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NATIONAL NCHRPREPORT 563 COOPERATIVE HIGHWAY RESEARCH PROGRAM Development of LRFD Specifications for Horizontally Curved Steel Girder Bridges

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TRANSPORTATION RESEARCH BOARD EXECUTIVE COMMITTEE 2006 (Membership as of April 2006) OFFICERS Chair: Michael D. Meyer, Professor, School of Civil and Environmental Engineering, Georgia Institute of Technology Vice Chair: Linda S. Watson, Executive Director, LYNX--Central Florida Regional Transportation Authority Executive Director: Robert E. Skinner, Jr., Transportation Research Board MEMBERS MICHAEL W. BEHRENS, Executive Director, Texas DOT ALLEN D. BIEHLER, Secretary, Pennsylvania DOT JOHN D. BOWE, Regional President, APL Americas, Oakland, CA LARRY L. BROWN, SR., Executive Director, Mississippi DOT DEBORAH H. BUTLER, Vice President, Customer Service, Norfolk Southern Corporation and Subsidiaries, Atlanta, GA ANNE P. CANBY, President, Surface Transportation Policy Project, Washington, DC DOUGLAS G. DUNCAN, President and CEO, FedEx Freight, Memphis, TN NICHOLAS J. GARBER, Henry L. Kinnier Professor, Department of Civil Engineering, University of Virginia, Charlottesville ANGELA GITTENS, Vice President, Airport Business Services, HNTB Corporation, Miami, FL GENEVIEVE GIULIANO, Professor and Senior Associate Dean of Research and Technology, School of Policy, Planning, and Development, and Director, METRANS National Center for Metropolitan Transportation Research, USC, Los Angeles SUSAN HANSON, Landry University Professor of Geography, Graduate School of Geography, Clark University JAMES R. HERTWIG, President, CSX Intermodal, Jacksonville, FL GLORIA J. JEFF, General Manager, City of Los Angeles DOT ADIB K. KANAFANI, Cahill Professor of Civil Engineering, University of California, Berkeley HAROLD E. LINNENKOHL, Commissioner, Georgia DOT SUE MCNEIL, Professor, Department of Civil and Environmental Engineering, University of Delaware DEBRA L. MILLER, Secretary, Kansas DOT MICHAEL R. MORRIS, Director of Transportation, North Central Texas Council of Governments CAROL A. MURRAY, Commissioner, New Hampshire DOT JOHN R. NJORD, Executive Director, Utah DOT SANDRA ROSENBLOOM, Professor of Planning, University of Arizona, Tucson HENRY GERARD SCHWARTZ, JR., Senior Professor, Washington University MICHAEL S. TOWNES, President and CEO, Hampton Roads Transit, Hampton, VA C. MICHAEL WALTON, Ernest H. Cockrell Centennial Chair in Engineering, University of Texas at Austin MARION C. BLAKEY, Federal Aviation Administrator, U.S.DOT (ex officio) JOSEPH H. BOARDMAN, Federal Railroad Administrator, U.S.DOT (ex officio) REBECCA M. BREWSTER, President and COO, American Transportation Research Institute, Smyrna, GA (ex officio) GEORGE BUGLIARELLO, Chancellor, Polytechnic University of New York, and Foreign Secretary, National Academy of Engineering (ex officio) SANDRA K. BUSHUE, Deputy Administrator, Federal Transit Administration, U.S.DOT (ex officio) J. RICHARD CAPKA, Acting Administrator, Federal Highway Administration, U.S.DOT (ex officio) THOMAS H. COLLINS (Adm., U.S. Coast Guard), Commandant, U.S. Coast Guard (ex officio) JAMES J. EBERHARDT, Chief Scientist, Office of FreedomCAR and Vehicle Technologies, U.S. Department of Energy (ex officio) JACQUELINE GLASSMAN, Deputy Administrator, National Highway Traffic Safety Administration, U.S.DOT (ex officio) EDWARD R. HAMBERGER, President and CEO, Association of American Railroads (ex officio) WARREN E. HOEMANN, Deputy Administrator, Federal Motor Carrier Safety Administration, U.S.DOT (ex officio) JOHN C. HORSLEY, Executive Director, American Association of State Highway and Transportation Officials (ex officio) JOHN E. JAMIAN, Acting Administrator, Maritime Administration, U.S.DOT (ex officio) J. EDWARD JOHNSON, Director, Applied Science Directorate, National Aeronautics and Space Administration (ex officio) ASHOK G. KAVEESHWAR, Research and Innovative Technology Administrator, U.S.DOT (ex officio) BRIGHAM MCCOWN, Deputy Administrator, Pipeline and Hazardous Materials Safety Administration, U.S.DOT (ex officio) WILLIAM W. MILLAR, President, American Public Transportation Association (ex officio) SUZANNE RUDZINSKI, Director, Transportation and Regional Programs, U.S. Environmental Protection Agency (ex officio) JEFFREY N. SHANE, Under Secretary for Policy, U.S.DOT (ex officio) CARL A. STROCK (Maj. Gen., U.S. Army), Chief of Engineers and Commanding General, U.S. Army Corps of Engineers (ex officio) NATIONAL COOPERATIVE HIGHWAY RESEARCH PROGRAM Transportation Research Board Executive Committee Subcommittee for NCHRP MICHAEL D. MEYER, Georgia Institute of Technology (Chair) ROBERT E. SKINNER, JR., Transportation Research Board J. RICHARD CAPKA, Federal Highway Administration C. MICHAEL WALTON, University of Texas at Austin JOHN C. HORSLEY, American Association of State Highway LINDA S. WATSON, LYNX--Central Florida Regional and Transportation Officials Transportation Authority JOHN R. NJORD, Utah DOT

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NATIONAL COOPERATIVE HIGHWAY RESEARCH PROGRAM NCHRP REPORT 563 Development of LRFD Specifications for Horizontally Curved Steel Girder Bridges JOHN M. KULICKI WAGDY G. WASSEF DANIELLE D. KLEINHANS Modjeski and Masters, Inc. Harrisburg, PA CHAI H. YOO Auburn University Auburn, AL ANDRZEJ S. NOWAK University of Nebraska Lincoln, NE MIKE GRUBB Bridge Software Development International, Ltd. Coopersburg, PA S UBJECT A REAS Bridges, Other Structures, Hydraulics and Hydrology Research Sponsored by the American Association of State Highway and Transportation Officials in Cooperation with the Federal Highway Administration TRANSPORTATION RESEARCH BOARD WASHINGTON, D.C. 2006 www.TRB.org

