
Citation 
 Permanent Link:
 http://digital.auraria.edu/AA00000116/00001
Material Information
 Title:
 Behavior and analysis of highly skewed steel Igirder bridges
 Creator:
 Dobbins, Konlee Baxter ( author )
 Place of Publication:
 Denver, CO
 Publisher:
 University of Colorado Denver
 Publication Date:
 2013
 Language:
 English
 Physical Description:
 1 electronic file (96 pages). : ;
Thesis/Dissertation Information
 Degree:
 Master's ( Master of Science)
 Degree Grantor:
 University of Colorado Denver
 Degree Divisions:
 Department of Civil Engineering, CU Denver
 Degree Disciplines:
 Civil engineering
Subjects
 Subjects / Keywords:
 Girders ( lcsh )
Steel Ibeams ( lcsh )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Review:
 Skewed bridge supports for steel Igirder bridges, introduce complexities to the behavior of the girder system that can be difficult to accurately model and analyze. In addition there have been some reported shortfalls in the 2Dgrid analysis method typically used by engineers to design steel girder bridges with significant skews. So improvements have been suggested to bridge the gap in the inaccuracies of 2Dgrid and 2Dframe analyses. These improvements include overwriting the girder torsional stiffness to include warping effect overwriting the equivalent beam stiffness of crossframes using a mo accurate method of calculating the stiffness, including lockedin crossframe forces due to dead load fit detailing, and more accurate calculating flange lateral bending stresses with staggered crossframe layouts. This thesis examines these improvements, compares different levels of analysis, provides recommendations for these methods of analysis, and explains the behavior of the girder system during erection.
 Thesis:
 Thesis: (M.S.)University of Colorado Denver. Civil engineering
 Bibliography:
 Includes bibliographic references.
 System Details:
 System requirements: Adobe Reader.
 General Note:
 Department of Civil Engineering
 Statement of Responsibility:
 by Konlee Baxter Dobbins.
Record Information
 Source Institution:
 University of Colorado Denver
 Holding Location:
 Auraria Library
 Rights Management:
 All applicable rights reserved by the source institution and holding location.
 Resource Identifier:
 891217215 ( OCLC )
ocn891217215

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BEHAVIOR AND ANALYSIS OF
HIGHLY SKEWED STEEL IGIRDER BRIDGES
by
KONLEE BAXTER DOBBINS B.S., University of Virginia, 2001
A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Civil Engineering
2013
This thesis for the Master of Science degree by Konlee Baxter Dobbins has been approved for the Department of Civil Engineering by
Kevin Rens, Chair Cheng Yu Li Yail Jimmy Kim
November 12, 2013
Dobbins, Konlee B. (M.S., Civil Engineering)
Behavior and Analysis of Highly Skewed Steel IGirder Bridges Thesis directed by Professor Kevin Rens
ABSTRACT
Skewed bridge supports for steel Igirder bridges, introduce complexities to the behavior of the girder system that can be difficult to accurately model and analyze. In addition there have been some reported shortfalls in the 2Dgrid analysis method typically used by engineers to design steel girder bridges with significant skews. Some improvements have been suggested to bridge the gap in the inaccuracies of 2Dgrid and 2Dframe analyses. These improvements include overwriting the girder torsional stiffness to include warping effects, overwriting the equivalent beam stiffness of crossframes using a more accurate method of calculating the stiffness, including lockedin crossframe forces due to dead load fit detailing, and more accurately calculating flange lateral bending stresses with staggered crossframe layouts. This thesis examines these improvements, compares different levels of analysis, provides recommendations for these methods of analysis, and explains the behavior of the girder system during erection.
The form and content of this abstract are approved. I recommend its publication.
Approved: Kevin Rens
m
ACKNOWLEDGMENTS
I would like to thank my committee members Dr. Kevin Rens, Dr. Cheng Yu Li, and Dr. Jimmy Kim for reviewing my work. I would like to thank Parsons for funding my masters degree. I would also like to thank my coworkers who provided encouragement and guidance, especially Steve Haines, with his help providing information about the Geneva Road Bridge project. And most of all, I would like to thank my family and in particular my wife, Kirsi Petersen, for her patience and encouragement during the long nights and weekends spent away from the family.
IV
TABLE OF CONTENTS
CHAPTER
I. INTRODUCTION...................................................1
Effects of Skew................................................1
Continuation of Previous Thesis................................2
II. LITERATURE REVIEW..............................................4
Introduction...................................................4
Relevant Documents.............................................5
III. THEORETICAL BACKGROUND........................................10
Suggestions to Simplify Structure Geometry in Skewed Bridges..10
Framing Plan CrossFrame Layout.............................11
Rotations and Deflections.....................................12
Detailing NLF vs. SDLF vs. TDLF..............................14
Analysis Methods..............................................17
Improvements to 2D Modeling...................................19
Preferred Analysis Method for Straight Skewed Girders.........28
IV. ANALYTICAL PLAN...............................................33
Example Bridge Description....................................33
Analysis Models...............................................40
V. ANALYTICAL RESULTS COMPARING MODELS ........................47
v
VI. CONCLUSIONS..........................................75
Recommended Method of Analysis.......................75
General Recommendations for Future Work..............77
REFERENCES ..................................................80
APPENDIX A...................................................81
vi
LIST OF FIGURES
Figure
III. 1 Typical Fitup Procedure for Skewed IGirders...........................16
III.2 Lateral Bending Moment, Ml, in a Flange Segment Under Simply Supported and
FixedEnd Conditions......................................................24
ID.3 Conceptual Configurations Associated with Dead Load Fit (TDLF or SDLF)
Detailing................................................................26
III. 4 Matrix of Grades for Recommended Level of Analysis for IGirder Bridges.31
IV. 1 Typical Section of the Geneva Road Bridge...............................34
IV.2 Layout of the Geneva Road Bridge..........................................35
IV.3 Elevation Layout of the Geneva Road Bridge................................36
IV.4 Girder Elevation of the Geneva Road Bridge................................38
IV.5 Framing Plan of the Geneva Road Bridge....................................39
IV. 6 Underside of the Geneva Road Bridge......................................40
V. l MajorAxis Bending Stress of Girder 1 Top Flange ......................49
V.2 Axis Bending Stress of Girder 1 Bottom Flange.............................49
V.3 Vertical Girder Displacements Along Girder 1 and Girder 3................50
V.4 CrossFrame Axial Forces Along Bay 2 Between Girder 2 and Girder 3.......51
V.5 MajorAxis Bending Stress of Girder 1.....................................56
V.6 Vertical Girder Displacements Along Girder 1 and Girder 3................57
V.7 CrossFrame Axial Forces Along Bay 2 Between Girder 2 and Girder 3.......58
V.8 Flange Lateral Bending Stress.............................................62
vii
V.9 Girder Layover at Bearings................................................65
V.10 MajorAxis Bending Stress of Girder 1....................................66
V. 11 Vertical Girder Displacements along Girder 1 and Girder 3...............67
V. 12 CrossFrame Axial Forces Along Bay 2 Between Girder 2 and Girder 3......68
V. 13 Flange Lateral Bending Stress Along Girder 1............................71
V. 14 Girder Layover at Start and End Bearings for 3D Models..................73
V. 15 Girder Layover at Start and End Bearings for 3D TDLF Models and from Field
Data.....................................................................74
viii
CHAPTER I
INTRODUCTION Effects of Skews
Skewed bridge supports and horizontal curvature in steel Igirder bridges exhibit torsional forces that can introduce unexpected stress, displacements, and rotations during construction. As the skew angle or degree of curvature increases, the difficulty of constructing steel Igirder bridges increases. The sequence of erection and assumptions made during fabrication can introduce forces and deflections that were not accounted for during the design. In many cases, these forces and deflections are negligible; however, in some cases they can be significant and unaccounted for if following todays standard design practice and codes. Many of todays more commonly used structural software take into account the effects of horizontal curvature on steel superstructures. However, accurately capturing the effects of skewed supports seems to be lacking in these software (NCHRP, 2012).
Many reports and research papers lump the effects of horizontal curvature and skews together and tend to provide all encompassing guidelines that address both aspects. Many of the effects of horizontal curvature and skews are similar in nature; however, they can act in opposite directions or in different locations and affect the design differently. Therefore, it is important to understand the effects of each separately. This thesis will focus on the effects of skews only.
Skews at bridge supports alter the behavior of girders. Historically, skews were avoided whenever possible because the effects were not well understood. Over time, advances in structural analysis and results from case studies have made the effects a bit
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clearer. One report in particular, National Cooperative Highway Research Program (NCHRP) Report 725 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges, has taken great strides in highlighting the shortcomings of todays standard practice, specifications, and guidelines for highly skewed steel Igirder bridges. More is discussed on these shortcomings and how to accurately account for them in the Literature Review and Theoretical Background sections.
Continuation of Previous Thesis
This thesis follows up and expands on a fellow University of Colorado Denver graduate students thesis Crossframe Analysis of HighlySkewed and Curved Steel IGirder Bridges that touched on a variety of similar topics and provided a case study example. That thesis focused on crossframe design by looking at different framing plan and crossframe configurations (xframe vs. kframe) to find the most efficient design. It also included some background research, a literature review, theoretical background, and analysis of crossframes in highly skewed and curved steel Igirder example bridges (Schaefer, 2012).
The theoretical background is predominantly based on the American Association of State Highway Transportation Officials (AASHTO) and National Steel Bridge Alliance (NSBA) Steel Bridge Collaboration Document G13.1, Guidelines for Steel Girder Bridge Analysis. That thesis is a good source for background information on crossframe types, framing plan configurations, and specification requirements from AASHTO and can be used as a precursor. It also provides a list of several curved and/or
2
skewed bridges, with framing plans and member sizes included for each bridge listed, in the Denver, CO metro area (Schaefer, 2012).
The conclusions that can be taken from Crossframe Analysis of HighlySkewed and Curved Steel IGirder Bridges include:
Staggered crossframe configurations induced the least amount of forces within its crossframes.
Contiguous crossframe configurations induced the most forces within its crossframes.
A stiffer transverse system will accumulate more force than a flexible system.
The Kframe type crossframe performs better than the Xframe. The diagonal members in a Kframe crossframe absorb significantly less force than the diagonal members in an Xframe.
The double angle and WTmembers are less slender, more flexible, and thus attract fewer loads than single angles that have to meet slenderness requirements (Schaefer, 2012).
This thesis focuses on the effects that displacements and detailing have on the design of highly skewed steel Igirders and the most accurate design methods that should be used with common steel Igirder structural software.
3
CHAPTER II
LITERATURE REVIEW Introduction
The literature review in Crossframe Analysis of HighlySkewed and Curved Steel IGirder Bridges includes the history of design specifications that contributed to todays codes and standard practice for the design of skewed or horizontally curved steel girders. The list includes:
AASHTO Guide Specifications for Horizontally Curved Steel Girder Highway Bridges, 1980
NCHRP Project 1238, 1993
AASHTO Guide Specifications for Horizontally Curved Steel Girder Highway Bridges, 1993
NCHRP Project 1252, 1999
AASHTO Guide Specifications for Horizontally Curved Steel Girder Highway Bridges, 2003
AASHTO/NSBA G13.1 Guidelines for Steel Girder Bridge Analysis 1st Edition, 2011
One very important document missing from the list is from the research of NCHRP Project 1279, NCHRP Report 725 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges. NCHRP Report 725 points out several deficiencies in the latest AASHTO Load and Resistance Factor Design (LRFD) Bridge Design Specifications, the latest guidelines (AASHTO/NSBA 
4
G13.1 Guidelines for Steel Girder Bridge Analysis 1st Edition), and standard practices assumed with the most commonly used ID and 2D analysis software.
Relevant Documents
The following literature review and theoretical background focuses on the G13.1 Guidelines and NCHRP Report 725, while briefly discussing contributions from other research papers.
AASHTO/NSBA G13.1 Guidelines for Steel Girder Bridge Analysis
In the Forward of this document, it states the document is intended only to be a guideline, and only offers suggestions, insights, and recommendations but few, if any, rules. The purpose of the document is to provide engineers, particularly less experienced designers, with guidance on various issues related to the analysis of common steel girder bridges. The document focuses on presenting the various methods available for analysis of steel girder bridges and highlighting the advantages, disadvantages, nuances, and variations in the results. The guidelines are, to a certain extent, allencompassing for steel girder bridges, while briefly discussing the effects of different variations such as skews and horizontal curvature. The general behavior and suggested analysis methods are discussed; however, it does not go into great detail. At the time this document was released there had been very few guideline resources for the design of skewed and horizontally curved steel girders and their corresponding crossframes.
The contents include:
1. Modeling descriptions
2. History of steel bridge analysis
5
3. Issues, objectives, and guidelines common to all steel girder bridge analyses
4. Analysis guidelines for specific types of steel girder bridges
Of particular interest are the sections on skewed bridges. These sections include information on the behavior, constructability analysis issues, predicted deflections, detailing of crossframes and girders for the intended erected position, crossframe modeling in 2D, geometry considerations, and analysis guidelines for skewed steel Igirder bridges (AASHTO/NSBA, 2011).
NCHRP Report 725 Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges
This report contains guidelines on the appropriate level of analysis needed to
determine the constructability and constructed geometry of curved and skewed steel
girder bridges. The report also introduces improvements to ID and 2D analysis that
require little additional computational costs. The research for this report was performed
under NCHRP Project 1279. The objectives and scope of NCHRP Project 1279
include:
1. An extensive evaluation of when simplified ID or 2D analysis methods are sufficient and when 3D methods may be more appropriate.
2. A guidelines document providing recommendations on the level of construction analysis, plan detail, and submittals suitable for direct incorporation into specifications or guidelines.
Of particular interest for this thesis are the sections pointing out the deficiencies of ID or 2D analysis used in standard practice and the proposed improvements for analyzing skewed bridges. The report focuses on problems that can occur during, or
6
related to, the construction. The key construction engineering considerations for skewed steel girder bridges include:
1. The prediction of the deflected geometry at the intermediate and final stages of the construction,
2. Determination and assessment of cases where the stability of a structure or unit needs to be addressed,
3. Identification and alleviation of situations where fitup may be difficult during the erection of the structural steel, and
4. Estimation of component internal stresses during the construction and in the final constructed configuration.
AASHTO LRFD Bridge Design Specifications, Customary U.S. Units, 6th Edition
(2012)
This specification is used in every state throughout the United States as the national standard that engineers are required to follow for bridge design. Many states include their own amendments to this specification and additional guidelines, but its still the standard that the nations bridge designs are based upon. The specifications have also been adopted or referenced by other bridgeowning authorities and agencies in the United States and abroad. Since its first publication in 1931, the theory and practice have evolved greatly resulting in 17 editions of the Standard Specifications for Highway Bridges with the last edition appearing in 2002 and six editions to date of the loadandresistance factor design (LRFD) specifications (AASHTO, 2012).
As the national standard, the specifications are a bit lacking in providing requirements or guidance for designing highly skewed bridges. In Section 4 Structural
7
Analysis and Evaluation, equations are provided to adjust the live load distribution factors for moment and shears using approximate methods of analysis. The approximate method of analysis involves line girder or ID analysis of typical bridges within a set range of applicability for girder design. Section 6 Steel Structures, includes commentary on the effects that skews have on girder and crossframe deflections, rotations, and potential additional stresses. However, in many cases, it recommends performing a more refined analysis to more accurately capture the effects of skews and leaves a fair amount to engineering judgment to decide when a refined analysis is necessary and to what amount of detail. The AASHTO/NSBA G13.1 Guidelines, NCHRP Report 725, and several other reports and research papers help bridge that gap and provide more guidance.
Other Reports
There are many more research reports, presentations, and short articles on the effects of skews on steel Igirder bridges and experiences during construction. The authors include structural engineers, professors, fabricators, and construction managers. Several of these authors also contributed to NCHRP Report 725. Some articles, such as Design and Construction of Curved and Severely Skewed Steel IGirder EastWest Connector Bridge over 188, describe the challenges and lessons learned during the design and construction of a specific bridge. In the presentation Erection of Skewed Bridges: Keys to an Effective Project, the chief engineer for High Steel Structures Inc., presents three case studies of highly skewed steel girder bridges and the experiences from the point of view of the fabricator. The presence of large skews and the assumptions made on fitup detailing during erection affect all stages of design and construction.
8
Engineers, fabricators, and contractors all need to understand the movements, forces required for fitup, and corresponding lockedin stresses that occur during different stages of construction.
9
CHAPTER III
THEORETICAL BACKGROUND Suggestions to Simplify Structure Geometry in Skewed Bridges
Skews present complexities in design, detailing, fabrication, and erection that translate into increased costs for steel girder bridges. As per NSBA/AASHTO Steel Bridge Collaboration, skew angles should be eliminated or reduced wherever possible. The bridge designer should work closely with the roadway designer to improve and simplify roadway alignments. Once the alignment is set, a few suggestions for eliminating or reducing skews include:
Lengthening spans to locate the abutments far enough from the roadways below to allow for the use of radial abutments or bents while maintaining adequate horizontal clearance. Designers should consider the cost of a longer span versus the cost associated with the complications of skew in the bridge.
Retaining walls may allow the use of a radial abutment in place of a header slope. Typically these walls are of variable height and require oddshaped slope protection behind the wall. Designers should consider the cost of the walls versus the cost associated with the complications of skew in the bridge.
Use integral radial interior bent instead of a skewed traditional bent cap to maintain adequate vertical clearance in cases where a traditional radial bent would have insufficient vertical clearance and where the vertical profile of the bridge cannot be raised.
10
Use dapped girder ends with invertedtee bent caps to maintain adequate vertical clearance at expansion joint locations instead of an integral bent cap (AASHTO/NSBA 2011).
In many cases, highly skewed supports cannot be avoided for a number of reasons. Typically, geometry constraints in highly congested highway interchanges leave very little wiggle room to eliminate or reduce large skews. Where large skews cannot be avoided, design engineers, detailers, fabricators, and contractors all need to understand the stresses and deflections that occur during different stages of construction.
Framing Plan CrossFrame Layout
Crossframes or diaphragms should be placed at bearing lines that resist lateral force. Wind loads and other lateral forces are transferred from the deck and girders through the crossframes at supports to the bearings and down to the substructure. As per AASHTO LRFD Bridge Design Specifications 6.7.4.2 Diaphragms and CrossFrames for ISection Members, crossframes at supports can either be placed along the skew or perpendicular to the girder:
Where support lines are skewed more than 20 degrees from normal, intermediate diaphragms or crossframes shall be normal to the girders and may be placed in contiguous or discontinuous lines.
Where a support line at an interior pier is skewed more than 20 degrees from normal, elimination of the diaphragms or crossframes along the skewed interior support line may be considered at the discretion of the Owner. Where discontinuous intermediate diaphragm or crossframe lines are employed normal to the girders in the vicinity of that support line, a skewed or normal diaphragm or crossframe should be matched with each bearing that resists lateral force (AASHTO, 2012).
As research has shown, placing a crossframe normal to the girders and at the bearing location of a skewed support, provides an alternate load path and attracts a
11
significant amount of force in that crossframe. NCHRP Report 725 referred to these crossframes at or near the supports as providing nuisance stiffness transverse load paths and should be avoided if possible. Therefore, standard practice is to provide a crossframe along the skew at the supports.
Staggered crossframe layout configurations appear to induce the least amount of forces within its crossframes. This is especially true and desirable for interior crossframes closest to highly skewed supports. Staggered crossframes allow for more flexibility in the system and therefore attract less load. Also, Kframe type crossframes tend to be the better choice over Xframe type crossframes (Schaefer, 2012).
Rotations and Deflections
The root of the complications due to skew is the out of plane rotations and deflections at the skewed supports that cause twisting in the girders as vertical loads are applied. In ID line girder analysis, the effects of a skew are not captured. When analyzing a single girder in a single span, as vertical loads (such as dead loads from the steel selfweight, concrete deck weight, and miscellaneous superimposed dead loads and live loads from vehicular traffic) are applied, the girder deflects downward with the max deflection occurring at midspan. There are, theoretically, no lateral deflections or twisting. However, as crossframes are attached connecting skewed girders together, the girders start to twist near the supports. The differential deflection between two adjacent girders causes a twisting motion, also known as layover. This twisting motion can be counterbalanced by specifying certain detailing that essentially forces the girders to be twisted in the opposite direction when connecting the crossframes to the girders during erection. This is known as Steel Dead Load Fit (SDLF) and Total Dead Load Fit (TDLF)
12
detailing, where the goal is to have vertically plumb girder webs at the specified construction stage. This will be discussed in more detail in the next section.
Per the AASHTO/NSBA Steel Bridge Collaboration Document G12.1
Guidelines for Design Constructability:
The problem for crossframes at skewed piers or abutments is the rotation of the girders at those locations. In a square bridge, rotation of the girders at the bearings is in the same plane as the girder web. If supports are skewed, girder rotation due to noncomposite loads will be normal to the piers or abutments.
This rotation displaces the top flange transversely from the bottom flange and causes the web to be out of plumb (AASHTO/NSBA, 2003).
Where end crossframes are skewed parallel to the support, which is typically standard practice, these end crossframes contribute to the rotations and transverse movements described above. The end crossframes are very stiff in the axial direction along the skewed support and flexible in the weak axis direction, which allow these rotations normal to the support.
The movements of simple span straight girders on nonskewed supports are predictably uniform. With downward deflection between supports due to vertical dead loads, the top flange compresses. At the supports, the top flange deflects toward midspan. Conversely, the bottom flange is in tension and deflects away from midspan at the expansion supports that are free to move longitudinally. The ends of girders also rotate due to the length changes in the flanges. For girders on skewed supports the movement becomes more complex by adding transverse deflections and twisting rotations. The rotation normal to the pier as described in AASHTO/NSBA G12.1 is a bit of a generalization and is true for bearings that are at the same elevation at the given support. If the bearing elevations differ along the given support, the axis of rotation will be in the
13
plane including the actual centerline of bearing, but slope to intersect the centers of rotation at adjacent bearings. This describes the theoretical movements. Actual movements will vary slightly since the members are framed together and restrained by the deck and bearings, therefore some distortions will result (Beckmann and Medlock, n.d.).
The transverse movements and twisting causes the ends of girders to be out of plumb as vertical loads are applied if the girders are not detailed to counteract these movements during construction. For more detailed information on rotations and deflections in skewed steel girder bridges, the article Skewed Bridges and Girder Movements Due to Rotations and Differential Deflections is recommended.
Detailing NLF vs SDLF vs TDLF
As the skew angle increases, the transverse flange movement increases. For strength, serviceability, and aesthetic reasons, it is typically desirable to detail the girders with sizeable skews to counteract these girder end movements and be plumb at certain dead load cases. However, each bridge needs to be evaluated for several factors, including constructability and girder design at different stages of construction, to determine the most economic design. Fabrication and construction must follow the fitup condition assumed during the design of the girders and crossframes. Otherwise, unintended lockedin forces or movements that were not considered during design can arise.
The designer generally has three choices of conditions for which the girders and crossframes shall be designed:
14
NoLoad Fit (NLF) condition the girder webs are theoretically plumb/vertical before any load is applied.
