinhui, CHEN Fanghuai, ZHANG Haiping, et al. Calculation method of stress intensity factor for crack of rib-to-deck double-sided welded joints in steel bridge deck considering residual stress[J]. Journal of Central South University(Science and Technology), 2024, 55(2): 810−821.第 2 期肖新辉，等：考虑残余应力的钢桥面板−肋双面焊裂纹应力强度因子计算方法Abstract:A finite element model of welding analysis was established to simulate the welding process of rib-to-deck double−sided joints, and an empirical formula for the transverse residual stress distribution along the direction of plate thickness was obtained. The weight functions for the stress intensity factor at the deepest point and surface point of the fatigue crack at the welded toe of the deck were fitted using a unified weight function expression and numerical analysis results from fracture mechanics of steel bridge decks. The stress intensity factors calculated by the weight function were compared with those calculated by finite element analysis. The results show that the residual stress at the weld toe of the rib-to-deck double−sided welded joints exhibits a tension−compression−tension state along the plate thickness direction, and follows a sinusoidal function distribution. Under the secondary stress distribution, the maximum relative error of the stress intensity factor at the deepest point of the crack calculated using the weight function method and the finite element method is 7.4%, and the maximum relative error of the stress intensity factor at the surface point of the crack is 4.1%. Under the welding residual stress field, the maximum relative error of the stress intensity factor at the deepest point of the crack at the weld toe calculated using the weight function method and the finite element method is 7.6%, and the maximum relative error of the stress intensity factor at the surface point of the crack is 8.6%. The weight function method can effectively calculate the stress intensity factor of fatigue crack at the welded toe of rib-to-deck double−sided welded joints in orthotropic steel decks.Key words: orthotropic steel deck; weight function method; fatigue crack; stress intensity factor; welding residual stress正交异性钢桥面板(简称“钢桥面板”)具有质量小、强度高、适用性范围广和工厂化程度高等优点，在大跨径钢箱梁斜拉桥和悬索桥等缆索支承桥梁中得到了广泛应用，但由于受焊接残余应力、焊接缺陷以及服役环境等诸多因素影响，导致其疲劳开裂问题突出[1−4]。
Closed Orthotropic-Rib Bridge ConstructiontOrthotropic Bridge DesignOrthotropic is a term used to describe an ortho gonal-anisoropic structure. This type of structure exhibits different mechanicalproperties along different perpendicular axes. When used to describe a bridge structure, orthotropic simply means that the bridge’smechanical properties along the direction of travel are different from those across the width. There are several types of orthotropic designs. The one used by METTLER TOLEDO consists of a deck reinforced by closed ribs with a trapezoidal shape. A proven design that is widely used in bridge construction, it was chosen to replace the driving surface of the Golden Gate Bridge. It was also used to rebuild the infrastructure in postwar Germany. So why don’t more people use it? The answer is simple: $$$. It requires a large up-front investment to produce this type of structure. Not everyone is committed to providing this level of quality. Open Rib versus Closed RibThere are many ways to design an orthotropic bridge section. The two most common are the open-rib deck (Figure 1) and the closed-rib deck (Figure 2). Please note that the closed-rib design is constructed of individual ribs, not a sandwich of plates and beams. The sandwich design does not offer the same structural efficiency as the closed-rib design. Although the open-rib and closed-rib concepts are similar, the closed-rib design is significantly stronger. Let’s take a look at why.Figure 1: Open-Rib Orthotropic Design, No Load Figure 2: Closed-Rib Orthotropic Design, No LoadA truck’s weight is concentrated at the load points where its tires meet the surface of the bridge. To reduce the stress at those load points, the bridge design should spread as much of the load as possible to the ribs adjacent to each load point. Because the closed-rib design is better at distributing the load across several ribs, it is stronger and more efficient.Notice how each design reacts when a load is concentrated directly over one of the ribs (the deflections are exaggerated to show the effect more clearly). In the open-rib design (Figure 3), the I-beams adjacent to the load point are bent. Because the load is not transferred through these adjacent beams in a straight line, they support less of the load. In the closed-rib design (Figure 4), the torsional rigidity of the ribs resists the tendency of the load to deform them. As a result, the adjacent ribs provide significant supportand the load is distributed more evenly across the ribs.Figure 3: Open-Rib Orthotropic Design, Loaded Figure 4: Closed-Rib Orthotropic Design, LoadedClosed Orthotropic-Rib Bridge ConstructionAnother benefit of the closed-rib design is that the ribs resist buckling. Figure 5 shows an open rib that is directly under the load.The beam’s web can buckle to either the right or left, causing the rib to fail. To protect against buckling, you would need to add stiffeners to the beam. Figure 6 shows one leg of a closed rib and the direction in which it will always tend to buckle. Figure 7 shows a closed rib with a trapezoidal design. Since each leg tends to buckle toward the inside of the rib, the two forces act against each other. The flat section connecting the two legs transfers the load from one leg to the other, eliminating the possibility that the rib’s legs will buckle. What if the closed rib were square (Figure 8) instead of trapezoidal? As with the I-beam design, the legs canbuckle in either direction. The flat bottom of the rib is less effective because both legs could fail in the same direction, causing therib to collapse. Figure 5 Figure 6 Figure 7 Figure 8It is difficult to analyze the actual strength of an orthotropic bridge structure. The calculated strength is usually a small factor of the actual strength. Tests have shown that the actual strength of the open-rib design is 10.3 times greater than the computed strength. Similar testing on the closed-rib design could not be completed because the test equipment failed at 42 times the computed strength (Design Manual for Orthotropic Steel Plate Deck Bridges , American Institute of Steel Construction, 