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南宁市五象大桥施工图设计,南宁市,大桥,施工图,设计
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南宁市五象大桥施工图设计论文姚勇号：级：田仲初201018020116桥梁10-01班蒋田勇毕业设计(论文)开题报告题目：课 题 类 别： 设计 学 生 姓 名：学班专业（全称）： 土木工程专业（桥梁工程）指 导 教 师：2014年3月一、本课题设计（研究）的目的：(1)通过桥梁毕业设计，掌握桥梁的基本概念，学会桥梁的基本理论和基本计算方法；(2)通过独立完成桥梁毕业设计，提高学生独立分析问题和解决问题，综合运用各种所学知识的能力；(3)提高桥梁结构分析能力和运用电算的能力，掌握一到两种商业软件及计算机辅助设计（CAD）等基本技能(4)通过桥梁毕业设计,熟悉桥梁设计的整个过程，加强对规范手册的了解和应用；(5)掌握计算绘图、查阅文献、使用桥梁规范和设计手册、编写技术文件和运用办公软件等基本技能,树立正确的设计思想,逐渐掌握设计原则,设计方法和设计步骤；(6)学会设计的基本原则、设计方法和设计步骤，为毕业后从事桥梁技术工作打下坚实的基础；(7)加强严肃认真，一丝不苟的学习态度和培养刻苦钻研、勇于创新的科学精神；(8)学会查阅外文文献并对其进行翻译，了解论文的标准格式。二、设计（研究）现状和发展趋势（文献综述）：根据查阅的资料，五象大桥是连接新区核心区与柳沙片区、城市中心区的一条便捷交通通道。它南起五象新区8号路与五象大道交叉口，向北跨越堤园路、青环路后，终点为柳沙片区英华路。它的北岸离南宁大桥较近，大致位置在南宁大桥上游约2公里处。在本次设计中，以混凝土简支梁桥和混凝土连续梁桥为首选的两种方案，现对其现状及其未来发展综述如下：1、混凝土简支梁桥梁式桥是我国一种非常普遍的桥型，它的适用范围非常广泛。50年来，新中国桥梁建设取得了突飞猛进的发展，公路铁路两用桥梁向着大跨度、重荷载、高时速方向发展。从大桥主跨上来看，武汉长江大桥主跨为128m。就公路简支桥梁而言，我国自1956年建成第一座跨径20米的预应力混凝土梁桥后，于1988年在浙江省建成了跨径为62m的飞云江桥，这是国内跨度最大的预应力混凝土简支梁桥。国内已建的简支梁桥还有1956年建成的北京哑巴河桥，该桥位于北京至周口店的公路上，是我国第一座预应力混凝土桥。该桥为跨径20米的装配式简支T梁桥，桥宽净6m，由6片T梁组成。再有1989年建成的浙江飞云江桥，它是我国最大跨度的预应力混凝土简支梁桥。桥全长1721m，分跨为18*51+5*62+14*35m，最大跨度为62m，梁高2.85m，桥面宽13m，由5片主梁构成，翼缘宽2.5m，安装后下翼缘间设置12cm厚底板，形成4横箱截面。2006年建成通车的河南开封黄河大桥，此桥全长7.8公里，共108孔，其中77孔为50m的预应力混凝土简支T型梁。国外的简支梁桥有法国巴黎加里利亚诺桥，跨越巴黎塞纳河之上，由3跨组成，焊接到6片横梁上，只在靠近河岸边有两个桥墩；还有1966年在美国宾夕法尼亚州的费城建成的西江驱动桥等，随着新技术、新材料和新工艺的发展，梁桥的跨度也将有所增加。梁桥按受力体系大致可以分为：简支梁桥、连续梁桥和T型刚构桥。简支梁桥主梁以孔为单位，两端设有支座，属于静定结构，桥梁的最大弯矩发生在中跨，若地基不均匀沉降时，上部结构内力不受影响；若桥梁的一个孔遭受破坏，邻孔也不会受到牵连，它主要适用于中，小跨度的桥型设计。混凝土简支梁桥按承重结构的横截面形式，可分为板桥、肋梁桥和箱型梁桥。简支梁桥的优缺点：简支梁桥具有可以就地取材，施工耐久性好，适应性强，受力明确，整体性好以及构造简单美观等优点，是中小型跨度桥梁中应用最广泛的桥型。受材料、结构的影响，简支梁桥的跨径一般不能超过50米，且桥墩很多。简支梁桥的发展趋势：随着高速公路、铁路和高速铁路的建设，混凝土简支梁桥仍然是桥梁的主要桥型，并且会向整体大跨度方向发展，因此简支梁桥的施工设备和技术尚需进一步地研究。尤其是架桥机的发展，尚应研发能够适应较大纵横坡及曲线桥梁，以防止架桥机在工作过程中出现机毁人亡的事故。2、混凝土连续梁桥两跨或两跨以上连续的梁桥，属于超静定体系。连续梁在恒活载作用下，长胜的支点负弯矩对跨中正弯矩有卸载的作用，使内力状态均匀合理，因而梁高可以减小，节省材料，并且刚度大，超载能力大，整体性好，安全性能好，桥面伸缩缝少。连续梁是一种古老的结构体系。自60年代中期在德国莱茵河上采用悬臂浇注法建成Bendorf桥以来，悬臂浇注施工和悬臂拼装施工法得到不断改进和完善并且推广应用，从而使得预应力混凝土连续梁桥成为许多国家广泛采用的桥型之一。我国自50年代中期开始修建预应力混凝土梁桥，至今也已有40多年的历史，近年来发展迅速，在预应力混凝土连续桥梁的设计、结构分析、试验研究、施工工艺等都已达到很高地水平。而在50年代前，预应力混凝土连续梁虽然是常被采用的一种体系，但跨径都在百米以下。当时主要采用的是满堂支架施工，费时费工，限制了它的发展。50年代后，预应力混凝土桥梁采用悬臂施工方法后，加速了它的发展。随着悬臂浇筑施工法和悬臂拼装施工法的不断改进和完善及推广应用，在跨度为40200米范围内的桥梁中，连续梁桥逐步占据了主导地位。目前，无论是城市桥梁、高架道路还是跨河大桥，预应力混凝土连续梁都发挥了它独特的优势，成为优胜方案。国内已建成的有2003年通车的东陵浑河大桥，此桥位于沈阳，跨越浑河，是连接沈阳市区与浑南新区的交通动脉，桥宽49.5m，双向10车道；还有1983年建成的大光明桥，此桥位于天津，跨越海河，连接天津市河东区十一经路与和平村区阜道，全桥长133.3m，宽30.5m，具有古典欧式风格。国外已建成的有1973年建成的里约尼泰罗伊桥，该桥位于巴西里约热内卢的瓜那巴拉湾。目前，该桥是南半球最长的预应力混凝土桥梁，桥梁全长13290m；还有建于1966年跨越苏格兰泰河湾的泰河公路桥，全长2250m。预应力混凝土连续梁桥是预应力桥梁中的一种，它具有整体性能好、结构刚度大、变形小、抗震性能好，特别是主梁变形挠曲线平缓，桥面伸缩缝少，行车舒适等优点。加上这种桥型的设计施工均较成熟，施工质量和施工工期能得到控制，成桥后养护工作量小。预应力混凝土连续梁的适用范围一般在150m以内，上述种种因素使得这种桥型在公路、城市和铁路桥梁工程中得到广泛采用。近20年来，我国已建成的具有代表意义的连续梁桥主要有跨径90m的哈尔滨松花江大桥，跨径84m+120m+120m+120m+84m湖南常德沅水大桥和跨径为85m+154m+85m的云南六库怒江大桥等。连续梁桥的优势:内力状态较均匀，可以减小梁高，节省材料，养护简易；变形小，结构刚度大，超载能力大，安全度高；特别是主梁变形挠曲线平缓，抗震能力强，桥面伸缩缝少、行车平顺舒适，整体性好等优点。(二）设计方案比选2.1 设计标准：(1)公路等级：高速公路(2)设计行车速度：120公里/小时(3)设计荷载：公路-I级方案1：采用的是不等跨变截面的预应力混凝土连续80+130+80的三跨布置梁桥，主桥采用单箱单室薄臂箱形截面。主桥上部结构采用混凝土箱形梁，支点处高7.5m，跨中梁高3.0m，悬臂长度取3.0m，箱梁底宽8.0m，箱梁顶宽14m，箱梁顶板厚度30cm。车行道为2%双向横坡，人行道为双向1.5横坡。桥面全宽（分两幅设计）：0.25m（人行道栏杆）+2m(人行道)+33.75m(车行道)+0.5m防撞墙，全宽14m。桥面铺装层厚度为10cm的沥青混凝土。立面图尺寸（mm）80060 40300150150图2.1140030040图2.1 主梁跨中截面尺寸图（单位：cm）80015030014003001509060 40图2.2 主梁支点截面尺寸图（单位：cm）方案2：采用的是简支T梁桥是静定结构，构造简单，内力不受基础不均匀沉降等附加变形影响，对基础要求较低，设计计算方便，跨径不大，各梁手里较为独立，行车舒适，线型简洁美观。简支简支T梁桥可分为整体式和装配式两种结构。这里考虑采用装配式简支结构，其具有建桥速度快、工期短、模板支架少等优点而应用广泛。T 形主梁是使用最普遍的结构形式 ，其优点是制造简单、整体性好、接头施工也方便。同时做预应力结构适用跨径为 20m50m 之间，符合此设计跨径要求。主梁纵断面尺寸（mm）主梁横断面尺寸（mm）方案一80+130+80m混凝土连续梁桥方案二40+5*44+40m简支梁桥图2.3图2.42.2 方案比选根据设计宗旨，桥型方案应从安全、适用、经济、美观等原则出发，考虑多方面的因素来进行比选。最终所选桥型力求技术先进，安全可靠，适用耐久，美观经济；另一方面，所选桥型应满足所需施工设备少、工艺简单的要求，以减小施工难度、加快施工进度、保证施工质量。综合以上因素，比选结果见表2.2.1如下：表2.2.1方案序号桥型290m 290m公路一级 公路一级1.单孔130m主跨跨越主车道， 1.适用于中小跨径桥梁；1.主梁采用不等跨截面设计， 1.T形梁桥，构造简单，受力桥梁外形美观，曲线线形与周 桥梁线型简单单调，但也不缺经济 较经济选用 弃用设计跨径荷载标准适用性与车道适应性好； 2.在城市立交桥中广泛使用；2.路边填挖方较少； 3.构造简单，跨径较小的桥型3.两岸副孔为80m，伸缩缝少， 4.横隔梁保证各根主梁相互便于截面连续； 连接成整体，以提高桥梁的整4.使用变截面可以提高桥下 体刚度；净空；安全性可以降低跨中的设计弯矩，也 明确；有余力适应支点截面剪力很 2.施工采用吊装施工，需要一大的要求； 定的吊装设备来保证工期；2.采用悬臂法施工，施工阶段主梁刚度大，且内力与运营阶段一致；3.采用变截面梁高，可以节省材料，减轻自重，更加安全可靠；4.中间无伸缩装置，行车平顺舒适；美观性围环境更加协调； 乏这特有的简单之美；经济性结论2.3最终方案选定综上所述，决定使用方案1（80m+130m+80m）预应力混凝土连续梁桥作为设计方案。三、设计（研究）的重点与难点，拟采用的途径（研究手段）：3.1.重点与难点本设计桥梁的形式为混凝土连续梁桥，适合采用悬臂浇筑和悬臂拼装，但是大跨径采用悬臂施工时，存在墩梁临时固结和体系转换的工序，结构稳定性应予以重视，施工较为复杂；此外，主墩需要布置大型橡胶支座，存在养护上甚至更换上的麻烦。实际施工中受各种因素的干扰，可能导致合拢困难，使成桥线形与内力状态偏离设计要求，给桥梁施工安全、外形、可靠性、行车条件和经济性等方面带来不同程度的影响。3.2研究手段因此在施工过程中，必须实施有效的施工控制，主要包括线形控制、应力控制、结构安全控制、温度观测和有关材料试验控制等。从某种意义上讲，线形控制尤其是悬浇施工中挠度控制和预拱度控制是保证桥梁线形符合要求的必不可少的措施。施工挠度的计算与控制是桥梁悬臂施工中的难点，科学合理地确定悬臂浇筑梁段的预拱度设置是至关重要的。只有预拱度设置合理，才能保证一个跨径内将要合拢的两个悬臂端在同一水平线上，也才能保证桥梁上部结构在施工和运营一定时间后能达到设计所期望的标高线形。四、设计（研究）进度计划：1、英语翻译：1-2周；2、方案必选：3-4周；3、结构计算（计算软件学习）：1-7周；4、结构计算（成桥阶段计算）：7-9周；5、施工方法设计：10周；6、成桥阶段计算分析：11周；7、荷载组合承载能力分析：12-13周；8、绘图：14周；9、完成设计递交修改：15周；10、完成修改进行答辩：16-19周。五、参考文献：1 JTG B01-2003, 公路工程技术标准S. 北京：人民交通出版社，2004.2 JTG D60-2004, 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(2010).AASHTO LRFD bridge design specifications, 5thEd.,Washington, DC.19 ABAQUSComputer software. Providence, RI, Dassault Systmes SimuliaCorp.20FiggBridgeEngineers.(2008).I-35WSt.AnthonyFallsBridgeinspectionand maintenance manual, Figg Bridge Engineers, Dallas.21French,C.E.W.,Shield,C.K.,Stolarski,H.K.,Hedegaard,B.D.,andJilk, B. J. (2012).“Instrumentation, Monitoring, and Modeling of the I-35WBridge.”Rep.MN/RC2012-24,MinnesotaDept.ofTransportation,Minneapolis.22Minnesota Dept. ofTransportation. (2008).“As-built construction planfor bridge nos. 27409 and 27410. ” State Proj. 2783-120, Fed.Proj.E.R.MN07(300), Minneapolis.23R.J.Watson,Inc.(2008).“St.AnthonyFalls(I-35W)bridgereplacement.”24 State Proj. 2783-120, Fed. Proj. E.R.MN07(300), Amherst, NY.日日指导教师意见签名：月教研室（学术小组）意见教研室主任（学术小组长）（签章）：月毕业设计毕业设计( (论文论文) )开题报告开题报告题目题目：南宁市五象大桥施工图设计南宁市五象大桥施工图设计课课 题题 类类 别：别： 设计设计 论文论文学学 生生 姓姓 名：名：姚勇姚勇学学号：号：201018020116201018020116班班级：级：桥梁桥梁 10-0110-01 班班专业（全称专业（全称） ： 土木工程专业（桥梁工程）土木工程专业（桥梁工程）指指 导导 教教 师：师：田仲初田仲初蒋田勇蒋田勇20142014 年年 3 3 月月 一、一、本课题设计（研究）的目的本课题设计（研究）的目的：(1)通过桥梁毕业设计，掌握桥梁的基本概念，学会桥梁的基本理论和基本计算方法；(2)通过独立完成桥梁毕业设计， 提高学生独立分析问题和解决问题，综合运用各种所学知识的能力；(3)提高桥梁结构分析能力和运用电算的能力， 掌握一到两种商业软件及计算机辅助设计（CAD）等基本技能(4)通过桥梁毕业设计,熟悉桥梁设计的整个过程，加强对规范手册的了解和应用；(5)掌握计算绘图、查阅文献、使用桥梁规范和设计手册、编写技术文件和运用办公软件等基本技能,树立正确的设计思想,逐渐掌握设计原则,设计方法和设计步骤；(6)学会设计的基本原则、 设计方法和设计步骤，为毕业后从事桥梁技术工作打下坚实的基础；(7)加强严肃认真，一丝不苟的学习态度和培养刻苦钻研、勇于创新的科学精神；(8)学会查阅外文文献并对其进行翻译，了解论文的标准格式。二、设计（研究）现状和发展趋势（文献综述设计（研究）现状和发展趋势（文献综述） ：根据查阅的资料，五象大桥是连接新区核心区与柳沙片区、城市中心区的一条便捷交通通道。它南起五象新区 8 号路与五象大道交叉口，向北跨越堤园路、青环路后，终点为柳沙片区英华路。它的北岸离南宁大桥较近，大致位置在南宁大桥上游约 2 公里处。在本次设计中，以混凝土简支梁桥和混凝土连续梁桥为首选的两种方案，现对其现状及其未来发展综述如下：1 1、混凝土简支梁桥、混凝土简支梁桥梁式桥是我国一种非常普遍的桥型，它的适用范围非常广泛。50年来，新中国桥梁建设取得了突飞猛进的发展，公路铁路两用桥梁向着大跨度、重荷载、高时速方向发展。从大桥主跨上来看，武汉长江大桥主跨为128m。就公路简支桥梁而言， 我国自1956年建成第一座跨径20米的预应力混凝土梁桥后，于1988年在浙江省建成了跨径为62m 的飞云江桥，这是国内跨度最大的预应力混凝土简支梁桥。