Steel Dead Load Fit (SDLF) condition the girder webs are theoretically plumb/vertical under the steel dead after the crossframes are installed and before the concrete deck is poured.
Total Dead Load Fit (TDLF) condition the girder webs are theoretically plumb/vertical under the total dead load in the final condition (Beckmann and Medlock, n.d.).
Each detailing method affects deflected geometry, can create fitup issues, produce stability effects and secondorder amplification, and affect component internal stresses during construction. Construction plans and submittals for these complex geometries with high skews need to clearly state the fitup method assumed during design and construction (NCHRP, 2012).
In SDLF and TDLF detailing methods, the crossframes do not fitup with the connection work points on the initially fabricated girders. During fitup of crossframes with the girders, the girders are forced into place by twisting the girders. A girder is much more flexible twisting about its longitudinal axis than a crossframe deforming axially. As the dead load is applied, the girders deflect and rotate back to plumb. AASHTO/NSBA G 12.1 Guidelines for Design Constructability describes the process of SDLF or TDLF fitup as seen in Figure III. 1.
15
G1
G4
Stage 1:
Erected position prior to attaching crossframes
G1
G2
Erect crossframes and pin top hole on each girder. Girders will remain in vertical position.
stiffener hole lines up with crossframe hole.
stiffener hole lines up with crossframe hole.
Erect crossframe and pin top and bottom holes on G2. Pin top hole on G3, push bottom of G3 until bottom hole lines up.
G1 G2
Pm top hole on G4_ push bottom of G4 until bottom hole lines up.
G3 G4
Figure III.l Typical Fitup Procedure for Skewed IGirders (AASHTO/NSBA, 2003)
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For the designer, the biggest concern is the presence of any unaccounted forces and correctly modeling the structure at different stages of construction. SDLF and TDLF detailing introduces lockedin forces during erection when the girders are forced to fit up with the stiffer crossframes. In many cases, especially for straight girders on skewed supports, lockedin forces are relieved as the dead loads are applied. However, it can be dangerous to assume that this occurs in all cases. For example, as with curved girders with radial/nonskewed supports, the lockedin forces from fitup and forces due to differential deflections between adjacent girders can be additive. Or in highly skewed straight bridges, if the first intermediate crossframes are too close to the bearing line, the lockedin crossframe forces near the acute corners tend to be additive with the dead load effects (NCHRP, 2012).
Analysis Methods
The level of detail for the girder and crossframe analysis is an important decision to make and is often left to engineering judgment. 3D finite element analysis (FEA) provides the most accurate results when done correctly. However, it is by far the most complex and timeconsuming and with a large number of variables, it leaves a lot of room for error. ID and 2D simplified analysis are much less timeconsuming and therefore preferred by engineers for the design of noncomplex structures. What constitutes a structure to be complex and where to draw the line is often a topic of debate among engineers. AASHTO LRFD Bridge Design Specifications provide criteria for determining if using a simplified method of linear analysis is acceptable. When a refined method of analysis is required or recommended, there are still a good number of methods to choose from including 2Dgrid and 3DFEA. It is ultimately left up to engineering
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judgment to choose an appropriate refined method of analysis and understand the basic assumptions and methodology of the software used (AASHTO, 2012). Even in cases where ID or 2D methods of analysis are deemed acceptable, NCHRP Report 725 has made light of some assumptions that can turn out to be quite erroneous. NCHRP Report 725 has also exposed some assumptions typically made by most ID or 2D analysis software that can significantly alter the results. The following sections provide a brief overview of the different methods of analysis.
ID Line Girder Analysis Method
Line girder analysis, as the name suggests, isolates and analyzes one single girder line. Loads are distributed to each girder by way of distribution factors. Effects on girder moments and shear from skews no greater than 60 degrees are accounted for with additional factors in AASHTO LRFD Bridge Specifications. The effects of the crossframes are not taken into account. This method is adequate for fairly simple structures with little to no skew angle.
2D Grid Analysis Method
In plan grid or grillage analysis, the structure is divided into plan grid elements with three degrees of freedom at each node. This method is most often used in steel bridge design and analysis (AASHTO/NSBA, 2011). The effects of the crossframes are taken into account; however, most common 2D software, such as DESCUS and MDX, use equivalent beam element properties when modeling the crossframes. As discussed in NCHRP Report 725, how these common 2D software compute the equivalent beam element properties for the crossframes and the equivalent torsional constant properties of the girders, isnt typically accurate especially in cases of high skews or high degrees of
18
horizontal curvature. These inaccuracies and how to account for them will be explained in greater detail in later sections. As the most commonly used method of analysis, it is vital to keep the inaccuracies to a minimum.
3D Finite Element Analysis Method
In the 3DFEA method, the bridge superstructure is fully modeled in all three dimensions. The model typically includes modeling the girder flanges as beam elements or plate/shell elements; modeling the web as plate/shell elements; modeling each member of the crossframes as beam or truss elements; and modeling the deck as plate/shell elements. This method is arguably the most accurate; however, it is typically very timeconsuming and complicated. Therefore, it is mostly only used for very complex structures or for performing refined local stress analysis of a complex detail. There are other complicating factors, such as the output reporting the stresses in each element instead of moments and shears that the engineer typically checks against the required limits in AASHTO or local state specifications. The engineer would need to convert the stresses into moments and shears if so desired. When and how to use refined 3D finite element analysis is a controversial issue, and this method has not been fully incorporated into the AASHTO specifications to date (AASHTO/NSBA, 2011).
Improvements to 2D Modeling
CrossFrame Modeling
Most designers use the methods described in the AASHTO/NSBA (2011) G13.1 document for finding the equivalent beam stiffness of crossframes in 2D analysis models. There are two approaches here:
19
1. Calculate the equivalent moment of inertia based on the flexural analogy method. In a model of the crossframe, a unit force couple is applied to one end to find the equivalent rotation that is then used to backcalculate the equivalent moment of inertia.
2. Calculate the equivalent moment of inertia based on the shear analogy method. In a model of the crossframe, a unit vertical force is applied to one end to find the equivalent deflection that is then used to backcalculate the equivalent moment of inertia (AASHTO/N SB A, 2011).
Both methods use EulerBemoulli beam theory equations. The issue with using one of these methods is the flexural analogy method only accounts for the flexural stiffness, while the shear analogy only accounts for shear stiffness. In cases where either the flexure or shear is considered negligible, using the appropriate method above is acceptable. However, in cases where both flexure and shear are present, the equivalent moment of inertia should account for both flexural and shear stiffness. Differential deflection of adjacent girders might primarily engage the shear stiffness of the crossframes, while differential rotation of adjacent girders might be more likely to engage the flexural stiffness of the crossframes (AASHTO/NSBA, 2011).
NCHRP Report 725 recommends a more accurate approach for calculating the crossframe equivalent beam stiffness. This approach includes an equivalent shear area for a sheardeformable beam element representation (Timoshenko beam theory) of the crossframe. In the report, it compares the equivalent stiffness results from the flexural analogy method, shear analogy method, pure bending (Timoshenko) method, and 3DFEA calibrated to a test bridge and finds that the pure bending (Timoshenko) method
20
provides the most accurate overall results. This is due to the fact that the Timoshenko beam theory element is able to represent both flexure and shear deformations.
In the pure bending (Timoshenko) method, the equivalent moment of inertia is determined first based on pure flexural deformation. This is similar to the flexural analogy method except that the constraints are modeled differently and the corresponding end rotation is equated from the beam pure flexure solution M/(EIeq/L) versus the EulerBemoulli beam rotation equation M/(4EIeq/L) used in the flexural analogy method. This results in a substantially larger equivalent moment of inertia and that EIeq represents the true flexural rigidity of the crossframe. The crossframe is supported as a cantilever at one end and is subjected to a force couple at the other end, producing a constant bending moment and corresponding end rotation. In the second step of this method, the crossframe is still supported as a cantilever but is subjected to a unit transverse load at its tip. The Timoshenko beam equation for the transverse displacement is:
A =
VL3
3Â£7^ +
VL
GAeq
which is used to find the equivalent shear area (NCHRP, 2012).
As per NCHRP Report 725, the Timoshenko beam element provides a closer approximation of the physical crossframe behavior compared to the EulerBernoulli beam for all types of crossframes (including X and K type crossframes) that are typically used in Igirder bridges. Not only are the calculated forces more accurate but the deflections and rotations are more accurate. Predicting deflections and rotations
21
during construction becomes much more important as skew angles increase (NCHRP, 2012).
The fabricator can more accurately fabricate the girders for the appropriate final orientation and fitup method. The contractor more accurately understands the deflections and rotations to expect during construction and the forces necessary for the chosen fitup method. The engineer can more accurately and efficiently design the girders and crossframes for the expected movements and lockedin forces from fitup and final condition loads.
IGirder Torsion Modeling
Current practice in 2Dgrid models substantially underestimates the girder torsional stiffness. This is due to software only considering St. Venant torsional stiffness of the girders while neglecting warping torsional stiffness. This practice tends to discount the significant transverse load paths in highly skewed bridges, since the girders are so torsionally soft that they are unable to accept any significant load from the crossframes causing torsion in the girders. As a result, the crossframe forces can be significantly underestimated (NCHRP, 2012).
NCHRP Report 725 provides some equations to calculate an equivalent torsional constant, Jeq that includes both the St. Venant and warping torsional stiffness. It should be noted that these equations were based in part on prior research developments by Ahmed and Weisgerber (1996), as well as the commercial implementation of this type of capability within the software RISA3D. In this approach, an equivalent torsional constant must be calculated for each unbraced length and girder sectional property. The equation for the equivalent torsion constant for the opensection thinwalled beam
22
associated with warping fixity as each end of a given unbraced length (crossframe spacing) is:
_
sinh (pLb) [cosh (pLb) 1]21
pLb pLb sinh(pLb)
Where Lb is the unbraced length between the crossframes, J is the St. Venant torsional
constant, andp2 is defined as GJ/ECW. Assuming warping fixity at the intermediate
crossframe locations leads to a reasonably accurate characterization of the girder
torsional stiffness (NCHRP, 2012).
J eq(fxfx) ~ J
IGirder Flange Lateral Bending Modeling
AASHTO LRFD Bridge Specifications section C4.6.1.2.4b provides a simplified equation to calculate the lateral moment for a horizontally curved girder based on the radius, majoraxis bending moment, unbraced length, and web depth. For other conditions that produce torsion, such as skew, AASHTO suggests other analytical means which generally involve a refined analysis. However, Section C6.10.1 provides a coarse estimate by stating:
The intent of the Article 6.10 provisions is to permit the Engineer to consider flange lateral bending effects in the design in a direct and rational manner should they be judged to be significant. In absence of calculated values of fi from a refined analysis, a suggested estimate for the total unfactored fi in a flange at a crossframe or diaphragm due to the use of discontinuous crossframe or diaphragm lines is 10.0 ksi for interior girders and 7.5 ksi for exterior girders. These estimates are based on a limited examination of refined analysis results for bridges with skews approaching 60 degrees from normal and an average D/bf ratio of approximately 4.0. In regions of the girders with contiguous crossframes or diaphragms, these values need not be considered. Lateral flange bending in the exterior girders is substantially reduced when crossframes or diaphragms are placed in discontinuous lines over the entire bridge due to the reduced crossframe or diaphragm forces. A value of 2.0 ksi is suggested for f for the exterior girders in such cases, with the suggested value of 10 ksi retained for the interior girders.
In all cases, it is suggested that the recommended values of fi be proportioned to dead and live load in the same proportion as the unfactored majoraxis dead and
23
live load stresses at the section under consideration. An examination of crossframe or diaphragm forces is also considered prudent in all bridges with skew angles exceeding 20 degrees (AASHTO, 2012).
NCHRP Report 725 recommends a more accurate but simplified method of calculating lateral bending stress than the coarse estimates provided above. Their method includes a local calculation in the vicinity of each crossframe, utilizing the forces delivered to the flanges from the crossframes placed in discontinuous lines. The approximate calculation takes the average of pinned and fixed end conditions as shown in Figure III.2 below.
Pab2/L2
M/nax
Pa~b2/L'
\J Pa:b/L2
a h
L
Averaged Moments
Pab (I + ablL:)flL
Pah2111?ucfTl._.rT_^_VJ Pa2bl2L2
Figure III.2 Lateral Bending Moment, Ml, in a Flange Segment Under Simply Supported and FixedEnd Conditions (NCHRP, 2012)
Calculation of LockedIn Forces Due to CrossFrame Detailing
Regardless the type of analysis used (2Dgrid, 2Dframe, or 3DFEA), the analysis essentially assumes a NLF condition unless the lockedin forces are accounted for in the model. Any lockin forces, due to the lack of fit of the crossframes with the girders in the undeformed geometry in SDLF or TDLF, add to or subtract from the forces determined from the analysis. Typically for straight skewed bridges, the lockedin forces
24
tend to be opposite in sign to the internal forces due to dead loads. Therefore the 2Dgrid or 3DFEA analysis solutions for crossframe forces and flange lateral bending stresses are conservative when SDLF or TDLF initial fitup forces are neglected. However, these solutions can be prohibitively conservative for highly skewed bridges (NCHRP, 2012).
TDLF or SDLF detailing is first and foremost a geometrical calculation for the detailer and fabricator. Yet, they can significantly affect the lockedin crossframe forces. Figure III.3 shows four configurations that visually explain how the lockedin forces can be calculated. Configurations 1 and 4 are used by structural detailers. Configurations 2 and 3 are theoretical geometries that technically never take place in the physical bridge, but are used to calculate the internal lockedin forces. The differential camber shown in Configuration 1 is detailed to counterbalance the eventual differential deflection that occurs under the corresponding dead load. This differential camber induces the twisting shown in Configuration 3 from the crossframes being forced into place and released. The deflections due to the twisting are approximately equal and opposite to the deflections at these locations under the corresponding total or steel dead load (NCHRP, 2012).
For cases where the initial lackoffit effects are important, the designer can simply include an initial stress or strain similar to a thermal stress or strain. Calculating the initial strains and stresses associated with SDLF or TDLF detailing of the crossframes involves finding the nodal displacements between Configurations 2 and 4 and applying the corresponding stresses to the crossframe ends. In 3DFEA, the calculated axial strains from the nodal displacements are converted into stresses simply by multiplying the strains by the elastic modulus of the material. The stresses are then
25
multiplied by the crossframe member areas to determine the axial forces. In 2Dgrid models that use equivalent beam elements for the crossframes, the displacements calculated above are converted into beam end displacements and end rotations.
Assuming fixedend conditions, the end displacements are used to calculate the fixedend forces, which are then applied to the equivalent crossframe beam element.
(b) Configuration 2 Girders locked in the initial noload, plumb and cambered geometry, crossframes subjected to initial strains and initial stresses to connect them to the girders
Figure III.3 Conceptual Configurations Associated with Dead Load Fit (TDLF or SDLF) Detailing (NCHRP, 2012)
26
(c) Configuration 3 Theoretical geometry under noload (dead load not yet applied I. alter resolving the initial lack of fit by connecting the crossframes to the girders, then "releasing the girders to deflect under the lackoffit effects from the crossframes
Id) Configuration 4 Geometry under the combined effects of the total tor steel) dead load phis the loekedin internal forces due to (he dead load fit detailing
Figure III.3 (Continued) Conceptual Configurations Associated with Dead Load Fit (TDLF or SDLF) Detailing (NCHRP, 2012)
27
The behavior of the end crossframes at skewed bearing lines is slightly different, however the lockedin forces due to crossframe fitup is calculated in the same manner following the configurations in Figure III.3. The girders cannot displace vertically at the bearings and the skewed crossframes impose a twist in the girder ends. The top flange of the girders at the bearing line can only displace significantly in the direction normal to the plane of the crossframe. In order for the skewed end crossframe to fit up with the girders in Configuration 2, the crossframe has to rotate about its longitudinal axis and be strained into position to connect them with the rotated connection plates in the initial cambered noload, plumb geometry of the girders (NCHRP, 2012). Again, this is a theoretical configuration that technically would not occur in the physical bridge. It is used to calculate the displacements and the corresponding forces.
Preferred Analysis Method for Straight Skewed Girders
NCHRP Report 725 provides recommendations on the analysis and detailing method that should be used for various levels of skews and horizontal curvature. For straight skewed steel Igirder bridges, the recommendations are prominently based on the skew index, Is. The skew index is a measure of the severity of the skew based on the skew angle, the span length, and the bridge width measured between fascia girders.
wn tan 6
Straight skewed Igirder bridges are divided into three groups: Low (Is< 0.30), Moderate (0.30 0.65). Bridges with a low skew index of less than 0.30 are not as sensitive to the effects of skews. As the skew index increases above 0.30, responses associated with lateral bending of the girder flanges becomes significant. At
28
this point, the stress ratio of flange lateral bending stress over majoraxis bending stress ifi/fb) increases above 0.30 where crossframes are staggered. This is considered a large flange bending effect. As the skew index increases into the High category above 0.65, the skew effects can significantly influence the majoraxis bending responses. Below this level the vertical components of the forces from the crossframes are too small to noticeably influence the majoraxis bending response (NCHRP Appendix C, 2012).
NCHRP Report 725 provides a matrix of grades for traditional 2Dgrid and IDline girder analysis for several different levels of skew and horizontal curvature as seen in the Figure III.4. For straight skewed bridges with a high skew index (Is > 0.65), 2Dgrid and IDline girder analysis receive really poor grades. However, it should be noted that the recommended improvements to 2Dgrid analysis, as described in previous sections, dramatically improve grades and percentage of error, especially for solutions of crossframe forces and flange lateral bending stresses. The grades are based on the percentage of normalized mean error of the results for each structure response. The breakdown of grades include:
A: 6% or less normalized mean error, reflecting excellent accuracy;
B: between 7% and 12% normalized mean error, reflecting reasonable agreement;
C: between 13% and 20% normalized mean error, reflecting significant deviation from the accurate benchmark;
D: between 21% and 30% normalized mean error, reflecting poor accuracy; and
29
F: over 30% normalized mean error, reflecting unreliable accuracy and inadequate for design (NCHRP, 2012).
NCHRP Report 725 also provides recommendations for crossframe detailing methods for straight skewed Igirder bridges based on the skew index. In general TDLF detailing is preferred in order to keep layover to a minimum and ensure the web is plumb in the final TDL condition. Layover is defined as the relative lateral deflection of the flanges from the twisting motion of the girders. For Is < 0.30, TDLF is typically the preferred option.
The total dead load (TDL) crossframe forces and girder flange lateral bending stresses will essentially be canceled out by the TDLF lockedin forces. With a low skew index level, the forces required for crossframe fitup during steel erection are very manageable. Ensuring that the first intermediate crossframes are a minimum distance offset from centerline of bearing, will help alleviate nuisance stiffness effects and reduce fitup forces by providing enough flexibility at the end of girder to force the girders into position with the relatively stiff crossframes. The recommended minimum offset distance from the bearing centerline is:
a>max(1.5Z), OAb)
where D is the girder depth and b is the second unbraced length within the span from the bearing line (NCHRP, 2012).
30
Response
Geometry
WorstCase Scores
Traditional
2DGrid
IDLine
Girder
Mode of Scores
Traditional 2D Grid
IDLine
Girder
C {Ic< 1)
B
B
MajorAxis
Bending
Stresses
C (I c > 1)
D
S (Is < 0.30)
B
S (0.30
B
S Us > 0.65)
Vertical
Displacements
C&S (Ic > 05 & Is > 0.1)
C (/c< 1)
C Uc > 1)
CrossFrame
Forces
s (rs < 0.30)
S (0.30
S Us >0.65)
C&S (Ic > 0.5 & Is> 0.1)
C(/c
Flange Lateral Bending
Stresses
C (/c > 1)
S (Jv < 0.30)
S (0.30
S Us > 065)
C&S (/c > 0.5 & Is > 0.1)
Girder Layover at Bearings
C (/c< 1)
C (/1 > 1)
S [Is < 0.30)
S (0.30 0.65)
C&S (Ic > 0.5 & Is> 0.1)
B
B
Magnitudes should be negligible for bridges that are properly designed & detailed. The crossframe design is likely to be controlled by considerations other than gravityload forces.
b Results are highly inaccurate due to modeling deficiencies addressed In Ch, 6 of the NCHRP 1279 Task 8 report. The Improved 2Dgrld method discussed In tills Ch. 6 provides an accurate estimate of these forces. r Finegirder analysis provides no estimate of crossframe forces associated with skew,
J The flange lateral bending stresses tend to be small. AASHTO Article C6.10.1 may be used as a conservative estimate of the flange lateral bending stresses due to skew.
1 Linegirder analysis provides no estimate of girder flange lateral bending stresses associated with skew.
' Magnitudes should be negligible for bridges that are properly designed & detailed.
Figure III.4 Matrix of Grades for Recommended Level of Analysis for IGirder Bridges (NCHRP, 2012)
31
For straight skewed Igirder bridges with a higher skew index of Is > 0.30, TDLF, SDLF, or detailing between SDLF and TDLF are typically good options. As the skew index increases, the force required for crossframe fitup increases and becomes much more difficult to erect. If SDLF detailing is used, excessive layover in the final TDL condition may become a concern for bridges with large skews and long spans. Besides TDLF crossframe detailing, layover can be addressed with the use of beveled sole plates and/or using bearings with a larger rotational capacity (NCHRP, 2012).
32
CHAPTER IV
ANALYTICAL PLAN
The analytical plan involves applying the theories and recommendations discussed in the Theoretical Background chapter of this thesis towards an example bridge. Several different models of the example bridge superstructure were created and analyzed and then the results are compared. The models include a conventional 2Dgrid base model, an improved 2Dgrid model, a 2Dframe base model, an improved 2Dframe model, a 3DFEA NLFdetailing model, and a 3DFEA TDLFdetailing model. The results for majoraxis bending stresses, vertical displacements, crossframe forces, flange lateral bending stresses, and girder layover at bearings are all compared in the Analytical Results chapter.
Example Bridge Description
The Geneva Road Bridge in Utah was analyzed as the example bridge used in this thesis. The bridge is a part of the SR114 Geneva Road DesignBuild Project which was undertaken to improve travel between Provo and Pleasant Grove, Utah. The project involved reconstruction and widening work of about four miles of SR114 and new construction of a bridge over the Union Pacific Railroad and Utah Transit Authority tracks. Parsons served as the designer and teamed with the contractor, Kiewit, to design and build the bridge. The owner is the Utah Department of Transportation.
The Geneva Road Bridge has a 1034 wide deck that includes four lanes of traffic (two in each direction), two 10 shoulders, a 14 median, and a sidewalk on each side. The single span bearing to bearing length is 2545 Vi. The skew angle is almost 62 degrees and its skew index, Is = 0.65, puts it right on the edge of the most severe
33
category as per NCHRP Report 725 and as seen in Figure III.4. There are nine steel plate Igirders spaced at 110 % on center and T5 overhangs. All structural steel conforms to AASHTO M 27050W, which is a weathering steel with a yield stress of 50 ksi. See Figure IV. 1 for the typical superstructure section and Figures IV.2 and IV.3 for the plan and elevation layouts.