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NATIONAL COOPERATIVE HIGHWAY RESEARCH PROGRAM NCHRP REPORT 563 Systematic, well-designed research provides the most effective Price $33.00 approach to the solution of many problems facing highway Project 12-52 administrators and engineers. Often, highway problems are of local ISSN 0077-5614 interest and can best be studied by highway departments individually or in cooperation with their state universities and ISBN 0-309-09855-6 others. However, the accelerating growth of highway transportation Library of Congress Control Number 2006926471 develops increasingly complex problems of wide interest to 2006 Transportation Research Board highway authorities. These problems are best studied through a coordinated program of cooperative research. In recognition of these needs, the highway administrators of the American Association of State Highway and Transportation COPYRIGHT PERMISSION Officials initiated in 1962 an objective national highway research Authors herein are responsible for the authenticity of their materials and for obtaining program employing modern scientific techniques. This program is written permissions from publishers or persons who own the copyright to any previously published or copyrighted material used herein. supported on a continuing basis by funds from participating member states of the Association and it receives the full cooperation Cooperative Research Programs (CRP) grants permission to reproduce material in this publication for classroom and not-for-profit purposes. Permission is given with the and support of the Federal Highway Administration, United States understanding that none of the material will be used to imply TRB, AASHTO, FAA, Department of Transportation. FHWA, FMCSA, FTA, or Transit Development Corporation endorsement of a The Transportation Research Board of the National Academies particular product, method, or practice. It is expected that those reproducing the was requested by the Association to administer the research material in this document for educational and not-for-profit uses will give appropriate program because of the Board's recognized objectivity and acknowledgment of the source of any reprinted or reproduced material. For other uses of the material, request permission from CRP. understanding of modern research practices. The Board is uniquely suited for this purpose as it maintains an extensive committee structure from which authorities on any highway transportation subject may be drawn; it possesses avenues of communications and NOTICE cooperation with federal, state and local governmental agencies, The project that is the subject of this report was a part of the National Cooperative universities, and industry; its relationship to the National Research Highway Research Program conducted by the Transportation Research Board with the Council is an insurance of objectivity; it maintains a full-time approval of the Governing Board of the National Research Council. Such approval research correlation staff of specialists in highway transportation reflects the Governing Board's judgment that the program concerned is of national importance and appropriate with respect to both the purposes and resources of the matters to bring the findings of research directly to those who are in National Research Council. a position to use them. The members of the technical committee selected to monitor this project and to review The program is developed on the basis of research needs this report were chosen for recognized scholarly competence and with due identified by chief administrators of the highway and transportation consideration for the balance of disciplines appropriate to the project. The opinions and departments and by committees of AASHTO. Each year, specific conclusions expressed or implied are those of the research agency that performed the areas of research needs to be included in the program are proposed research, and, while they have been accepted as appropriate by the technical committee, to the National Research Council and the Board by the American they are not necessarily those of the Transportation Research Board, the National Research Council, the American Association of State Highway and Transportation Association of State Highway and Transportation Officials. Officials, or the Federal Highway Administration, U.S. Department of Transportation. Research projects to fulfill these needs are defined by the Board, and Each report is reviewed and accepted for publication by the technical committee qualified research agencies are selected from those that have according to procedures established and monitored by the Transportation Research submitted proposals. Administration and surveillance of research Board Executive Committee and the Governing Board of the National Research contracts are the responsibilities of the National Research Council Council. and the Transportation Research Board. The needs for highway research are many, and the National Cooperative Highway Research Program can make significant contributions to the solution of highway transportation problems of mutual concern to many responsible groups. The program, however, is intended to complement rather than to substitute for or duplicate other highway research programs. Published reports of the NATIONAL COOPERATIVE HIGHWAY RESEARCH PROGRAM are available from: Transportation Research Board Business Office NOTE: The Transportation Research Board of the National Academies, the 500 Fifth Street, NW National Research Council, the Federal Highway Administration, the American Washington, DC 20001 Association of State Highway and Transportation Officials, and the individual and can be ordered through the Internet at: states participating in the National Cooperative Highway Research Program do http://www.national-academies.org/trb/bookstore not endorse products or manufacturers. Trade or manufacturers' names appear herein solely because they are considered essential to the object of this report. Printed in the United States of America

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The National Academy of Sciences is a private, nonprofit, self-perpetuating society of distinguished schol- ars engaged in scientific and engineering research, dedicated to the furtherance of science and technology and to their use for the general welfare. On the authority of the charter granted to it by the Congress in 1863, the Academy has a mandate that requires it to advise the federal government on scientific and techni- cal matters. Dr. Ralph J. Cicerone is president of the National Academy of Sciences. The National Academy of Engineering was established in 1964, under the charter of the National Acad- emy of Sciences, as a parallel organization of outstanding engineers. It is autonomous in its administration and in the selection of its members, sharing with the National Academy of Sciences the responsibility for advising the federal government. The National Academy of Engineering also sponsors engineering programs aimed at meeting national needs, encourages education and research, and recognizes the superior achieve- ments of engineers. Dr. William A. Wulf is president of the National Academy of Engineering. The Institute of Medicine was established in 1970 by the National Academy of Sciences to secure the services of eminent members of appropriate professions in the examination of policy matters pertaining to the health of the public. The Institute acts under the responsibility given to the National Academy of Sciences by its congressional charter to be an adviser to the federal government and, on its own initiative, to identify issues of medical care, research, and education. Dr. Harvey V. Fineberg is president of the Institute of Medicine. The National Research Council was organized by the National Academy of Sciences in 1916 to associate the broad community of science and technology with the Academy's purposes of furthering knowledge and advising the federal government. Functioning in accordance with general policies determined by the Acad- emy, the Council has become the principal operating agency of both the National Academy of Sciences and the National Academy of Engineering in providing services to the government, the public, and the scientific and engineering communities. The Council is administered jointly by both the Academies and the Institute of Medicine. Dr. Ralph J. Cicerone and Dr. William A. Wulf are chair and vice chair, respectively, of the National Research Council. The Transportation Research Board is a division of the National Research Council, which serves the National Academy of Sciences and the National Academy of Engineering. The Board's mission is to promote innovation and progress in transportation through research. In an objective and interdisciplinary setting, the Board facilitates the sharing of information on transportation practice and policy by researchers and practitioners; stimulates research and offers research management services that promote technical excellence; provides expert advice on transportation policy and programs; and disseminates research results broadly and encourages their implementation. The Board's varied activities annually engage more than 5,000 engineers, scientists, and other transportation researchers and practitioners from the public and private sectors and academia, all of whom contribute their expertise in the public interest. The program is supported by state transportation departments, federal agencies including the component administrations of the U.S. Department of Transportation, and other organizations and individuals interested in the development of transportation. www.TRB.org www.national-academies.org