国内已建的简支梁桥还有1956年建成的北京哑巴河桥，该桥位于北京至周口店的公路上，是我国第一座预应力混凝土桥。该桥为跨径20米的装配式简支 T 梁桥，桥宽净6m，由6片 T 梁组成。再有1989年建成的浙江飞云江桥，它是我国最大跨度的预应力混凝土简支梁桥。桥全长1721m，分跨为18*51+5*62+14*35m，最大跨度为62m，梁高2.85m，桥面宽13m，由5片主梁构成，翼缘宽2.5m，安装后下翼缘间设置12cm 厚底板，形成4横箱截面。2006年建成通车的河南开封黄河大桥，此桥全长7.8公里，共108孔，其中77孔为50m 的预应力混凝土简支 T 型梁。国外的简支梁桥有法国巴黎加里利亚诺桥，跨越巴黎塞纳河之上，由3跨组成，焊接到6片横梁上，只在靠近河岸边有两个桥墩；还有1966年在美国宾夕法尼亚州的费城建成的西江驱动桥等，随着新技术、新材料和新工艺的发展，梁桥的跨度也将有所增加。梁桥按受力体系大致可以分为：简支梁桥、连续梁桥和 T 型刚构桥。简支梁桥主梁以孔为单位，两端设有支座，属于静定结构，桥梁的最大弯矩发生在中跨，若地基不均匀沉降时，上部结构内力不受影响；若桥梁的一个孔遭受破坏，邻孔也不会受到牵连，它主要适用于中，小跨度的桥型设计。混凝土简支梁桥按承重结构的横截面形式，可分为板桥、肋梁桥和箱型梁桥。简支梁桥的优缺点：简支梁桥具有可以就地取材，施工耐久性好，适应性强，受力明确， 整体性好以及构造简单美观等优点， 是中小型跨度桥梁中应用最广泛的桥型。受材料、结构的影响，简支梁桥的跨径一般不能超过50米，且桥墩很多。简支梁桥的发展趋势：随着高速公路、铁路和高速铁路的建设，混凝土简支梁桥仍然是桥梁的主要桥型，并且会向整体大跨度方向发展，因此简支梁桥的施工设备和技术尚需进一步地研究。尤其是架桥机的发展，尚应研发能够适应较大纵横坡及曲线桥梁，以防止架桥机在工作过程中出现机毁人亡的事故。2 2、混凝土连续梁桥、混凝土连续梁桥两跨或两跨以上连续的梁桥，属于超静定体系。连续梁在恒活载作用下，长胜的支点负弯矩对跨中正弯矩有卸载的作用，使内力状态均匀合理，因而梁高可以减小，节省材料，并且刚度大，超载能力大，整体性好，安全性能好，桥面伸缩缝少。连续梁是一种古老的结构体系。自 60 年代中期在德国莱茵河上采用悬臂浇注法建成 Bendorf 桥以来，悬臂浇注施工和悬臂拼装施工法得到不断改进和完善并且推广应用，从而使得预应力混凝土连续梁桥成为许多国家广泛采用的桥型之一。我国自 50年代中期开始修建预应力混凝土梁桥，至今也已有 40 多年的历史，近年来发展迅速，在预应力混凝土连续桥梁的设计、结构分析、试验研究、施工工艺等都已达到很高地水平。而在 50 年代前，预应力混凝土连续梁虽然是常被采用的一种体系，但跨径都在百米以下。当时主要采用的是满堂支架施工，费时费工，限制了它的发展。50 年代后，预应力混凝土桥梁采用悬臂施工方法后，加速了它的发展。随着悬臂浇筑施工法和悬臂拼装施工法的不断改进和完善及推广应用，在跨度为 40200 米范围内的桥梁中，连续梁桥逐步占据了主导地位。 目前，无论是城市桥梁、高架道路还是跨河大桥，预应力混凝土连续梁都发挥了它独特的优势，成为优胜方案。国内已建成的有 2003 年通车的东陵浑河大桥，此桥位于沈阳，跨越浑河，是连接沈阳市区与浑南新区的交通动脉，桥宽 49.5m，双向 10 车道；还有 1983 年建成的大光明桥，此桥位于天津，跨越海河，连接天津市河东区十一经路与和平村区阜道，全桥长 133.3m，宽 30.5m，具有古典欧式风格。国外已建成的有 1973 年建成的里约尼泰罗伊桥，该桥位于巴西里约热内卢的瓜那巴拉湾。目前，该桥是南半球最长的预应力混凝土桥梁，桥梁全长 13290m；还有建于 1966 年跨越苏格兰泰河湾的泰河公路桥，全长 2250m。预应力混凝土连续梁桥是预应力桥梁中的一种， 它具有整体性能好、 结构刚度大、变形小、抗震性能好，特别是主梁变形挠曲线平缓，桥面伸缩缝少，行车舒适等优点。加上这种桥型的设计施工均较成熟，施工质量和施工工期能得到控制，成桥后养护工作量小。预应力混凝土连续梁的适用范围一般在150m 以内，上述种种因素使得这种桥型在公路、城市和铁路桥梁工程中得到广泛采用。近20年来，我国已建成的具有代表意义的连续梁桥主要有跨径90m 的哈尔滨松花江大桥，跨径84m+120m+120m+120m+84m湖南常德沅水大桥和跨径为85m+154m+85m 的云南六库怒江大桥等。连续梁桥的优势:内力状态较均匀， 可以减小梁高， 节省材料， 养护简易； 变形小，结构刚度大，超载能力大，安全度高；特别是主梁变形挠曲线平缓，抗震能力强，桥面伸缩缝少、行车平顺舒适，整体性好等优点。(二）设计方案比选2.12.1 设计标准：设计标准：(1)公路等级：高速公路(2)设计行车速度：120公里/小时(3)设计荷载：公路-I 级方案方案1 1：采用的是不等跨变截面的预应力混凝土连续80+130+80的三跨布置梁桥， 主桥采用单箱单室薄臂箱形截面。主桥上部结构采用混凝土箱形梁，支点处高7.5m，跨中梁高3.0m，悬臂长度取3.0m，箱梁底宽8.0m，箱梁顶宽14m，箱梁顶板厚度30cm。车行道为2%双向横坡， 人行道为双向1.5横坡。 桥面全宽 （分两幅设计） ： 0.25m （人行道栏杆） +2m(人行道)+33.75m(车行道)+0.5m 防撞墙， 全宽14m。 桥面铺装层厚度为10cm 的沥青混凝土。图2.1立面图尺寸（mm）14002050230300300800300303060 4015015040图 2.1 主梁跨中截面尺寸图（单位：cm）14009060 4030300800300205068075085150150图 2.2 主梁支点截面尺寸图（单位：cm）方案方案2 2：采用的是简支 T 梁桥是静定结构，构造简单，内力不受基础不均匀沉降等附加变形影响，对基础要求较低，设计计算方便，跨径不大，各梁手里较为独立，行车舒适，线型简洁美观。简支简支 T 梁桥可分为整体式和装配式两种结构。这里考虑采用装配式简支结构，其具有建桥速度快、工期短、模板支架少等优点而应用广泛。T 形主梁是使用最普遍的结构形式 ，其优点是制造简单、整体性好、接头施工也方便。同时做预应力结构适用跨径为 20m50m 之间，符合此设计跨径要求。图2.3主梁纵断面尺寸（mm）图2.4主梁横断面尺寸（mm）2.22.2 方案比选方案比选根据设计宗旨，桥型方案应从安全、适用、经济、美观等原则出发，考虑多方面的因素来进行比选。最终所选桥型力求技术先进，安全可靠，适用耐久，美观经济；另一方面，所选桥型应满足所需施工设备少、工艺简单的要求，以减小施工难度、加快施工进度、保证施工质量。综合以上因素，比选结果见表2.2.1如下：表表.1方案序号方案一方案二桥型80+130+80m 混凝土连续梁桥40+5*44+40m 简支梁桥设计跨径290m290m荷载标准公路一级公路一级适用性1.单孔 130m 主跨跨越主车道，与车道适应性好；2.路边填挖方较少；3.两岸副孔为 80m， 伸缩缝少，便于截面连续；4.使用变截面可以提高桥下净空；1.适用于中小跨径桥梁；2.在城市立交桥中广泛使用；3.构造简单，跨径较小的桥型4.横隔梁保证各根主梁相互连接成整体，以提高桥梁的整体刚度；安全性1.主梁采用不等跨截面设计，可以降低跨中的设计弯矩，也有余力适应支点截面剪力很大的要求；2.采用悬臂法施工，施工阶段主梁刚度大，且内力与运营阶段一致；3.采用变截面梁高，可以节省材料，减轻自重，更加安全可靠；4.中间无伸缩装置，行车平顺舒适；1.T 形梁桥，构造简单，受力明确；2.施工采用吊装施工，需要一定的吊装设备来保证工期；美观性桥梁外形美观，曲线线形与周围环境更加协调；桥梁线型简单单调，但也不缺乏这特有的简单之美；经济性经济较经济结论选用弃用2.32.3 最终方案选定最终方案选定综上所述，决定使用方案 1（80m+130m+80m）预应力混凝土连续梁桥作为设计方案。三、三、设计（研究）的重点与难点，拟采用的途径（研究手段设计（研究）的重点与难点，拟采用的途径（研究手段）：.重点与难点重点与难点本设计桥梁的形式为混凝土连续梁桥，适合采用悬臂浇筑和悬臂拼装，但是大跨径采用悬臂施工时，存在墩梁临时固结和体系转换的工序，结构稳定性应予以重视，施工较为复杂；此外，主墩需要布置大型橡胶支座，存在养护上甚至更换上的麻烦。实际施工中受各种因素的干扰， 可能导致合拢困难， 使成桥线形与内力状态偏离设计要求，给桥梁施工安全、外形、可靠性、行车条件和经济性等方面带来不同程度的影响。3.23.2 研究手段研究手段因此在施工过程中，必须实施有效的施工控制，主要包括线形控制、应力控制、结构安全控制、温度观测和有关材料试验控制等。从某种意义上讲，线形控制尤其是悬浇施工中挠度控制和预拱度控制是保证桥梁线形符合要求的必不可少的措施。施工挠度的计算与控制是桥梁悬臂施工中的难点，科学合理地确定悬臂浇筑梁段的预拱度设置是至关重要的。只有预拱度设置合理，才能保证一个跨径内将要合拢的两个悬臂端在同一水平线上，也才能保证桥梁上部结构在施工和运营一定时间后能达到设计所期望的标高线形。四、设计（研究）进度计划：四、设计（研究）进度计划：1、英语翻译：1-2 周；2、方案必选：3-4 周；3、结构计算（计算软件学习） ：1-7 周；4、结构计算（成桥阶段计算） ：7-9 周；5、施工方法设计：10 周；6、成桥阶段计算分析：11 周；7、荷载组合承载能力分析：12-13 周；8、绘图：14 周；9、完成设计递交修改：15 周；10、完成修改进行答辩：16-19 周。五、五、参考文献：参考文献：1 JTG B01-2003, 公路工程技术标准S. 北京：人民交通出版社，2004.2 JTG D60-2004, 公路桥涵设计通用规范S. 北京：人民交通出版社，2004.3 JTJ022-85, 公路砖石及混凝土桥涵设计规范S. 北京： 人民交通出版社， 2004.4 JTG D62-2004, 公路钢筋混凝土及预应力混凝土桥涵设计规范S. 北京：人民交通出版社，2004.5 JTJ 024-85.公路桥涵地基与基础设计规范S.北京：人民交通出版社，1985.6 叶见曙. 结构设计原理M. 北京：人民交通出版社，2005.7 易建国. 桥梁计算示例丛书混凝土简支梁（板）桥M. 北京：人民交通出版社, 2006.8 周念先. 桥梁方案比选M. 上海：同济大学出版社, 1997.9 范立础. 预应力混凝土连续梁桥M. 北京：人民交通出版社，1996.10 邵旭东. 桥梁工程 M. 北京：人民交通出版社，200611 邵旭东. 桥梁设计百问M. 北京：人民交通出版社，200312 陈忠延. 土木工程专业毕业设计指南（桥梁工程）M. 北京：人民交通出版社, 2002.13 贺拴海，谢仁物.公路桥梁荷载横向分布计算方法M.北京：人民教育出版社，1996.14 贺拴海.桥梁结构理论与计算方法M.北京：人民交通出版社，2003.15 徐光辉.桥梁计算示例集预应力混凝土刚构桥M.北京：人民交通出版社，1995.16 江祖铭，王崇礼.公路桥涵设计手册丛书墩台与基础M.北京：人民交通出版社，1994.17 徐岳.预应力混凝土连续梁桥设计M.北京：人民交通出版社，2000.18AASHTO.(2010).AASHTOLRFDbridgedesignspecifications,5thEd.,Washington, DC.19 ABAQUSComputer software. Providence, RI, Dassault Systmes SimuliaCorp.20 Figg Bridge Engineers. (2008).I-35W St. Anthony Falls Bridge inspectionand maintenance manual, Figg Bridge Engineers, Dallas.21 French, C. E. W., Shield, C. K., Stolarski, H. K., Hedegaard, B. D., andJilk, B. J. (2012).“Instrumentation, Monitoring, and Modeling of the I-35WBridge.”Rep. MN/RC 2012-24, Minnesota Dept. of Transportation, Minneapolis.22Minnesota Dept. of Transportation. (2008).“As-built construction planforbridgenos.27409and27410. ” StateProj.2783-120,Fed.Proj.E.R.MN07(300), Minneapolis.23 R.J. Watson, Inc. (2008).“St. Anthony Falls (I-35W) bridge replacement.”24 State Proj. 2783-120, Fed. Proj. E.R.MN07(300), Amherst, NY.指导教师意见签名：月日教研室（学术小组）意见教研室主任（学术小组长） （签章） ：月日Approximate method for estimation of collapse loads of steel cable-stayed bridgesHoon Yooa, Ho-Sung Nab, Dong-Ho Choib,aR&D Center, Hyundai Engineering and Construction, 102-4 Mabuk-dong, Kiheung-gu, Yongin, 446-716, South KoreabDepartment of Civil and Environmental Engineering, Hanyang University, 17 Haengdang-dong, Seoungdong-gu, Seoul, 133-791, South Koreaa b s t r a c ta r t i c l ei n f oArticle history:Received 20 September 2011Accepted 3 December 2011Available online 28 January 2012Keywords:Cable-stayed bridgeStructural stabilityCollapse loadsBucklingInelastic buckling analysisThis paper proposes a new and simple method for estimating the collapse load of a steel cable-stayed bridge.A new convergence criterion for iterative eigenvalue computations is suggested to consider the beamcolumn effect of a cable-stayed bridge system. The collapse loads of two example bridges representingmedium and long-span models are evaluated by the proposed method and compared to a nonlinearinelastic analysis. The results demonstrate that the proposed method is a good substitute for a complexnonlinear inelastic analysis to approximately evaluate the collapse loads as well as failure modes of steelcable-stayed bridges. 