The deck overhangs appear to be a bit large compared to the girder spacing; however the plans explicitly state that the sidewalks shall never be converted to travelled lanes. The analysis in this thesis focuses on the behavior of the structure due to dead loads during construction; therefore the exterior girders appear to take a larger amount of load in the analytical results. The relatively small amount of live load due to pedestrian loads distributed to the exterior girder compared to the much larger vehicular live loads distributed to the interior girders, balances out the total end design load among all girders. Typically, the preferred ratio of overhang length to girder spacing is between 0.3 and 0.5 for overhangs that could potentially see large vehicular live loads.
SECTION THROUGH STRUCTURE
(NORMAL TO CENTER LNE OF GENEVA ROAD)
Figure IV.l Typical Section of the Geneva Road Bridge (Parsons, 2011)
34
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Figure IV.2 Plan Layout of the Geneva Road Bridge (Parsons, 2011)
7 EMC *SL1 K j !LMV
UNO.
uH; iwt* r:irhil jmmi i
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Figure IV.3 Elevation Layout of the Geneva Road Bridge (Parsons, 2011)
The girder sections and lengths are the same for all nine steel plate Igirders. Girder 1 only differs by location of the splice; however, splice location is irrelevant for the purposes of this thesis. The web is constant at 105 x Va\ The top flange width is a constant 30 and the thickness varies from 1 V2 at the ends to 1 % at the middle section. The bottom flange width remains constant at 32 with a thickness that varies from 1 % at the ends to 2 Vi at midspan. See Figure IV.4 for Girder Elevations.
The crossframes are Ktype crossframes with WT members for the bottom, top, and diagonal chords. The interior crossframes are continuous where possible as seen in the framing plan in Figure IV.5. After an initial analysis in the original design, the first interior crossframe near each obtuse corner of the framing plan was removed. Those crossframes attracted a significant amount of load due to the behavior of wide and highly skewed bridges tending to find an alternate load path by spanning between the obtuse corners in addition to spanning along the centerline of the girders. These crossframes at
36
or near the supports provide nuisance stiffness transverse load paths especially at the obtuse corners (NCHRP, 2012). The idea to remove the first interior crossframe at the obtuse corners came from the article, Design and Construction of the Curved and Severely Skewed Steel IGirder EastWest Connector Bridges over 188 The article explains how nonskewed crossframes that frame directly into skewed supports provide alternate load paths and also refer to these effects as nuisance stiffness. These crossframes were removed to mitigate these effects (Chavel et al, 2010).
The next few crossframes that are inline with the removed crossframes on the Geneva Road Bridge, still experienced significant loads in the analysis and required larger member sizes. See the framing plan in Figure IV. 5 for the location of the stiffer type 2 crossframes. As per recommendations from NCHRP Report 725, AASHTO/NSBA G13.1, and Schaefers thesis, all of which were published after the Geneva Road Bridge was designed, the crossframes could have been staggered (discontinuous) and pushed back a distance a > max( 1 .5/7, QAb) offset from the bearing line to the first interior crossframe in order to reduce the crossframe loads and mitigate nuisance stiffness effects. However, arranging the crossframes in continuous lines could significantly reduce the lateral flange bending stresses.
The Geneva Road Bridge has already been designed and constructed. The designers used the commonly used 2Dgrid steel girder structural analysis software, MDX, for the majority of the superstructure analysis. The design has been checked and construction occurred without any issues that would have compromised the integrity of the structure. The bridge is open to traffic and there have been no reported issues to date.
37
00
GIRDERS G2 G9 GIRDER ELEVATION
Figure IV.4 Girder Elevation of the Geneva Road Bridge (Parsons, 2011)
FRAMING PI AS 1
Figure IV.5 Framing Plan of the Geneva Road Bridge (Parsons, 2011)
39
The intent of using this bridge in this thesis isnt to recommend a better layout or a better design method but rather to gain a better understanding of the behavior of the girders and crossframes during construction. The method of construction is known and the behavior was witnessed with some recorded field data, which helped verify the modeled behavior.
Figure IV.6 Underside of the Geneva Road Bridge
After precast panel and deck rebar installation and before the castinplace concrete deck pour (with permission from Kiewit).
Analysis Models
Six different analysis models were created and analyzed. The steel girder analysis software, MDX, is used for two 2Dgrid models and the 3D structural analysis software,
40
LARSA 4D with the Steel Bridge Module, is used for the other four models that include 2Dframe and 3DFEA. The results are compared in the next chapter Analytical Results. As described in the Theoretical Background chapter, improvements to the 2D models include:
Adjusting the equivalent beam stiffness assumed for crossframes,
Adjusting the torsional stiffness to include warping stiffness, and
Calculating more accurate lateral flange bending stresses.
All 2D and 3D models assume NLF detailing by default, meaning no initial lockedin crossframe forces are included in the analysis. The final improvement includes adding the lockedin crossframe forces due to TDLF detailing for the 3DFEA model.
The theory behind calculating more accurate lateral flange bending stresses is based on assuming a staggered crossframe layout is used. Since the crossframes are continuous, the lateral flange bending stresses will not be calculated as per the outlined 2Dgrid improvements in the analysis of the example bridge. This improvement would have been a postprocessing step and will continue to be one unless 2Dgrid software companies choose to rewrite their code and implement it directly into the software.
2DGrid Base Model MDX
This model was used for the original design and is left unchanged without any improvements implemented for comparison purposes. In the MDX software program, the user runs through a wizard to input various geometric and load parameters. The user runs through five modes or input phases in the process of creating a girder system design model:
41
1. Layout Mode the user provides general layout information to establish the framing plan.
2. Preliminary Analysis Mode the user provides the loading.
3. Preliminary Design Mode the user provides design controls to be enforced on the generation of a set of girder designs based on the preliminary design forces.
4. Design Mode the user defines the bracing and can generate bracing and girder designs after setting up certain parameters.
5. Rating Mode this final mode is used for tuning the design (MDX, 2013).
The output includes forces, stresses, and displacement results for each girder and for the girder system that includes the crossframes. The results are also checked against the latest AASHTO bridge specifications.
2DGrid Improved Model MDX
This model includes any possible recommended improvements to a 2Dgrid analysis. The issue is, given the constraints of the input wizard, theres very little that can be manipulated to improve the analysis and better represent the behavior of the girder and crossframe system. The software automatically calculates the torsional stiffness, J, based on the St. Venant pure torsional stiffness by using the section dimensions input. Warping stiffness is not included in the torsional stiffness and theres no way to overwrite this sectional property. In addition, there is no way to add the lockedin crossframe forces for TDLF or SDLF detailing.
42
That leaves adjusting the equivalent beam stiffness assumed for crossframes as the only improvement that can be implemented in the 2Dgrid model. The user has the option of inputting the crossframe type (Ktype, Xtype, or diaphragm) and the associated member sizes or manually input the equivalent crossframe properties. If the first option is chosen, the software automatically converts the crossframe into an equivalent beam and calculates the equivalent stiffness using the flexural analogy method. This method does not account for the shear stiffness. The improved method as described in the Theoretical Background chapter is implemented in this model. See Appendix A for calculations.
2DFrame Base Model LARSA 4D
This model does not include any improvements and is used as a base model for comparison purposes. LARSA 4D allows much more flexibility in modeling a structure compared to commonly used 2Dgrid software. There are two methods of modeling a steel girder structure: 2Dframe and 3DFEA. 2Dframe models create the structure in one horizontal plane with each girder modeled as a beam element offset from the deck and connected with rigid links. The deck is modeled as plate elements and the crossframes are modeled as truss or beam elements as appropriate with the connection points offset from the deck.
LARSA 4D includes a design tool called the Steel Bridge Module that helps significantly reduce the time required to model the structure and apply the appropriate loads. The user goes through the module in similar fashion as the MDX wizard to set up the model, and has the flexibility to adjust the model and add loads manually as deemed
43
appropriate by the user. LARSA 4D also includes a construction staging analysis function. Typically this function is used to analyze material time effects (time is considered the fourth dimension in the name) such as creep and shrinkage of concrete and relaxation of stressed tendons. However, time is irrelevant in steel girder design, except for considering fatigue but that is based on total stress cycles. The construction staging analysis can still be a useful tool for steel girder design to determine stresses and movements as loads are applied and as cross section properties change (composite vs. noncomposite) at each construction stage.
2DFrame Improved Model LARSA 4D
This model includes improvements for the girder torsional stiffness. See Appendix A for calculations on the equivalent girder torsional stiffness. Other potential 2D improvements were not included in this model. The crossframes are modeled in 3D, therefore computing the equivalent beam stiffness is unnecessary. The flange lateral bending stresses are automatically computed. Lockedin crossframe forces due to TDLF detailing are only analyzed in the 3DFEA TDLF model for ease of comparison with the 3DFEA NLF model.
3DFEA NLF Model LARSA 4D
3DFEA truly models the structure in three dimensions. The girders are modeled as a combination of beam elements for the flanges and plate elements for the web. The crossframes are again modeled as truss or beam elements and are connected to the corresponding top and bottom flange beam elements. The warping component of the girder torsional stiffness is automatically included. The crossframes are modeled in 3D
44
and therefore do not need to be converted to equivalent beam elements. As with all 2D models, this model assumes NLF detailing by default and does not include any initial lockedin crossframe forces that would be present for TDLF or SDLF detailing methods. By assuming the NLF detailing method, the results can be compared directly against the 2D models. Further research would need to be conducted to validate the accuracy of this model with a fullsize test bridge. However, this is out of the scope of this thesis and the 3DFEA NLF model is used as the benchmark and assumed to be the most accurate.
The majoraxis bending stresses and lateral bending stresses in the flanges are determined from the member stresses results. The axial stress at the centroid of the flange beam members resembles the stress due to majoraxis bending. The lateral flange bending stresses are determined from taking the difference between the axial stress at the centroid and the average of the top and bottom stress points at one side of the rectangular flange section. The vertical deflections are taken from the joint displacements results in the vertical direction along the bottom flange of the girders. The crossframe axial forces are taken from the member end forces results in the local member coordinates. The girder layovers are taken from the lateral joint displacements at the top of the girder ends. The bottom of the girder is restrained in the lateral direction at the bearings.
3DFEA TDLF Model LARSA 4D
The only improvement needed for this model is including the initial lockedin crossframe forces due to TDLF detailing. The lockedin crossframe forces are calculated by determining the axial strain of the truss type members of the crossframes due to the camber differences for total dead load differential deflections. These initial
45
strains are inputted into the model as an equivalent thermal strain load. See Appendix A for example calculations.
46
CHAPTER V
ANALYTICAL RESULTS COMPARING MODELS
The primary goal of this thesis is to find the most efficient method of analysis that accurately models the behavior of highly skewed steel plate Igirder bridges. Six models were created, analyzed, and results compared. The results include:
Majoraxis bending stresses
Vertical displacements
Crossframe forces
Flange lateral bending stresses
Girder layover at bearings
These are the same results used in NCHRP Report 725 to grade the accuracy of traditional 2Dgrid and IDlinear analysis as seen in Figure III.4. The results of the example bridge models in this thesis are compared to the average and worst case results reported by NCHRP. With a skew index, Is = 0.65, the example bridge is compared to bridges in the highest skew index category.
The 3DFEA NLF LARSA 4D model is assumed to be the most accurate and therefore used as the benchmark against which all other 2Dgrid and 2Dframe models are compared. The 2Dgrid MDX models (base and improved) are first compared to the 3DFEA NLF LARSA 4D model. Next, the 2Dframe LARSA 4D models (base and improved) are compared to the 3DFEA NLF LARSA 4D model. Finally, the 3DFEA NLF LARSA 4D model is compared to the 3DFEA TDLF LARSA 4D model.
47
2DGrid Models
Majoraxis bending stresses. The average grade of traditional 2Dgrid analyses for majoraxis girder bending stresses as reported by NCHRP is a C and the worstcase grade is a D. A grade of C means the normalized mean error is between 13% and 20%, reflecting a significant deviation from the accurate benchmark. A grade of D means the normalized mean error is between 21% and 30%, reflecting poor accuracy.
Figures V.l and V.2 compare the unfactored majoraxis bending stresses in the top and bottom flanges, respectively. The stresses are due to the dead loads, including the weight of the deck, on the noncomposite steel section. The 2Dgrid MDX models (base and improved) are compared to the 3DFEA NLF LARS A 4D model. Both figures show similar patterns for the bending stresses along the length of the girder. The normalized mean error was not calculated due to the jagged lines in the 3DFEA NLF LARSA 4D model results; however the results appear to be within 6% error, which results in a grade of A. The improved MDX model with the updated crossframe beam stiffness appears to resemble the benchmark pattern slightly more accurately.
The reasoning behind the jagged line display, which is more prominent in the top flange, is unknown for the 3D benchmark model. It is most likely due to the influence of the crossframes. The reasoning behind the small but noticeable jump at the girder end in the 3D benchmark model is also not completely known but not unexpected either. The results in NCHRP Report 725 show a similar spike at the obtuse end of the exterior girders, but do not explain the reasoning for this spike. The end crossframes along the high skew may be providing some equivalent continuity at the ends of the girder and therefore cranking in a moment. However, this is purely speculation. The results are
48
deemed acceptable and further analysis into the reasoning of the jagged line pattern and spike at the obtuse end is considered outside of the scope of this thesis.
Figure V.l MajorAxis Bending Stress of Girder 1 Top Flange
Figure V.2 MajorAxis Bending Stress of Girder 1 Bottom Flange
49
Vertical Displacements. The average grade of traditional 2Dgrid analysis for girder vertical displacements as reported by NCHRP is a C and the worstcase grade is a D. Figure V.3 compares the vertical displacements due to noncomposite dead loads among the 2Dgrid MDX models (base and improved) and the 3DFEA NLF LARSA 4D model along Girder 1 and Girder 3.
Figure V.3 Vertical Girder Displacements Along Girder 1 and Girder 3
50
The vertical displacements all follow the same pattern and all have a normalized
mean error of 2% or less, which is a grade A level. This is a much better result than the
average grade reported by NCHRP for bridges with similar skew indexes.
CrossFrame Forces. The average and worstcase grade of traditional 2Dgrid
analysis models for crossframe forces as reported by NCHRP is an F. A grade of F
means the normalized mean error is over 30%, reflecting unreliable accuracy and making
the results inadequate for design. Figure V.3 compares the unfactored crossframe forces
due to noncomposite dead loads for the 2Dgrid MDX base model and 3DFEA LARSA
4D model for each member of the crossframes. The crossframes along bay 2 between
girders 2 and 3 are shown.
Top Chord
100
50
0
wT
Q.
i 50
a
ro
0
1 loo 1
150
200
250
Crossframe Number
Figure V.4 CrossFrame Axial Forces Along Bay 2 Between Girder 2 and Girder 3
51
Left Diagonal
Crossframe Number
MDX Base
Larsa 3D
Left Bottom Chord
MDX Base Larsa 3D
Figure V.4 (Continued) CrossFrame Axial Forces Along Bay 2 Between Girder 2
and Girder 3
52
Right Diagonal
MDX Base Larsa 3D
Right Bottom Chord
MDX Base Larsa 3D
Figure V.4 (Continued) CrossFrame Axial Forces Along Bay 2 Between Girder 2
and Girder 3
53
The pattern of crossframe member forces along bay 2 is similar for both analysis models. A spike in axial load in all members at crossframes 11 and 12 can clearly been seen. This illustrates the increased loads that occur at the obtuse corners. However, a significant difference in crossframe forces can clearly be seen between the two models from the graphs. The normalized mean error in comparison to the 3DFEA NLF benchmark model ranges from 7.3% to 21.2% as seen in Table V.l. This would suggest a grade of D for the worst case. However, some of the worst errors occur at the controlling crossframes with the highest forces. With percentages of error well over 30% for these critical crossframes, the grade should be an F and the method deemed unacceptable for calculating crossframe forces.
Normalized Mean Error
Top Chord 14.6%
Left Diagonal 13.9%
Left Bottom Chord 21.2%
Right Diagonal 14.0%
Right Bottom Chord 7.3%
Table V.l Normalized Mean Error for CrossFrame Forces in the 2DGrid MDX Model
Flange Lateral Bending Stresses. The average and worstcase grade of traditional 2Dgrid analysis models for girder flange lateral bending stresses as reported by NCHRP is an F. Responses to flange lateral bending are not provided in the MDX results. In order to determine the flange lateral bending stresses, postprocessing using the crossframe forces would need to be completed. Since the crossframe forces results received a grade of F for the 2Dgrid model, calculation results for flange lateral bending stresses would be inaccurate as well. Therefore, these calculations were not performed.
54
Girder Layover at Bearings. The average grade of traditional 2Dgrid analysis models for girder layover at bearings as reported by NCHRP is a C and the worstcase grade is a D. MDX does not produce this output, therefore there is nothing to compare. Girder layover at bearings would be calculated by hand using the differential deflection output. Because girder layover and vertical displacements are directly related, NCHRP gave them the same grades.
2DFrame Models
Majoraxis bending stresses. Figure V.5 compares the unfactored majoraxis bending stresses in the top and bottom flanges. The stresses are due to the dead loads, including the weight of the deck, on the noncomposite steel section. The 2Dframe LARSA 4D models (base and improved) are compared to the 3DFEA NLF LARSA 4D model. Both graphs show similar patterns for the bending stresses along the length of the girder. The normalized mean error was not calculated due to the jagged lines in the 3DFEA NLF LARSA 4D model results; however the results appear to be within 12% error, which results in a grade of B. It is surprising that the 2Dframe models have a higher percentage of error; however, with a grade of B, they are considered acceptable for analysis results.
Vertical Displacements. Figure V.6 compares the vertical displacements due to noncomposite dead loads among the 2Dframe LARSA 4D models (base and improved) and the 3DFEA NLF LARSA 4D model along Girder 1 and Girder 3.
55
Bottom Flange
Normalized Length
Figure V.5 MajorAxis Bending Stress of Girder 1
56
The vertical displacements all follow the same pattern and all have a normalized mean error of less than 8%, which is a grade level of A to B. The critical maximum deflection near midspan is off by as much as 14% for both girders. However, with a maximum difference of 2.37, the error can be made up by specifying a large enough haunch in the plans. These results are considered acceptable for the example bridge.
Figure V.6 Vertical Girder Displacements Along Girder 1 and Girder 3
57
CrossFrame Forces. Figure V.7 compares the unfactored crossframe forces due to noncomposite dead loads for the 2Dframe LARSA 4D models (base and improved) and 3DFEA LARSA 4D model for each member of the crossframes. The crossframes along bay 2 (between girders 2 and 3) are shown.
The pattern of crossframe member forces in the graph along bay 2 is similar for all models, except for the end crossframes in the 2Dframe models. The spike in axial load in all members at crossframes 11 and 12 illustrates the increased loads that occur at the obtuse corners and is much more accurately represented in both 2Dframe LARSA 4D models versus the 2Dgrid MDX models.
Figure V.7 CrossFrame Axial Forces Along Bay 2 Between Girder 2 and Girder 3
58
Figure V.7 (Continued) CrossFrame Axial Forces Along Bay 2 Between Girder 2 and Girder 3
59
Figure V.7 (Continued) CrossFrame Axial Forces Along Bay 2 Between Girder 2
and Girder 3
60
A significant difference in crossframe forces can clearly be seen for the end and first interior crossframes between the improved 2Dframe LARSA 4D model and the 3D benchmark. The improvements to the girder torsional stiffness appears to provide inaccurate results for crossframe forces near the girder ends and provide no noticeable improvement over the 2Dframe base model. The normalized mean error ranges from 3.1% to 6.8% for the 2Dframe base model and from 6.0% to 12.0% for the 2Dframe improved model as seen in Table V.2. This results in a grade of A and B for the two models respectively.
Normalizec Mean Error
Member Base Improved
Top Chord 3.1% 7.8%
Left Diagonal 3.1% 12.0%
Left Bottom Chord 6.8% 6.0%
Right Diagonal 3.2% 12.0%
Right Bottom Chord 4.1% 11.7%
Table V.2 Normalized Mean Error for CrossFrame Forces in the 2DFrame LARSA 4D Models
Flange Lateral Bending Stresses. Figure V.8 compares the unfactored flange lateral bending stresses due to noncomposite dead loads for the 2Dframe LARSA 4D base model and 3DFEA LARSA 4D model for Girders 1 and 3. The pattern of lateral bending stresses is similar for both models in the top flange; however, they appear to differ significantly in the second half of the bottom flange near the obtuse corner.
61
Figure V.8 Flange Lateral Bending Stress
62
Girder 3 Top Flange
Normalized Length
Figure V.8 (Continued) Flange Lateral Bending Stress
63
The normalized mean error for the top flange is 13.2% for Girder 1 and 13.8% for Girder 3, which results in a grade of C. In contrast, the normalized mean error for the bottom flange is much worse at 90.3% for Girder 1 and 54.5% for Girder 3 and results in a grade of F. It is important to note that the lateral bending stresses are relatively small, making the percentage of error a bit inconsequential. The lateral bending stresses are less than 2 ksi everywhere except near the end of girders near the obtuse end. Because the first interior crossframe in bay 1 near the obtuse corner was removed, this area could be considered as having discontinuous crossframes. As per AASHTO LRFD Bridge Design Specifications C6.10.1, in absence of calculated values of fi from a refined analysis, a suggested estimate for the total unfactored fi in a flange at a crossframe or diaphragm due to the use of discontinuous crossframe or diaphragm lines is 10.0 ksi for interior girders and 7.5 ksi for exterior girders. It continues, in regions of the girders with contiguous crossframes or diaphragms, these values need not be considered (AASHTO, 2012). Therefore, 7.5 ksi for exterior and 10 ksi for the interior could conservatively be assumed for the flange lateral bending stress near the obtuse corners. The rest of the crossframes are considered contiguous and therefore the flange lateral bending stress can be considered negligible.
Girder Layover at Bearings. Figure V.9 compares the girder layover at bearings under dead loads on the noncomposite girders for both 2Dframe models (base and improved) and the 3DFEA NLF model. The girder layover is the horizontal transverse displacement measured at the top of the girder web with respect to the bottom of the girder web. The bearings are fixed in the transverse direction.
64
The normalized mean error for the 2Dframe base model is 7.3% at the start
bearing and 9.1% at the end bearing, which results in a grade of B. The normalized mean error for the 2Dframe improved model is 10.9% at the start bearing and 14.5% at the end bearing, which results in a grade of B and C respectively. The significance of the percentage of error depends on the crossframe fitup detailing method used and the rotational capacity in the bearings. If NLF detailing is used, the bearings would need to
be able to handle the large transverse rotations.
a>
Q
E
ai
c
19
2 0 2 Girder Layover (in)
3D End 3D Start
2DFrame Imp End 2DFrame Imp Start
2DFrame Base End
2DFrame Base Start
Figure V.9 Girder Layover at Bearings
65
3DFEA Model with TDLF Detailing
The 3DFEA TDLF model includes the initial lockedin crossframe forces due to TDLF detailing. Results are compared to the 3DFEA NLF model that by default assumes no initial crossframe forces and the girder webs are plumb in the noload case. Results at different construction stages are also compared.
Majoraxis bending stresses. As seen in Figure V.10, the majoraxis bending stress due to dead loads on the noncomposite girder sections are very similar. The patterns are almost identical and the results at each data point differ very slightly.