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COOPERATIVE RESEARCH PROGRAMS STAFF FOR NCHRP REPORT 563 ROBERT J. REILLY, Director, Cooperative Research Programs CRAWFORD F. JENCKS, Manager, NCHRP DAVID B. BEAL, Senior Program Officer EILEEN P. DELANEY, Director of Publications BETH HATCH, Editor NCHRP PROJECT 12-52 Field of Design--Area of Bridges EDWARD P. WASSERMAN, Tennessee DOT (Chair) RALPH E. ANDERSON, Illinois DOT KARL FRANK, University of Texas at Austin GREGG C. FREDRICK, Wyoming DOT THOMAS P. MACIOCE, Pennsylvania DOT VASANT C. MISTRY, FHWA MARK RENO, Quincy Engineering, Inc., Sacramento, CA KEVIN WESTERN, Minnesota DOT WILLIAM WRIGHT, FHWA SHEILA RIMAL DUWADI, FHWA Liaison STEPHEN F. MAHER, TRB Liaison AUTHOR ACKNOWLEDGMENTS The research reported herein was performed under NCHRP Proj- to the preparation of two curved girders design examples, which are ect 12-52. Modjeski and Masters, Inc., was the contractor. Dr. John available online at http://www.transportation.org/sites/bridges/ M. Kulicki was the principal investigator. Other co-authors, acting docs/Box%20Girder.pdf and http://www.transportation.org/sites/ as sub-consultants to Modjeski and Masters, and their main contri- bridges/docs/I-Girder.pdf. bution to the work are as follows: The research team would like to acknowledge the support of the AASHTO Technical Committee on Steel Bridges, T14. The support Dr. Chai H. Yoo, Auburn University: Literature search and technical contributions of the committee's chair, Mr. Ed Dr. Andrzej S. Nowak, University of Nebraska: Statistical Wasserman, Civil Engineering Director, Structures Division, Ten- calibration nessee Department of Transportation, are gratefully acknowledged. Mr. Mike Grubb, Bridge Software Development International, Dr. Donald White, Georgia Institute of Technology, provided sig- Ltd.: Preparation of the specifications articles. nificant technical support in incorporating the results of his work on many other research projects into the recommended specifications. The research team would like to acknowledge the contributions Dr. Dennis Mertz, University of Delaware, reviewed and com- of many of Modjeski and Masters's staff. Particularly acknowledged mented on the recommended specification articles. His contribu- are the contributions of Mr. Christopher Smith and Mr. Kevin Johns tions to the work are greatly appreciated.

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FOREWORD By David B. Beal Staff Officer Transportation Research Board This report contains the findings of research performed to develop design specifications for horizontally curved steel girder bridges. The developed specifications have been adopted by AASHTO and are included in the 2005 Interims to the third edition of the AASHTO LRFD Bridge Design Specifications. Detailed examples showing the application of the speci- fications to the design of an I-girder bridge and a box-girder bridge are available from AASHTO. The material in this report will be of immediate interest to bridge designers. In the late 1980s and early 1990s, AASHTO and the FHWA recognized the need to address problems with design and construction of horizontally curved steel girder highway bridges. AASHTO, through the NCHRP, embarked on an overhaul of the AASHTO Guide Specifications for Horizontally Curved Highway Bridges. NCHRP Project 12-38, "Improved Design Specifications for Horizontally Curved Steel Girder Highway Bridges," provided rec- ommended load factor design (LFD) and construction specifications based on the state of the art that addressed many of the problems associated with the design and construction of these structures. While the LFD specifications were under development, the FHWA and others began conducting significant research that enhanced the understanding of steel bridge behavior in general and horizontally curved bridges in particular. The FHWA research included tests on a full-scale I-girder bridge, and knowledge of the moment and shear capacities of hor- izontally curved I-girder bridges resulted from this research. Analytical work, also funded by the FHWA, resulted in the unification of the design equations for straight and curved steel girders. NCHRP Project 12-52 was performed by Modjeski & Masters, Inc., with the assistance of Mike Grubb, Andrzej Nowak, Don White, and Chai Yoo. The report fully documents the effort leading to the specifications and contains an extensive compilation of abstracts of horizontally curved girder bridge research reports. Appendix C, "Calibration of LRFD Design Specifications for Steel Curved Girder Bridges," and Appendix D, "Comparison of Curved Steel I-Girder Bridge Design Specifications," can be downloaded from http://trb.org/ news/blurb_detail.asp?id=5965 or from the Project 12-52 website at trb.org/nchrp.

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CONTENTS 1 Summary 3 Chapter 1 Introduction and Research Approach 3 1.1 Introduction 4 1.2 Research Objective 4 1.3 Scope of the Study 4 1.4 Research Approach 6 Chapter 2 Findings 6 2.1 Literature Search 6 2.2 Design Specifications 6 2.3 Calibration 6 2.3.1 Scope 7 2.3.2 Study Bridges 7 2.3.3 Calibration Procedure 8 2.3.4 Load Models 8 2.3.4.1 Load Components 8 2.3.4.2 Dead Load 8 2.3.4.3 Live Load 9 2.3.4.4 Dynamic Load 10 2.3.4.5 Load Ratios 11 2.3.5 Resistance Models 11 2.3.6 Calibration Results 11 2.4 Design Comparisons 11 2.4.1 Objective 11 2.4.2 Application of the NCHRP 12-50 Process 13 2.4.3 Methodology 17 2.4.4 Shear Design 19 2.4.5 Flexural Design 20 2.4.6 Summary of Comparison Results 21 2.5 Design Examples 23 Chapter 3 Conclusions 24 References A-i Appendix A Literature Search B-i Appendixes BD Related Materials