1963, pp.19-20). METTLER TOLEDO Closed Orthotropic RibThe METTLER TOLEDO closed-rib orthotropic steel deck design offers several other advantages. The three main ones are aimed at extending the life of the scale by reducing metal fatigue and internal corrosion.1. The closed-rib design resists metal fatigue because there are no welds in the areas of the bridge that experience the greatest stress. The welds are located as close as possible to the neutral axis. What is the neutral axis? When a vehicle is driven onto a scale module, the module bends. Its top surface is pushed together (placed in compression), and its bottom surface is pulled apart(placed in tension). As you move downward from the top surface, the amount of compression decreases. As you move upward from the bottom surface, the amount of tension decreases. At somepoint near the center of the structure, the stress point is zero (it is in neither compression nor tension). This point is called the neutral axis. The actual position of the neutral axis will vary, depending on thegeometry of the module. In the closed-rib design, the neutral axis iscloser to the top surface of the deck (Figures 9 and 10). The greateststress is at the surface that is farthest from the neutral axis, in this casethe bottom of the rib. The welds are located in the lower stress regionwhere the ribs meet the underside of the deck plate, near theneutral axis and far from the bottom of the rib.Figure 9: Scale Cross Section 2. The design also reduces metal fatigue by using continuouswelds to join the rib to the deck plate. A start or stop in a weldincreases local stresses and the potential for failure. For thisreason, we do not use intermittent welds.Figure 10: Scale Side ViewClosed Orthotropic-Rib Bridge Construction 3. The closed-rib design helps extend the life of the scale by reducing the possibility of internal corrosion. Each rib is completely sealed to limit the amount of moisture inside the rib chamber. Once the small amount of moisture sealed inside the rib has reacted with the metal to form iron oxide (rust), the rusting process stops. Since no more moisture can penetrate the chamber, there is no possibility of the scale rusting from the inside out.METTLER TOLEDO has invested the time and money to develop and manufacture the industry’s premier vehicle scale weighbridge. If you have questions about our weighbridges or about any of the information in this data sheet, please do not hesitate to contact us for clarification.Contact your local METTLER TOLEDO authorized distributor or sales office for more information.。
桥梁建设2021年第51卷第1期（总第269期）Bridge Construction, Vol. 51# No. 1 #2021 (Totally No. 269)101文章编号！003 — 4722(2021)01 — 0101 — 08重庆土湾大桥主桥方案设计孟杰12，陈晓虎2，邓宇2，赖亚平2(1.东南大学土木工程学院，江苏南京210096;2.林同梭国际工程咨询（中国）有限公司，重庆401121)摘要：重庆土湾大桥为城市公轨两用桥，结合该桥建设条件，经比选，采用跨径布置为（95 +90 + 690 + 90 + 95)m的斜拉一自锚式悬索协作体系桥梁。
主桥中跨采用正交异性钢桥面板桁架结构，边跨采用叠合混凝土桥面板桁架结构，钢一混结合段区域正交异性钢桥的U肋间增加了板式加劲肋进行刚度过波。
主桁标准节段长为15 m，2片桁横向间距为14. 7 m，桁高13. 19 m。
梁端设置混凝土主缆型钢锚固系统。
大桥中跨中央225 m范围内主桁由16对主缆吊索支撑，其余主桁由斜拉索（共28对）支撑。
下部结构采用钻石形桥塔，塔高197 m，圆端形承台接群桩基础。
主桥边跨桁架采用顶推施工，中跨桁架采用临时拉索悬臂施工。
主缆吊索区先施工直至主桁合龙，再通过体系转换完成主缆架设。
关键词！公轨两用桥；斜拉一自锚式悬索协作体系；钢桁梁；桥塔；主缆；斜拉索；桥梁设计中图分类号：U448. 12;U442. 5 文献标志码：ADesign of Main Bridge of Chongqing Tuwan BridgeM EN G Jie12, CHEN Xiao-hu2, DENG Yu2, L A I Ya-ping2(1. School of Civil Engineering, Southeast University, Nanjing 210096 , China；2. T. Y. Lin InternationalEngineering Consulting (China) C o. L td. Chongqing 401121, China)Abstract：The Chongqing Tuwan Bridge was expected to be a road-light rail bridge. Given the construction condition, a solution of combined cabl--stayed and sel--anchored suspension systemwas determined. The bridge has five spans of 95, 90, 690, 90 and 95 m. The deck of the centralspan consists of steel trusses and orthotropic steel plates, and the decks of the side spans areformed of steel trusses and concrete slabs, with section steel-concrete composite anchorages for themain cables at the deck ends. In the steel-concrete joint zone, plate stiffeners are added betweenthe orthotropic steel plates and U ribs for stiffness transition. The trusses are measure depth and two lateral trusses are distanced 14. 7 m along bridge width. Typical truss segment15 m long. The 225 m-long steel trusses in the midspan of central span are supported by 16 pairs ofhanger cables, and the remaining steel trusses are supported by stay cables (totaling 28 pairs).The towers are d iamond-shaped towers rising 197 m, seated on round-cornered pile caps that aresupported by group pile foundations. The trusses in side spans were constructed launching, and the trusses in central span was constructed by cantilever assembly, takingadvantage of the stay cables. The steel trusses in the hanger cables region were first assembled tothe closure of the main truss, and then the systematic transformation was conducted to completethe erection of the main cables.Key words：road-light rail bridge；combined cable-stayed and self-anchored suspension system；steel truss girder；tower；main cable；stay cable；bridge design收稿日期！2020 — 02 — 26基金项目：重庆市院士牵头科技创新引导专项项目（cstc2017zdcy-yszxX0003)Academician of Chongqing Leads the Special Project of Guiding Scientific and Technological Innovation (cstc2017zdcy-yszxX0003)作者筒介：孟杰，正高级工程师，E-mail:mengfe@tylin. com. cn&研究方向：桥梁结构设计。
附录1 英文文献翻译斜拉桥斜拉桥特别适合于200m～500m的跨径范围，可以说是从连续箱梁桥到加劲悬索桥的过渡。
它起源于战后德国，当时主要是为了节约刚材。
高强钢材的应用，先进的焊接技术和电子计算机对高度超静定结构的严密分析，极大地促进了斜拉桥的发展。
斜拉桥的主要承载构件是：斜拉索、索塔和正交异性板。
简而言之，锚固于桥面板和塔上的斜拉索取消了中间桥墩，因而为航行提供了更大的桥下净宽。
桥面板通过斜拉索以扇形（集中于塔顶）或琴形（多点锚于塔）来形成对桥面板的多点支承。
密索体系减小了桥面板支承点之间的距离，并减小了结构高度，这样设计和张拉的拉索使结构的工作行为类似于刚性支承的连续梁。
由于斜拉索的阻尼作用，斜拉桥的桥面板不致像悬索桥那样容易风致摆动。
拉索可以像诺德利贝桥那样设置成单索面，可以像斯特伦松德桥或杜塞尔多夫桥那样设置成垂直双索面，也可以像塞韦林桥设置成双斜索面。
单索面体系的优点是：桥面板上的锚固区分布于交通中线上，使桥面总宽度最小。
对于双索面体系，则需要额外的桥面宽度来调节塔梁的锚固。
从美学上看，单索面体系更吸引人，因为这为两侧的司机提供了更广阔的视野。
对双索面而言，桥梁一侧的拉索会给人以交叉的感觉。
双斜索面体系，像塞韦林桥一样索从A型框架的顶尖辐射下来，使上部结构具有三维结构的性能，减小了风引起的桥面板扭振。
对单索面体系，偏心集中荷载引起的扭矩使箱形截面正交异性板的应用成为必要。
双索面体系的桥面板一般是正交异性的箱梁，也可以是预应力混凝土梁，如委内瑞拉的马拉开波桥和西德美因河上的赫希斯特桥。
拉索是预张拉的锁丝结构，这些索不易腐蚀，抗拉性强。
最后一个特点对斜拉桥是重要的，因为伸长会引起更大的弯矩和增加结构高度。
斜拉索的倾角会影响塔的高度。
塔为锚固索而在锚固点以上增加高度是正常的，就像诺德利贝桥（这种情况下，作为城市的礼物）。
不仅要考虑结构，在塔形的选择中，美学也占有突出的地位。
例如，由于塞韦林桥接近科洛涅大教堂，故采用A型框架。