Girder 1 Top Flange
Normalized Length
Girder 1 Bottom Flange
Normalized Length
Figure V.10 MajorAxis Bending Stress of Girder 1
66
Vertical Displacements. The vertical displacements due to dead loads on noncomposite girder sections are also very similar between the 3DFEA NLF and 3DFEA TDLF models. There is no noticeable difference between the exterior girders. There is a slight difference between the interior girders that is most noticeable closest to the centerline of the bridge (Girder 5). This is to be expected as per the results from NCHRP Report 725. Girder 1 and Girder 3 vertical displacements are compared in Figure V.ll.
Figure V.ll Vertical Girder Displacements along Girder 1 and Girder 3
67
CrossFrame Forces. Figure V.12 compares the unfactored crossframe forces due to the girder, crossframe, and deck noncomposite dead loads for the 3DFEA NLF and TDLF models, and the forces due to only the steel girder and crossframe noncomposite dead loads for the 3DFEA TDLF model. For most of the crossframe members with TDLF detailing, the maximum force in the crossframes occurs during the fitup of the crossframes with the girders. The girders are twisted and forced into an out of plumb orientation and as dead loads are applied, the crossframe forces are offset or relieved as the girders twist back into the vertically plumb position under total dead loads.
Figure V.12 CrossFrame Axial Forces Along Bay 2 Between Girder 2 and Girder 3
68
Figure V.12 (Continued) CrossFrame Axial Forces Along Bay 2 Between Girder 2 and Girder 3
69
Figure V.12 (Continued) CrossFrame Axial Forces Along Bay 2 Between Girder 2 and Girder 3
70
Flange Lateral Bending Stresses. Figure V.13 compares the unfactored flange lateral bending stresses due to noncomposite dead loads along Girder 1 for the 3DFEA NLF and 3DFEA TDLF models. The TDLF model results include two construction stages steel girder and crossframe dead loads only and steel plus deck dead loads.
Bottom Flange
6
8
0.0 0.2 0.4 0.6 0.8 1.0
Normalized Length
Figure V.13 Flange Lateral Bending Stress Along Girder 1
71
As expected the flange lateral bending stress is minimal at the total dead load case when using TDLF detailing. The results for the crossframe forces and flange lateral bending stress, clearly shows the advantage of using TDLF detailing.
Girder Layover at Bearings. Figure V.14 compares the girder layover at bearings under dead loads on the noncomposite girders for the 3DFEA NLF and TDLF models with girders, crossframes, and deck dead loads and the 3DFEA TDLF model with the steel girders and crossframe dead loads only.
The girder layover under total noncomposite dead load using the TDLF detailing is close to zero indicating the girder webs are nearly plumb in the total dead load condition. Figure V.14 also indicates that the initial layover with the initial lockedin crossframe forces due to TDLF detailing is in the opposite direction than the layover due to dead loads.
Field data was gathered during construction that indicates the girder layover. A 4 level was used to measure how far out of plumb the girder webs were over the height of the level. The data was used to determine the girder layover in relation to the full height of the girder. The field data is believed to have been taken just after girder and crossframe erection; however, additional dead loads may have been present, such as the precast deck panels. Figure V. 15 compares the girder layover taken from field data against the 3DFEA TDLF models at the steel only load case and steel plus deck load case. The field data mostly follows the same pattern along the bearing lines and appears to be slightly less than the 3DFEA TDLF steel only load case. The method of measuring the field data leaves room for human error that would affect the accuracy; however, the
72
intent of this figure is to validate the behavior found in the 3DFEA TDLF models with the actual behavior of the bridge.
Figure V.14 Girder Layover at Start and End Bearings for 3D Models
73
Start Bearing Line
Girder Layover (in)
3D TDLF Steel+Deck 3DTDLF Steel Only Field Data
Figure V.15 Girder Layover at Start and End Bearings for 3D TDLF Models and from Field Data
74
CHAPTER VI
CONCLUSIONS
Recommended Method of Analysis
The primary goal of this thesis is to determine the most efficient method of analysis that accurately models the behavior of highly skewed steel plate Igirder bridges. By implementing some improvements to 2D methods of analysis as described in NCHRP Report 725, the hope is that 2D type methods could provide very accurate results. The improvements appear to be fairly straightforward and simple in theory, but the application ended up being far from simple.
NLF detailing is assumed by default in all software. However, for straight steel Igirder bridges, TDLF detailing is the preferred option. If TDLF detailing is chosen, it is very important to include any lockedin forces, which will counterbalance to a certain extent the crossframe loads, lateral deflections, and rotations caused by the dead loads.
If these lockedin forces are not included, the designer is assuming NLF detailing, which can lead to overly conservative crossframes forces and lateral flange bending stresses in cases of very large skews and with the presence of nuisance stiffness crossframes close to the bearing line. Correctly modeling the behavior for the chosen detailing method is important to develop an accurate and efficient design.
The 3DFEA method is considered the most accurate method; however it still comes with its own limitations. The biggest limitation is its complexity and the amount of detail that is required to create a 3DFEA model. Creating the model, running through the analysis, and sorting through the massive amount of output data can be very time
75
consuming. Modeling an already complex structure with a complex method and interpreting the output can also increase the chance for human error.
The 3DFEA method was assumed to be the most accurate for analyzing the example and was used as the benchmark to compare all other models. The field data measurements of girder layovers during construction provided a very loose validation of the software. The results for deflections, rotations, stresses, and general overall behavior of the highly skewed steel Igirder example bridge in the 3DFEA model were as expected. The girder ends twisted and rotated about the centerline of the bearing support and the framing system generally behaved as previously described in the Theoretical Background section. However, a fullsized test bridge with stress gauges and with similar geometry is needed to truly validate the assumptions made in the creating and analyzing the 3DFEA model. This was considered outside the scope of this thesis.
The 2Dgrid method of analysis for highly skewed Igirder bridges appears to be the least accurate for calculating crossframe forces for nuisance stiffness crossframes and lateral flange bending stresses. 2Dgrid software, such as MDX and DESCUS, are very powerful and useful tools when used in the right context. But until the software companies update their software to include better equivalent estimations of the girder torsional stiffness and equivalent beam stiffness of the crossframes, designers need to do a significant amount of postprocessing calculations to check for additional crossframe forces and later flange bending stresses that may not have been captured in the 2Dgrid software analysis. NCHRP Report 725 repeatedly encouraged the bridge software industry to implement these improvements into the software.
76
The 2Dframe method using the LARSA 4D software or something similar, currently appears to be the most efficient method to use for highly skewed steel Igirder bridge design. LARSA 4D contains a steel bridge module that makes creating the model much easier, similar to the 2Dgrid software models. Software like LARSA 4D that has 3D capabilities has the flexibility to manually override certain section properties and add user specified loads to better model the behavior of highlyskewed steel Igirder bridges. With this flexibility, the necessity of postprocessing can be eliminated or greatly reduced. Compared to 3DFEA models, 2Dframe model output is much more manageable and therefore less timeconsuming and efficient for the designer.
General Recommendations and Future Work Current Recommendations
Recommendations for designers currently analyzing highly skewed steel Igirder bridges include:
Use the 2Dframe method with software that includes a bridge module for easily creating the geometry.
Include improvements to the 2Dframe model in certain situations as outlined by NCHRP Report 725. Adjusting the equivalent girder torsional constant that includes warping capacity should be used for staggered crossframe layouts. When contiguous crossframes are used throughout each span, a more detailed analysis should be used to analyze the crossframe forces and lateral flange bending stresses near the girder ends.
77
Include initial lockedin crossframe forces when TDLF or SDLF detailing is used to get a more efficient design.
The 3DFEA method should be used on a limited basis to verify behavior and check localized stresses. It is a good tool to use to check crossframe forces and lateral flange bending stresses; however it is too cumbersome to use as an allencompassing analysis.
The 2Dgrid method is not recommended for steel Igirder bridges with a high skew index until improvements are made to the software or the designer decides to accompany this analysis with extensive postprocessing.
Future Work Considerations
Future work considerations and recommendations include:
Encourage the bridge software industry to implement the recommended improvements to 2Dgrid software.
Encourage the 3Dcapable bridge software industry to implement improvements to steel bridge design modules to accurately and easily include appropriate lockedin crossframe forces for SDLF or TDLF detailing and to automatically update the internally calculated equivalent girder torsional stiffness.
Research and analyze more highly skewed bridges with different crossframe layouts. Specifically, analyze the girders and crossframes with contiguous crossframe layouts and nuisance stiffness crossframe near bearing supports. Adjust the 2D analysis method improvements accordingly.
78
Research fitup practices typically used in highlyskewed steel Igirder erection. Determine at what point the fitup forces for TDLF detailing become too large.
Research and analyze more innovative crossframe configurations, including partially skewed crossframes, leanon bracing, temporary bracing, and different connection and bearing plate detailing.
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REFERENCES
AASHTO. American Association of State Highway and Transportation Officials (2012). AASHTO LRFD Bridge Design Specifications, Customary U.S. Units, 6th Edition with 2012 and 2013 Interim Revisions and 2012 Errata.
AASHTO/NSBA G12.1. American Association of State Highway and Transportation Officials / National Steel Bridge Alliance Steel Bridge Collaboration (2003). G
12.1 Guidelines for Design for Constructability.
AASHTO/NSBA G13.1. American Association of State Highway and Transportation Officials / National Steel Bridge Alliance Steel Bridge Collaboration (2011). G
13.1 Guidelines for Steel Girder Bridge Analysis, 1st Edition.
Ahmed, M.Z. and Weisberger, F.E. (1996). Torsion Constant for Matrix Analysis of Structures Including Warping Effect, International Journal of Solids and Structures, Elsevier, 33(3), 361374.
Beckmann, F., and Medlock, R.D. Skewed Bridges and Girder Movements Due to Rotations and Differential Deflections.
Chavel, B., Peterman, L., and McAtee, C. (2010). Design and Construction of the
Curved and Severely Skewed Steel IGirder EastWest Connector Bridges over I88. 27th Annual International Bridge Conference 2010. IBC1024.
MDX(2013). MDXNetHelp. http://www.mdxsoftware.com/. April 2013.
NCHRP Report 725. Transportation Research Board of the National Academy of
Sciences (2012). National Cooperative Highway Research Program Report 725: Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges. Project 1279.
Parsons Corporation (2011). Design Plans for SR114 Geneva Road, Roadway
Widening: Geneva Road over UPRR & UTA. Signed by registered Professional Engineer: Haines, Steve. Owner: Utah Department of Transportation.
Schaefer, A.L. (2012). Crossframe Analysis of HighlySkewed and Curved Steel IGirder Bridges. Thesis submitted to the University of Colorado Denver. ProQuest LLC.
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APPENDIX A
Appendix A includes calculations for the equivalent beam stiffness of the crossframes and the steel plate girder design calculations that include the equivalent girder torsional stiffness constant used in the 2D models.
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UCD
Master's Thesis Skewed Steel IGirders
Equivalent CrossFrame Stiffness Calculations
Calculations By: K. Dobbins
10/18/2013
Equivalent Beam Stiffness for CrossFrames
Constants:
Es =
29000
ksi
Crossframe Type 1
in in plf plf plf kips
height btwn working pts width btwn working pts weight top chord weight bott chord weight diagonals weight connection plates Total weight
____85
127
26.5
26.5
26.5 0.726 1.548
STAAD Output Nodal Displacements
Florizontal Vertical Resultant Rotational
Node L/C X (in) Y (in) z (in) (in) rX (rad) rY (rad) rZ (rad)
1 1UNIT LOAD COUPLE 0 0 0 0 0 0 0
2 1 UNIT LOAD COUPLE 0 0 0 0 0 0 0
3 1 UNIT LOAD COUPLE 0.00056 0.00084 0 0.00101 0 0 0
4 1 UNIT LOAD COUPLE 0.00056 0.00084 0 0.00101 0 0 0
5 1 UNIT LOAD COUPLE 0.00028 0.00021 0 0.00035 0 0 0
1 2 UNIT SHEAR 0 0 0 0 0 0 0
2 2 UNIT SHEAR 0 0 0 0 0 0 0
3 2 UNIT SHEAR 0.00042 0.0024 0 0.00244 0 0 0
4 2 UNIT SHEAR 0.00042 0.0024 0 0.00244 0 0 0
5 2 UNIT SHEAR 0.00042 0.00104 0 0.00113 0 0 0
0
Uq
Aseq
0.00001318
28250
7.268
rad
. 4
in
. 2 in
82
UCD
Master's Thesis Skewed Steel IGirders
Equivalent CrossFrame Stiffness Calculations
Calculations By: K. Dobbins
10/18/2013
Crossframe Type 2
in in plf plf plf kips
height btwn working pts width btwn working pts weight top chord weight bott chord weight diagonals weight connection plates Total weight
83.5 104
59.5
59.5
59.5 1.633 3.359
Nodal Displacements
Horizontal Vertical Resultant Rotational
Node L/C X (in) Y (in) z (in) (in) rX (rad) rY (rad) rZ (rad)
1 1UNIT LOAD COUPLE 0 0 0 0 0 0 0
2 1 UNIT LOAD COUPLE 0 0 0 0 0 0 0
3 1 UNIT LOAD COUPLE 0.0002 0.00025 0 0.00033 0 0 0
4 1 UNIT LOAD COUPLE 0.0002 0.00025 0 0.00033 0 0 0
5 1 UNIT LOAD COUPLE 0.0001 0.00006 0 0.00012 0 0 0
1 2 UNIT SHEAR 0 0 0 0 0 0 0
2 2 UNIT SHEAR 0 0 0 0 0 0 0
3 2 UNIT SHEAR 0.00013 0.00077 0 0.00078 0 0 0
4 2 UNIT SHEAR 0.00013 0.00077 0 0.00078 0 0 0
5 2 UNIT SHEAR 0.00013 0.00035 0 0.00037 0 0 0
0
Uq
Aseq
0.00000479
62510
16.557
rad
. 4
in
. 2 in
83
UCD
Master's Thesis Skewed Steel IGirders
Equivalent CrossFrame Stiffness Calculations
Calculations By: K. Dobbins
10/18/2013
Crossframe Type 3
in in plf plf plf kips
height btwn working pts width btwn working pts weight top chord weight bott chord weight diagonals weight connection plates Total weight
____85
127
26.5
____34
____34
0.726
1.713
Nodal Displacements
Horizontal Vertical Resultant Rotational
Node L/C X (in) Y (in) z (in) (in) rX (rad) rY (rad) rZ (rad)
1 1UNIT LOAD COUPLE 0 0 0 0 0 0 0
2 1 UNIT LOAD COUPLE 0 0 0 0 0 0 0
3 1 UNIT LOAD COUPLE 0.00056 0.00075 0 0.00093 0 0 0
4 1 UNIT LOAD COUPLE 0.00044 0.00075 0 0.00087 0 0 0
5 1 UNIT LOAD COUPLE 0.00022 0.00016 0 0.00027 0 0 0
1 2 UNIT SHEAR 0 0 0 0 0 0 0
2 2 UNIT SHEAR 0 0 0 0 0 0 0
3 2 UNIT SHEAR 0.00042 0.00194 0 0.00199 0 0 0
4 2 UNIT SHEAR 0.00033 0.00194 0 0.00197 0 0 0
5 2 UNIT SHEAR 0.00033 0.00081 0 0.00088 0 0 0
0
Uq
Aseq
0.00001176
31641
9.521
rad
. 4
in
. 2 in
84
UCD
Master's Thesis Skewed Steel IGirders
Equivalent CrossFrame Stiffness Calculations
Calculations By: K. Dobbins
10/18/2013
Crossframe Type 4 (End)
in in plf plf plf kips
height btwn working pts width btwn working pts weight top chord weight bott chord weight diagonals weight connection plates Total weight
______85
230.125
26.5
26.5
26.5 0.818 2.372
Nodal Displacements
Horizontal Vertical Resultant Rotational
Node L/C X (in) Y (in) Z (in) (in) rX (rad) rY (rad) rZ (rad)
1 1UNIT LOAD COUPLE 0 0 0 0 0 0 0
2 1 UNIT LOAD COUPLE 0 0 0 0 0 0 0
3 1 UNIT LOAD COUPLE 0.00102 0.00275 0 0.00294 0 0 0
4 1 UNIT LOAD COUPLE 0.00102 0.00275 0 0.00294 0 0 0
5 1 UNIT LOAD COUPLE 0.00051 0.00069 0 0.00086 0 0 0
1 2 UNIT SHEAR 0 0 0 0 0 0 0
2 2 UNIT SHEAR 0 0 0 0 0 0 0
3 2 UNIT SHEAR 0.00138 0.00918 0 0.00928 0 0 0
4 2 UNIT SHEAR 0.00138 0.00918 0 0.00928 0 0 0
5 2 UNIT SHEAR 0.00138 0.00366 0 0.00391 0 0 0
0
Uq
Aseq
0.00002400
28104
4.917
rad
. 4
in
. 2 in
85
UCD
Masters Thesis Geneva Road
Steel Plate Girder Design
By: Konlee Dobbins
10/7/2013
Steel Girder Layout and Section Properties
Overview
Skewed straight steel plate girder bridge Simple span bridge
Design Code: AASHTO LRFD Bridge Design Specifications, 5th edition, 2010
Live Loads: HL93 and Tandem as per AASHTO (no permit trucks considered)
Design Parameters Roadway width Barrier width Left Sidewalk Width Right Sidewalk Width Deck width
Number of Design Lanes Span 1 Length Haunch and Top Flange Assumed Avg Haunch Deck thickness
Overhang deck thickness at edge
FWS Asphalt overlay
sacrificial deck thickness
Design deck thickness
Barrier area
Barrier weight
Concrete strength, f'c
Reinf steel fy
Structural steel fy
Reinf Cone unit weight
Cone Unit weight for Ec
Asphalt unit weight
Steel unit weight
Es
Ec
n = Es/Ec Future ADTT
Layout
Number of girders Girder spacing Overhang overhang/spacing
Value
Unit
Comments
82
2.000
6.667
10.667
103.333
254.4375
4.5
8.5
8.5
3.43
0.5
8
4.667
0.7
____60
____50
0.15
0.145
0.14
0.49
29000
3644
8.0
2500
11.0625
7.417
0.67
ft ft ft ft ft integer part of roadway width/12'
ft Brg to Brg
in Bott of deck to bott of top flange, constant
in
in
in
in 40 psf
in
in
ft2
klf includes 0.05 klf for chainlink fence
ksi
ksi
ksi M270 Grade 50W
kef includes extra 0.005 for rebar
kef
kef
kef
ksi
ksi 33000*wcA1.5*(f'c)A0.5
trucks/day
ft
ft
86
UCD
Section Properties Section 1 at ends Section 2 between SI & S3 Section 3 at midspan
Steel eirder only
top flange width
top flange thickness
web thickness
web height
Bottom flange width
Bottom flange thickness
Girder depth
Girder Area
top flange cog, y
web cog, y
bottom flange cog, y
Girder COG from bott, y
Girder COG from top of deck, y
Major Moment of inertia, lx
Stop
Sbot
Top Flange Moment of Inertia, 1^ Bott Flange Moment of Inertia, lybf Minor Moment of Inertia, ly Torsional Constant, J Warping Constant, Cw Shear Modulus, G P
Masters Thesis Geneva Road
Steel Plate Girder Design
By: Konlee Dobbins
10/7/2013
SI S2 S3
30 30 30
1.5 1.75 1.75
0.75 0.75 0.75
105 105 105
32 32 32
1.75 2 2.25
108.25 108.75 109
179.75 195.25 203.25
107.5 107.875 108.125
54.25 54.5 54.75
0.875 1 1.125
50.95 51.32 49.54
65.548 65.685 67.709
357558 403157 423492
6240 7019 7122
7017 7856 8548
3375 3938 3938
4779 5461 6144
8157 9403 10085
105.7 153.7 189.9
22487732 26133499 27473512
11154 11154 11154
0.001344 0.001504 0.001630
D/tw<150
140.0
EA*y/SA
87
UCD
Masters Thesis Geneva Road
Steel Plate Girder Design
By: Konlee Dobbins
10/7/2013
Eauivalent Torsional Constant SI S2 S3
Brace length 1 Lb 25 N/A N/A in
Equivalent Torsional Constant, Jeq 1122714 N/A N/A in4
Brace length 2 Lb 272 N/A N/A in
Equivalent Torsional Constant, Jeq 9610 N/A N/A . 4 in
Brace length 3 Lb 247 247 247 in
Equivalent Torsional Constant, Jeq 11627 13549 14278 in4
Brace length 4 Lb 240 240 240 in
Equivalent Torsional Constant, Jeq 12308 14340 15109 . 4 in
Brace length 5 Lb 227 227 227 in
Equivalent Torsional Constant, Jeq 13743 16008 16863 in4
Brace length 6 Lb 293 293 293 in
Equivalent Torsional Constant, Jeq 8299 9682 10212 . 4 in
Brace length 7 Lb 248 248 248 in
Equivalent Torsional Constant, Jeq 11534 13442 14165 in4
Brace length 8 Lb 245 245 245 in
Equivalent Torsional Constant, Jeq 11816 13768 14508 . 4 in
Brace length 9 Lb 251 251 251 in
Equivalent Torsional Constant, Jeq 11263 13126 13833 in4
Brace length 10 Lb 276 276 276 in
Equivalent Torsional Constant, Jeq 9337 10888 11480 . 4 in
Brace length 11 Lb 45 N/A N/A in
Equivalent Torsional Constant, Jeq 346604 N/A N/A in4
Brace length 12 Lb 41 N/A N/A in
Equivalent Torsional Constant, Jeq 417508 N/A N/A . 4 in
Brace length 13 Lb 37 N/A N/A in
Equivalent Torsional Constant, Jeq 512630 N/A N/A in4
Brace length 14 Lb 33 N/A N/A in
Equivalent Torsional Constant, Jeq 644403 N/A N/A . 4 in
Brace length 15 Lb 29 N/A N/A in
Equivalent Torsional Constant, Jeq 834392 N/A N/A in4
88

Full Text 
PAGE 1
BEHAVIOR AND ANALYSIS OF HIGHLY SKEWED STEEL IGIRDER BRIDGES by KONLEE BAXTER DOBBINS B.S., University of Virginia, 2001 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Civil Engineering 2013
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ii This thesis for the Mast er of Science degree by Konlee Baxter Dobbins has been approved for the Department of Civil Engineering by Kevin Rens, Chair Cheng Yu Li Yail Jimmy Kim November 12, 2013
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iii Dobbins, Konlee B. (M.S., Civil Engineering) Behavior and Analysis of Highl y Skewed Steel IGirder Bridges Thesis directed by Professor Kevin Rens ABSTRACT Skewed bridge supports for steel Igirde r bridges, introduce co mplexities to the behavior of the girder system that can be difficult to accurately model and analyze. In addition there have been some reported s hortfalls in the 2Dg rid analysis method typically used by engineers to design steel gi rder bridges with signi ficant skews. Some improvements have been suggested to bridge the gap in the inaccura cies of 2Dgrid and 2Dframe analyses. These improvements include overwriting the girder torsional stiffness to include warping effects, overwri ting the equivalent beam stiffness of crossframes using a more accurate method of calculating the stiffness, including lockedin crossframe forces due to dead load fit deta iling, and more accurately calculating flange lateral bending stresses with staggered crossframe layouts. This thesis examines these improvements, compares different levels of analysis, provides recommendations for these methods of analysis, and expl ains the behavior of the gi rder system during erection. The form and content of this abstract are approved. I recommend its publication. Approved: Kevin Rens
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iv ACKNOWLEDGMENTS I would like to thank my committee memb ers Dr. Kevin Rens, Dr. Cheng Yu Li, and Dr. Jimmy Kim for reviewing my work. I would like to thank Parsons for funding my masterÂ’s degree. I would also lik e to thank my coworkers who provided encouragement and guidance, especially Steve Haines, with his help providing information about the Geneva Road Bridge pr oject. And most of all, I would like to thank my family and in particular my wife, Kirsi Petersen, for her patience and encouragement during the long nights and weekends spent away from the family.