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1 SUMMARY Development of LRFD Specifications for Horizontally Curved Steel Girder Bridges AASHTO's Guide Specifications for Horizontally Curved Highway Bridges (hereafter referred to as the "Guide Specifications") was first published in 1980. The specifications were based on work conducted in the late 1960s and early 1970s by a group of researchers called the "Con- sortium of University Research Teams" (CURT). The research work resulted in guidance on the analysis of curved bridges and equations for determining the strength and checking the stability of curved girders. An updated version of the Guide Specifications was published in 1993. The 1980 Guide Specifications was written in the allowable stress design (ASD) format. The 1993 Guide Specifications was written in both the ASD and the load factor design (LFD) format. As a result of the work on the National Cooperative Highway Research Program (NCHRP) 12-38 project, the Guide Specifications were updated again and the updated version, written in the LFD format, was published in 2003. In 1999, the NCHRP 12-52 project was initiated to develop design provisions for curved bridges in the AASHTO load and resistance factor design (LRFD) format. These provisions were intended to be incorporated into the specifications to extend the specifications' cov- erage to curved bridges. Statistically calibrating the curved bridge design provisions was required to ensure smooth merging of these provisions into the then-existing straight girder design provisions. The original organization of the NCHRP 12-52 project called for a two-phase approach. Phase I was intended to produce curved bridge design provisions that were based on the information available at that time. These specifications were intended to be revised in Phase II based on the results of the then-ongoing research on curved bridges. This research was funded by the Federal Highway Administration (FHWA). Several universities collabo- rated with the FHWA in conducting this research. Phase I of the NCHRP 12-52 project produced curved bridge design provisions as planned. It also produced two design examples, one of a box-girder bridge and the other of an I-girder bridge. However, at that time it became clear that the FHWA-sponsored research would produce a new set of design provisions that would be applicable to both straight and curved bridges and that would have some terms of the equations "dropping out" when applied to straight bridges. The new set of provisions was considered to be a significant improvement toward streamlining the design provisions. It was decided not to publish the design specifications developed in Phase I of the project and to develop a new set of specifi- cations and design examples based on the results of the FHWA-sponsored research in Phase II. These provisions were approved by ballot of the AASHTO Highway Subcommittee on Bridges and Structures (HSCOBS) in 2003 and 2004 for straight girders and curved girders, respectively. The straight girder provisions were published in the third edition of AASHTO LRFD specifications in 2004. The curved girder provisions were published in the 2006 interim to the AASHTO LRFD specifications.

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2 In addition to the recommended specifications that were subsequently adopted by AASHTO, the NCHRP 12-52 project resulted in the following: The statistical calibration of the load and resistance factors for curved bridges. This calibration indicated that the factors developed for straight girders are applicable to curved girders. The comparison of resistance analysis conducted using the AASHTO Guide Specifica- tions for curved bridges to those conducted using the new LRFD-based design provi- sions. Twenty-one existing bridges provided by several state DOTs and 11 simulated bridges were used in this comparison. The comparison indicated that member propor- tions will not be significantly altered in unanticipated ways and that anticipated changes manifested themselves in the example bridges. The updating of the I-girder and box-girder bridge design examples. These examples, originally produced in the NCHRP 12-38 project, were updated in Phase I of the project and then updated again in Phase II based on the new design provisions developed in Phase II of the project. The curved bridge design provisions, the statistical calibration work, and the comparison between the existing designs and those conducted using the new provisions are included in this report. The two design examples are available on the AASHTO website at http://www.transportation.org/sites/bridges/docs/Box%20Girder.pdf and http://www. transportation.org/sites/bridges/docs/I-Girder.pdf.

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3 CHAPTER 1 Introduction and Research Approach 1.1 Introduction augment a limited number of experimental tests with hun- dreds of analytic investigations. The history of specification development related to hori- An update to the 1993 Guide Specifications was prepared zontally curved girder bridges spans more than 30 years. Dur- under NCHRP Project 12-38 (7, 8) and was published in ing the late 1960s and early 1970s, a group of researchers called 2003 (9). The state of the art of curved girder specifications the "Consortium of University Research Teams" (CURT) dev- and a review of the intervening 10 years' advances in under- eloped guidance on the analysis of curved girder bridges and standing the resistance of curved sections resulted in a more characterizations of the strength and stability of curved girders. modern specification, which was written in the LFD format. This work lead to software products and the design provisions The provisions for I-girders retained some of the features that became codified in the 1980 AASHTO Guide Specifications introduced in the 1980 Guide Specifications, but made sig- for Horizontally Curved Highway Bridges (hereafter referred to nificant advances in the recognition of the need to directly as the "Guide Specifications") (1). These specifications were interrelate the lateral flange bending stress, or "warping" initially produced in the allowable stress design (ASD) format. stress, with the vertical bending stress. This interrelation was In 1993, an updated version of the 1980 Guide Specifications done by subtracting part or all of the lateral flange bending was released (2). The 1993 Guide Specifications was written stress from the resistance of the cross-section. The need for in both allowable stress design (ASD) and load factor design additional stud connectors in composite sections due to the (LFD) format. radial component of shear between the deck and girders was Recognizing the need to update the technology in these also recognized. earlier design specifications, a major research effort was initi- Recognizing the need to include curved girder bridges in ated by the FHWA that involved both experimental and ana- the AASHTO LRFD Bridge Design Specifications (10),AASHTO lytic investigations of curved steel bridges. The experimental asked the NCHRP to initiate Project 12-52 to develop rec- work undertaken in the FHWA's Turner-Fairbank Highway ommended state-of-the-art design specifications for hor- Research Laboratory involved a three-girder, single-span struc- izontally curved steel bridges. The recommended design ture. The experimental program has been widely reported specifications were required to be statistically calibrated and has been ongoing for over a decade (3, 4, 5). The FHWA's and to be written in the load and resistance factor design experimental program was augmented by tests of single (LRFD) format. The recommended specifications were also girders at several universities and by large-scale finite element required to incorporate research results accumulated over analysis with nonlinear materials, plate out-of-flatness, and the years, including the results of the FHWA research. To nonlinear geometries (6). Comparisons were made between ensure a smooth transition to the new provisions, the design the results of tests on the curved girder test frame and analytic example for an I-girder bridge and the design example for results. a box-girder bridge (which were both developed under the One of the pivotal features of the finite element analysis NCHRP 12-38 project) would have to be updated to reflect was its ability to accurately represent the capacities due to the application of the new provisions. In addition, to fur- local flange buckling and lateral torsional buckling, both in ther investigate the effect of the new design provisions terms of the resistance of the cross-section and in terms of the on the required girder sections, design comparisons were deflected shape. The agreement between the analytical and conducted for a large number of existing and simulated experimental results was excellent and made it possible to bridges.