2011 Elsevier Ltd. All rights reserved.1. IntroductionA cable-stayed bridge is distinguished from conventional highwaybridges by special features, including the use of a longer center spanwithout intermediate piers, as well as light-weight girders and highstrength cables. In a cost sense, bridge engineers and researchershad considered this kind of a bridge to be superior to medium andshort-span bridges with several intermediate piers, for constructinglong-span bridge crossings. As the center span of cable-stayed bridgesincreases,twomajorissues arenaturallyaddressed:bucklinginstabilityand wind instability. The buckling instability of steel girder and towermembers is caused by the large axial forces that are transmitted bycables under dead and live loads, whereas the wind instability of steelgirder members comes from lateral wind loads. In particular, thebucklinginstabilityofgirderandtowermembersmaybeafundamentalproblem that should be checked in the preliminary design of a cable-stayed bridge because it directly controls the geometric dimensions ofstructural members and the practical limitation of the center spanlength.The buckling instability problem of steel cable-stayed bridges hadbeen traditionally evaluated by elastic buckling analysis based on thebifurcation-point stability concept. Since Tang 1 calculated thebucklingloadofthetwo-dimensionalcable-stayedbridgebythesimpleenergy method in his seminal paper, many researchers have studiedvarious aspects of the buckling instability problem of cable-stayedbridges, such as the overall safety by eigenvalue analysis 2, the effectof cable numbers and the center-span length 3,4, the span lengthsandthecablearrangement5,andtheeffectoftheratiobetweengirderandtowerstiffnessandloadrating6,7.Theapplicationofthesestudiesis currently discarded to estimate the collapse loads of a cable-stayedbridge because they cannot take into account both geometric andmaterial nonlinearities that largely affect the load-carrying capacity ofsuch a slender bridge system. Relevant references show that an elasticbuckling analysis based on the bifurcation-point stability concept isinadequate to estimate the collapse loads of steel framed structures8,9 and steel cable-stayed bridges 10,11.As computer resources and algorithms have improved, rigorousnonlinear inelastic analysis based on the limit-point stability conceptbecame common among many researchers for obtaining the collapseloads of cable-stayed bridge systems. Corresponding studies includemodeling techniques for structural members using nonlinear beamsand plates 12, a sophisticated analysis method considering bothgeometric and material nonlinearities 13, the individual effect ofnonlinearities 14,15, and a feasibility study for a super long-spanbridge with respect to static and dynamic instabilities 16. Becausethe nonlinear inelastic analysis used in these studies could rigorouslyconsider both geometric and material nonlinearities of structuralmembers, this method has gained general acceptance as an exactapproach to predict the collapse load of a cable-stayed bridge.However, this method obviously requires the preparation of adequateanalysis tools and efficient computing machines as well as a widerange of understanding of the complex nonlinear theory. In practicalsituation, especially in the preliminary design stage, the stability ofseveral bridge systems should be checked for various load cases asmuch as possible in order to determine the optimum bridge systemamong proposed designs. Frequent use of this nonlinear inelasticanalysis seems to still be a burdensome task for most engineers inthis practical situation due to limitations of time and cost. In fact,Journal of Constructional Steel Research 72 (2012) 143154 Corresponding author. Tel.: +82 2 2220 0328; fax: +82 2 2220 4322.E-mail addresses: hoonyoohdec.co.kr (H. Yoo), saintnahanyang.ac.kr (H.-S. Na),csmilehanyang.ac.kr (D.-H. Choi).0143-974X/$ see front matter 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.jcsr.2011.12.003Contents lists available at SciVerse ScienceDirectJournal of Constructional Steel Researchthe need for a simpler and faster method to estimate the collapse loadof these special bridges has been reasonably claimed by manyengineers without sacrificing much in accuracy.Consequently, a simple alternative for the complex nonlinearinelastic analysis was proposed and studied by several researchersin order to approximately obtain the collapse load of a steel cable-stayed bridge 10,11,17. These studies adopted the inelastic bucklinganalysis, which utilized the columnstrength curve in design specifi-cations in order to take into account the material inelasticity ofstructural members. Iterative eigenvalue computations were adoptedin the method for determining the tangent modulus of each structuralmember. The inelastic buckling analysis could only partially considerthe geometrical nonlinearities of a cable-stayed bridge. According tothe corresponding references 10,14,16,18, however, the ultimatebehavior of a steel cable-stayed bridge is mainly affected by materialnonlinearity rather than by geometrical nonlinearities, even near thecollapse state of the bridge system. The results of these studies seemto be evidence of the validity of the inelastic buckling analysis. Ofcourse, there is also the criticism that the method of inelastic bucklinganalysis is not valid for obtaining the collapse load of a cable-stayedbridge due to its failure in capturing geometrical nonlinearities ofthe bridge system 19.This paper proposes a new method for simple estimation of thecollapse loads of steel cable-stayed bridges that can be effectivelyused for checking many candidate designs with various load casesin the preliminary design stage. Based on the fundamental conceptof the inelastic buckling analysis previously established by theauthors 9,10,18, we widen the range of application for the methodby suggesting a new criterion of each structural member in the bridgesystem. The proposed method determines the tangent stiffness ofeach structural member in the bridge system by iterative eigenvaluecomputations with the classical tangent modulus theory. In addition,an improved convergence criterion for girder and tower members isproposed to take into account the beamcolumn interactions. Aftersummarizing theoretical approaches, we analyze the two examplebridges representing medium and long-span models with differentgirder depths. To show the validity and applicability of the method,the results of the proposed method are compared with those of theestablishedinelasticbucklinganalysisandnonlinearinelasticanalysis.Some discussionsare alsomadeabout the failure modes oftheexamplebridges.2. Nonlinear inelastic analysis: Exact approach2.1. Beamcolumn element for girder and tower membersSince the large axial force