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v TABLE OF CONTENTS CHAPTER I. INTRODUCTION .............................................................................................1 Effects of Skew ..................................................................................................1 Continuation of Previous Thesis ........................................................................2 II. LITERATURE REVIEW ..................................................................................4 Introduction ........................................................................................................4 Relevant Documents ..........................................................................................5 III. THEORETICAL BACKGROUND .................................................................10 Suggestions to Simplify Structure Geometry in Skewed Bridges ...................10 Framing Plan Â– CrossFrame Layout ...............................................................11 Rotations and Deflections ................................................................................12 Detailing Â– NLF vs. SDLF vs. TDLF ..............................................................14 Analysis Methods.............................................................................................17 Improvements to 2D Modeling ........................................................................19 Preferred Analysis Method for Straight Skewed Girders ................................28 IV. ANALYTICAL PLAN ....................................................................................33 Example Bridge Description ............................................................................33 Analysis Models...............................................................................................40 V. ANALYTICAL RESULTS Â– COMPARING MODELS ...............................47
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vi VI. CONCLUSIONS .............................................................................................75 Recommended Method of Analysis .................................................................75 General Recommendations for Future Work ...................................................77 REFERENCES .................................................................................................................80 APPENDIX A ................................................................................................................... 81
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vii LIST OF FIGURES Figure III.1 Typical Fitup Procedure for Skewed IGirders. ..................................................... 16 III.2 Lateral Bending Moment, Ml, in a Fl ange Segment Under Simply Supported and FixedEnd Conditions. .............................................................................................. 24 III.3 Conceptual Configurations Associat ed with Dead Load Fit (TDLF or SDLF) Detailing .................................................................................................................... 26 III.4 Matrix of Grades for Recommended Le vel of Analysis for IGirder Bridges ......... 31 IV.1 Typical Section of th e Geneva Road Bridge ........................................................... 34 IV.2 Layout of the Geneva Road Bridge ......................................................................... 35 IV.3 Elevation Layout of the Geneva Road Bridge ......................................................... 36 IV.4 Girder Elevation of the Geneva Road Bridge .......................................................... 38 IV.5 Framing Plan of the Geneva Road Bridge ............................................................... 39 IV.6 Underside of the Geneva Road Bridge .................................................................... 40 V.1 MajorAxis Bending Stress of Girder 1 Top Flange ............................................... 49 V.2 Axis Bending Stress of Girder 1 Bottom Flange ...................................................... 49 V.3 Vertical Girder Displacements Along Girder 1 and Girder 3 ................................... 50 V.4 CrossFrame Axial Forces Along Ba y 2 Between Girder 2 and Girder 3 ................ 51 V.5 MajorAxis Bending Stress of Girder 1 .................................................................... 56 V.6 Vertical Girder Displacements Along Girder 1 and Girder 3 ................................... 57 V.7 CrossFrame Axial Forces Along Ba y 2 Between Girder 2 and Girder 3 ................ 58 V.8 Flange Lateral Bending Stress .................................................................................. 62
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viii V.9 Girder Layover at Bearings ....................................................................................... 65 V.10 MajorAxis Bending Stress of Girder 1 .................................................................. 66 V.11 Vertical Girder Displacements along Girder 1 and Girder 3 .................................. 67 V.12 CrossFrame Axial Forces Along Ba y 2 Between Girder 2 and Girder 3 .............. 68 V.13 Flange Lateral Bendi ng Stress Along Girder 1 ....................................................... 71 V.14 Girder Layover at Start and End Bearings for 3D Models ..................................... 73 V.15 Girder Layover at Start and End Beari ngs for 3D TDLF Models and from Field Data .......................................................................................................................... 74
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1 CHAPTER I INTRODUCTION Effects of Skews Skewed bridge supports and horizontal curv ature in steel Igirder bridges exhibit torsional forces that can introduce unexpected stress, displacements, and rotations during construction. As the skew angle or degree of curvature increases, the difficulty of constructing steel Igirder bri dges increases. The sequence of erection and assumptions made during fabrication can introduce forces and deflections that were not accounted for during the design. In many cases, these forces and deflections are ne gligible; however, in some cases they can be significant and unaccounted for if following todayÂ’s standard design practice and codes. Many of todayÂ’s more commonly used structural software take into account the effects of horizontal cu rvature on steel superstr uctures. However, accurately capturing the effects of skewed supports seems to be lacking in these software (NCHRP, 2012). Many reports and research papers lump the effects of horizontal curvature and skews together and tend to provide all encomp assing guidelines that address both aspects. Many of the effects of horizon tal curvature and skews are similar in nature; however, they can act in opposite directions or in different locations an d affect the design differently. Therefore, it is important to unde rstand the effects of each separately. This thesis will focus on the effects of skews only. Skews at bridge supports alter the behavior of girders. Historically, skews were avoided whenever possible because the eff ects were not well understood. Over time, advances in structural analys is and results from case studie s have made the effects a bit
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2 clearer. One report in particular, Nati onal Cooperative Highway Research Program (NCHRP) Report 725 Â“Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder BridgesÂ”, has taken great strides in highlighting the shortcomings of todayÂ’s standard practice, specifications, and guidelines for highly skewed steel Igirder bridges. More is discussed on these shortcomings and how to accurately account for them in the Lite rature Review and Theoretical Background sections. Continuation of Previous Thesis This thesis follows up and expands on a fellow University of Colorado Denver graduate studentÂ’s thesis Â“Crossframe Anal ysis of HighlySkewed and Curved Steel IGirder BridgesÂ” that touched on a variety of similar topics an d provided a case study example. That thesis focused on crossfram e design by looking at different framing plan and crossframe configurations (xframe vs. kfr ame) to find the most efficient design. It also included some background research, a literature review theoretical background, and analysis of crossframes in highly skewed and curved st eel Igirder example bridges (Schaefer, 2012). The theoretical background is predominan tly based on the American Association of State Highway Transportation Official s (AASHTO) and Nati onal Steel Bridge Alliance (NSBA) Steel Bridge Collaborat ion Document G13.1, Guidelines for Steel Girder Bridge Analysis. That thesis is a good source for background information on crossframe types, framing plan configura tions, and specification requirements from AASHTO and can be used as a precursor. It also provides a list of several curved and/or
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3 skewed bridges, with framing plans and member sizes included for each bridge listed, in the Denver, CO metro area (Schaefer, 2012). The conclusions that can be taken from Â“Crossframe Analysis of HighlySkewed and Curved Steel IGir der BridgesÂ” include: Staggered crossframe configurations induced the least amount of forces within its crossframes. Contiguous crossframe configurations indu ced the most forces within its crossframes. A stiffer transverse system will accumulate more force than a flexible system. The Kframe type crossframe performs better than the Xframe. The diagonal members in a Kframe crossframe absorb significantly less force than the diagonal members in an Xframe. The double angle and WTmembers are less slender, more flexible, and thus attract fewer loads than single angles that have to meet slenderness requirements (Schaefer, 2012). This thesis focuses on the effects that displacements and detailing have on the design of highly skewed steel Igirders and th e most accurate design methods that should be used with common steel Ig irder structural software.
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4 CHAPTER II LITERATURE REVIEW Introduction The literature review in Â“Crossframe Analysis of HighlySkewed and Curved Steel IGirder BridgesÂ” includes the history of design specific ations that contributed to todayÂ’s codes and standard practice for the desi gn of skewed or horiz ontally curved steel girders. The list includes: AASHTO Guide Specifications for Horizont ally Curved Steel Girder Highway Bridges, 1980 NCHRP Project 1238, 1993 AASHTO Guide Specifications for Horizont ally Curved Steel Girder Highway Bridges, 1993 NCHRP Project 1252, 1999 AASHTO Guide Specifications for Horizont ally Curved Steel Girder Highway Bridges, 2003 AASHTO/NSBA Â– G13.1 Guidelines for Steel Girder Bridge Analysis 1st Edition, 2011 One very important document missing from the list is from the research of NCHRP Project 1279, Â“NCHRP Report 725 Â– Guidelines for Analysis Methods and Construction Engineering of Curved and Sk ewed Steel Girder Bridges.Â” NCHRP Report 725 points out several deficiencies in the latest AASHTO Load a nd Resistance Factor Design (LRFD) Bridge Design Specifications the latest guidelines (AASHTO/NSBA Â–
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5 G13.1 Guidelines for Steel Girder Bridge Anal ysis 1st Edition), and standard practices assumed with the most commonly used 1D and 2D analysis software. Relevant Documents The following literature review and th eoretical background focuses on the G13.1 Guidelines and NCHRP Report 725, while briefl y discussing contributions from other research papers. AASHTO/NSBA Â– G13.1 Guidelines for Steel Girder Bridge Analysis In the Forward of this document, it states Â“the document is intended only to be a guideline, and only offers suggestions, insi ghts, and recommendations but few, if any, Â‘rules.Â’Â” The purpose of the document is to provide engineers, particularly less experienced designers, with gui dance on various issues relate d to the analysis of common steel girder bridges. The document focuses on presenting the various methods available for analysis of steel girder bridges and highlighting the advantages, disadvantages, nuances, and variations in the results. Th e guidelines are, to a certain extent, allencompassing for steel girder bridges, while briefly discussing the effects of different variations such as skews and horizontal curvature. The ge neral behavior and suggested analysis methods are discussed; however, it does not go into great detail. At the time this document was released there had been very few guideline resources for the design of skewed and horizontally curved steel gird ers and their corresp onding crossframes. The contents include: 1. Modeling descriptions 2. History of steel bridge analysis
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6 3. Issues, objectives, and guid elines common to all steel girder bridge analyses 4. Analysis guidelines for specific types of steel girder bridges Of particular interest are the sections on skewed bridges. These sections include information on the behavior, constructability analysis issues, predicted deflections, detailing of crossframes and girders for the intended erected position, crossframe modeling in 2D, geometry considerations, a nd analysis guidelines for skewed steel Igirder bridges (AAS HTO/NSBA, 2011). NCHRP Report 725 Â– Guidelines for An alysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges This report contains guidelines on the a ppropriate level of analysis needed to determine the constructability and construc ted geometry of curved and skewed steel girder bridges. The report also introduces improvements to 1D and 2D analysis that require little additional computational costs. The research for this report was performed under NCHRP Project 1279. The objectiv es and scope of NCHRP Project 1279 include: 1. An extensive evaluation of when simplif ied 1D or 2D analysis methods are sufficient and when 3D methods may be more appropriate. 2. A guidelines document providing recomme ndations on the level of construction analysis, plan detail, and submittals suitable for direct incorporation into specifications or guidelines. Of particular interest for this thesis ar e the sections pointing out the deficiencies of 1D or 2D analysis used in standard practice and the proposed improvements for analyzing skewed bridges. The report fo cuses on problems that can occur during, or
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7 related to, the construction. The key construc tion engineering considerations for skewed steel girder bridges include: 1. The prediction of the deflected geometry at the intermediate and final stages of the construction, 2. Determination and assessment of cases wh ere the stability of a structure or unit needs to be addressed, 3. Identification and alleviati on of situations where fitup may be difficult during the erection of the structural steel, and 4. Estimation of component internal stresses during the construction and in the final constructed configuration. AASHTO LRFD Bridge Design Specification s, Customary U.S. Units, 6th Edition (2012) This specification is used in every st ate throughout the United States as the national standard that engin eers are required to follow for bridge design. Many states include their own amendments to this specifica tion and additional guideli nes, but itÂ’s still the standard that the nationÂ’s bridge designs are based upon. The specifications have also been adopted or referenced by other bridgeown ing authorities and agencies in the United States and abroad. Since its first public ation in 1931, the theory and practice have evolved greatly resulting in 17 editions of the Standard Specifications for Highway Bridges with the last edition appear ing in 2002 and six editions to date of the loadandresistance factor design (LRFD) specifications (AASHTO, 2012). As the national standard, the specific ations are a bit lacking in providing requirements or guidance for designing highly skew ed bridges. In Section 4 Â– Structural
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8 Analysis and Evaluation, equations are provi ded to adjust the live load distribution factors for moment and shears using approximate methods of analysis. The approximate method of analysis involves line girder or 1D analysis of Â“t ypicalÂ” bridges within a set range of applicability for gi rder design. Section 6 Â– Steel Structures, includes commentary on the effects that skews have on girder and crossframe deflections, rotations, and potential additional stresses. However, in many cases, it recommends performing a more refined analysis to more accurately capture the effects of skews and leaves a fair amount to engi neering judgment to decide when a refined analysis is necessary and to what amount of deta il. The AASHTO/NSBA G13.1 Guidelines, NCHRP Report 725, and several othe r reports and research pape rs help bridge that gap and provide more guidance. Other Reports There are many more research reports, pr esentations, and shor t articles on the effects of skews on steel Igirder bridges and experiences during construction. The authors include structural engi neers, professors, fabricators, and construction managers. Several of these authors also contributed to NCHRP Report 725. Some articles, such as Â“Design and Construction of Curved and Se verely Skewed Steel IGirder EastWest Connector Bridge over I88Â”, describe the challenges and lessons learned during the design and construction of a sp ecific bridge. In the presen tation Â“Erection of Skewed Bridges: Keys to an Effective ProjectÂ”, the chief engineer for High Steel Structures Inc., presents three case studies of highly skewed st eel girder bridges and the experiences from the point of view of the fabricator. The presence of large skews and the assumptions made on fitup detailing during erection affect all stages of design and construction.
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9 Engineers, fabricators, and contractors all need to understand the movements, forces required for fitup, and correspond ing lockedin stresses that occur during different stages of construction.
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10 CHAPTER III THEORETICAL BACKGROUND Suggestions to Simplify Structu re Geometry in Skewed Bridges Skews present complexities in design, detailing, fabrication, and erection that translate into increased costs for steel gi rder bridges. As per NSBA/AASHTO Steel Bridge Collaboration, skew angles should be eliminated or reduced wherever possible. The bridge designer should work closely with the roadway designer to improve and simplify roadway alignments. Once the alignment is set, a few suggestions for eliminating or reducing skews include: Lengthening spans to locate the abutme nts far enough from the roadways below to allow for the use of radial abutment s or bents while maintaining adequate horizontal clearance. Designers should cons ider the cost of a longer span versus the cost associated with the comp lications of skew in the bridge. Retaining walls may allow the use of a radial abutment in place of a header slope. Typically these walls are of variable height and require oddshaped slope protection behind the wall. Designers should consider the cost of the walls versus the cost associated with the comp lications of skew in the bridge. Use integral radial interior bent instead of a skewed traditional bent cap to maintain adequate vertical clearance in cases where a traditional radial bent would have insufficient vertical clearance and wh ere the vertical prof ile of the bridge cannot be raised.
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11 Use dapped girder ends with invertedtee bent caps to ma intain adequate vertical clearance at expansion jo int locations instead of an integral bent cap (AASHTO/NSBA 2011). In many cases, highly skewed supports cannot be avoided for a number of reasons. Typically, geometry constraints in highly congested highway interchanges leave very little wiggle room to eliminate or redu ce large skews. Where large skews cannot be avoided, design engineers, detailers, fabricator s, and contractors all need to understand the stresses and deflections that occur during different stages of construction. Framing Plan Â– CrossFrame Layout Crossframes or diaphragms should be placed at bearing lines that resist lateral force. Wind loads and other lateral forces are transferred from the deck and girders through the crossframes at suppor ts to the bearings and down to the substructure. As per AASHTO LRFD Bridge Design Specifications 6.7.4.2 Â– Diaphragms and CrossFrames for ISection Members, crossframes at supports can either be placed along the skew or perpendicular to the girder: Where support lines are skewed more than 20 degrees from normal, intermediate diaphragms or crossframes shall be nor mal to the girders and may be placed in contiguous or discontinuous lines. Where a support line at an interior pier is skewed more than 20 degrees from normal, elimination of the diaphragms or crossframes along the skewed interior support line may be considered at the discretion of the Owner. Where discontinuous intermediate diaphragm or crossframe lines are employed normal to the girders in the vicinity of that s upport line, a skewed or normal diaphragm or crossframe should be matched with each bearing that resists lateral force (AASHTO, 2012). As research has shown, placing a crossfr ame normal to the girders and at the bearing location of a skewed support, provide s an alternate load path and attracts a
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12 significant amount of force in that crossframe. NCHRP Report 725 referred to these crossframes at or near the supports as pr oviding Â“nuisance stiffne ssÂ” transverse load paths and should be avoided if possible. Th erefore, standard pract ice is to provide a crossframe along the skew at the supports. Staggered crossframe layout configurations appear to induce the least amount of forces within its crossframes. This is esp ecially true and desira ble for interior crossframes closest to highly skewed supports Staggered crossframes allow for more flexibility in the system and therefore attract less load. Al so, Kframe type crossframes tend to be the better choice over Xframe type crossframes (Schaefer, 2012). Rotations and Deflections The root of the complications due to sk ew is the out of plane rotations and deflections at the skewed supports that cause twisting in the girders as vertical loads are applied. In 1D line girder analysis, the effects of a sk ew are not captured. When analyzing a single girder in a single span, as vertical loads (such as dead loads from the steel selfweight, concrete deck weight, and miscellaneous superimposed dead loads and live loads from vehicular traffic) are applied, the girder deflects downward with the max deflection occurring at midspan. There are, theoretically, no lateral deflections or twisting. However, as crossframes are att ached connecting skewed girders together, the girders start to twist near the supports. The differentia l deflection betwee n two adjacent girders causes a twisting motion, also known as layover. This twisting motion can be counterbalanced by specifying certa in detailing that essentiall y forces the girders to be twisted in the opposite direction when connecting the cros sframes to the girders during erection. This is known as Steel Dead Load Fit (SDLF) and Total Dead Load Fit (TDLF)
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13 detailing, where the goal is to have verti cally plumb girder webs at the specified construction stage. This will be discussed in more detail in the next section. Per the AASHTO/NSBA Steel Brid ge Collaboration Document G12.1 Â“Guidelines for Design ConstructabilityÂ”: The problem for crossframes at skewed pi ers or abutments is the rotation of the girders at those locations. In a square bridge, rotation of the girders at the bearings is in the same plane as the gi rder web. If supports are skewed, girder rotation due to noncomposite loads will be normal to the piers or abutments. This rotation displaces the top flange transversely from the bottom flange and causes the web to be out of plumb (AASHTO/NSBA, 2003). Where end crossframes are skewed parall el to the support, which is typically standard practice, these end crossframes contribute to the rotations and transverse movements described above. The end crossfram es are very stiff in the axial direction along the skewed support and fl exible in the weak axis direction, which allow these rotations normal to the support. The movements of simple span strai ght girders on nonskewed supports are predictably uniform. With downward deflecti on between supports due to vertical dead loads, the top flange compresse s. At the supports, the top flange deflects toward midspan. Conversely, the bottom flange is in tens ion and deflects away from midspan at the expansion supports that are free to move longitudinally. The e nds of girders also rotate due to the length changes in the flanges. For girders on skewed supports the movement becomes more complex by adding transverse deflections and twisting rotations. The rotation normal to the pier as describe d in AASHTO/NSBA G12.1 is a bit of a generalization and is true for bearings that ar e at the same elevation at the given support. If the bearing elevations diffe r along the given support, the axis of rotation will be in the
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14 plane including the actual cen terline of bearing, but slope to intersect the centers of rotation at adjacent bearings. This descri bes the theoretical m ovements. Actual movements will vary slightly since the member s are framed together and restrained by the deck and bearings, therefore some dist ortions will result (Beckmann and Medlock, n.d.). The transverse movements a nd twisting causes the ends of girders to be out of plumb as vertical loads are applied if the girders are not detailed to counteract these movements during construction. For more detailed information on rotations and deflections in skewed steel girder bridge s, the article Â“Skewed Bridges and Girder Movements Due to Rotations and Diffe rential DeflectionsÂ” is recommended. Detailing Â– NLF vs SDLF vs TDLF As the skew angle increases, the transv erse flange movement increases. For strength, serviceability, and aesthetic reasons, it is typically desirable to detail the girders with sizeable skews to count eract these girder end moveme nts and be plumb at certain dead load cases. However, each bridge n eeds to be evaluated for several factors, including constructability a nd girder design at different stages of construction, to determine the most economic design. Fabric ation and construction must follow the fitup condition assumed during the design of the girders and crossframes. Otherwise, unintended lockedin forces or movements th at were not considered during design can arise. The designer generally has three choices of conditions for which the girders and crossframes shall be designed:
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15 NoLoad Fit (NLF) condition Â– the girder webs are theoretically plumb/vertical before any load is applied. Steel Dead Load Fit (SDLF) condition Â– the girder webs are theoretically plumb/vertical under the steel dead after the crossframes are installed and before the concrete deck is poured. Total Dead Load Fit (TDLF) condition Â– the girder webs are theoretically plumb/vertical under the total dead lo ad in the final condition (Beckmann and Medlock, n.d.). Each detailing method affects deflected geometry, can create fitup issues, produce stability effects and s econdorder amplification, and affect component internal stresses during construction. Constructi on plans and submittals for these complex geometries with high skews n eed to clearly state the fitup method assumed during design and construction (NCHRP, 2012). In SDLF and TDLF detailing methods, the crossframes do not fitup with the connection work points on the initi ally fabricated girders. During fitup of crossframes with the girders, the girders are forced into place by twisting the girders. A girder is much more flexible twisting about its longitudinal axis th an a crossframe deforming axially. As the dead load is applied, the girders deflect and rota te back to plumb. AASHTO/NSBA G 12.1 Guidelines for Design Constructability describes the process of SDLF or TDLF fitup as seen in Figure III.1.