Design and Construction of Movable Bridges in the USABeile Yin, Ph.D. Paul M. Skelton, P.E.Associate Principal PartnerEngineer Hardesty & HanoverHardesty & Hanover New York, NY, USANew York, NY, USABeile Yin received Paul M. Skelton receivedhis BS degree in his BEME degree in.1968 from Tsinghu 1985 from StateUniversity, China; University of New YorkPh.D.in 1986 from NC at Stony Brook, USAState University,USARobert S. Moses, PEPartnerHardesty & HanoverNew York, NY,USARobert S. Moses receivedhis BSEE degree in1991 from BucknellUniversity, New YorkUSASummaryMovable bridges are an important part of the transportation infrastructure in the USA and around the world. Some movable bridges have been in service for 60-100 years and remain in good condition. New movable bridges continue to be built yet and existing bridges must be maintained, retrofitted and upgraded to meet current transportation requirements. Engineers face unique challenges during all phases of movable bridge work from analysis to design and throughout construction, including structural, mechanical and electrical engineering.Keywords: movable, bridge, bascule, lift, swing, electrical, mechanical, trunnion, counterweight. 1INTRODUCTIONThere are many types and sub-types of movable bridges but the most popular are the simple trunnion bascule, the center bearing swing bridge, and the vertical lift bridge. The motions of the movable spans are as follows: simple trunnion bascule – rotation about a fixed horizontal axis; center bearing swing bridge – rotation about a fixed vertical axis; and vertical lift – translation along a fixed vertical axis. Descriptions of many types of movable bridges that have been constructed or proposed, including the machinery to move or stabilize the spans, may be found elsewhere. [3] [4] [5] There are a number of concerns arising from features of the movable bridge. The concept of movable bridges can be traced back to a fairly early dates in Europe and Asia. However, the earliest movable bridges, in the modern sense that serve today’s transportation needs, are better defined as later 19 century. [3]The first notable vertical lift build in the USA was the South Halstead Street Bridge at Chicago. This bridge, designed by Doctor Waddell in 1892, was constructed soon thereafter, with a span of 130 ft. and a maximum vertical clearance of 155 ft.The Van Buren Street Bridge in Chicago, a Scherzer rolling bascule design, was completed in 1893, and the famous tower bridge in London, a roller bearing, trunion bascule constructed about the same time may be re-garded as the fore-runners of the modern bascule bridge. Construction of Willis Avenue Swing Bridge soon fol-lowed and was built in 1901.2 DESIGN AND MODELINGThe AASHTO document that does cover movable bridges is the Standard Specification for Movable Bridges, 1988 (ASD) [1]. In addition, some states have supplements to AASHTO that contain provisions for movable bridges. The bridge engineer shall determine the following: 1. the type of movable bridges (swing, bascule or vertical bridge); 2. for swing bridges, Fig. 1: South Halstead Street Bridge at Chicago the type of center (swing bridges shall preferably be the center bearing type); 3. for bascule bridges, the type of bascule; 4. for vertical-lift bridges, the type of tower, the location of the prime mover or movers and the provisions for keeping the moving span level; 5. the system of emergency operation, if any, and the standby power system, if any. Loading conditions and loading combinations are described in [1], for determining the maximum and minimum stresses. Lateral loading such as wind load and seismic load shall also be given due consideration. Particularly, the seismic load is a relatively new issue, which needs engineer’s attention. The AASHTO LRFD Specification for Movable Bridges [2] contains significantly more information of the types of structural and mechanical issues than its counterpart [1]. Movable bridges can be analyzed with three-dimensional finite element software such as SAP or ADINA. Models are constructed primarily in beam elements, supplemented with shell elements, as necessary. Soil-structure interaction is modeled with foundation springs and dampers. Modeling of nonlinear elements, necessary for movable bridge analysis, is a comprehensive and time-consuming task. Fig. 2: Computer Model in ADINA This is the computer model using SAP2000 software for Roosevelt Island Lift Bridge in New York. Recently, H & H is performing rehabilitation design for electrical and mechanical system as well as seismic study, because this bridge is the only link between the island and the mainland. Fig. 3: Computer Model for Lift Span Bridge 3 UNIQUE ISSUES FOR MOVABLE BRIDGE3.1 StructuralFoundation and Substructure-Typically, both foundation and substructure of a movable bridge are very massive ensuring long-term performance of the superstructure. With the strong support, thesuperstructures can be operated in a more stable platform. The substructures with large inertias Fig. 1 South Halstead Street Bridge Fig. 1 South Halstead Street Bridgerequire the foundation having proper stiffness. In turn, foundation forces and the overall structural response are strongly affected by foundation stiffness. Both foundation capacity and foundation stiffness are strongly affected by scour. Movable bridge foundations are frequently located in a navigable stream or waterway, and therefore are more likely to be exposed to strong currents. Most of the horizontal loads from the leaf and counterweight are transferred to the foundations through the trunnion tower. As such, the trunnion tower is a critical component of the structure. Failure of the trunnion tower can represent complete failure of the structure.Superstructure - Movable superstructures are designed to be able to open and close, and as such often have unique structural systems. Of particular concern is the load path that vertical and lateral forces from the superstructure and counterweight are taken to the foundations. Bascule bridges, for example, have several key points of vulnerability along the load path from the counterweights to the foundation including the trunnion bearing which supports the movable span in the closed and open positions and the toe joints at the span end. The movable spans are designed to be lightweight, and, typically, have lightweight open steel grid decks, partially filled grid decks, or orthotropic steel plate decks.Open and closed positions - Movable bridges are analyzed in both their open and closed positions, and occasionally are analyzed for positions in between. Highway bridges are generally left in their closed position most of the time. The