and bending moments occur in girderand tower members in a typical cable-stayed bridge under serviceloads, these structural members are usually considered to be beamcolumn members. It is well known that the beamcolumn interaction,called the second-order effect, is observed in a beamcolumn member8,2023. In general, the second-order effect can be convenientlytaken intoaccount by using thestability functions 8,20. Frommanip-ulation of the slope-deflection equation for a beamcolumn member,which is based on the assumption of EulerBernoulli beam theory andSaint-Vernant torsion, the tangent stiffness of a three-dimensionalprismatic beamcolumn element shown in Fig. 1 may be described as8,9uavawaxayazaubvbwbxbybzbktan? abzby000abzby000cz000dzbzcz000gzcy0dy000cy0gz0e00000e00fy000dy0hy0fz0dz000hzabzby000cz000gzSymm:cy0gy0e00iy0iz26666666666666666664377777777777777777751where the terms are defined by a=EtA/L, by=(Mya+Myb)/L2, bz=(Mza+Mzb)/L2, cy=(kiiy+2kijy+kjjy)/L2+P/L, cz=(kiiz+2kijz+kjjz)/L2+P/L, dy=(kiiy+kijy)/L, dz=(kiiz+kijz)/L, e=GJ/L, fy=kiiy, fz=kiiz,gy=(kijy+kjjy)/L, gz=(kijz+kjjz)/L, hy=kijy, hz=kijz, iy=kjjy, and iz= kjjz.In the tangent stiffness in Eq. (1), the terms Et, G and J indicate the tan-gent modulus, shear modulus, and torsional constant, respectively. Theterms P, Myand Mzare the axial force and bending moments of the y-and z-axes at the current state of loadings, respectively. The tangentmodulus Etof a member is simply approximated from the columnstrength curve with respect to the magnitude of axial force of eachmember at the current state of loadings as described in Refs. 8,9. Thestiffness terms of k are defined with the stability functions for geometricnonlinearities and gradual yielding parameters for material yielding asgiven in Table 1.In Table 1, the terms S1, S2, S3and S4represent the conventionalstability functions, which consider the second moment effect for abeamcolumn member. The explicit form of these functions wasomitted here for brevity and can be found in Refs. 8,9,18,23.The terms aand bin Table 1 are scalar parameters that allow forgradual yielding of an element associated with plasticization at theend of nodes. These terms are equal to 1.0 when a member is fullyelastic and zero when a plastic hinge is formed in a member. Theterms aand bare assumed to vary according to the parabolic functionthat is derived with the form of the column strength curve as 8,9,23 1:0when b 0:392:7243 ln when 0:39?2where is a forcestate parameter that measures the magnitude ofaxial force and bending moment at the end of the member. The term may be described in AISC-LRFD 24 as PPy89MyMpyMzMpz !whenPPy29MyMpyMzMpz !P2PyMyMpyMzMpzwhenPPyb29MyMpyMzMpz !8:3where Py, Mpyand Mpzare theyield load and full plastic moments of they- and z-axes of an element, respectively. The stability functions andmaterial yielding parameters are changed at a certain incrementalstep of applied loads in the incremental load analysis.Fig. 1. Degrees of freedom of a beamcolumn element.Table 1The stiffness terms of k in the tangent stiffness of a prismatic beamcolumn.y-axisz-axiskiiy=a(S1S22/S1(1b)EtIy/Lkiiz=a(S3S42/S3(1b)EtIz/Lkijy=abS2EtIy/Lkijz=abS4EtIz/Lkjjy=b(S1S22/S1(1a)EtIy/Lkjjz=b(S3S42/S3(1a)EtIz/L144H. Yoo et al. / Journal of Constructional Steel Research 72 (2012) 1431542.2. Cable elementWhen a cable is suspended between the girder and towermembers,it sags into the shape of a catenary due to its own weight 15,25. Thepeculiar nonlinear behavior in cables, called the sag effect, resultsfrom this phenomenon. The tensile axial force in a cable is affected bynot only the deformation of the cable as a usual tension member butalso the cable sag. Therefore, the axial stiffness of the cable varies non-linearly as a function of the amount of the cable sag as well as thechangeinthelengthofcables.Ifatensileforceinacableislargeenoughto neglect the amount of the cable sag, the cable acts as a generaltension member. However, as a tensile force in the cable decreases,the cable sag increases considerably and the axial stiffness of the cableis dramatically reduced. Furthermore, the cable is only capable ofresisting a tensile axial force and there is no stiffness against the axialcompressionforce.Toconsiderthesenonlinearbehaviors,anequivalentbar element incorporating the equivalent tangent modulus is usuallyused for modeling cables in a cable-stayed bridge system 26,27. Thetangent stiffness matrix of cables may be written asuavawaubvbwbktan? a00a00cz00cz0cy00cya00Symm:cz0cy26666664377777754where the terms are defined by a=Et,cAc/Lcand cy=cz=P/Lc. Theterms Et,c, Ac, and Lcare the equivalent tangent modulus, sectionalarea, and the length of a cable member, respectively. The equivalenttangent modulus of a cable may be given as 18,26,28Et;cEc1 w Lch2AcEc12T3c#when 0 b c y;c0when cb 0 or c y;c8:5whereEc,w and Tcare theoriginalmodulusofelasticity,theunit weightand theaxialtensionof a cable,respectively. ThetermLchis a horizontalprojected length of a cable. The terms cand y,care the current cablestress including initial stress, and the yield stress of a cable member,respectively.2.3. Incremental load analysis considering large displacementsSince a cable-stayed bridge system is inherently flexible andundergoes large displacements under service loads, the global tangentstiffness of the system changes as external loads are applied. Therefore,a step-by-step incremental load analysis should be conducted to tracethe response of the bridge system. The global tangent stiffness of thebridge system is required to be updated at each load level throughoutthe incremental load analysis.By performing conventional matrix manipulations in finite elementprocedures, the element tangent stiffness matrices of girder, tower andcable members in Eqs. (1) and (4) may be combined into the globaltangent stiffness matrix Ktan. The basic system equation of the incre-mental load analysis is then described as 9,18,25,26Ktan? dfg dPfg6where d and dP are the incremental displacement vector and theincremental force vector, respectively. A detailed derivation of Eq. (6)and the procedures of incremental load analysis can be found inRefs. 