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16 Figure III.1 Typical Fitup Procedure for Skewed IGirders (AASHTO/NSBA, 2003)
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17 For the designer, the biggest concern is the presence of any unaccounted forces and correctly modeling the structure at differe nt stages of construction. SDLF and TDLF detailing introduces lockedin fo rces during erection when the girders are forced to fit up with the stiffer crossframes. In many cases especially for strai ght girders on skewed supports, lockedin forces are relieved as the dead loads are applied. However, it can be dangerous to assume that this occurs in all cases. For example, as with curved girders with radial/nonskewed supports, the lockedi n forces from fitup and forces due to differential deflections between adjacent girder s can be additive. Or in highly skewed straight bridges, if the first intermediate cro ssframes are too close to the bearing line, the lockedin crossframe forces near the acute corn ers tend to be additive with the dead load effects (NCHRP, 2012). Analysis Methods The level of detail for the girder and crossframe analys is is an important decision to make and is often left to engineering j udgment. 3D finite element analysis (FEA) provides the most accurate results when done co rrectly. However, it is by far the most complex and timeconsuming and with a large number of variables, it leaves a lot of room for error. 1D and 2D simplified analysis are much less timeconsuming and therefore preferred by engineers for the design of noncomplex structures. What constitutes a structure to be complex and where to draw the line is often a topic of debate among engineers. AASHTO LRFD Bridge De sign Specifications provide criteria for determining if using a simplified method of lin ear analysis is acceptable. When a refined method of analysis is require d or recommended, there are st ill a good number of methods to choose from including 2Dgrid and 3DFEA. It is ultimately left up to engineering
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18 judgment to choose an appropriate refined method of analysis and understand the basic assumptions and methodology of the software used (AASHTO, 2012). Even in cases where 1D or 2D methods of analysis are deemed acceptable, NCHRP Report 725 has made light of some assumptions that can tu rn out to be quite erroneous. NCHRP Report 725 has also exposed some assumptions typi cally made by most 1D or 2D analysis software that can significantly alter the resu lts. The following sect ions provide a brief overview of the different methods of analysis. 1D Â– Line Girder Analysis Method Line girder analysis, as the name suggests isolates and analyzes one single girder line. Loads are distributed to each girder by wa y of distribution factor s. Effects on girder moments and shear from skews no greater than 60 degrees are accounted for with additional factors in AASHTO LRFD Bridge Specifications. The effects of the crossframes are not taken into account. This method is adequate for fairly simple structures with little to no skew angle. 2D Â– Grid Analysis Method In plan grid or grillage analysis, the stru cture is divided into plan grid elements with three degrees of freedom at each node. This method is most often used in steel bridge design and analysis (AASHTO/NSBA, 2011). The eff ects of the crossframes are taken into account; however, most common 2D software, such as DESCUS and MDX, use equivalent beam element properties when modeling the crossframes. As discussed in NCHRP Report 725, how these common 2D so ftware compute the equivalent beam element properties for the crossframes and the equivalent torsional c onstant properties of the girders, isnÂ’t typically accu rate especially in cases of high skews or high degrees of
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19 horizontal curvature. These inaccuracies a nd how to account for them will be explained in greater detail in later sections. As the most commonly used method of analysis, it is vital to keep the inaccu racies to a minimum. 3D Â– Finite Element Analysis Method In the 3DFEA method, the bridge superstr ucture is fully modeled in all three dimensions. The model typically includes mode ling the girder flanges as beam elements or plate/shell elements; modeling the web as plate/shell elements; modeling each member of the crossframes as beam or truss elem ents; and modeling the deck as plate/shell elements. This method is arguably the most accurate; however, it is typically very timeconsuming and complicated. Therefore, it is mostly only used for very complex structures or for performing re fined local stress analysis of a complex detail. There are other complicating factors, such as the out put reporting the stresses in each element instead of moments and shears that the engineer typically checks against the required limits in AASHTO or local state specifications The engineer would need to convert the stresses into moments and shears if so desire d. When and how to use refined 3D finite element analysis is a controversial issue, a nd this method has not been fully incorporated into the AASHTO specifications to date (AASHTO/NSBA, 2011). Improvements to 2D Modeling CrossFrame Modeling Most designers use the methods descri bed in the AASHTO/ NSBA (2011) G13.1 document for finding the equivalent beam sti ffness of crossframes in 2D analysis models. There are two approaches here:
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20 1. Calculate the equivalent mo ment of inertia based on th e flexural analogy method. In a model of the crossframe, a unit force couple is applied to one end to find the equivalent rotation that is then used to backcalculate the equivalent moment of inertia. 2. Calculate the equivalent mo ment of inertia based on the shear analogy method. In a model of the crossframe, a unit vertical force is applied to one end to find the equivalent deflection that is then used to backcalculate the equivalent moment of inertia (AASHTO/NSBA, 2011). Both methods use EulerBernoulli beam th eory equations. The issue with using one of these methods is the flexural analogy method only accounts for the flexural stiffness, while the shear analogy only accounts fo r shear stiffness. In cases where either the flexure or shear is considered neglig ible, using the approp riate method above is acceptable. However, in cases where both fle xure and shear are present, the equivalent moment of inertia should account for both fle xural and shear stiffness. Differential deflection of adjacent girders might primarily engage the shear stiffness of the crossframes, while differential rotation of adjacent girders might be more likely to engage the flexural stiffness of the crossframes (AASHTO/ NSBA, 2011). NCHRP Report 725 recommends a more accu rate approach for calculating the crossframe equivalent beam stiffness. This approach includes an equivalent shear area for a sheardeformable beam element repres entation (Timoshenko beam theory) of the crossframe. In the report, it compares the equivalent stiffness results from the flexural analogy method, shear analogy method, pure bending (Timoshenko) method, and 3DFEA calibrated to a test bridge and finds that the pure bending (Timoshenko) method
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21 provides the most accurate overall results. This is due to the fact that the Timoshenko beam theory element is able to represen t both flexure and shear deformations. In the pure bending (Timoshenko) method, th e equivalent moment of inertia is determined first based on pure flexural defo rmation. This is similar to the flexural analogy method except that the constraints ar e modeled differently and the corresponding end rotation is equated from the beam pure flexure solution M/(EIeq/L) versus the EulerBernoulli beam rotation equation M/(4EIeq/L) used in the flexural analogy method. This results in a substantially larger equi valent moment of inertia and that EIeq represents the Â“trueÂ” flexural rigidity of the crossframe. The crossframe is supported as a cantilever at one end and is subjected to a force couple at the other end, produci ng a constant bending moment and corresponding end rotation. In the second step of this method, the crossframe is still supported as a cantilever but is s ubjected to a unit transver se load at its tip. The Timoshenko beam equation for the transverse displacement is: which is used to find the equi valent shear area (NCHRP, 2012). As per NCHRP Report 725, the Timoshenko beam element provides a closer approximation of the physical crossframe behavior compared to the EulerBernoulli beam for all types of crossframes (incl uding X and K type crossframes) that are typically used in Igir der bridges. Not only are the cal culated forces more accurate but the deflections and rotations are more accura te. Predicting deflections and rotations
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22 during construction becomes much more impor tant as skew angles increase (NCHRP, 2012). The fabricator can more accurately fabric ate the girders for th e appropriate final orientation and fitup method. The cont ractor more accurately understands the deflections and rotations to expect during c onstruction and the forces necessary for the chosen fitup method. The engineer can more accurately and efficiently design the girders and crossframes for the expected movements and lockedin forces from fitup and final condition loads. IGirder Torsion Modeling Current practice in 2Dgrid models subs tantially underestimates the girder torsional stiffness. This is due to software only considering St. Venant torsional stiffness of the girders while neglecting warping torsio nal stiffness. This practice tends to discount the significant transverse load paths in highly skewed bridges, since the girders are so torsionally soft that they are unable to accept any significant load from the crossframes causing torsion in the girders. As a result, the crossframe forces can be significantly underestim ated (NCHRP, 2012). NCHRP Report 725 provides some equations to calculate an equivalent torsional constant, Jeq that includes both the St. Venant and warping torsiona l stiffness. It should be noted that these equations were based in part on prior research developments by Ahmed and Weisgerber (1996), as well as the commercial implementation of this type of capability within the software RISA3D. In this approach, an equivalent torsional constant must be calculated for each unbraced length and girder sectional property. The equation for the equivalent torsion consta nt for the opensection thinwalled beam
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23 associated with warping fixity as each end of a given unbraced length (crossframe spacing) is: Where Lb is the unbraced length between the crossframes, J is the St. Venant torsional constant, and p2 is defined as GJ/ECw. Assuming warping fixity at the intermediate crossframe locations leads to a reasonably accurate characterization of the girder torsional stiffness (NCHRP, 2012). IGirder Flange Lateral Bending Modeling AASHTO LRFD Bridge Specifications section C4.6.1.2.4b provides a simplified equation to calculate the lateral moment fo r a horizontally curved girder based on the radius, majoraxis bending moment, unbraced length, and web depth. For other conditions that produce torsion, such as sk ew, AASHTO suggests other analytical means which generally involve a refi ned analysis. However, Se ction C6.10.1 provides a coarse estimate by stating: The intent of the Article 6.10 provisions is to permit the Engineer to consider flange lateral bending effects in the desi gn in a direct and rational manner should they be judged to be significant. In absence of calculated values of fl from a refined analysis, a suggested es timate for the total unfactored fl in a flange at a crossframe or diaphragm due to the use of discontinuous crossframe or diaphragm lines is 10.0 ksi for interior gi rders and 7.5 ksi for exterior girders. These estimates are based on a limited examin ation of refined analysis results for bridges with skews approaching 60 degr ees from normal and an average D/bf ratio of approximately 4.0. In regions of the girders with contiguous crossframes or diaphragms, these values need not be c onsidered. Lateral flange bending in the exterior girders is substantially reduced when crossframes or diaphragms are placed in discontinuous lines over the entire bridge due to the reduced crossframe or diaphragm forces. A value of 2.0 ksi is suggested for fl for the exterior girders in such cases, with the suggested value of 10 ksi retained for the interior girders. In all cases, it is suggested that the recommended values of fl be proportioned to dead and live load in the same propor tion as the unfactored majoraxis dead and
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24 live load stresses at the section under c onsideration. An examination of crossframe or diaphragm forces is also consid ered prudent in all bridges with skew angles exceeding 20 degrees (AASHTO, 2012). NCHRP Report 725 recommends a more accurate but simplified method of calculating lateral bending stress than the co arse estimates provided above. Their method includes a local calculation in the vicinity of each crossframe, utilizing the forces delivered to the flanges from the crossframes placed in discontinuous lines. The approximate calculation takes th e average of pinned and fixed end conditions as shown in Figure III.2 below. Figure III.2 Lateral Bending Moment, Ml in a Flange Segment Under Simply Supported and FixedEnd Conditions (NCHRP, 2012) Calculation of LockedIn Forces Due to CrossFrame Detailing Regardless the type of analysis used (2Dgrid, 2Dframe, or 3DFEA), the analysis essentially assumes a NLF condition unless the lock edin forces are accounted for in the model. Any lockin forces, due to the lack of fit of the crossframes with the girders in the undeformed geometry in SDLF or TDLF, add to or subtract from the forces determined from the analysis. Typically for st raight skewed bridges, the lockedin forces
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25 tend to be opposite in sign to the internal forc es due to dead loads. Therefore the 2Dgrid or 3DFEA analysis solutions for crossfram e forces and flange lateral bending stresses are conservative when SDLF or TDLF initial fitup forces are neglected. However, these solutions can be prohibitively conservativ e for highly skewed bridges (NCHRP, 2012). TDLF or SDLF detailing is first and fo remost a geometrical calculation for the detailer and fabricator Yet, they can significantly a ffect the lockedin crossframe forces. Figure III.3 shows four configurati ons that visually expl ain how the lockedin forces can be calculated. Configurations 1 and 4 are used by structural detailers. Configurations 2 and 3 are theo retical geometries that techni cally never take place in the physical bridge, but are used to calculate the internal locked in forces. The differential camber shown in Configuration 1 is detailed to counterbalance the eventual differential deflection that occurs unde r the corresponding dead load. This differential camber induces the twisting shown in Configuration 3 from the crossframes being forced into place and released. The deflections due to the twisting are approximately equal and opposite to the deflections at these locations under the corresponding total or steel dead load (NCHRP, 2012). For cases where the initial lackoffit effects are important, the designer can simply include an initial stress or strain sim ilar to a thermal stress or strain. Calculating the initial strains and stresses associated w ith SDLF or TDLF detailing of the crossframes involves finding the nodal displacemen ts between Configurations 2 and 4 and applying the corresponding stresses to the cros sframe ends. In 3DFEA, the calculated axial strains from the nodal displacements are converted into stresses simply by multiplying the strains by the elastic modulus of the material. The stresses are then
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26 multiplied by the crossframe member areas to determine the axial forces. In 2Dgrid models that use equivalent beam elements for the crossframes, the displacements calculated above are converted into beam end displacements and end rotations. Assuming fixedend conditions, the end displace ments are used to calculate the fixedend forces, which are then applied to the e quivalent crossframe beam element. Figure III.3 Conceptual Configurations A ssociated with Dead Load Fit (TDLF or SDLF) Detailing (NCHRP, 2012)
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27 Figure III.3 (Continued) Conceptual Configurations Associated with Dead Load Fit (TDLF or SDLF) Detailing (NCHRP, 2012)
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28 The behavior of the end crossframes at skewed bearing lines is slightly different, however the lockedin forces due to crossfram e fitup is calculated in the same manner following the configurations in Figure III.3. Th e girders cannot displ ace vertically at the bearings and the skewed crossframes impose a twist in the girder ends. The top flange of the girders at the bearing lin e can only displace significantly in the direction normal to the plane of the crossframe. In order for the skewed end crossframe to fit up with the girders in Configuration 2, the crossframe has to rotate about its l ongitudinal axis and be strained into position to connect them with the rotated connection plates in the initial cambered noload, plumb geometry of the girders (NCHRP, 2012). Again, this is a theoretical configuration that technically would not occur in the physical bridge. It is used to calculate the displacemen ts and the corresponding forces. Preferred Analysis Method for Straight Skewed Girders NCHRP Report 725 provides recommendations on the analysis and detailing method that should be used for various levels of skews and horizontal curvature. For straight skewed steel Igird er bridges, the recommendations are prominently based on the skew index, IS. The skew index is a measure of th e severity of the skew based on the skew angle, the span length, and the bridge width measured betw een fascia girders. Straight skewed Igirder bridges ar e divided into three groups: Low ( IS < 0.30), Moderate (0.30 IS < 0.65), and High ( IS 0.65). Bridges with a low skew index of less than 0.30 are not as sensitive to the e ffects of skews. As the skew index increases above 0.30, responses associated with lateral bending of the girder flanges becomes significant. At
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29 this point, the stress ra tio of flange lateral bending st ress over majoraxis bending stress ( fl/fb) increases above 0.30 where crossframes ar e staggered. This is considered a large flange bending effect. As the skew inde x increases into the High category above 0.65, the skew effects can significantly influence th e majoraxis bending responses. Below this level the vertical components of the forces from the crossframes are too small to noticeably influence the majoraxis be nding response (NCHRP Appendix C, 2012). NCHRP Report 725 provides a matrix of gr ades for traditional 2Dgrid and 1Dline girder analysis for several different levels of skew and horizontal curvature as seen in the Figure III.4. For straight skewed bridges with a high skew index (IS 0.65), 2Dgrid and 1Dline girder analysis receive really poo r grades. However, it should be noted that the recommended improvements to 2Dgrid anal ysis, as described in previous sections, dramatically improve grades and percentage of error, especially for solutions of crossframe forces and flange lateral bending stress es. The grades are based on the percentage of normalized mean error of the results for each structure response. The breakdown of grades include: A: 6% or less normalized mean e rror, reflecting excellent accuracy; B: between 7% and 12% normalized mean error, reflecting reasonable agreement; C: between 13% and 20% normalized mean error, reflecting significant deviation from the accurate benchmark; D: between 21% and 30% normalized mean error, reflecting poor accuracy; and
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30 F: over 30% normalized mean error, reflecting unreliable accuracy and inadequate for design (NCHRP, 2012). NCHRP Report 725 also provides recomm endations for crossframe detailing methods for straight skewed Igirder bridges based on the skew index. In general TDLF detailing is preferred in orde r to keep layover to a minimum and ensure the web is plumb in the final TDL condition. Layover is define d as the relative lateral deflection of the flanges from the twisting motion of the girders. For IS < 0.30, TDLF is typically the preferred option. The total dead load (TDL) crossframe fo rces and girder flange lateral bending stresses will essentially be canceled out by the TDLF lockedi n forces. With a low skew index level, the forces required for crossframe fitup during steel erection are very manageable. Ensuring that the first intermed iate crossframes are a minimum distance offset from centerline of bear ing, will help alleviate nuisanc e stiffness effects and reduce fitup forces by providing enough flexibility at th e end of girder to force the girders into position with the relatively stiff crossfr ames. The recommended minimum offset distance from the bearing centerline is: a max(1.5 D 0.4 b ) where D is the girder depth and b is the second unbraced length within the span from the bearing line (NCHRP, 2012).
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31 Figure III.4 Matrix of Grades for Reco mmended Level of Analysis for IGirder Bridges (NCHRP, 2012)
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32 For straight skewed Igirder bridge s with a higher skew index of IS > 0.30, TDLF, SDLF, or detailing between SDLF and TDLF are typically good options. As the skew index increases, the force required for cros sframe fitup increases and becomes much more difficult to erect. If SDLF detailing is used, excessive layover in the final TDL condition may become a concern for bridges wi th large skews and l ong spans. Besides TDLF crossframe detailing, layover can be addre ssed with the use of beveled sole plates and/or using bearings with a larg er rotational capacity (NCHRP, 2012).
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33 CHAPTER IV ANALYTICAL PLAN The analytical plan involves applyi ng the theories and recommendations discussed in the Theoretical Background chapter of this thesis towards an example bridge. Several different models of the exam ple bridge superstructu re were created and analyzed and then the results are compared. The models include a conventional 2Dgrid base model, an improved 2Dgrid model, a 2D frame base model, an improved 2Dframe model, a 3DFEA NLFdetailing model, a nd a 3DFEA TDLFdetailing model. The results for majoraxis bending stresses, vertic al displacements, crossframe forces, flange lateral bending stresses, and girder layover at be arings are all compared in the Analytical Results chapter. Example Bridge Description The Geneva Road Bridge in Utah was analy zed as the example bridge used in this thesis. The bridge is a part of the SR114 Geneva Road DesignBuild Project which was undertaken to improve travel between Pr ovo and Pleasant Grove, Utah. The project involved reconstruction and wi dening work of about four miles of SR114 and new construction of a bridge over the Union Paci fic Railroad and Utah Transit Authority tracks. Parsons served as the designer and teamed with the contra ctor, Kiewit, to design and build the bridge. The owner is th e Utah Department of Transportation. The Geneva Road Bridge has a 103Â’4Â” wide deck that includes four lanes of traffic (two in each direction), two 10Â’ s houlders, a 14Â’ median, and a sidewalk on each side. The single span bearing to bearing leng th is 254Â’5 Â”. The skew angle is almost 62 degrees and its skew index, IS = 0.65, puts it right on the edge of the most severe
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34 category as per NCHRP Report 725 and as seen in Figure III.4. There are nine steel plate Igirders spaced at 11Â’0 Â” on center and 7Â’5Â” overhangs. A ll structural steel conforms to AASHTO M 27050W, which is a weathering st eel with a yield stress of 50 ksi. See Figure IV.1 for the typical superstructure se ction and Figures IV.2 and IV.3 for the plan and elevation layouts. The deck overhangs appear to be a bit large compared to the girder spacing; however the plans explicitly state that the side walks shall never be converted to travelled lanes. The analysis in this thesis focuses on the behavior of the structure due to dead loads during construction; therefore the exterior girders appear to ta ke a larger amount of load in the analytical results. The relativel y small amount of live load due to pedestrian loads distributed to the exterior girder comp ared to the much larger vehicular live loads distributed to the interior girders, balances out the total end design lo ad among all girders. Typically, the preferred ratio of overhang length to girder spacing is between 0.3 and 0.5 for overhangs that could potentially see large vehicular live loads. Figure IV.1 Typical Section of th e Geneva Road Bridge (Parsons, 2011)
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Figure IV.2 Plan Layout of the Geneva Road Bridge (Parsons, 2011)
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36 Figure IV.3 Elevation Layout of th e Geneva Road Bridge (Parsons, 2011) The girder sections and lengths are the sa me for all nine steel plate Igirders. Girder 1 only differs by location of the splice; however, splice location is irrelevant for the purposes of this thesis. The web is consta nt at 105Â” x Â”. The top flange width is a constant 30Â” and the thickness varies from 1 Â” at the ends to 1 Â” at the middle section. The bottom flange width remains constant at 32Â” with a thickness that varies from 1 Â” at the ends to 2 Â” at midspan. See Figure IV.4 for Girder Elevations. The crossframes are Ktype crossframes with WT members for the bottom, top, and diagonal chords. The interior crossfram es are continuous where possible as seen in the framing plan in Figure IV.5. After an init ial analysis in the or iginal design, the first interior crossframe near each obtuse corner of the framing plan was removed. Those crossframes attracted a significant amount of lo ad due to the behavior of wide and highly skewed bridges tending to find an alternat e load path by spanni ng between the obtuse corners in addition to spanning along the centerline of the girders. These crossframes at
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37 or near the supports provide Â“ nuisance stiffnessÂ” transverse load paths especially at the obtuse corners (NCHRP, 2012). The idea to rem ove the first interior crossframe at the obtuse corners came from the article, Â“Des ign and Construction of the Curved and Severely Skewed Steel IGirder EastWest Connector Bridges over I88.Â” The article explains how nonskewed crossframes that frame directly into skewed supports provide alternate load paths and also refer to these effects as Â“nuisance stiffness.Â” These crossframes were removed to mitigate these effects (Chavel et al, 2010). The next few crossframes that are inl ine with the removed crossframes on the Geneva Road Bridge, still experienced signi ficant loads in the analysis and required larger member sizes. See the framing plan in Figure IV.5 for the location of the stiffer type 2 crossframes. As per re commendations from NCHRP Report 725, AASHTO/NSBA G13.1, and SchaeferÂ’s thesis all of which were published after the Geneva Road Bridge was designed, the cr ossframes could have been staggered (discontinuous) and pushed back a distance a max(1.5 D 0.4 b ) offset from the bearing line to the first interior crossframe in orde r to reduce the crossframe loads and mitigate nuisance stiffness effects. However, arrangi ng the crossframes in continuous lines could significantly reduce the latera l flange bending stresses. The Geneva Road Bridge has already been designed and constructed. The designers used the commonly used 2Dgrid st eel girder structural analysis software, MDX, for the majority of the superstructure analysis. The design has been checked and construction occurred without any issues that would have compromised the integrity of the structure. The bridge is open to traffic a nd there have been no repor ted issues to date.