AASHTO Movable Bridge Specification and other codes specify that the seismic load used for the open position may be reduced by 50 percent if the bridge is in that position for less than 10 percent of the time.Counterweight - Counterweights are unique for movable bridge which balance the moving span in order to reduce the operating power. Usually found on bascule and lift bridges, these elements require special design considerations.3.2Machinery and Mechanical Devices3.2.1Span Drive MachinerySpan drive machinery moves the movable span and auxiliary machinery stabilizes or facilitates the span as a live load carrying structure when it is at rest. For some bridges, span drive machinery also serves to stabilize the movable span in certain positions. Many combinations of electrical, mechanical, and hydraulic components may be assembled to form span drives. Most US bridges have electro-mechanical span drives. The function of electro-mechanical conventional gear drives is to convert the high-speed low-torque input of an electrical motor to a low-speed high-torque output suitable for moving a heavy movable span. There are many arrangements of equipment, but usually an electric motor powers the input shaft of a primary speed reducer (which may contain a differential to permit the output shaft speeds to differ while equalizing output torque). The output shafts of the primary reducer are connected to two secondary gear reduction units. The output of each secondary reducer rotates a pinion that drives the leaf open or closed, or holds the leaf stationary by means of brakes in the drive train. On older bridges, some or all the gearings may be “open” i.e., not encased in housings.3.2.2Auxiliary Drive MachineryAuxiliary machinery supports the movable span to carry live load when it is at rest. The components are usually eletro-mechanical, but fluid power is also used. General examples of auxiliary machinery for various types of movable bridges are:●Bascule Bridges- toe and tail locks, centering devices, buffers●Swing Bridges- center and end wedges●Rim Bearing Swing Bridges-tread fastened to undersides of drum girders, tapered roll-ers, tracks, stools and racks, live rings with spiders, pivot posts, center latches ormechanisms, end lifts, rail lifts, buffers, and rigid stops●Vertical Lift Bridges- buffers.Auxiliary machinery, such as locks, wedges, bumpers, balance wheels and centering devices, are used in movable bridges to fix the structure in place either before or after it has transitioned from one position to the other. The exact behavior of mechanical device depends on the details of its design and configuration. The use of shear locks or span locks varies in purpose. On double leaf bascules, these devices transfer live loads between leaves. Most locks are positioned between the two leaves to simply transfer vertical shear. Sometimes, as in the case of some European bridges and a recent Hardesty & Hanover design, they are positioned so as to enable transfer of live load moment also. On lift bridges or single leaf bascules, these devices assist in keeping the span seated. On swing bridges, wedges or lift systems perform specific live load function to permit the span to perform reliably.3.2.3Structural/Mechanical Interface ComponentsFor the various bridges the interfacing details included are:●Bascule Bridges- trunnion bearingsRolling Lift Bridge- segmental girder/track●Vertical Lift Bridges- counterweight trunnion bearings, span guides (tower and span)●Swing Bridges- pivot bearing/rim bearingFor bascule bridges of any trunnion type, the critical connection for transmittal of loads is the trunnion bearing and trunnion shaft system. The issue critical to our case is the translation of the span in its bearing, designed to permit opening about the span’s axis of rotation. The computed horizontal load is imparted via the trunnion shaft or trunnion hub into the bearing. Most bearings are the split journal type but even roller bearings will have a jointed bearing base. This requires that the base connections to the support are designed for the full lateral force.For center bearing swing bridges all dead load goes through the pivot bearings. The bearing, whether bronze disk or roller, permits horizontal rotation of the span and permits some out of normal ‘rocking’ due to spherical surfaces. For rim-bearing swing bridges, the dead load is carried either exclusively, or mainly by a series of heavy tapered rollers centered on a post. This assembly, often known as a ‘roller nest’, will have spokes of various designs connecting their rollers to the central pivot post.3.3 Electrical SystemThe electrical systems on movable bridges consist of power distribution systems and control systems. Movable bridges powered by electrical motors are serviced by industrial-type utility power or self-contained generator sets, or both. Electrical power is distributed to the motors, lighting panels, and control components via commercially-available electrical components such as circuit breakers, fuses, transformers, and motor control centers.The principal considerations for movable bridge electrical design involve the control system and the span motor controller. Span motor controllers, also known as span drives, provide proper speed and torque control for the electric motors which operate movable bridges. Four examples of span motor controllers are reviewed: Secondary Resistance Control, Primary Thyristor (SCR) Drive, Regenerative Direct Current Drive, and Flux Vector Drive.Secondary resistance control is a non-regulated drive system that employs an AC wound rotor motor and contactors. The SCR drive offers speed control, torque control and countertorque capabilities. The DC drive provides stepless, programmable speed control of a DC motor with similar reduced seating torque, counter-torque control, and feedback loops as the AC SCR drive. Flux vector drive represents the latest in variable frequency drive technology with increasing number of applications in movable bridges.Electrical system redundancy is analyzed by deciding which subsystems are most critical, and providing alternate means of operation. Such subsystems are the power source, the span drives, motors, and their control systems. Redundancy of such subsystems improve operational reliability for waterborne users, vehicular/rail users or both. A brief overview of common components, suchas: Generator Set, Transfer