8,9,18,23. The collapse load of a bridge system is determined atthe maximum load point where the tangent stiffness matrix in Eq. (6)is nearly indefinite, i.e., the slope of a loaddisplacement path in theglobal bridge system is nearly zero as given in Eq. (7). The critical loadfactor (CLF) is calculated as the ratio of the collapse load to the serviceload initially applied to the bridge models.det Ktan?jj 0:73. Proposed inelastic buckling analysis: Approximate approach3.1. Criterion for a column memberThe tangent modulus theory proposed by Engesser 29 considerstheinelasticity of a column asthe tangent modulus, that is, the gradientataspecific pointofthestressstraincurve.Theinelastic criticalloadPcrmay be written asPcr2EtIL2eEtEPe8where Le, and Peare the effective length, and the elastic critical load,respectively. The modulus of elasticity of a column is unique for agiven material of a column, and the elastic critical load may be easilycalculated by conventional eigenvalue computation. Therefore, we canapproximate the inelastic critical load of a column without tracing acomplex path of material inelastic behavior if we know the accuratetangent modulus of a column.As expected, the tangent modulus cannot be obtained by a one-step analysis because the accurate inelastic critical load of a columnis also an unknown value in Eq. (8). Therefore, a simple iterativescheme is needed to obtain the tangent modulus and the inelasticcritical load of a column, simultaneously. Eq. (8) can be transformedintoEEtPePcr:9Iterative eigenvalue computations are the modification process ofthe tangent modulus such that the elastic critical load Pe(the Eulerbuckling load) is equalized with the inelastic critical load Pcrfromthe column strength curve. In other words, the ratio of the currenttangent modulus to the previous tangent modulus of a columnapproaches the value of 1.0 at the end of the iterationsteps of iterativeeigenvalue computations asEi1tEitPiePicr 1:010where superscript i denotes the number of iterations.In Eq. (10), the elastic critical load of a column is obtained byconventional eigenvalue analysis in conjunction with the theoreticalformula as Pei=iP0where is the eigenvalue at the i-th iterationand P0is the axial force of each member determined from a linearstress analysis. The inelastic critical load of a column is approximatedfrom the column strength curve in combination with the slendernessratio calculated from the elastic critical load. Eq. (11), derived bysubstitutingPei=iP0intoEq.(10),isthecriterionforacolumnmemberused in the established inelastic buckling analysis described inRefs. 10,11,17,19.EitPicriP0Ei1t:11The eigenvalue computation should be repeated in order to deter-mine the tangent modulus, and the inelastic critical load of a columnuntil Eq. (10) is satisfied for all members in a bridge system. After theconvergenceiscompleted,theconvergedeigenvalueconvmayrepresent145H. Yoo et al. / Journal of Constructional Steel Research 72 (2012) 143154theinelastic criticalload factor of the system. The critical load factormaybe accepted as the index of the collapse load of a cable-stayed bridgesystem provided that the bridge system fails due to the bucklinginstability.ThecriterionforacolumnmemberinEq.(11)wasalreadyappliedtosome cable-stayed bridge models by several researchers 10,11,17,19.In these papers, they insisted that the inelastic buckling analysis withthecriterionforacolumninEq.(11)gavesatisfactoryresultstoapprox-imately determine the collapse load of long-span steel cable-stayedbridges 10,11,17, but others did not 19. In theory, it is certain thatthe convergence criterion of Eq. (11) is sufficient only for the analysisof the column-dominated structural system such as the isolated toweror the high-rise frames. For a complex bridge system such as a cable-stayed bridge, the validity of Eq. (11) is questionable because this crite-rioncannottakeintoaccounttheeffectoftheprimarybendingmomentin each member. The convergence criterion of Eq. (11) may need somemodification for the analysis of the complex bridge systems.3.2. Proposed criterion for a beamcolumn memberFor a typical beamcolumn member, the stability of a member isusually checked by the axialflexural interaction equation (Eq. (12),which is a fundamental form of Eq. (3). 2022PPnMyMpyMzMpz 1:012where the terms Pnrepresent the axial resistance of a beamcolumnmember. After some modifications, we may use Eq. (12) as a newcriterion for the inelastic buckling analysis instead of using Eq. (11).At a specific load state, the terms of axial force and moments of amember in Eq. (12) may be written with the eigenvalue from aconventional eigenvalue analysis asP iP0;My iM0yand Mz iM0z13where the terms P0, My0and Mz0are the member force and momentsthat are calculated from a linear stress analysis. The axial resistancePnof a member is also described as Pn=Pcri, where the term Pcriis theinelasticcriticalloadofamembercalculatedfromthecolumnstrengthcurveatthei-thiteration.Weassumedthatthefullplasticmomentsofa member are not affected by the iterations in iterative eigenvaluecomputations. By substituting Eq. (13) and the term of the axialresistance into Eq. (12), and comparing it with the case for a columnmemberofEq.(10),weobtainthecriterionforabeamcolumnmemberasEi1tEitiP0PicriM0yMpyiM0zMpz1:0:14By reversing the numerator terms and denominator terms ofEq. (14), the tangent modulus of a beamcolumn member can beobtained at i-th iteration steps asEitPicrMpyMpziP0MpyMpz PicriM0yMpz PicrMpyiM0zEi1t:15Fig. 2. Flow chart of inelastic buckling analysis.146H. Yoo et al. / Journal of Constructional Steel Research 72 (2012) 143154It can be seen that the criterion of Eq. (15) for a beamcolumn isequivalent to the criterion of a column member when the momentsare not exerted on a member. Therefore, we can say that Eq. (15) isthe general convergence criterion for a column and beamcolumnmember in inelastic buckling analysis, and may be used for structuralmembers in a steel cable-stayed bridge system without any theoreticaldiscrepancy.3.3. Procedures of the inelastic buckling analysisThe inelastic buckling analysis proposed in this paper utilizesiterative eigenvalue computations based on the bifurcation-pointstability concept. The basic equation of the inelastic buckling analysisis similar to that of conventional elastic buckling analysis 10,18except for the elastic stiffness matrix term of the structure, which isdet KeEt? Kg? 016where Ke(Et) and Kg are the modified stiffness matrix and geometricstiffness matrix of the global bridge system corresponding to eigen-values of , respectively. Eq. (16) is conceptually placed along thesame line with Eq. (7) of the nonlinear inelastic analysis.By the same definitions of the degrees of freedom in nonlinearinelastic analysis (Fig. 1), the modified stiffness matrix ke(Et) for abeamcolumn element may be explicitly given as 9,18uavawaxayazaubvbwbxbybzbkeEt? a00000a00000bz000cz0bz000czby0cy000by0cy0d00000d00ey000cy0fy0ez0cz000fza00000bz0000Symm:by0cy0d00ey0ez266666666666666666643777777777777777777517where the individual terms are defined by a=EtA/L, by=12EtIy/L3,bz=12EtIz/L3, cy=6EtIy/L2, cz=6EtIz/L2, d=GJ/L, ey=4EtIy/L, ez=4EtIz/L, fy=2EtIy/L and fz=2EtIz/L.The geometric stiffness matrix for a beamcolumn element kg isalso given asuavawaxayazaubvbwbxbybzbkghi0bzby0000bzby000a0cyadebza0cybdeaczaedby0aczbedfgzgy0cyaczafgzgyh00degzijh0edgyji0bzby000a0cybdeSymm:aczbedfgzgyh0h266666666666666666643777777777777777777518where the terms in Eq. (18) are defined by a=6P/(5L), by=(Mya+Myb)/L2,bz=(Mza+Mzb)/L2,cya=Mya/L,cyb=Myb/L,cza=Mza/L,czb=Mzb/L,d=Mx/L,e=P/10,f=PJ/(AL),gy=(Mya+Myb)/6,gz=(Mza+Mzb)/6, h=(2PL)/15, i=(PL)/30 and j=Mx/2.For a cable element, the element and geometric stiffness matricesareextractedfromthetangentstiffnessofEq.