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17 Among the simulated bridges, the minimum radius was be noted that when a reference to an equation number is made approximately 120 feet. in this section (i.e., Section 2.4), the equation number is that The webs varied in depth from 33 inches to 94 inches. used in the specifications. Lateral bending stresses were not submitted for some of the bridges in the study. For one of these bridges, the orig- inal design ignored the lateral stresses. Another bridge 2.4.4 Shear Design consisted of straight girders with a curved deck. The lat- Appendix D contains a series of flowcharts that outline the eral stresses of the other bridges were estimated using the shear design protocols and variables according to each of the V-load method. three specifications--the 1993 Guide Specifications, the 2003 Guide Specifications, and the 12-52 recommended specifica- For each bridge, the critical positive flexure, critical negative tions (which have subsequently been adopted by AASHTO flexure, and critical shear locations were either submitted by and published in the 2006 interim specifications). The reader is the agency or determined by Modjeski and Masters, Inc., from encouraged to consult the appropriate specification for clarifi- the documentation submitted. cation and further information regarding the design protocols. The critical-to-applied stress ratios according to each spec- In general, the shear design protocols for the three analysis ification were plotted for each bridge. Submitted load effects specifications are very similar. This is true for both transversely (e.g., moments and shears) were assumed to be service loads, stiffened and unstiffened members. The main differences and the applied stress calculations used the load factors for among the three specifications are as follows: the LRFD Strength I combination as a basis for comparison. An attempt to remove the effects of the load factors was made by The maximum transverse stiffener spacing has been progres- also plotting the critical stress calculated by each specification sively increased from D, the depth of the web, in the 1993 normalized to the critical stress calculated by the 1993 Guide Guide Specifications to 3D in the 12-52 recommended Specifications. These comparisons of normalized stress are the specifications. primary focus of the analyses outlined in this section. Addi- The 12-52 recommended specifications allow for the con- tional analyses, which are outlined in Appendix D (available sideration of the additional post-buckling strength from online at http://trb.org/news/blurb_detail.asp?id=5965), were tension-field action in the shear critical stress calculations. performed to compare the LRFD Service II load provisions with the corresponding provisions in both the 1993 Guide Figure 6 shows the shear critical-to-applied stress ratio for the Specifications and the 2003 Guide Specifications. It should sample of bridges according to the three specifications, while 9.00 8.00 Critical-to-Applied Stress Ratio 7.00 6.00 Existing Bridges Simulated Bridges 5.00 4.00 3.00 2.00 1.00 0.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Bridge ID LFD 12-38 12-52 Note: Stiffened bridges are denoted with a box around the number identifier. Figure 6. Shear critical-to-applied stress ratio.

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18 Figure 7 shows the shear critical stress normalized to that cal- post-buckling strength is also recognized in the 1993 Guide culated using the 1993 Guide Specifications. In these figures, as Specifications and AASHTO LRFD specifications straight girder in the shear-related figures in Appendix D, the stiffened bridges provisions. are denoted with a box around the number identifier. The fig- According to the 12-52 recommended specifications, ures also differentiate the existing bridges from the simulated Eq. 6.10.9.2-1, which is identical to Eq. 6-4 in the 2003 Guide bridges. In general, the critical-to-applied stress ratios for the Specifications, is used for both stiffened and unstiffened webs three analysis specifications are relatively consistent among to account for the shear-yielding or shear-buckling strength specifications and within the sample. of the web. Note that this equation is unnumbered in the 1993 All bridges except for Bridges 4 and 22 exhibited equal or Guide Specifications. slightly higher shear critical stress according to the 1993 Guide Specifications than according to the 2003 Guide Specifications. Vcr = CVp Eq. 6.10.9.2-1 The change in maximum transverse stiffener spacing among the three specifications is evident in the critical stress values for Bridges 4 and 22. These two bridges are the only ones that The 12-52 recommended specifications use an additional have a transverse stiffener spacing value greater than the equation, Eq. 6.10.9.3.2-2, to account for the post-buckling depth of the web and are the only ones that exhibit a higher strength of the interior panels of stiffened webs that satisfy critical stress according to the 2003 Guide Specifications than certain geometric requirements. The use of this equation pro- according to the 1993 Guide Specifications. This is because duces a higher multiplier on Vp for these webs. Eq. 6.10.9.3.2-2 the sections are analyzed as unstiffened according to the 1993 is used in the 12-52 recommended specifications for both Guide Specifications when the transverse stiffener spacing straight and curved steel girders and is applicable for Bridges 2, exceeds the maximum value of D. 4, 1214, 16, and 2231.Among the bridges with stiffened webs, The analysis of transversely stiffened members with the only Bridge 7 does not meet the geometric requirements for 1993 Guide Specifications and the 2003 Guide Specifications the use of this equation and, subsequently, uses a more conser- is more conservative than with the 12-52 recommended spec- vative variation, identified as Eq. 6.10.9.3.2-8. The flowcharts ifications in several instances, as evidenced by the lower nor- in Appendix D (available online at http://trb.org/news/blurb_ malized critical stress values. These cases are for Bridges 2, detail.asp?id=5965) outline the number of bridges in the sam- 4, 7, 1214, 16, and 2228. For these bridges, the 12-52 rec- ple that meet the requirements for each classification according ommended specifications allow the additional post-buckling to each of the specifications for the stiffened and unstiffened strength from tension-field action to be considered. This critical stress equations. 2.50 Critical Stress/LFD Critical Stress 2.00 Simulated Bridges Existing Bridges 1.50 1.00 0.50 0.00 1 2 3 4 5 6 7 8 9 10 1112 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Bridge ID 12-38 12-52 Note: Stiffened bridges are denoted with a box around the number identifier. Figure 7. Shear critical stress divided by 1993 Guide Specifications shear critical stress.