Switch, Transformers, Motor Control Centers, Motors and Drives, other electrical distribution equipment, and their applications in the movable bridges, is included in the 2000 LRFD AASHTO Standard Specification for Movable Highway Bridges [2].4 BASCULE BRIDGESHardesty & Hanover designed the new 12-lane bascule bridge to replace the existing Woodrow Wilson Bridge that crosses the Potomac River in Washington DC. The new bridge comprises four side-by-side double-leaf bascule spans, each with a 270-foot center-to-center trunnion spacing and an overall bridge width of 249 feet. The bascule span is supported on V-shaped concrete bascule piers. Features of the span include a composite concrete deck, moment-resisting span locks, tail Fig. 4: New Woodrow Wilson Bascule Bridge locks and the option of independent or group leaf operation. The bridge has been designed to accommodate future plans for a transit system. Contracted by VDOT, H&H performed studies, preliminary and final design for the widening and design of a new double-leaf bascule bridge. The successfully completed project doubles the traffic capacity of I-264. Large naval vessels can now navigate through the 150-ft. wide channel with ease . Fig. 5: Berkley Bridge – Norfolk, VA 5 LIFT BRIDGE H&H.Fig. 6: New Tomlinson Bridge and its Construction, New Haven, CT Marine Parkway Vertical Lift Bridge located in New York is a beautiful signature project linking Brooklyn and Coney Island. Fig. 7: Marine Parkway Lift Bridge – Brooklyn, NY6SWING BRIDGESThis is a railroad swing bridge in Connecticut showing fullopen for navigation.Fig. 8: Railroad Swing BridgeWillis Avenue Swing Bridge, origi-nally built in 1901, connects Man-hattan and Bronx in the city of NewYork. This bridge carries 70,000vehicles per day. H & H is currentlydesigning a replacement bridge.This project covers a length ap-proximately 1 mile including Mainline, FDR Ramp and Bruckner Blvd.Ramp. The left picture is the pro-posed replacement plans.Fig. 9: Design plans for replacement of Willis Avenue BridgeThe photos to the right are the existing of ThirdAvenue Bridge and its replacementconstruction designed by H&H.ACKNOWLEDGEMENTThe authors wish to thank and acknowledgemany Hardesty & Hanover engineers for theircontributions and participation in thepreparation of this paper.Fig. 10: Construction of Third Avenue Bridge - NY REFERENCES[1] AASHTO, 1988. AASHTO Standard Specifications for Movable Highway Bridges. Washington, DC. American Association of State Highway and Transportation Officials.[2] AASHTO, 2000. AASHTO LRFD Movable Highway Bridge Design Specifications. Washington, DC. American Association of State Highway and Transportation Officials.[3] Hool, G.A. & Kinne, W.S., 1943. Movable and Long-Span Steel Bridges. New York, NY. McGraw-Hill.[4] Koglin, T.L., 2003. Movable Bridge Engineering. Hoboken, NJ. John Wiley & Sons.[5] Waddell, J.A.L., 1916. Bridge Engineering. Vols 1 & 2, New York, NY. McGraw-Hill[6] Altebrando, N.J., Yin, B., Birnstiel, C., & Ludvik, M., 2003. Seismic Analysis of Movable Bridges. 2nd New York City Bridge Conference.。
Evaluation of effective flange width by shear lag model for orthotropic FRP bridge decksBin Zou a ,An Chen a ,⇑,Julio F.Davalos a ,Hani A.Salim ba Department of Civil and Environmental Engineering,West Virginia University,Morgantown,WV 26506-6103,United States bDepartment of Civil and Environmental Engineering,University of Missouri-Columbia,Columbia,MO 65211,United Statesa r t i c l e i n f o Article history:Available online 16September 2010Keywords:Effective flange width FRP bridge deck Orthotropic plate Shear lag modelFinite Element analysisa b s t r a c tIn bridge engineering,the three-dimensional behavior of a bridge system is usually reduced to the anal-ysis of a T-beam section with a reduced width of deck in relation to center-to-center spacing of stringers,over which the longitudinal normal stresses are assumed to be uniformly distributed,which is termed as effective flange width.It is defined in the AASHTO specifications primarily for concrete slabs and has inherent applicable limitations because of its empirical nature.This paper provides an analytical shear lag model for effective flange width for orthotropic bridge decks,applicable to various materials including Fiber-Reinforced Polymer (FRP)and concrete decks.To verify this solution,a Finite Element (FE)paramet-ric study is conducted on 44simply-supported FRP deck-on-steel girder bridges.The results from the shear lag model correlate well with the FE results.The accuracy of this model is further verified by close correlations with an existing empirical solution.It is also illustrated that the shear lag model,with the introduction of a reduction factor,can be applied to predict effective flange width for FRP deck-on-steel girder bridges with partial composite action,by favorable comparisons between the analytical and test-ing results for a T-beam section cut from a one-third scaled bridge model,which consists of an FRP sand-wich deck attached to steel girders by mechanical connectors.Ó2010Elsevier Ltd.All rights reserved.1.IntroductionIn bridge engineering,a deck-and-girder system acting compos-itely is usually reduced to the analysis of a T-beam section charac-terized by elementary beam theory,and called beam-line analysis.In this approach,and consistent with beam theory,the longitudinal normal stress on the deck section is assumed to be constant.How-ever,due to in-plane shear flexibility of the deck,the longitudinal normal stress over a center-to-center bridge deck section is non-uniform along its transverse cross section,with the maximum value occurring at the mid-line junction with the girder;and grad-ually decreasing towards the center spacing line.This non-uniform distribution of stress is known as shear lag.The stress distribution depends on several factors such as cross sectional dimensions and stiffness of the deck (flange)and girder and loading conditions,resulting in analytical solutions not easily applicable in practice.Therefore,an effective flange width is used in design practice to simplify the problem.The effective width is defined as a reduced width of deck,in relation to center-to-center spacing of girders,over which the normal or longitudinal stresses are assumed to be uniformly distributed,based on the premise that the stress resul-tant over the effective width should be equal to the stress resultant over the