(4)innonlinearinelasticanalysis. By splitting Eq. (4) into conventional material terms and120m1420m=280m120m1420m=280m1420m=280m1420m=280m600m40m120m68m134m=52mSC1SC2160m160mxz(a) Center span length of 600 m300m2920m=580m280m2920m=580m2920m=580m2920m=580m1200m40m240m128m284m=112mSC1SC2280m300mxz(b) Center span length of 1200 mFig. 3. Numerical models of the example cable-stayed bridges.Table 2Cross sections and geometric properties of girder members in example cable-stayedbridges.Cross sectionDepth H(m)Area Ag(m2)Second moment ofcross sections Iy(m4)30mvaries(1-6m)yz0.02m0.05m1.01.2960.2962.01.3961.2393.01.4962.8804.01.5965.2705.01.6968.4576.01.79612.492147H. Yoo et al. / Journal of Constructional Steel Research 72 (2012) 143154geometric variation terms, the element and geometric stiffnessmatrices for a cable element may be given byke?cuavawaubvbwba00a00000000000a00Symm:000?andkghicuavawaubvbwb000000cz00cz0cy00cy000Symm:cz0cy?19where the component terms are the same with those of Eq. (4). It isnoted that the equivalent tangent modulus and the axial force of acable element are obtained by linear stress analysis under serviceloads.The first step of inelastic buckling analysis is to set up the initialshape of a cable-stayed bridge system. After setting up the initialshape of a bridge system and obtaining the sectional forces by a linearstressanalysis,wecanbegintheiterativeprocedureofinelasticbucklinganalysis with conventional eigenvalue computation. The eigenvalueanalysis at the first iteration step of inelastic buckling analysis is equiva-lent to that of theconventionalelastic bucklinganalysis,sincethemem-bers in a bridge system have the elastic modulus at the first step. Thetangent modulus of eachmember is determined by thecriteria for a col-umn or a beamcolumn as given in Eq. (11) or Eq. (15). Since the tan-gentmodulusofeachmembermaynotbechangedattheconvergence state, the modified stiffness matrix of the bridge systemat the current iteration step is nearly the same as that of the previous it-eration step. As a result, the eigenvalue of the bridge system does notchange when the convergence is satisfied. If the convergence is not sat-isfied, the tangent modulus of each member at this iteration step issubstituted into the modified stiffness matrix of the bridge system forthe next iteration step. Fig. 2 presents the flow-chart for the computerprogram developed in this study. In addition, the detailed proceduresof the inelastic buckling analysis are summarized as follows:1) Set up the initial shape of a cable-stayed bridge system under deadloads. Calculate the plastic moments of each member consideringthe sectional dimensions and material yield stress.2) Perform a linear stress analysis for the cable-stayed bridge systemin order to calculate the sectional force and the moments in eachelement.CreateanelasticstiffnessmatrixKeandgeometricstiffnessmatrix Kg for the model.3) Solve the eigenvalue problem designated as det|Ke(Eti)+iKg(Pj0,Mj0)|=0 and calculate the minimum eigenvalue i. Calculate theelastic buckling load and the current moments of each elementas iPj0, iMy,j0and iMz,j0. Subscripts i and j denote the number ofiterations and the number of elements, respectively.4) Obtain the inelastic buckling load Pcr,jiof each element from thecolumn strength curve using the effective length of each elementas Lie;j ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEi1t;jIj= iP0j?r.5) Calculate the tangent modulus of each element as the criteria of acolumn member or a beamcolumn member as given in Eq. (11)or Eq. (15). Check for convergence withii1ib , where isthe convergence limit (0.001). If convergence is not reached, goback to step 3.6) When convergence is satisfied, obtain the converged eigenvalueconvas the index of a collapse load for the cable-stayed bridgesystem.4. Numerical models of the example cable-stayed bridges4.1. Example bridge modelsIn order to validate the proposed inelastic buckling analysis, weanalyzed two different example bridges, which represent mediumand long-span models, using both nonlinear inelastic analysis andthe proposed method. Fig. 3 presents the elevation view of thenumerical models of example bridges used in this study. The numer-ical models considered in this paper are almost the same as thosein Refs. 10,18 except for some minor variations in the supportconditions.Table 3Cross sections and geometric properties of tower and cable members in example cable-stayed bridges.TowerCablesModelCross sectionArea At(m2)Second moment ofcross sections Iy(m4)Area Ac(102m2)600-m6m8m0.04m0.04myx1.1147.067SC1: 2.686SC2: 1.4401200-m10m12m0.04m0.04myx1.75430.316SC1: 3.056SC2: 1.624(a) LC1 (b) LC2 (c) LC3 Fig. 4. Three live load cases for example cable-stayed bridges.Table 4Service loads of the example cable-stayed bridges.LoadMemberModelsDepth of girderValueH (m)(kN/m)Dead load (1.2DC)GirdersAll models1.0239.602.0252.543.0265.484.0278.405.0291.346.0304.27Towers600-m144.071200-m226.84Live loads (1.7LL)GirdersAll models129.54148H. Yoo et al. / Journal of Constructional Steel Research 72 (2012) 1431544.2. Material and sectional propertiesTables 23 show the cross sections and geometric properties ofgirder, tower and cable members in example bridges. The crosssections of girder members are assumed to be single box sectionswith variation of girder depths. The tower members are also assumedto be single box sections and their cross sections are different withrespect to the length of the center span of models as shown inTable 3. Two types of cables are used for example bridge models.Themodulusofelasticityforgirderandtowermembersis2.1105MPa. For cables, it is assumed to be 2.0105MPa. Theyield strength of the steel for girder and tower members is 450 MPaand it is assumed to be 1600 MPa for cables.4.3. LoadsDead loads (DC) and design live loads (LL) are applied to examplebridges. Dead loads include the self-weight of the members, initialcable forces, and other permanent loads. The self-weight and otherpermanent loads of girder and tower members are determined byconsideringsectiongeometriesandadditionalattachmentsasdescribed in Refs. 10,18. The value of the uniform lane load76.2 kN/m is applied on girder members as a live load (LL). Threelive load scenarios are considered as illustrated in Fig. 4. The uniformlane load is applied on girder members: (1) on the center span only(LC1), (2) on the center span and one side span (LC2), (3) over fulllength of span (LC3). The