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Figure IV.4 Girder Elevation of the Geneva Road Bridge (Parsons, 2011)
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39 Figure IV.5 Framing Plan of the Geneva Road Bridge (Parsons, 2011)
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40 The intent of using this bridge in this th esis isnÂ’t to recommend a better layout or a better design method but rather to gain a better understanding of the behavior of the girders and crossframes during constructi on. The method of construction is known and the behavior was witnessed with some reco rded field data, which helped verify the modeled behavior. Figure IV.6 Underside of the Geneva Road Bridge After precast panel and deck rebar inst allation and before the castinplace concrete deck pour (with permission from Kiewit) Analysis Models Six different analysis models were created and analyzed. The st eel girder analysis software, MDX, is used for two 2Dgrid models and the 3D structural analysis software,
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41 LARSA 4D with the Steel Bridge Module, is us ed for the other four models that include 2Dframe and 3DFEA. The results are comp ared in the next chapter Â– Analytical Results. As described in the Theoretical Background chapter, im provements to the 2D models include: Adjusting the equivalent beam s tiffness assumed for crossframes, Adjusting the torsional stiffness to include warping stiffness, and Calculating more accurate late ral flange bending stresses. All 2D and 3D models assume NLF detailing by default, meaning no initial lockedin crossframe forces are included in the anal ysis. The final improvement includes adding the lockedin crossframe forces due to TDLF detailing for the 3DFEA model. The theory behind calculating more accura te lateral flange bending stresses is based on assuming a staggered crossframe layout is used. Since the crossframes are continuous, the lateral flange bending stresses will not be calculated as per the outlined 2Dgrid improvements in the analysis of th e example bridge. This improvement would have been a postprocessing step and will co ntinue to be one unless 2Dgrid software companies choose to rewrite their code a nd implement it directly into the software. 2DGrid Base Model Â– MDX This model was used for the original de sign and is left unc hanged without any improvements implemented for comparison purp oses. In the MDX software program, the user runs through a wizard to input various geometric and load parameters. The user runs through five modes or input phases in the process of creating a girder system design model:
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42 1. Layout Mode Â– the user provides genera l layout information to establish the framing plan. 2. Preliminary Analysis Mode Â– th e user provides the loading. 3. Preliminary Design Mode Â– the user provi des design controls to be enforced on the generation of a set of girder designs based on the preliminary design forces. 4. Design Mode Â– the user defines the bracing and can generate bracing and girder designs after setting up certain parameters. 5. Rating Mode Â– this final mode is us ed for tuning the design (MDX, 2013). The output includes forces, stresses, and di splacement results for each girder and for the girder system that includes the crossframes. The results are also checked against the latest AASHTO br idge specifications. 2DGrid Improved Model Â– MDX This model includes any possible reco mmended improvements to a 2Dgrid analysis. The issue is, given th e constraints of the input wiza rd, thereÂ’s very little that can be manipulated to improve the analysis an d better represent the behavior of the girder and crossframe system. The software automatic ally calculates the torsional stiffness, J, based on the St. Venant pure torsional stiffne ss by using the section dimensions input. Warping stiffness is not included in the torsi onal stiffness and thereÂ’s no way to overwrite this sectional property. In addition, there is no way to add the lockedin crossframe forces for TDLF or SDLF detailing.
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43 That leaves adjusting the equivalent beam stiffness assumed for crossframes as the only improvement that can be implemented in the 2Dgrid model. The user has the option of inputting the crossframe type (Ktype, Xtype, or diaphragm) and the associated member sizes or manually input the equivalent crossframe properties. If the first option is chosen, the software automa tically converts the crossframe into an equivalent beam and calculates the equiva lent stiffness using the flexural analogy method. This method does not account for the shear stiffness. The improved method as described in the Theoretical Background chap ter is implemented in this model. See Appendix A for calculations. 2DFrame Base Model Â– LARSA 4D This model does not include any improvements and is used as a base model for comparison purposes. LARSA 4D allows much more flexibility in modeling a structure compared to commonly used 2Dgrid softwa re. There are two methods of modeling a steel girder structure: 2Dframe and 3DFEA 2Dframe models cr eate the structure in one horizontal plane with each girder modeled as a beam element offset from the deck and connected with rigid links. The deck is modeled as plate elements and the crossframes are modeled as truss or beam elements as appropriate with the connection points offset from the deck. LARSA 4D includes a design tool called the Steel Bridge Module that helps significantly reduce the time required to mode l the structure and apply the appropriate loads. The user goes through the module in similar fashion as the MDX wizard to set up the model, and has the flexibility to adjust the model and add loads manually as deemed
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44 appropriate by the user. LA RSA 4D also includes a cons truction staging analysis function. Typically this func tion is used to analyze material time effects (time is considered the fourth dimension in the name) such as creep and shrinkage of concrete and relaxation of stressed tendons. However, time is irrelevant in st eel girder design, except for considering fatigue but that is based on total stress cycles. The construction staging analysis can still be a useful tool for st eel girder design to determine stresses and movements as loads are applied and as cro ss section properties change (composite vs. noncomposite) at each construction stage. 2DFrame Improved Model Â– LARSA 4D This model includes improvements for the girder torsional stiffness. See Appendix A for calculations on the equivalent gi rder torsional stiffness. Other potential 2D improvements were not included in this m odel. The crossframes are modeled in 3D, therefore computing the equiva lent beam stiffness is unnecessary. The flange lateral bending stresses are automatically computed. Lockedin crossframe forces due to TDLF detailing are only analyzed in the 3DFEA TDLF model for ease of comparison with the 3DFEA NLF model. 3DFEA NLF Model Â– LARSA 4D 3DFEA truly models the structure in th ree dimensions. The girders are modeled as a combination of beam elements for the flanges and plate elements for the web. The crossframes are again modeled as truss or beam elements and are connected to the corresponding top and bottom flange beam elements. The warping component of the girder torsional stiffness is automatically in cluded. The crossframes are modeled in 3D
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45 and therefore do not need to be converted to equivalent beam elemen ts. As with all 2D models, this model assumes NLF detailing by default and does not include any initial lockedin crossframe forces that would be pr esent for TDLF or SDLF detailing methods. By assuming the NLF detailing method, the resu lts can be compared directly against the 2D models. Further research would need to be conducted to validate the accuracy of this model with a fullsize test bridge. However, this is out of the scope of this thesis and the 3DFEA NLF model is used as the benchmark and assumed to be the most accurate. The majoraxis bending stresses and latera l bending stresses in the flanges are determined from the member stresses results. The axial stress at the centroid of the flange beam members resembles the stress due to majoraxis bending. The lateral flange bending stresses are determined from taking the difference between the axial stress at the centroid and the average of the top and bottom stress points at one side of the rectangular flange section. The vertical deflections are taken from the joint displacements results in the vertical direction al ong the bottom flange of the girders. The crossframe axial forces are taken from the member end forces result s in the local member coordinates. The girder layovers are taken from the lateral joint displacements at the top of the girder ends. The bottom of the girder is restrained in the lateral directi on at the bearings. 3DFEA TDLF Model Â– LARSA 4D The only improvement needed for this m odel is including the initial lockedin crossframe forces due to TDLF detaili ng. The lockedin crossframe forces are calculated by determining the axial strain of the truss type members of the crossframes due to the camber differences for total dead load differential deflections. These initial
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46 strains are inputted into the model as an equi valent thermal strain load. See Appendix A for example calculations.
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47 CHAPTER V ANALYTICAL RESULTS COMPARING MODELS The primary goal of this thesis is to find the most efficient method of analysis that accurately models the behavior of highly skewed steel plate Igirder bridges. Six models were created, analyzed, and result s compared. The results include: Majoraxis bending stresses Vertical displacements Crossframe forces Flange lateral bending stresses Girder layover at bearings These are the same results used in NCHR P Report 725 to grade the accuracy of traditional 2Dgrid and 1Dlinear analysis as seen in Figure III.4. The results of the example bridge models in this thesis are co mpared to the average and worst case results reported by NCHRP. W ith a skew index, IS = 0.65, the example bridge is compared to bridges in the highest skew index category. The 3DFEA NLF LARSA 4D model is a ssumed to be the most accurate and therefore used as the benchmark against wh ich all other 2Dgrid and 2Dframe models are compared. The 2Dgrid MDX models (bas e and improved) are first compared to the 3DFEA NLF LARSA 4D model. Next, th e 2Dframe LARSA 4D models (base and improved) are compared to the 3DFEA NL F LARSA 4D model. Finally, the 3DFEA NLF LARSA 4D model is compared to th e 3DFEA TDLF LARS A 4D model.
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48 2DGrid Models Majoraxis bending stresses. The average grade of traditional 2Dgrid analyses for majoraxis girder bending stresses as re ported by NCHRP is a C and the worstcase grade is a D. A grade of C means the normalized mean error is between 13% and 20%, reflecting a significant deviation from the accu rate benchmark. A grade of D means the normalized mean error is between 21% and 30%, reflecting poor accuracy. Figures V.1 and V.2 compare the unfactor ed majoraxis bending stresses in the top and bottom flanges, respectively. The stre sses are due to the dead loads, including the weight of the deck, on the noncomposite steel section. The 2Dgrid MDX models (base and improved) are compared to the 3DFEA NLF LARSA 4D model. Both figures show similar patterns for the bending stre sses along the length of the girder. The normalized mean error was not calculated due to the jagged lines in the 3DFEA NLF LARSA 4D model results; however the results appear to be within 6% error, which results in a grade of A. The improved MDX model with the updated crossframe beam stiffness appears to resemble the benchmark pattern slightly more accurately. The reasoning behind the jagged line displa y, which is more prominent in the top flange, is unknown for the 3D benchmark model. It is most likely due to the influence of the crossframes. The reasoning behind the sma ll but noticeable jump at the girder end in the 3D benchmark model is also not comple tely known but not unexpected either. The results in NCHRP Report 725 show a similar spike at the obtuse end of the exterior girders, but do not explain the reasoning for this spike. The end crossframes along the high skew may be providing some equivalent co ntinuity at the ends of the girder and therefore cranking in a moment. However, th is is purely speculat ion. The results are
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49 deemed acceptable and further analysis into th e reasoning of the jagged line pattern and spike at the obtuse end is considered out side of the scope of this thesis. Figure V.1 MajorAxis Bending Stress of Girder 1 Top Flange Figure V.2 MajorAxis Bending Stress of Girder 1 Bottom Flange 35 30 25 20 15 10 5 0 5 00.20.40.60.81Stress (ksi)Normalized Length MDX base MDX Improved Larsa 3D 5 0 5 10 15 20 25 30 00.20.40.60.81Stress (ksi)Normalized Length MDX base MDX Improved Larsa 3D
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50 Vertical Displacements. The average grade of traditional 2Dgrid analysis for girder vertical displacements as reported by NCHRP is a C and the worstcase grade is a D. Figure V.3 compares the vertical disp lacements due to noncomposite dead loads among the 2Dgrid MDX models (base and im proved) and the 3DFEA NLF LARSA 4D model along Girder 1 and Girder 3. Figure V.3 Vertical Girder Displacem ents Along Girder 1 and Girder 3 20 18 16 14 12 10 8 6 4 2 0 00.20.40.60.81Vertical Deflection (in)Normalized LengthVertical Girder Deflection Girder 1 MDX base MDX Improved Larsa 3D 20 18 16 14 12 10 8 6 4 2 0 00.20.40.60.81Vertical Deflection (in)Normalized LengthVertical Girder Deflection Girder 3 MDX base MDX Improved Larsa 3D
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51 The vertical displacements all follow the same pattern and all have a normalized mean error of 2% or less, which is a grade A le vel. This is a much better result than the average grade reported by NCHRP for br idges with similar skew indexes. CrossFrame Forces. The average and worstcase grade of traditional 2Dgrid analysis models for crossframe forces as reported by NCHRP is an F. A grade of F means the normalized mean error is over 30% reflecting unreliable accuracy and making the results inadequate for design. Figure V.3 compares the unfactored crossframe forces due to noncomposite dead loads for the 2D grid MDX base model and 3DFEA LARSA 4D model for each member of the crossfr ames. The crossframes along bay 2 between girders 2 and 3 are shown. Figure V.4 CrossFrame Axial Forces Along Bay 2 Between Girder 2 and Girder 3 250 200 150 100 50 0 50 100 1234567891011121314Axial Load (kips)Cross frame NumberTop Chord MDX Base Larsa 3D
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52 Figure V.4 (Continued) CrossFrame Axia l Forces Along Bay 2 Between Girder 2 and Girder 3 200 150 100 50 0 50 100 1234567891011121314Axial Load (kips)Cross frame NumberLeft Diagonal MDX Base Larsa 3D 100 50 0 50 100 150 200 250 1234567891011121314Axial Load (kips)Cross frame NumberLeft Bottom Chord MDX Base Larsa 3D
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53 Figure V.4 (Continued) CrossFrame Axia l Forces Along Bay 2 Between Girder 2 and Girder 3 100 50 0 50 100 150 200 1234567891011121314Axial Load (kips)Cross frame NumberRight Diagonal MDX Base Larsa 3D 100 50 0 50 100 150 200 250 1234567891011121314Axial Load (kips)Cross frame NumberRight Bottom Chord MDX Base Larsa 3D
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54 The pattern of crossframe me mber forces along bay 2 is similar for both analysis models. A spike in axial load in all member s at crossframes 11 and 12 can clearly been seen. This illustrates the increased loads th at occur at the obtuse corners. However, a significant difference in crossframe forces can clearly be seen between the two models from the graphs. The normalized mean error in comparison to the 3DFEA NLF benchmark model ranges from 7.3% to 21.2% as seen in Table V.1. This would suggest a grade of D for the worst case. However, some of the worst errors occur at the controlling crossframes with the highest fo rces. With percenta ges of error well over 30% for these critical crossframes, the gr ade should be an F and the method deemed unacceptable for calculating crossframe forces. Normalized Mean Error Top Chord 14.6% Left Diagonal 13.9% Left Bottom Chord 21.2% Right Diagonal 14.0% Right Bottom Chord 7.3% Table V.1 Normalized Mean Error for CrossFrame Forces in the 2DGrid MDX Model Flange Lateral Bending Stresses. The average and worstcase grade of traditional 2Dgrid analysis models for girder flange lateral bending stresses as reported by NCHRP is an F. Responses to flange lateral bending are not provided in the MDX results. In order to determine the flange lateral bending stresses postprocessing using the crossframe forces would need to be comp leted. Since the crossframe forces results received a grade of F for the 2Dgrid model, calculation results for flange lateral bending stresses would be inaccurate as well. Theref ore, these calculations were not performed.
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55 Girder Layover at Bearings. The average grade of traditional 2Dgrid analysis models for girder layover at bearings as reported by NCHRP is a C and the worstcase grade is a D. MDX does not produce this output, therefore there is nothing to compare. Girder layover at bearings would be calcul ated by hand using the differential deflection output. Because girder layover and vertical displacements are directly related, NCHRP gave them the same grades. 2DFrame Models Majoraxis bending stresses. Figure V.5 compares the unfactored majoraxis bending stresses in the top and bottom flanges. The stresses are due to the dead loads, including the weight of the deck, on the noncomposite steel section. The 2Dframe LARSA 4D models (base and improved) are compared to the 3DFEA NLF LARSA 4D model. Both graphs show similar patterns for the bending stresses along the length of the girder. The normalized mean error was not cal culated due to the jagged lines in the 3DFEA NLF LARSA 4D model results; however the results appear to be within 12% error, which results in a grade of B. It is surp rising that the 2Dframe models have a higher percentage of error; however, with a grad e of B, they are considered acceptable for analysis results. Vertical Displacements. Figure V.6 compares the vertical displacements due to noncomposite dead loads among the 2Dframe LARSA 4D models (base and improved) and the 3DFEA NLF LARSA 4D mode l along Girder 1 and Girder 3.
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56 Figure V.5 MajorAxis Bending Stress of Girder 1 35 30 25 20 15 10 5 0 5 0.00.20.40.60.81.0Stress (ksi)Normalized LengthTop Flange Larsa 2D base Larsa 2D Improved Larsa 3D 5 0 5 10 15 20 25 30 0.00.20.40.60.81.0Stress (ksi)Normalized LengthBottom Flange Larsa 2D base Larsa 2D Improved Larsa 3D
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57 The vertical displacements all follow the same pattern and all have a normalized mean error of less than 8%, which is a grad e level of A to B. The critical maximum deflection near midspan is off by as much as 14% for both girders. However, with a maximum difference of 2.37Â”, the error can be made up by specifying a large enough haunch in the plans. These results are considered acceptable for the example bridge. Figure V.6 Vertical Girder Displacem ents Along Girder 1 and Girder 3 20 18 16 14 12 10 8 6 4 2 0 0.00.20.40.60.81.0Vertical Deflection (in)Normalized LengthVertical Girder Deflection Girder 1 Larsa 2D base Larsa 2D improved Larsa 3D 20 18 16 14 12 10 8 6 4 2 0 0.00.20.40.60.81.0Vertical Deflection (in)Normalized LengthVertical Girder Deflection Girder 3 Larsa 2D base Larsa 2D improved Larsa 3D
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58 CrossFrame Forces. Figure V.7 compares the unfact ored crossframe forces due to noncomposite dead loads for the 2Dframe LARSA 4D models (base and improved) and 3DFEA LARSA 4D model for each member of the crossframes. The crossframes along bay 2 (between girders 2 and 3) are shown. The pattern of crossframe me mber forces in the graph along bay 2 is similar for all models, except for the end crossframes in the 2Dframe models. The spike in axial load in all members at crossframes 11 and 12 illustrates the increased loads that occur at the obtuse corners and is much more accura tely represented in both 2Dframe LARSA 4D models versus the 2Dgrid MDX models. Figure V.7 CrossFrame Axial Forces Along Bay 2 Between Girder 2 and Girder 3 250 200 150 100 50 0 50 100 1234567891011121314Axial Load (kips)Cross frame NumberTop Chord Larsa 2D Base Larsa 2D Improved Larsa 3D
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59 Figure V.7 (Continued) CrossFrame Axia l Forces Along Bay 2 Between Girder 2 and Girder 3 200 150 100 50 0 50 100 1234567891011121314Axial Load (kips)Cross frame NumberLeft Diagonal Larsa 2D Base Larsa 2D Improved Larsa 3D 150 100 50 0 50 100 150 200 250 300 1234567891011121314Axial Load (kips)Cross frame NumberLeft Bottom Chord Larsa 2D Base Larsa 2D Improved Larsa 3D
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60 Figure V.7 (Continued) CrossFrame Axia l Forces Along Bay 2 Between Girder 2 and Girder 3 100 50 0 50 100 150 200 1234567891011121314Axial Load (kips)Cross frame NumberRight Diagonal Larsa 2D Base Larsa 2D Improved Larsa 3D 200 150 100 50 0 50 100 150 200 250 1234567891011121314Axial Load (kips)Cross frame NumberRight Bottom Chord Larsa 2D Base Larsa 2D Improved Larsa 3D
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61 A significant difference in crossframe for ces can clearly be seen for the end and first interior crossframes between the im proved 2Dframe LARSA 4D model and the 3D benchmark. The Â“improvementsÂ” to the gird er torsional stiffne ss appears to provide inaccurate results for crossframe forces near the girder ends and provide no noticeable improvement over the 2Dframe base model. The normalized mean error ranges from 3.1% to 6.8% for the 2Dframe base model and from 6.0% to 12.0% for the 2Dframe improved model as seen in Table V.2. This results in a grade of A and B for the two models respectively. Normalized Mean Error Member Base Improved Top Chord 3.1% 7.8% Left Diagonal 3.1% 12.0% Left Bottom Chord 6.8% 6.0% Right Diagonal 3.2% 12.0% Right Bottom Chord 4.1% 11.7% Table V.2 Normalized Mean Error for Cr ossFrame Forces in the 2DFrame LARSA 4D Models Flange Lateral Bending Stresses. Figure V.8 compares the unfactored flange lateral bending stresses due to noncomposite dead loads for the 2Dframe LARSA 4D base model and 3DFEA LARSA 4D model for Girders 1 and 3. The pattern of lateral bending stresses is similar for both models in the top flange; howev er, they appear to differ significantly in the second half of the bottom flange near the obtuse corner.
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62 Figure V.8 Flange Lateral Bending Stress 8 6 4 2 0 2 4 0.00.20.40.60.81.0Stress (ksi)Normalized LengthGirder 1 Top Flange Larsa 2D base Larsa 3D 8 6 4 2 0 2 4 0.00.20.40.60.81.0Stress (ksi)Normalized LengthGirder 1 Bottom Flange Larsa 2D base Larsa 3D
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63 Figure V.8 (Continued) Flange Lateral Bending Stress 4 3 2 1 0 1 2 3 4 5 6 0.00.20.40.60.81.0Stress (ksi)Normalized LengthGirder 3 Top Flange Larsa 2D base Larsa 3D 4 3 2 1 0 1 2 3 4 5 6 0.00.20.40.60.81.0Stress (ksi)Normalized LengthGirder 3 Bottom Flange Larsa 2D base Larsa 3D
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64 The normalized mean error for the top fla nge is 13.2% for Girder 1 and 13.8% for Girder 3, which results in a gr ade of C. In contrast, the normalized mean error for the bottom flange is much worse at 90.3% for Gird er 1 and 54.5% for Girder 3 and results in a grade of F. It is important to note that the lateral bending stresses are relatively small, making the percentage of error a bit inconseque ntial. The lateral be nding stresses are less than 2 ksi everywhere except near the end of girders near the obtuse end. Because the first interior crossframe in bay 1 near the ob tuse corner was removed, this area could be considered as having discontinuous crossframes. As per AASHTO LRFD Bridge Design Specifications C6.10.1, Â“in abse nce of calculated values of fl from a refined analysis, a suggested estimate for the total unfactored fl in a flange at a crossframe or diaphragm due to the use of discontinuous cr ossframe or diaphragm lines is 10.0 ksi for interior girders and 7.5 ksi for exterior girders. Â” It continues, Â“in regions of the girders with contiguous crossframes or diaphrag ms, these values need not be consideredÂ” (AASHTO, 2012). Therefore, 7.5 ksi for exte rior and 10 ksi for the interior could conservatively be assumed for the flange late ral bending stress near th e obtuse corners. The rest of the crossframes are considered contiguous and therefor e the flange lateral bending stress can be co nsidered negligible. Girder Layover at Bearings. Figure V.9 compares the gi rder layover at bearings under dead loads on the noncomposite girder s for both 2Dframe models (base and improved) and the 3DFEA NLF model. The gi rder layover is the horizontal transverse displacement measured at the top of the gird er web with respect to the bottom of the girder web. The bearings are fixe d in the transverse direction.