actual flange width.This concept was adopted in both AASHTO Standard [1]and AASHTO LRFD specifications [2],defined primarily for concrete decks in composite steel bridges.The Cana-dian highway bridge design code CSA [13]defined the effective flange width in a similar manner as AASHTO [1],with formulas pri-marily developed by Cheung and Chan [12].Both AASHTO and Canadian specifications consider span length and girder spacing as the most important parameters that can affect effective flange width.The AASHTO LRFD code additionally includes slab thickness and girder dimensions.Numerous studies have been conducted on this topic,most of which were focused on concrete decks.Moffat and Dowling [22]studied the effective flange width for steel box girder bridges using Finite Element (FE)analysis.The bridges were loaded with both uniform load and point load.They concluded that the ratio of gir-der spacing to span length was the most significant factor;and loading types and positions were other factors that affected the effective flange width.Moffat and Dowling [23]later studied the effective flange width provisions in the British bridge code and pointed out that the nondimensional ratio of flange width to span length was the most dominant parameter,and girder size and deck thickness had little effect on the effective flange width for most0263-8223/$-see front matter Ó2010Elsevier Ltd.All rights reserved.doi:10.1016/pstruct.2010.08.033⇑Corresponding author.Tel.:+13042939926;fax:+13042937109.E-mail addresses:an.chen@ (A.Chen),julio.davalos@ (J.F.Davalos).practical bridges.Cheung and Chan[12]used thefinite strip meth-od to study a wide range of steel bridges and box girder bridges.It was also concluded that girder spacing and bridge span length were major factors,while slab thickness and girder sections had little effect on effectiveflange width.Also,the effectiveflange width was found to be independent of the number of traffic lanes; and upper and lower bound values could be provided from models for multiple girder bridges under uniformly distributed load and single T-beam sections under point load,respectively.Ahn et al.[6]used a simply-supported reference bridge to com-pare the values of effective width from several design specifica-tions/codes including AASHTO,BS5400,Canadian code,Japanese code,and Eurocode4.Amadio et al.[7]evaluated the effective flange width according to Eurocode4at ultimate strength state, by testing four composite T-beams until failure.They found that the effectiveflange width approached the whole slab width when the T-beam was close to failure.They concluded that it was conser-vative to define the same effectiveflange width for both service limit and ultimate limit states,but provided separate formulas for service limit and ultimate limit states.For analytical solutions,Adekola[4]developed a method that accounted for both plane stress and bending stress effects in shear lag phenomenon,where the effective breadth was subsequently di-vided into shear effective breadth and bending effective breadth. Adekola[5]proposed a more rational basis for defining effective flange width,based on girder deflection rather thanflange stress. In his definition,the deflection response of the equivalent T-sec-tion was the same as that of the actual T-section.By adopting this new definition,he studied shear lag phenomenon with partial interaction for concrete deck on steel-stringer bridges,and the re-sults showed that effectiveflange width increased with increase in degree of interaction or composite action.Song and Scordelis [25,26]conducted harmonic shear lag analysis using plane stress for theflanges of simple or continuous beams with different girder cross sections,and he presented simplified empirical formulas and diagrams for determining the shear lag effects in simple beams un-der various loading conditions.For timber bridges,Davalos and Salim[15]studied effective flange width for125stress laminated timber bridges by FE analy-sis.Empirical equations for effectiveflange width were proposed, considering major variables including girder spacing,bridge span length,ratio of girder depth to deck thickness,and ratio of longitu-dinal girder elastic modulus to deck elastic modulus.Since the mid-1990s,a new type of bridge deck consisting of Fiber-Reinforced Polymer(FRP)panels has been implemented in the United States.Although a lot of studies have been conducted on stiffness and strength evaluations of various types of FRP deck panels[8,16,10,11],there are only limited studies available for effectiveflange width accounting for its orthotropic nature.Ten-chev[27]conducted FE parametric study on shear lag phenomenon for orthotropic plates,based on which empirical equations for effectiveflange width were proposed.Keelor et al.[18]conducted afield study on a short span bridge in Pennsylvania,USA.The bridge was12.6m long withfive steel girders equally spaced at 1.8m,and had a pultruded FRP deck with a thickness of19.5cm; and the design was based on full-composite action.Their results showed that under service load condition,the effectiveflange widths corresponded to about90%of girder spacing for interior girders,and75%of half girder spacing for exterior girders,respec-tively.Keller and Gurtler[19,20]conducted lab tests on two large scale T-sections to study composite action and effectiveflange width.Each test model was7.5m long with a pultruded FRP deck section of1.5m wide adhesively bonded to the topflange of a steel supporting beam.The normal strain distribution across the width of the FRP section was recorded at both top and bottom FRP facesheets.The results showed that under service limit state,the normal stress was almost uniform across the panel section.While under failure limit state,the normal stress decreased towards the panel edges,indicating a more pronounced effect of shear lag.Considering the increasingfield implementation of FRP decks,it is advantageous to develop an analytical model to calculate effec-tiveflange width for orthotropic bridge decks,which is the objec-tive of this ing a harmonic analysis developed for FRP thin-walled sections[24],a shear lag model is proposed,as de-scribed next.2.Shear lag modelFor an orthotropic deck supported by steel girders with full-composite action,the deck panel can be assumed to be stiffened by the two girders,as shown in Fig.1.The following assumptions are adopted in this model to simplify the