service loads for analyses are obtained bythe load combination of 1.2DC+1.7LL. The final values of serviceloads for example bridges are summarized in Table 4.4.4. Initial configuration of example bridgesThe initial cable forces and initial configurations of all numericalmodels usedinthisstudyareobtained byusing thezero-displacement method 30. For an illustration, the distributions ofaxial forces and bending moments in girder members for the 600-mand 1200-m models with 2 m girder depth are presented in Fig. 5. Alarge axial force is exerted on the girder member at the intersectionbetween the girder and the tower, whereas it is negligible on thegirder member at the end of both side spans and the center of thebridge. On the other side, the distribution of bending moments ingirder members is identical to those of the equivalent continuousbeams.Fig. 6 shows the distributions of cable stresses of the same bridgemodels at the initial configuration. The dotted line in Fig. 6 indicatesthe allowable stress level of cables calculated in terms of the yieldstress of cables and the safety factor of 2.5. It can be seen that allcable stresses at the initial configuration are below the allowablestress level of cables. The initial configurations of the bridge modelsshown in Figs. 5 and 6 are the base state in order to analyze furtherbehavior of the bridges under the live loads by nonlinear inelasticanalysis and the proposed inelastic buckling analysis.-120400-40-80Axialforce(MN)Bendingmoment(MN-m)200-20-40Axialforce(MN)Bendingmoment(MN-m)0-80-160-24040200-20-40(a) 600-m model (girder depth=2m)(b) 1200-m model (girder depth=2m)Fig. 5. Axial force and bending moment distributions in girder members of example bridges at the initial configuration.149H. Yoo et al. / Journal of Constructional Steel Research 72 (2012) 1431545. Collapse load analysis of example bridges5.1. Collapse loads of example bridgesThe collapse loads of all example bridges were evaluated by theproposed inelastic buckling analysis and compared to those of non-linear inelastic analysis. The inelastic buckling analyses were carriedout using both the column criterion (Eq. (11) and the proposedcriterion for a beamcolumn (Eq. (15) for comparison.Firstly, the collapse loads by nonlinear inelastic analysis wereestimated from the loaddisplacement curve of the global bridgesystems as illustrated in Fig. 7. The vertical axis in Fig. 7 indicatesthe load factor that means the ratio of the current load values to theservice load values. We chose the vertical displacement of the middleof the center span as the corresponding variable in the horizontal axisin Fig. 7. The collapse loads of example bridges were determined bythe maximum load factors (i.e., CLFs) where the slope of the loaddisplacement curveswere nearlyzero.For eachgirder depth,differentsymbols were used to designate the collapse loads of example bridgesin Fig. 7.The loaddisplacement curves in Fig. 7 exhibit typical cases of thebifurcationfailure,thatis,theequilibriumpathoftheloaddisplacementabruptly changes at the critical load except for the longest modelwith a deep depth of girder members. It can also be seen that theloaddisplacement curves are nearly linear between the initialconfiguration and the collapse load. It means that the effects of non-linearities caused by both geometrical and material sources are smallenough to be neglected until the load level reaches critical load evenfor the longest models of example bridges. Although the collapseloads of example bridge models tend to increase as the depth of girdermembersincreases,thedifferenceofthecollapseloadsisnotconsiderableforthebridgemodelsthathavecomparablydeepergirdermembersmorethan 4 m (solid symbols).In a second phase, the collapse loads of the same bridge modelswere also obtained by inelastic buckling analysis. Because inelasticbuckling analysis is inherently based on eigenvalue computations(see Eq. (16), the method cannot produce the incremental loaddisplacement path such as Fig. 7. Instead, inelastic buckling analysiscan givetheconvergedeigenvaluesintermsofexplicitnumbersaswellas the collapse modes of the system at the collapse state. In addition,inelasticbucklinganalysisisconsiderablyfasterthannonlinearinelasticanalysis in that the method does not require iterative processes such asupdating geometry, re-evaluating the stiffness matrix, re-solvingsystem equation and so on. The collapse loads for each bridge modelwere obtained by inelastic buckling analysis only in 10 iterations orless. Table 5 summarizes collapse load factors of example bridgesexplicitly with respect to analysis methods, girder depths, centerspans and live load cases for the sake of completeness.Table 6 shows the post-buckling state and buckling mode shapesof example bridges with a girder depth of 2 m predicted by bothmethods. It should be noted that the buckling mode shapes byinelastic buckling analysis in Table 6 do not indicate the realdeformed state at the collapse of the bridge system in that thebuckling analysis with eigenvalue computations merely gives theinformation of shapes in the post-buckling state, but does not deter-mine the amplitudes of shapes. Consistently, the post-buckling statefrom nonlinear inelastic analysis is defined as the displacementmeasured from the critical point in the loaddisplacement curves.As can be seen in Table 6, the buckling mode shapes predicted bythe proposed method are in good agreement with those of nonlinearinelastic analysis.5.2. Validity of the proposed methodIn order to prove the validity of the proposed method, wecompared the results by inelastic buckling analyses to those ofAllowable stress levelAllowable stress level80064048032016008006404803201600Stress (MN/m2)Stress (MN/m2)(a) 600-m model (girder depth=2m)(b) 1200-m model (girder depth=2m)Fig. 6. Stress distribution in cable members of example bridges at the initial configuration.150H. Yoo et al. / Journal of Constructional Steel Research 72 (2012) 143154nonlinear inelastic analysis that are usually regarded as an exactapproach. Fig. 8 shows the errors of the inelastic buckling analysesusing both the column criterion (Eq. (11) and the proposed criterion(Eq. (15) for example cable-stayed bridge models with differentcenter spans. The horizontal and vertical axes in Fig. 8 indicate thedepth of the girders and the errors of inelastic buckling analyses onthe basis of the results of nonlinear inelastic analysis.The inelastic buckling analysis using the established criterion for acolumn has an average error of 30% for the medium-span models (i.e.,the 600-m models), whereas the errors of this method are reduced toless than 20% for the long-span models (i.e., the 1200-m models). Wemay explain these results by investigating the effect of axial forces ingirder and tower members