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19 0.87 (1 - C ) For the 12-52 recommended specifications, the ratios are Vn = Vp C + Eq. 6.1 10.9.3.2-2 slightly higher than for the 2003 Guide Specifications. 2 dO The following trends, as they relate to the studied bridges, 1+ D are worth noting from the flowcharts: For the 12-52 recommended specifications related to com- 0.87 (1 - C ) posite sections in positive flexure (Section 6.10.7), there are Vn = Vp C + Eq q. 6.10.9.3.2-8 2 d d two bridges in the sample for which the section is regarded 1+ O + O D D as noncompact; it should be noted, however, that because plastic design is not permitted, all composite curved steel girders in positive flexure must be analyzed as noncompact Bridges 29, 30, and 31 also qualified for the use of the post- according to 12-52 Section 6.10.6.2.2. buckling strength provision. However, no additional strength According to the 12-52 recommended specifications, the is realized for cases when the ratio of the shear-buckling resist- majority of the bridges in the sample were classified as com- ance to the shear yield strength, C, is equal to unity; in this pact for flange local buckling (FLB) considerations and as situation, Eq. 6.10.9.3.2-2 becomes Eq. 6.10.9.2-1 and only noncompact for lateral-torsional buckling (LTB) consider- the buckling strength is considered. The equations used to ations for all design conditions. determine C for each of the three specifications are listed in For the composite negative flexure and noncomposite section Appendix D and are essentially equivalent. recommended specifications of 12-52 (Section 6.10.8), the Although also stiffened, Bridges 3, 20, and 32 were analyzed case where the compression flange is continuously braced at the end panels and are therefore not able to rely on the for- is not represented in the sample because such a case would mation of a tension field or to account for any additional post- be atypical. The bottom flange is usually braced at discrete buckling strength. These observations are evident in Figure 7. points where the cross-frames exist. Generally, the LTB considerations controlled the critical stress of the bridges in the sample, which is logical given the 2.4.5 Flexural Design classifications pertaining to their buckling behavior. None of the specifications reduce the capacity of a contin- The flowcharts in Appendix D outline the number of bridges uously braced flange below the yield strength, Fy. in the sample that met the requirements for each classifica- In a sense, the magnification factor in the 12-52 recom- tion according to each of the specifications for the composite mended specifications, defined in Section 6.10.1.6, replaces positive flexure (C+), composite negative flexure (C-), non- the factors from the 1993 Guide Specifications and the composite positive flexure (NC+), and noncomposite nega- 2003 Guide Specifications. A comparison reveals that both tive flexure (NC-), respectively. Furthermore, they outline the factors increase the tendency of the section to deform because design protocols and variables used for each of the three spec- of secondary effects. ifications. The reader is encouraged to reference the appro- Hybrid sections are not allowed by the 2003 Guide Specifi- priate specification for clarification and further information cations, but are allowed by the 1993 Guide Specifications and regarding the design protocols. the 12-52 recommended specifications, which have similar Table 5 outlines the width-to-thickness (i.e., slenderness) provisions for the consideration of stresses in hybrid sections. ratio limits for the various classifications of the flanges accord- The provisions for hybrid sections in the 1993 Guide Spec- ing to the three specifications. For a yield strength of 50 ksi, ifications consider only yielding of the tension flange in the limits for compact and slender flanges are lower with the positive bending regions or only yielding of the compres- 1993 Guide than with the 2003 Guide, but only marginally so. sion flange in negative bending regions, while the 12-52 Table 5. Flange classifications by slenderness ratio. Specifications Compact Flange Noncompact Flange Slender Flange 1993 Guide bf bf bf tf 14.31 14.31 19.68 f Specifications 2003 Guide bf bf bf tf 18 18 23 Specifications 12-52 bf bf bf tf 18.3 18.3 24 Recommended Specifications

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20 recommended specifications, in Eq. 6.10.1.10.1-1, have indicates that the compression flange is the critical flange in been adapted to include all positions of the neutral axis and unbraced or discretely braced conditions. all combinations of yield strengths for the various portions of the girder. 2.4.6 Summary of Comparison Results One noticeable difference among the three specifications is Based on the analyses, the following observations and con- the fact that while the 1993 Guide Specifications and the 2003 clusions can be made regarding the shear design protocols in Guide Specifications consider the lateral bending stress to the 1993 Guide Specifications, the 2003 Guide Specifications, reduce the critical stress, the 12-52 recommended specifications and the 12-52 recommended specifications (which have sub- consider the lateral bending stress as a load. In order to alleviate sequently been adopted by AASHTO and published in the this difference, the appropriate portion of the lateral bending 2006 interim specifications): stress was deducted from the critical stress calculated accord- Strength I load case, shear design: ing to the 12-52 recommended specifications to obtain the bending resistance, Fbu. The maximum transverse stiffener spacing has been progres- sively increased from D, the depth of the web, in the 1993 1 Guide Specifications to 3D in the 12-52 recommended Fbu = f Fn - fl 3 specifications. All three specifications use the ratio of the shear-buckling The flexural capacity figures in Appendix D (which is avail- resistance to the shear yield strength, C. A comparison of the able online at http://trb.org/news/blurb_detail.asp?id=5965) equations used to calculate C in each of the specifications use the bending resistance values, with the exception of the reveals that the constants for the equations are equivalent. In general, the critical stress values for the three specifica- critical-to-applied stress plots, which consider the flange crit- ical stress, Fn, as outlined in the recommended specifications tions are consistent except that the 12-52 recommended directly. Note that the gaps occurring in the negative flexure specifications allow for the consideration of the additional figures for Bridges 3, 6, 20, and 32 are because those bridges post-buckling strength from tension-field action in the shear critical stress calculations. This post-buckling strength is also are simple-span structures. Furthermore, Bridges 24, 25, 29, and recognized in the 1993 Guide Specifications and AASHTO 30 use Grade 36 steel throughout and Bridges 1 and 22 have LRFD specifications straight girder provisions and results hybrid sections in the negative moment regions. in a higher critical stress for the majority of the stiffened In spite of the differences between the flexural design spec- bridges in the sample. ifications, the calculated capacity values are, for the most part, very similar. The main causes of major differences in capacity Strength I load case, flexural design: among the three specifications are as follows: Flexural analysis, according to all of the specifications, is Changes in the classification of the flange (e.g., compact, divided between composite and noncomposite sections, noncompact, and slender) among the three specifications; positive and negative flexure, and compression and tension Lateral stress consideration (including magnification) flanges. according to the 12-52 recommended specifications as Only modest changes are made to the slenderness limits compared with the reductions due to the r factors used in for the various classifications of the flanges (e.g., compact, the 1993 Guide Specifications and the 2003 Guide Specifi- noncompact, and slender) according to the three specifica- cations; and tions. These changes, however, are often enough to change Noncomposite design as a temporary condition. the classification of a flange and, therefore, the critical stress of the section. Again, specific information regarding the capacity of the All three of the specifications account for an increased ten- bridges for each of the design conditions is outlined in dency of the curved girders to deform due to secondary Appendix D. bending effects. The 1993 Guide Specifications and the 2003 Although the proportion varies based on the design speci- Guide Specifications consider factors, and the 12-52 fication selected, most composite positive flexure sections are recommended specifications consider a magnification factor controlled by the tension flange, while most negative flexure for the lateral bending stress. For the 1993 Guide Specifica- sections are controlled by the compression flange. For the non- tions and the 2003 Guide Specifications, the factors act to composite section design checks, the compression flange con- reduce the critical stress of the section based on lateral trols both positive and negative flexural sections. This control bending stress and geometry resulting in Fcr1. The magni-