problem:(1)the axial force N y and moment M y are assumed to be zero and(2)the twist-ing moment in the plate M xy=0.When the bridge is subjected to out-of-plane load,the stress distribution due to out-of-plane mo-ment is shown in Fig.2a,where the deck is under compression and the steel girder is under tension,and the compression force in the deck is transferred through shear connections at the inter-face between the deck and steel girder.Therefore,only edge shear tractions N xy and axial force N x are acting on the panel,as shown in Fig.3,and the constitutive and compliance matrices can be given asN xN xyM x8><>:9>=>;¼A11000A66000D11264375e xcxyj x8><>:9>=>;ð1Þe xcxyj x8><>:9>=>;¼1A110001A660011126643775N xN xyM x8><>:9>=>;ð2ÞBased on force equilibrium for an infinitesimal section of the panel as shown in Fig.4,the equilibrium equations can be expressed as @N x@xþ@N xy@y¼0ð3ÞThe compatibility equation is given as@2e x@y2þ@2e y@x2¼@2c xy@x@yð4ÞNeglecting transverse normal strain e y and assuming that M x re-mains constant along the y-direction,the governing differential equation can be obtained by substituting Eqs.(3)and(4)into Eq.(2)as1A11@2N x@y2þ1A66@2N xy@x2¼0ð5ÞB.Zou et al./Composite Structures93(2011)474–482475Eq.(5)can be reduced to an ordinary differential equation by using harmonic analysis proposed by Salim and Davalos[24],which was used to analyze shear lag for thin-walled open and closed compos-ite beams.The panel in Fig.1is simply supported at x=0,a.Thus, the axial panel force can be defined asN xðx;yÞ¼X1j¼1N jðyÞsinj p xað6Þwhere N j(y)is an amplitude function.Substituting Eq.(6)into(5) leads to2ffiffiffiffiffiffiffis!@N xy¼ÀX1j¼1j p½C1j coshðn j yÞþC2j sinhðn j yÞ cosj p xð9ÞThe in-plane shear variation can be defined as[9]@N xy@y¼ÀVðxÞDðyÞþB cos/h ið10Þwhere A is the extensional stiffness of the cross section;B is thebending-extension coupling stiffness,which can be neglected sincethe orthotropic FRP panel is usually designed as symmetric and bal-anced;e(y)is the distance between the neutral axis of the cross sec-tion and the middle surface of theflange;V(x)is the resultant shearforce acting on the cross section;D is the cross section bending stiff-@N xy @y ¼X 1j ¼1Q j cos j p xa ð11ÞAnd the coefficient in Eq.(11)can be defined asQ j ¼2aZa@N xy @ycos j p xadx ð12ÞSubstituting Eq.(10)into (12),we have:Q j ¼À2Ae ðy ÞaDI j ð13Þwhere I j depends on loading condition.If the origin of the y -axis is located at the center of the cross section,then C 2j =0due to symme-try,and Eq.(8)can be reduced toN j ðy Þ¼C 1j cosh ðn j y Þð14ÞBy ensuring compatibility of shear flow at the junction of flange and web (y =Àb /2,b /2),C 1j can be obtained by equating Eqs.(9)and(11).Therefore,the normal force resultant and normal stress alongthe panel can be obtained asN x ðx ;y Þ¼ÀX 1j ¼1a j p Q jcosh ðn j y Þcosh n j b 22435sin j p xð15Þr x ðx ;y Þ¼N x ðx ;y ÞAð16ÞBased on definition of effective flange width,the longitudinal nor-mal stress is assumed to be uniformly distributed along the panel section,as shown in Fig.2b.The effective flange width can be ex-pressed as the integral of normal stress distribution divided by the maximum stress r max at the panel-stiffener intersection asb e ¼R b =2Àb =2r x dxr max¼2R b =2r x dxr maxð17ÞFinally,substituting Eqs.(15)and (16)into (17),and taking only the first-term approximation for simplicity,the expression for effective flange width is given asb e ¼R b =2cosh ðn 1y Þdycosh n 1b 23.Finite Element study 3.1.FE model descriptionsTo verify the shear lag model developed FE models for 44simple-span FRP deck-over-steel Fig.5.Typical cross section of bridge model.CS2CS3CS49.39.313.08.538.5312.22.60 1.98 3.510.770.69 1.230.200.200.24454parameters.Fig.6.AASHTO HS20truck live load.were constructed using ABAQUS[3].The bridge deck was an FRP honeycomb panel produced by Kansas Structural Composites,Inc. (KSCI),which can be idealized as a structurally orthotropic panel with homogeneous equivalent engineering properties,as shown in Table1[17].The bridge configurations considered are listed in Fig.5and Table2.Two and three lanes were considered,with widths of9.31m and12.97m,respectively.Other varied geometric parameters included span length,with11lengths selected from 15.24m to91.44m at increments of7.62m;and girder spacing, with1.98m,2.59m,and3.51m for two-lane bridges,and3.51m for three-lane bridges.The FRP panel and steel I-beam components were modeled using shell elements(S4R),and beam elementsThe following assumptions are adopted in the FE study to sim-plify the analysis effort while retaining adequate accuracy:(1)The bridge deck is idealized as a homogeneous,elastic and orthotropic slab with uniform thickness;(2)the slab is supported by equallyFig.7.Stress integration alongflange width.shear lag model and FE.Span(m)S/L b e/S B e/SShear lag FE Table5Comparison between shear lag model and empirical function for E/G=10.b/L b e/b Diff.(%)Empirical Shear lag0.880.2440.229À6.200.780.2700.258À4.58Fig.8.Photo(a)and details(b)of scaled bridge model.478 B.Zou et al./Composite Structures93(2011)474–482spaced I-shaped steel girders;(3)the edges of the slab-and-girder ends are simply supported at the abutments;(4)full-composite ac-tion is assumed between the supporting girders and slab,i.e.,no interface slip at the girder–slab interface;and(5)based on design guidelines,no truck wheel load can be placed closer than0.61m from the roadway edge.For simplicity,the assumption of full-composite action is adopted in the FE modeling,with the purpose of verifying the accuracy of the shear lag model developed in the previous section.In general,however,and as noted by Machado et al.[21],most FRP deck-on-steel girder systems are characterized by partial composite action.In that case,a reduction factor needs to be applied to full-composite action in order to apply the shear location to induce maximum moment in the bridge models.For cross section CS1,CS2,and CS3,one-and two-lane loaded cases were selected.For cross section CS4,one-,two-,and three-lane loaded cases were selected,as shown in Fig.6.3.3.Data reduction from FE resultsSimilar to the shear lag model,Eq.