on the collapse loads of the bridgesystems. In the case of the medium-span models (the 600-m models),bending moments as well as the axial force considerably contribute tothe limit state of the axialflexural interaction equation (Eq. (3). Inother words,thegirdermembersinthe600-mmodelsshouldobviouslybe treated as beamcolumn members. The criterion for a column(Eq. (11) is not valid evidently for this case, thus the inelastic bucklinganalysis with the criterion for a column cannot provide reasonablecollapseloadsofthesemodels.Onthecontrary,inthelong-spanmodels(the 1200-m models), the axial forces of girder members are very largeand the effect of the bending moments on the axialflexural interactionequation is trivial, i.e., the girder members in the 1200-m models seemto be approximately considered as simple column members. Therefore,the criterion for a column (Eq. (11) is adequate in this case and theinelastic buckling analysis with the criterion for a column producessimilar results to those of nonlinear inelastic analysis. Consequently,we can conclude that the criterion for a column (Eq. (11) isadequate for comparably longer span models, but not for shorterspan models in order to estimate the collapse loads of the bridgesystems.As expected, the inelastic buckling analysis with the proposedcriterion for a beamcolumn gives acceptable results irrespective ofthe length of the center span. The average errors of the method inFig. 8 are less than 10% for all bridge models. Consequently, the pro-posedcriterionfora beamcolumnis avalidmethodtoapproximatelydetermine the collapse loads of cable-stayed bridge systems for allspans and girder depth ranges. The inelastic buckling analysis usingthe proposed criterion for a beamcolumn is a good substitute forcomplex nonlinear inelastic analysis.5.3. Failure modes of bridge systemsIn order to test the validity of the proposed method, configurationof forming plastic hinges in members should be additionally checkedat the collapse state as well as the value of critical loads. Fig. 9 showsthe failure modes of the example cable-stayed bridges predicted bynonlinear inelastic analysis and the inelastic buckling analysis usingthe proposed criterion for a beamcolumn. The yielding parameters() of the individual members were plotted for nonlinear inelasticanalysis, whereas the tangent modulus were plotted for the case ofthe proposed inelastic buckling analysis. As mentioned, the yieldingparameter represents the gradual inelastic stiffness reduction of amember associated with the plastic hinges at both ends of the member(see Eq. (2). In addition, yielded cables were plotted by thickdotted lines for nonlinear inelastic analysis. Inelastic buckling analysis1200 m models1 m2 m3 m4 m5 m6 mGirder depth1 m2 m3 m4 m5 m6 mGirder depth600 m models43210432100246802468Vertical displacement (m)Vertical displacement (m)Load factorLoad factor(a) 600-m models(b) 1200-m modelsFig. 7. Loaddisplacement curves of example bridges by nonlinear inelastic analysis.Table 5Collapse load factors of example cable-stayed bridges.ModelGirderdepth(m)Nonlinear inelastic analysisInelastic buckling analysisColumn criterion (Eq. (11)The proposed beamcolumn criterion(Eq. (15)LC1LC2LC3LC1LC2LC3LC1LC2LC3600-m12.3752.3802.6672.9832.9923.1102.5832.5932.76922.9552.9593.3093.9884.0683.8982.5522.7513.23733.1223.1273.1894.1364.0194.0033.0913.0143.01943.1113.1282.7934.0343.9543.9382.8742.9692.97453.0833.1413.1453.9373.8833.8672.8422.9252.93063.0523.0702.7163.8593.8233.8082.7152.8582.8631200-m10.9560.9641.1731.1681.1851.2191.0521.0561.18321.5151.5481.6521.7841.7681.8231.6381.6411.78331.7251.7261.9542.0192.0242.0051.8741.8661.78141.8592.0151.8492.1352.1732.0741.9771.9791.81451.9631.9502.0712.2262.1812.1492.0432.0451.86961.9881.9752.0702.2792.2542.2221.8561.8621.855151H. Yoo et al. / Journal of Constructional Steel Research 72 (2012) 143154used in this study cannot catch the yielding of cables throughout theanalysis.As shown in Fig. 9, the tangent modulus of the member where thelarge axial force and the moment are concentrated (e.g., the membersat the intersection between the girder and the tower and at the lowerpart ofthe tower)is comparablysmallerthan thoseof othermembers.In particular, the calculated tangent modulus of the member by theinelastic buckling analysis using the proposed criterion for a beamcolumn is nearly zero for the member at the locations where theyielding parameter vanish, predicted by nonlinear inelastic analysis.The zero value of the tangent modulus of a member means that themember has no stiffness at current load levels and is perfectly yielded.As can be seen in Fig. 9, the distribution of the tangent modulus ofgirder and tower members by the proposed inelastic buckling analysisis in good agreement with those predicted by nonlinear inelasticanalysis.Consequently, it is probable that the inelastic buckling analysisusing the proposed criterion can predict the failure modes as wellas the collapse loads of the cable-stayed bridge models in acceptableranges. In addition, the calculations of the proposed inelastic bucklinganalysis is performed quickly, even on some types of under poweredcomputers because there is no need to solve incremental systemequations repeatedly. We believe that the inelastic buckling analysisusing the proposed criterion can be efficiently and frequently appliedto practical design stages for the estimation of the collapse load andthe overall safety of steel cable-stayed bridges.6. ConclusionsNonlinear inelastic analysis is an exact approach to determine thecollapse loads of steel cable-stayed bridges because it can take intoaccount all geometric and material nonlinear aspects of the bridgesystem. However, this sophisticated analysis method has limitedapplications in the preliminary design stages in that there are manydesign candidates to be checked. This paper proposed a simple alter-native for complex nonlinear inelastic analysis to estimate thecollapse load of a steel cable-stayed bridge. The fundamental theoriesof nonlinear inelastic analysis and inelastic buckling analysis werebriefly reviewed, and the established criterion for a column memberwas illustrated with comments. Afterward, a new criterion for abeamcolumn member was proposed based on the axialflexuralinteraction equationin combination with the classical tangentmodulustheory and the columnstrength curve. The example bridge models,whichhad differentcenter-spans withdifferentgirder depths,were an-alyzed for verification. By some parametric studies, we arrived at thefollowing conclusions:1. The established criterion for a column is obviously not valid todetermine the collapse loads of steel cable-stayed bridge models,although it can give partly acceptable results for compara
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