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21 fier on the lateral bending stress for the 12-52 recommended 2.5 Design Examples specifications focuses on the magnitude of the longitudi- nal bending stress, resulting in a final combined reduction Two design examples, one for an I-girder bridge and one in the stress limit due to the effects of geometry (i.e., either for a box-girder bridge, were developed in the NCHRP 12-38 FLB or LTB considerations) and bending. project. As part of Phase I of the NCHRP 12-52 project, these design examples were updated to conform to Phase I recom- The 1993 Guide Specifications and the 2003 Guide Speci- mended specifications. The work plan for Phase II called for fications consider the lateral bending stress to reduce the updating the two design examples to conform to the design critical stress, while the 12-52 recommended specifications provisions approved by AASHTO. This work was conducted, consider the lateral bending stress as a load. The final effect and the two design examples are available online at http:// of including the lateral bending stress in all of the specifi- www.transportation.org/sites/bridges/docs/Box%20Girder. cations is to reduce the useable stress limit for gravity loads. pdf and http://www.transportation.org/sites/bridges/docs/ For the bridges considered, the noncomposite design was a I-Girder.pdf. temporary condition, and the noncomposite dead loads that Following is a brief description of the bridges used for the were provided for the final structure were used to analyze two examples. the noncomposite structure. Information was not provided I-Girder bridge example: regarding any temporary support points or the construction sequence for a more inclusive analysis. Three continuous spans: 160 feet, 210 feet, and 160 feet. Hybrid sections are not allowed by the 2003 Guide Specifi- Centerline radius: 700 feet. cations, but are allowed by the 1993 Guide Specifications and Girder spacing: 11 feet, 0 inches. the 12-52 recommended specifications, which have similar Overhang width: 3 feet, 9 inches. provisions for the consideration of stresses in hybrid sec- Out-to-out deck width: 40 feet, 6 inches. tions. Minor reductions in the critical stress occur when the Three 12-foot design lanes. yield strength of the web is less than the yield strength of Total deck thickness: 9.5 inches (includes a 1/2-inch inte- one or both of the flanges. gral wearing thickness). In spite of the differences between the flexural design spec- ifications, the calculated critical stress values are largely very Figure 8 shows a cross-section of the I-girder example similar. bridge. Out to Out = 40'-6" Roadway = 37'-6" 3 Lanes @ 12'-0" Single Structural t = 9" Angles Slope = 5% G4 G3 Intermediate Cross Frame G2 At Simple G1 and Interior Supports Support 3'-9" 11'-0" 11'-0" 11'-0" 3'-9" 4 Girders total = 33'-0" Deck concrete f'c = 4,000 psi E = 3.6x106 psi Haunch 20 in. wide, 4 in. deep measured from top of web Permanent deck forms are present Total deck thickness = 9.5 in., structural thickness = 9.0 in. Figure 8. I-girder bridge example, cross-section.

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22 Out to Out = 40'-6" Roadway = 37'-6" 3 Lanes @ 12'-0" Slope = 5% t = 9 1/2" Angles Typ. Section at Interior Typ. Plate Diaphragm Cross-frame at Bearings 4'-0" 10'-0" 12'-6" 10'-0" 4'-0" Deck concrete f'c = 4,000 psi E = 3.6x106 psi Haunch 20 in. wide, 4 in. deep measured from top of web Permanent deck forms are present Total deck thickness = 9.5 in. Figure 9. Box-girder bridge example, cross-section. Box-girder bridge example: Out-to-out deck width: 40 feet, 6 inches. Three 12-foot design lanes. Total deck thickness: 9.5 inches (no provision for integral Three continuous spans: 160 feet, 210 feet, and 160 feet. Centerline radius: 700 feet. wearing thickness). Individual tub girder web spacing: 10 feet, 0 inches. Figure 9 shows a cross-section of the box-girder example Interior web spacing: 12 feet, 6 inches. bridge.

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23 CHAPTER 3 Conclusions Based on the results of the work on this project, the follow- Specifications, and the 12-52 recommended specifications, ing conclusions may be drawn: the three specifications produced very similar flexural resistance. For the design cases considered in the calibration, using the The shear resistance values calculated by the three specifi- same load and resistance factors used in the development cations are similar except that the 1993 Guide Specifications of AASHTO LRFD specifications resulted in an adequate and the 12-52 recommended specifications produce higher reliability factor (3.7 to 4.51). shear resistance of stiffened girders due to consideration of In spite of the differences between the flexural design pro- the post-buckling behavior by these two specifications. visions in the 1993 Guide Specifications, the 2003 Guide