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65 The normalized mean error for the 2Dframe base model is 7.3% at the start bearing and 9.1% at the end bear ing, which results in a grade of B. The normalized mean error for the 2Dframe improved model is 10.9 % at the start beari ng and 14.5% at the end bearing, which results in a grade of B a nd C respectively. The significance of the percentage of error depends on the crossframe fitup detailing method used and the rotational capacity in the bearings. If NLF detailing is used, the bearings would need to be able to handle the large transverse rotations. Figure V.9 Girder Layover at Bearings 6 4 20246 1 2 3 4 5 6 7 8 9 Girder Layover (in)Girder Number 3D End 3D Start 2D Frame Imp End 2D Frame Imp Start 2D Frame Base End 2D Frame Base Start
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66 3DFEA Model with TDLF Detailing The 3DFEA TDLF model incl udes the initial lockedin cr ossframe forces due to TDLF detailing. Results are compared to the 3DFEA NLF model that by default assumes no initial crossframe forces and the gi rder webs are plumb in the noload case. Results at different constructi on stages are also compared. Majoraxis bending stresses. As seen in Figure V.10, the majoraxis bending stress due to dead loads on the noncomposite girder sections are very similar. The patterns are almost identical and the results at each data point di ffer very slightly. Figure V.10 MajorAxis Bending Stress of Girder 1 35 25 15 5 5 0.00.20.40.60.81.0Stress (ksi)Normalized LengthGirder 1 Top Flange Larsa 3D TDLF Larsa 3D NLF 0 5 10 15 20 25 30 0.00.20.40.60.81.0Stress (ksi)Normalized LengthGirder 1 Bottom Flange Larsa 3D TDLF Larsa 3D NLF
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67 Vertical Displacements. The vertical displacements due to dead loads on noncomposite girder sections are also very similar between the 3DFEA NLF and 3DFEA TDLF models. There is no noticeable di fference between the exterior girders. There is a slight difference betw een the interior girders that is most noticeable closest to the centerline of the bridge (Girder 5). This is to be expected as per the results from NCHRP Report 725. Girder 1 and Girder 3 vertical displacements are compared in Figure V.11. Figure V.11 Vertical Girder Displacements along Girder 1 and Girder 3 20 15 10 5 0 0.00.20.40.60.81.0Vertical Deflection (in)Normalized LengthGirder 1 Larsa 3D TDLF Larsa 3D NLF 20 15 10 5 0 0.00.20.40.60.81.0Vertical Deflection (in)Normalized LengthGirder 3 Larsa 3D TDLF Larsa 3D NLF
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68 CrossFrame Forces. Figure V.12 compares the unfactored crossframe forces due to the girder, crossframe, and deck noncomposite dead loads for the 3DFEA NLF and TDLF models, and the forces due to only the steel gird er and crossframe noncomposite dead loads for the 3DFEA TDLF model. For most of the crossframe members with TDLF detailing, the maximum fo rce in the crossframes occurs during the fitup of the crossframes with the girders. The girders are twisted and forced into an out of plumb orientation and as dead loads are applied, the crossframe forces are offset or relieved as the girders twist back into th e vertically plumb pos ition under total dead loads. Figure V.12 CrossFrame Axial Forces Alon g Bay 2 Between Girder 2 and Girder 3 250 200 150 100 50 0 50 100 1234567891011121314Axial Load (kips)Cross frame NumberTop Chord 3D TDLF Steel Only 3D TDLF Steel+Deck 3D NLF Steel+Deck
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69 Figure V.12 (Continued) CrossFrame Axia l Forces Along Bay 2 Between Girder 2 and Girder 3 200 150 100 50 0 50 100 1234567891011121314Axial Load (kips)Cross frame NumberLeft Diagonal 3D TDLF Steel Only 3D TDLF Steel+Deck 3D NLF Steel+Deck 100 50 0 50 100 150 200 250 1234567891011121314Axial Load (kips)Cross frame NumberLeft Bottom Chord 3D TDLF Steel Only 3D TDLF Steel+Deck 3D NLF Steel+Deck
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70 Figure V.12 (Continued) CrossFrame Axia l Forces Along Bay 2 Between Girder 2 and Girder 3 100 50 0 50 100 150 200 1234567891011121314Axial Load (kips)Cross frame NumberRight Diagonal 3D TDLF Steel Only 3D TDLF Steel+Deck 3D NLF Steel+Deck 150 100 50 0 50 100 150 200 250 1234567891011121314Axial Load (kips)Cross frame NumberRight Bottom Chord 3D TDLF Steel Only 3D TDLF Steel+Deck 3D NLF Steel+Deck
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71 Flange Lateral Bending Stresses. Figure V.13 compares the unfactored flange lateral bending stresses due to noncomposite de ad loads along Girder 1 for the 3DFEA NLF and 3DFEA TDLF models. The TDLF model results incl ude two construction stages Â– steel girder and crossframe dead load s only and steel plus deck dead loads. Figure V.13 Flange Lateral Bending Stress Along Girder 1 8 6 4 2 0 2 4 6 0.00.20.40.60.81.0Stress (ksi)Normalized LengthTop Flange 3D TDLF Steel Only 3D TDLF Steel+Deck 3D NLF Steel+Deck 8 6 4 2 0 2 4 6 0.00.20.40.60.81.0Stress (ksi)Normalized LengthBottom Flange 3D TDLF Steel Only 3D TDLF Steel+Deck 3D NLF Steel+Deck
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72 As expected the flange lateral bending stress is minimal at the total dead load case when using TDLF detailing. The results for the crossframe forces and flange lateral bending stress, clearly s hows the advantage of us ing TDLF detailing. Girder Layover at Bearings. Figure V.14 compares th e girder layover at bearings under dead loads on the noncompos ite girders for the 3DFEA NLF and TDLF models with girders, crossframes, and d eck dead loads and th e 3DFEA TDLF model with the steel girders and crossframe dead loads only. The girder layover under total noncomposite dead load using the TDLF detailing is close to zero indicating the girder webs are nearly plumb in the total dead load condition. Figure V.14 also indicates that the initial layover with the initial lockedin crossframe forces due to TDLF detailing is in the opposite direction than the layover due to dead loads. Field data was gathered duri ng construction that indicates the girder layover. A 4Â’ level was used to measure how far out of plum b the girder webs were over the height of the level. The data was used to determine th e girder layover in rela tion to the full height of the girder. The field data is believed to have been taken just after girder and crossframe erection; however, additional dead load s may have been present, such as the precast deck panels. Figure V.15 compares the girder layover taken from field data against the 3DFEA TDLF models at the steel only load case and steel plus deck load case. The field data mostly follows the sa me pattern along the bearing lines and appears to be slightly less than the 3DFEA TDLF steel only load cas e. The method of measuring the field data leaves room for human error that would affect the accuracy; however, the
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73 intent of this figure is to validate the beha vior found in the 3DFEA TDLF models with the actual behavior of the bridge. Figure V.14 Girder Layover at Start and End Bearings for 3D Models 6 4 2024 1 2 3 4 5 6 7 8 9 Girder Layover (in)Girder NumberStart Bearing Line 3D TDLF Steel+Deck 3D TDLF Steel Only 3D NLF Steel+Deck 4 3 2 101234 1 2 3 4 5 6 7 8 9 Girder Layover (in)Girder NumberEnd Bearing Line 3D TDLF Steel+Deck 3D TDLF Steel Only 3D NLF Steel+Deck
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74 Figure V.15 Girder Layover at Start and E nd Bearings for 3D TDLF Models and from Field Data 101234 1 2 3 4 5 6 7 8 9 Girder Layover (in)Girder NumberStart Bearing Line 3D TDLF Steel+Deck 3D TDLF Steel Only Field Data 3.0 2.5 2.0 1.5 1.0 0.50.00.5 1 2 3 4 5 6 7 8 9 Girder Layover (in)Girder NumberEnd Bearing Line 3D TDLF Steel+Deck 3D TDLF Steel Only Field Data
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75 CHAPTER VI CONCLUSIONS Recommended Method of Analysis The primary goal of this thesis is to determine the most efficient method of analysis that accurately models the behavior of highly skewed steel plate Igirder bridges. By implementing some improvements to 2D methods of analysis as described in NCHRP Report 725, the hope is that 2D type methods could provide very accurate results. The improvements appear to be fairly strai ghtforward and simple in theory, but the application ended up being far from simple. NLF detailing is assumed by default in all so ftware. However, for straight steel Igirder bridges, TDLF detailing is the preferre d option. If TDLF detailing is chosen, it is very important to include any lockedin for ces, which will counterbalance to a certain extent the crossframe loads, lateral deflectio ns, and rotations caused by the dead loads. If these lockedin forces are not included, the designer is assuming NLF detailing, which can lead to overly conservative crossframes forces and lateral flange bending stresses in cases of very large skews and with the pres ence of nuisance stiffness crossframes close to the bearing line. Correctly modeling the behavior for the chosen detailing method is important to develop an accurate and efficient design. The 3DFEA method is considered the mo st accurate method; however it still comes with its own limitations. The biggest limitation is its complexity and the amount of detail that is required to create a 3DFEA model. Cr eating the model, running through the analysis, and sorting through the massive amount of output data can be very time
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76 consuming. Modeling an already comple x structure with a complex method and interpreting the output can also increa se the chance for human error. The 3DFEA method was assumed to be the most accurate for analyzing the example and was used as the benchmark to compare all other models. The field data measurements of girder layovers during cons truction provided a very loose validation of the software. The results for deflections, rota tions, stresses, and general overall behavior of the highly skewed steel Igirder example bridge in the 3DFEA model were as expected. The girder ends tw isted and rotated about the cent erline of the bearing support and the framing system generally behaved as previously described in the Theoretical Background section. However, a fullsized test bridge with stress gauges and with similar geometry is needed to truly validat e the assumptions made in the creating and analyzing the 3DFEA m odel. This was considered outside the scope of this thesis. The 2Dgrid method of analysis for highly skewed Igirder bridges appears to be the least accurate for calculating crossfram e forces for nuisance stiffness crossframes and lateral flange bending stresses. 2Dg rid software, such as MDX and DESCUS, are very powerful and useful tools when used in the right context. But until the software companies update their software to include be tter equivalent estimations of the girder torsional stiffness and equivalent beam stiffn ess of the crossframes, designers need to do a significant amount of postprocessing calculations to check for additional crossframe forces and later flange bending stresses that may not have been captured in the 2Dgrid software analysis. NCHRP Report 725 repeat edly encouraged the bridge software industry to implement these improvements into the software.
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77 The 2Dframe method using the LARSA 4D software or something similar, currently appears to be the most efficient me thod to use for highly skewed steel Igirder bridge design. LARSA 4D c ontains a steel bridge module that makes creating the model much easier, similar to the 2Dgrid software models. Software like LARSA 4D that has 3D capabilities has the flexib ility to manually override cert ain section properties and add user specified loads to better model the behavior of highlyskewed steel Igirder bridges. With this flexibility, the necessity of pos tprocessing can be eliminated or greatly reduced. Compared to 3DFEA models, 2Dframe model output is much more manageable and therefore less timecons uming and efficient for the designer. General Recommendations and Future Work Current Recommendations Recommendations for designers currently analyzing highly sk ewed steel Igirder bridges include: Use the 2Dframe method with software th at includes a bridge module for easily creating the geometry. Include improvements to the 2Dframe model in certain situations as outlined by NCHRP Report 725. Adjusting the equivale nt girder torsional constant that includes warping capacity should be used for staggered crossframe layouts. When contiguous crossframes are used throughout each span, a more detailed analysis should be used to analyze th e crossframe forces and lateral flange bending stresses near the girder ends.
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78 Include initial lockedin crossframe forces when TDLF or SDLF detailing is used to get a more efficient design. The 3DFEA method should be used on a limited basis to verify behavior and check localized stresses. It is a good tool to use to check crossframe forces and lateral flange bending stresses; however it is too cumbersome to use as an allencompassing analysis. The 2Dgrid method is not recommended for steel Igirder bridges with a high skew index until improvements are made to the software or the designer decides to accompany this analysis with extensive postprocessing. Future Work Considerations Future work considerations and recommendations include: Encourage the bridge software indus try to implement the recommended improvements to 2Dgrid software. Encourage the 3Dcapable bridge softwa re industry to implement improvements to steel bridge design modules to accu rately and easily include appropriate lockedin crossframe forces for SDLF or TDLF detailing and to automatically update the internally calculated equi valent girder torsional stiffness. Research and analyze more highly skew ed bridges with different crossframe layouts. Specifically, analyze the girders and crossframes with contiguous crossframe layouts and nuisance stiffness crossframe near bearing supports. Adjust the 2D analysis method improvements accordingly.
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79 Research fitup practices typically used in highlyskewed steel Igirder erection. Determine at what point the fitup forces for TDLF detailing become too large. Research and analyze more innovative crossframe configurations, including partially skewed crossframes, leanon bracing, temporary bracing, and different connection and bearing plate detailing.
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80 REFERENCES AASHTO. American Association of State Hi ghway and Transportati on Officials (2012). AASHTO LRFD Bridge Design Specificat ions, Customary U.S. Units, 6th Edition with 2012 and 2013 Interim Revisions and 2012 Errata. AASHTO/NSBA G12.1. American Associati on of State Highway and Transportation Officials / National Steel Br idge Alliance Steel Bridge Collaboration (2003). G 12.1 Guidelines for Design for Constructability. AASHTO/NSBA G13.1. American Associati on of State Highway and Transportation Officials / National Steel Br idge Alliance Steel Bridge Collaboration (2011). G 13.1 Guidelines for Steel Girder Bridge Analysis, 1st Edition. Ahmed, M.Z. and Weisberger, F. E. (1996). Â“Torsion Constant for Matrix Analysis of Structures Including Warping Effect,Â” International Jour nal of Solids and Structures, Elsevier, 33(3), 361374. Beckmann, F., and Medlock, R.D. Skewed Bridges and Girder Movements Due to Rotations and Differe ntial Deflections. Chavel, B., Peterman, L., and McAtee, C. (2010). Design and Construction of the Curved and Severely Skewed Steel IGi rder EastWest Connector Bridges over I88. 27th Annual International Bri dge Conference 2010. IBC1024. MDX (2013). MDX NetHelp. http://www.mdxsoftware.com/ April 2013. NCHRP Report 725. Transportation Research Board of the National Academy of Sciences (2012). National Cooperative Highway Research Program Report 725: Guidelines for Analysis Methods and C onstruction Engineering of Curved and Skewed Steel Girder Bridges. Project 1279. Parsons Corporation (2011). Design Plans for SR114 Geneva Road, Roadway Widening: Geneva Road over UPRR & UT A. Signed by registered Professional Engineer: Haines, Steve. Owner: Ut ah Department of Transportation. Schaefer, A.L. (2012). Crossframe Analysis of HighlySkewed and Curved Steel IGirder Bridges. Thesis submitted to the Univer sity of Colorado Denver. ProQuest LLC.
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81 APPENDIX A Appendix A includes calculations for the e quivalent beam stiffness of the crossframes and the steel plate girder design calcu lations that include the equivalent girder torsional stiffness constant used in the 2D models.
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UCDMaster's Thesis Skewed SteelIGirders Equivalent CrossFrame Stiffness Calculations Calculations By: K.Dobbins 10/18/2013 Equivalent BeamStiffnessforCrossFrame s Constants: Es= 29000ksi Crossframe Type1 heightbtwn workingpt s 85in width btwn workingpt s 127in weighttop chord 26.5plf weightbott chord 26.5plf weightdiagonals 26.5plf weightconnection plates 0.726kips Total weight 1.548 STAADOutput Nodal Displacements HorizontalVerticalResultant NodeL/CX (in)Y (in)Z (in)(in)rX (rad)rY(rad)rZ(rad) 11 UNIT LOADCOUPLE 0000000 21 UNIT LOADCOUPLE 0000000 31 UNIT LOADCOUPLE 0.000560.0008400.00101000 41 UNIT LOADCOUPLE0.000560.0008400.00101000 51 UNIT LOADCOUPLE0.000280.0002100.00035000 12 UNIT SHEAR 0000000 22 UNIT SHEAR 0000000 32 UNIT SHEAR 0.000420.002400.00244000 42 UNIT SHEAR0.000420.002400.00244000 52 UNIT SHEAR0.000420.0010400.00113000 0.00001318rad Ieq28250 in4Aseq 7.268 in2Rotational
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UCDMaster's Thesis Skewed SteelIGirders Equivalent CrossFrame Stiffness Calculations Calculations By: K.Dobbins 10/18/2013 Crossframe Type2 heightbtwn workingpt s 83.5in width btwn workingpt s 104in weighttop chord 59.5plf weightbott chord 59.5plf weightdiagonals 59.5plf weightconnection plates 1.633kips Total weight 3.359 Nodal Displacements HorizontalVerticalResultant Node L/C X (in)Y (in)Z (in)(in)rX (rad)rY(rad)rZ(rad) 11 UNIT LOADCOUPLE 0000000 21 UNIT LOADCOUPLE 0000000 31 UNIT LOADCOUPLE 0.00020.0002500.00033000 41 UNIT LOADCOUPLE0.00020.0002500.00033000 51 UNIT LOADCOUPLE0.00010.0000600.00012000 12 UNIT SHEAR 0000000 22 UNIT SHEAR 0000000 32 UNIT SHEAR 0.000130.0007700.00078000 42 UNIT SHEAR0.000130.0007700.00078000 52 UNIT SHEAR 0.000130.0003500.00037000 0.00000479rad Ieq62510 in4Aseq 16.557 in2Rotational
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UCDMaster's Thesis Skewed SteelIGirders Equivalent CrossFrame Stiffness Calculations Calculations By: K.Dobbins 10/18/2013 Crossframe Type3 heightbtwn workingpt s 85in width btwn workingpt s 127in weighttop chord 26.5plf weightbott chord 34plf weightdiagonals 34plf weightconnection plates 0.726kips Total weight 1.713 Nodal Displacements HorizontalVerticalResultant Node L/C X (in)Y (in)Z (in)(in)rX (rad)rY(rad)rZ(rad) 11 UNIT LOADCOUPLE 0000000 21 UNIT LOADCOUPLE 0000000 31 UNIT LOADCOUPLE 0.000560.0007500.00093000 41 UNIT LOADCOUPLE0.000440.0007500.00087000 51 UNIT LOADCOUPLE0.000220.0001600.00027000 12 UNIT SHEAR 0000000 22 UNIT SHEAR 0000000 32 UNIT SHEAR 0.000420.0019400.00199000 42 UNIT SHEAR0.000330.0019400.00197000 52 UNIT SHEAR 0.000330.0008100.00088000 0.00001176rad Ieq31641 in4Aseq 9.521 in2Rotational
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UCDMaster's Thesis Skewed SteelIGirders Equivalent CrossFrame Stiffness Calculations Calculations By: K.Dobbins 10/18/2013 Crossframe Type4(End) heightbtwn workingpt s 85in width btwn workingpt s 230.125in weighttop chord 26.5plf weightbott chord 26.5plf weightdiagonals 26.5plf weightconnection plates 0.818kips Total weight 2.372 Nodal Displacements HorizontalVerticalResultant Node L/C X (in)Y (in)Z (in)(in)rX (rad)rY(rad)rZ(rad) 11 UNIT LOADCOUPLE 0000000 21 UNIT LOADCOUPLE 0000000 31 UNIT LOADCOUPLE 0.001020.0027500.00294000 41 UNIT LOADCOUPLE0.001020.0027500.00294000 51 UNIT LOADCOUPLE0.000510.0006900.00086000 12 UNIT SHEAR 0000000 22 UNIT SHEAR 0000000 32 UNIT SHEAR 0.001380.0091800.00928000 42 UNIT SHEAR0.001380.0091800.00928000 52 UNIT SHEAR 0.001380.0036600.00391000 0.00002400rad Ieq28104 in4Aseq 4.917 in2Rotational
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UCDMasters Thesis Geneva Road SteelPlateGirderDesign By: KonleeDobbins 10/7/2013 SteelGirder Layout and Section Properties Overview Skewed straightsteelplategirder bridg e Simple span bridg e DesignCode: AASHTOLRFD BridgeDesignSpecifications, 5thedition, 201 0 LiveLoads: HL 93and TandemasperAASHTO(no permittrucks considered) DesignParameters ValueUnit Comments Roadwaywidth 82ft Barrier width 2.000ft LeftSidewalk Width 6.667ft RightSidewalk Width 10.667ft Deck width 103.333ft Number ofDesignLane s 6 integer partofroadwaywidth/12' Span 1Length 254.4375ftBrg toBrg Haunch and Top Flange 6inBott ofdeck tobott oftop flange,constant AssumedAvg Haunch 4.5in Deck thickness 8.5in Overhang deck thicknessatedg e 8.5in FWS Asphalt overla y 3.43in40 ps f sacrificial deck thickness 0.5in Designdeck thicknes s 8in Barrier area 4.667ft2 Barrier weight 0.7kl f includes 0.05klfforchain link fenc e Concrete strength, f'c 4ksi Reinf steelf y 60ksi Structural steelf y 50ksiM270 Grade 50W Reinf Conc unit weight 0.15kc f includes extra 0.005forreba r Conc UnitweightforEc 0.145kc f Asphalt unit weigh t 0.14kc f Steelunit weigh t 0.49kc f Es 29000ksi Ec 3644ksi33000*wc^1.5*(f'c)^0.5 n=Es/Ec 8.0 FutureADT T 2500trucks/day Layout Number ofgirders 9 Girderspacin g 11.0625ft Overhang7.417ft overhang/spacin g 0.67
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UCDMasters Thesis Geneva Road SteelPlateGirderDesign By: KonleeDobbins 10/7/2013 Section Properties Section 1atends Section 2between S1&S3 Section 3atmidspan Steelgirder onl y S1S2S3 top flangewidth 303030in top flangethickness 1.51.751.75in webthickness 0.750.750.75inD/tw<150140.0 webheight 105105105in Bottomflangewidth 323232in Bottomflangethickness 1.7522.25in Girderdepth 108.25108.75109in GirderArea 179.75195.25203.25 in2top flangecog, y 107.5107.875108.125in webcog, y 54.2554.554.75in bottom flangecog, y 0.87511.125in GirderCOGfrombott,y 50.9551.3249.54in A*y / A GirderCOGfromtop ofdeck, y 65.548 65.685 67.709in MajorMomentofinertia,Ix357558403157423492 in4Stop624070197122 in3Sbot701778568548 in3Top Flange MomentofInertia, Iytf337539383938 in4BottFlange MomentofInertia, Iybf477954616144 in4Minor MomentofInertia, Iy8157940310085 in4Torsional Constant, J 105.7153.7189.9 in4WarpingConstant, Cw224877322613349927473512 in6ShearModulus, G 111541115411154ksi p0.0013440.0015040.0016301/in
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UCDMasters Thesis Geneva Road SteelPlateGirderDesign By: KonleeDobbins 10/7/2013 Equivalent Torsional Constant S1S2S3 Brace length1Lb25N/AN/Ain Equivalent Torsional Constant, Jeq1122714N/AN/A in4Brace length2Lb272N/AN/Ain Equivalent Torsional Constant, Jeq9610N/AN/A in4Brace length3Lb247247247in Equivalent Torsional Constant, Jeq116271354914278 in4Brace length4Lb240240240in Equivalent Torsional Constant, Jeq123081434015109 in4Brace length5Lb227227227in Equivalent Torsional Constant, Jeq137431600816863 in4Brace length6Lb293293293in Equivalent Torsional Constant, Jeq8299968210212 in4Brace length7Lb248248248in Equivalent Torsional Constant, Jeq115341344214165 in4Brace length8Lb245245245in Equivalent Torsional Constant, Jeq118161376814508 in4Brace length9Lb251251251in Equivalent Torsional Constant, Jeq112631312613833 in4Brace length10Lb276276276in Equivalent Torsional Constant, Jeq93371088811480 in4Brace length11Lb45N/AN/Ain Equivalent Torsional Constant, Jeq346604N/AN/A in4Brace length12Lb41N/AN/Ain Equivalent Torsional Constant, Jeq417508N/AN/A in4Brace length13Lb37N/AN/Ain Equivalent Torsional Constant, Jeq512630N/AN/A in4Brace length14Lb33N/AN/Ain Equivalent Torsional Constant, Jeq644403N/AN/A in4Brace length15Lb29N/AN/Ain Equivalent Torsional Constant, Jeq834392N/AN/A in4