(17)was used to calculate effectiveflange width.The average normal stress,r x,which can be assumed uniform along its segmentally discrete width h,can be obtained from the output for each shell element,as shown in Fig.7.The integral of the normal stresses,or normal stress resul-Fig.10.T-beam test model.B.Zou et al./Composite Structures93(2011)474–482479boundary conditions;concentrated and uniformly distributed loading conditions;and different cross sections.A total of640FE models were analyzed to obtain the effectiveflange width,based on which empirical functions were proposed using regression tech-nique ask emp¼b eb¼0:57C1C2ðbLÞÀ0:85EGÀ0:416ð20ÞC1¼1þ5e X;X¼À6:4bLffiffiffiffiEGrð21ÞC2¼1þ0:31eÀ0:9Y Y¼EGbLÀ1ð22Þwhere b e is the effectiveflange width;b,the actual beamflange width;L,span length;E,beamflange Young’s modulus in longitudi-nal direction;and G is the beamflange in-plane shear modulus.Both the proposed shear lag model and the empirical function of Eq.(20)were used to study a simulated T-beam model with vary-ing modulus ratio E/G and aspect ratio b/L.Three modulus ratios were selected:1,10,and30.The aspect ratio was varied from 0.1to1for E/G=1;and from0.08to0.88for E/G=10;andfinally from0.03to0.60for E/G=30.The boundary condition was as-sumed to be simply supported and the loading was assumed to be uniformly distributed.The results are compared in Tables4–6 for E/G=1,10,and30,respectively.For E/G=1,the shear lag model consistently overestimates the effective width.The average differ-ence is about6.6%between the two models.The results are very close for b/L<0.5,as shown in Table4.For E/G=10and30,the average differences are1.5%and3.8%,respectively.Overall,there is a good correlation between the two models,which can further verify the accuracy of the shear lag model developed in this study.It is noted that,because of the nature of the regression technique, the applications of the empirical functions are limited,while the shear lag model can be applied to a wider range of problems and is more suitable for design and parametric study purposes.5.Application of shear lag model to FRP deckThe proposed shear lag model was further used to calculate effectiveflange width for a scaled FRP deck-on-steel girder bridge model.5.1.Bridge model descriptionA one-third scaled bridge model with a span of5.5m was con-structed consisting of three steel girders(W16Â36,Gr50)spaced at1.22m on center(Fig.8a and b).A5.5mÂ2.74mÂ0.13m FRP deck was used,consisting of three1.8m wide by2.74m long indi-vidual FRP honeycomb panels from KSCI,which were assembled using tongue-and-groove connections along the two2.74m trans-verse joints.The deck was attached to the girders using a prototype stud-sleeve connector[14].The longitudinal direction of the hon-eycomb core(Fig.9)was perpendicular to the traffic direction.5.2.T-section model description and test setupAfter completion of a series of static and fatigue tests on the scaled bridge,a T-section was cut-out from the center portion of the bridge.This section had aflange width of1.22m,as illustrated in Figs.8b and10,which was longitudinally supported by a center steel girder.Three brackets were placed on each side of theflangeto provide lateral support to the flange section.To study effective flange width,a patch load was applied at the mid-span of the T-beam over an area of 0.6Â0.25m 2using a 490kN actuator,as shown in Fig.10.The system was subjected to three-point bending with displacement control at a rate of 1mm/min within service load limit,and load–displacement relation was recorded.A total of 20strain gauges were attached at top and bottom surfaces of the deck,which 10at quarter-span and 10at mid-span,as shown in Fig.11,to measure the longitudinal normal strain of the FRP flange.The strain profiles at mid-span are plotted for both top and bot-tom surfaces in Figs.12and 13,respectively.These strain values are curve-fitted by approximate ing a similar tech-nique as that used for data reduction of FE results the effective flange width was calculated to be about 0.63m,which is about 50%of the actual flange ing the shear lag model,the effec-tive flange width is predicted to be 1.01m,which is higher than the testing results because full-composite action is assumed in the present model.Therefore,a Reduction Factor R is suggested in or-der to account for the effect of partial composite action.Accord-ingly,based on Eq.(18),we can define b e ,for partial composite action,asb e ¼RR b =2cosh ðn y Þdy cosh n b 2ÀÁn ¼p a ffiffiffiffiffiffiffiA 11A 66s !ð23ÞFor the scaled bridge model described above,R =0.62based ona correlation between testing and analytical results,corresponding to a 25%composite action as reported by Davalos et al.[14].Thus,B.Zou et al./Composite Structures 93(2011)474–482481based on an approximate regression of limited data,the following equation is proposed to calculate R for a given Degree of Composite Action (DCA)asR ¼1:0251À0:0244DCAð24Þwith the corresponding curve shown in Fig.14.More tests are needed in order to improve the accuracy of Eq.(24).6.ConclusionsIn this paper,a shear lag model is proposed to calculate effective flange width for orthotropic decks on steel girder bridges.To verify this solution,a Finite Element parametric study is conducted on 44simply-supported FRP deck-on-steel girder bridges assuming full-composite action.By comparing effective flange widths from FE analysis and shear lag model,it is found that the shear lag model predicts the effective flange width fairly well for interior girders,with an average difference of about 6%.The accuracy of this model is further verified by favorable correlation with an existing empir-ical solution,with average differences of 6.6%,1.5%,and 3.8%for E /G =1,20,and 30,respectively.A one-third scaled bridge model consisting of an FRP sandwich deck attached to three steel girders by mechanical connectors was constructed,which could achieve 25%composite action as reported in a separate study [14].A T-beam section was cut-out from the scaled bridge to be tested under three-point bending to obtain effective flange width.By comparison between the analytical and testing results for the T-beam model,it is illustrated that the pro-posed shear lag model can be further applied to predict effective flange width for bridges with partial Degree of Composite Action (DCA),with the introduction of a reduction factor R ,for a specified percent of composite action.To this effect,an empirical relation between DCA and R is suggested,which can be further refined by additional experimental data.AcknowledgementsWe gratefully acknowledge financial support from the NSF Part-nerships for Innovation 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