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24 References 1. AASHTO (1980). Guide Specifications for Horizontally Curved High- 14. Modjeski and Masters (2001). "Revised Proposed Draft Specifica- way Bridges. American Association of State Highway and Trans- tions." NCHRP 12-52 (Phase I) Quarterly Report, submitted to the portation Officials, Inc., Washington, DC. Transportation Research Board, Washington, DC, December. 2. AASHTO (1993). Guide Specifications for Horizontally Curved High- 15. Modjeski and Masters (2001). "LRFD Design ExampleHorizon- way Bridges. American Association of State Highway and Trans- tally Curved Steel I-Girder Bridge." NCHRP 12-52 Quarterly portation Officials, Inc., Washington, DC. Report, submitted to the Transportation Research Board, Washing- 3. Duwadi, S. R., Hall, D. H., Yadlosky, J. M., Yoo, C. H., and Zureick, ton, DC, September. A. H. (1995). "FHWA-CSBRP I-Girder Component Testing." Pro- 16. Modjeski and Masters (2001). "LRFD Design ExampleHorizon- ceedings of ASCE Structures Congress XIII, Boston, MA, April 25, tally Curved Steel Box Girder Bridges." NCHRP 12-52 Quarterly 1995, pp. 16951698. Report, submitted to the Transportation Research Board, Washing- 4. Grubb, M. A., and Hall, D. H. (2000). Philosophy and Design of the ton, DC, December. I-Girder Bending Component Tests. FHWA Curved Steel Bridge 17. Zureick, A., Naqib, R., and Yadlosky, J. M. (1994). "Curved Steel Research Project. Report submitted to the FHWA, Turner-Fairbank Bridge Research Project." Interim Report I,"synthesis," FHWA Con- Highway Research Center, McLean, VA. tract No. DTFH61-93-C-00136, FHWA, McLean, VA, December. 5. White, D. W., Jung, S. K., and Chang, C. J. (2002)."Design of Curved 18. Nowak, A. S. (1995). "Calibration of LRFD Bridge Code," ASCE Composite Test Bridge," Report to PSI Inc. and FHWA, December. Journal of Structural Engineering, Vol. 121, No. 8, pp. 12451251. 6. White, D. W., Zureick, A. H., Phoawanich, N. P., and Jung, S. K. 19. Nowak, A. S. (1999). NCHRP Report 368: Calibration of LRFD Bridge (2001). "Development of Unified Equations for Design of Curved Design Code, Transportation Research Board, Washington, DC. and Straight Steel Bridge I Girders," Final Report to AISI, PSI, Inc. 20. White D. W., Zureick, A. H., and Phoawanich, N. (2001)."Develop- and FHWA, October. ment of Unified Equations for Design of Curved and Straight Steel 7. Yoo, C. H., Hall, D. H., and Sabol, S. A. (1995). "Improved Design Bridge I-Girders," American Iron and Steel Institute and Federal Specifications for Horizontally Curved Steel-Girder Highway Highway Administration, presentation. Bridges." Proceedings of ASCE Structures Congress XIII, Boston, MA, 21. AASHTO (2002). Standard Specifications for Highway Bridges. 17th April 25, 1995, pp. 16991702. ed. American Association of State Highway and Transportation 8. Hall, D. H., Grubb, M. A., and Yoo, C. H. (1999). NCHRP Report 424: Officials, Inc., Washington, DC. Improved Design Specifications for Horizontally Curved Steel Girder Highway Bridges, Transportation Research Board, Washington, DC. 22. Ontario Ministry of Transportation and Communications (1991). 9. AASHTO (2003). Guide Specifications for Horizontally Curved Ontario Highway Bridge Design Code. OHBDC, 3rd Edition Downs- Highway Bridges. American Association of State Highway and view, Ontario. Transportation Officials, Inc., Washington, DC. 23. Ellingwood, B., Galambos, T. V., MacGregor, J. G., and Cornell, 10. AASHTO (2004). AASHTO LRFD Bridge Design Specifications. 3rd C. A. (1980). "Development of a Probability Based Load Criterion ed. American Association of State Highway and Transportation Offi- for American National Standard A58," NBS Special Publication 577, cials, Inc., Washington, DC. National Bureau of Standards, Washington, DC. 11. Yoo, C. H. (1996). "Progress Report on FHWA-CSBRP-Task D." 24. Nowak, A. S. (1993). "Live Load Model for Highway Bridges," Jour- FHWA Contract No. DTFH61-92-C-00136, Auburn University nal of Structural Safety, Vol. 13, Nos. 1+2, December, pp. 5366. Department of Civil Engineering Interim Report, submitted to HDR 25. Hwang, E. S., and Nowak, A. S. (1991). "Simulation of Dynamic Engineering, Inc., Pittsburgh, Office, Pittsburgh, PA, August 1996. Load for Bridges," ASCE Journal of Structural Engineering, Vol. 117, 12. Yoo, C. H., and Choi, B. H. (2000)."Literature SearchHorizontally No. 5, May, pp. 14131434. Curved Steel Girder Bridges," NCHRP 12-52 Quarterly Report, 26. Nassif, H. and Nowak, A. S. (1995). "Dynamic Load Spectra for submitted to the Transportation Research Board, Washington, DC, Girder Bridges," Transportation Research Record 1476, Transporta- March 2000. tion Research Board, pp. 6983. 13. Nowak, A. S., Szwed, A., and Galambos, T. V. (2001). "Calibration 27. Kim, S.-J., and Nowak, A. S. (1997)."Load Distribution and Impact of LRFD Design Code for Steel Curved Girder Bridges, NCHRP 12- Factors for I-Girder Bridges," ASCE Journal of Bridge Engineering, 52." University of Michigan. Vol. 2, No. 3, August, pp. 97104.

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APPENDIX A Literature Search

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A-iii CONTENTS A-iv List of Abbreviations and Symbols A-1 A1 Introduction A-1 A1.1 General A-1 A1.2 Objective A-1 A1.3 Research Procedure A-1 A1.4 Electronic Database A-1 A1.5 Synthesis A-1 A2 Overview of Literature Search A-1 A2.1 General A-2 A2.2 Analysis of Curved Bridges A-2 A2.3 Design A-3 A3 Curved I-Girder Bridges A-3 A3.1 Analysis A-3 A3.2 Design A-3 A3.2.1 Nominal Bending Strength A-5 A3.2.2 Curvature Effects on Elastic Lateral-Torsional Buckling A-5 A3.2.3 Cross-Frame Spacing and Lateral Bracing Effects A-5 A3.2.4 Local Buckling of Curved I-Girder Flanges A-5 A3.2.5 Strength of Curved I-Girder Web Panels under Pure Shear A-6 A3.2.6 Curved I-Girder Web Panels Subjected to Bending A-6 A3.2.7 I-Girder Webs Subjected to Combined Bending and Shear A-6 A3.2.8 Lifting of Slender Curved I-Girders A-6 A3.2.9 Constructibility Limit State A-6 A4 Curved Box-Girder Bridges A-6 A4.1 Analysis A-7 A4.2 Design A-7 A5 Conclusions and Recommendations for Further Study A-8 A5.1 Analysis Methods A-8 A5.2 I-Girders A-8 A5.3 Box-Girders A-8 A5.4 Constructibility A-8 A5.5 Extreme Event Limit State A-9 References A-13 Abstracts A-37 Background Research Pertaining to Updated AASHTO LRFD Specifications for Steel Structures, Third Edition

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A-iv LIST OF ABBREVIATIONS AND SYMBOLS a = distance between transverse stiffeners, b = width of flange, bf = width of compression flange, c = curvature parameter, C = shear strength constant, D = depth of the web panel, Dc = depth of the web panel in compression, fb = normal stress, Fb = allowable bending stress, fv = shear stress, Fv = allowable shear stress, Fy = minimum specified yield stress, fw = warping stress (flange lateral bending stress), k = elastic shear buckling coefficient, L = length of girder, Mu = ultimate vertical bending moment, R = radius of curvature, Rd = reduction factor of shear due to initial out of flatness or reduction factor of deflection, as appropriate, Rs = reduction factor of stress, t = thickness of flange, tf = thickness of flange, tw = thickness of the web panel, Vp = plastic shear capacity, Vu = ultimate shear capacity, x = subtended angle between adjacent cross frames, y = critical moment ratio, = unbraced length of compression flange of I-girder, = reduction factor of local buckling of compression flange, and w = parameter relating bend-buckling of curved I-girder web.