BEHAVIOR AND ANALYSIS OF HIGHLY SKEWED STEEL I-GIRDER 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
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
iii Dobbins, Konlee B. (M.S., Civil Engineering) Behavior and Analysis of Highl y Skewed Steel I-Girder Bridges Thesis directed by Professor Kevin Rens ABSTRACT Skewed bridge supports for steel I-girde 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 2D-g 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 2D-grid and 2D-frame 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 locked-in cross-frame 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
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.
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– Cross-Frame 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
vi VI. CONCLUSIONS .............................................................................................75 Recommended Method of Analysis .................................................................75 General Recommendations for Future Work ...................................................77 REFERENCES .................................................................................................................80 APPENDIX A ................................................................................................................... 81
vii LIST OF FIGURES Figure III.1 Typical Fit-up Procedure for Skewed I-Girders. ..................................................... 16 III.2 Lateral Bending Moment, Ml, in a Fl ange Segment Under Simply Supported and Fixed-End 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 I-Girder 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 Major-Axis 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 Cross-Frame Axial Forces Along Ba y 2 Between Girder 2 and Girder 3 ................ 51 V.5 Major-Axis Bending Stress of Girder 1 .................................................................... 56 V.6 Vertical Girder Displacements Along Girder 1 and Girder 3 ................................... 57 V.7 Cross-Frame Axial Forces Along Ba y 2 Between Girder 2 and Girder 3 ................ 58 V.8 Flange Lateral Bending Stress .................................................................................. 62
viii V.9 Girder Layover at Bearings ....................................................................................... 65 V.10 Major-Axis Bending Stress of Girder 1 .................................................................. 66 V.11 Vertical Girder Displacements along Girder 1 and Girder 3 .................................. 67 V.12 Cross-Frame 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
1 CHAPTER I INTRODUCTION Effects of Skews Skewed bridge supports and horizontal curv ature in steel I-girder 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 I-girder 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
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 I-girder 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 Highly-Skewed and Curved Steel IGirder BridgesÂ” that touched on a variety of similar topics an d provided a case study example. That thesis focused on cross-fram e design by looking at different framing plan and cross-frame configurations (x-frame vs. k-fr ame) to find the most efficient design. It also included some background research, a literature review theoretical background, and analysis of cross-frames in highly skewed and curved st eel I-girder 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 cross-frame 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
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 Highly-Skewed and Curved Steel I-Gir der BridgesÂ” include: Staggered cross-frame configurations induced the least amount of forces within its cross-frames. Contiguous cross-frame configurations indu ced the most forces within its crossframes. A stiffer transverse system will accumulate more force than a flexible system. The K-frame type cross-frame performs better than the X-frame. The diagonal members in a K-frame cross-frame absorb significantly less force than the diagonal members in an X-frame. The double angle and WT-members 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 I-girders and th e most accurate design methods that should be used with common steel I-g irder structural software.
4 CHAPTER II LITERATURE REVIEW Introduction The literature review in Â“Crossframe Analysis of Highly-Skewed and Curved Steel I-Girder 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 12-38, 1993 AASHTO Guide Specifications for Horizont ally Curved Steel Girder Highway Bridges, 1993 NCHRP Project 12-52, 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 12-79, 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–
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 cross-frames. The contents include: 1. Modeling descriptions 2. History of steel bridge analysis
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 cross-frames and girders for the intended erected position, cross-frame 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 12-79. The objectiv es and scope of NCHRP Project 12-79 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
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 fit-up 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 bridge-own 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 load-andresistance 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
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 cross-frame 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 I-girder 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 I-Girder East-West Connector Bridge over I-88Â”, 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 fit-up detailing during erection affect all stages of design and construction.
9 Engineers, fabricators, and contractors all need to understand the movements, forces required for fit-up, and correspond ing locked-in stresses that occur during different stages of construction.
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 odd-shaped 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.
11 Use dapped girder ends with inverted-tee 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– Cross-Frame Layout Cross-frames 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 cross-frames at suppor ts to the bearings and down to the substructure. As per AASHTO LRFD Bridge Design Specifications 22.214.171.124 Â– Diaphragms and Cross-Frames for I-Section Members, cross-frames 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 cross-frames 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 cross-frames along the skewed interior support line may be considered at the discretion of the Owner. Where discontinuous intermediate diaphragm or cross-frame lines are employed normal to the girders in the vicinity of that s upport line, a skewed or normal diaphragm or cross-frame should be matched with each bearing that resists lateral force (AASHTO, 2012). As research has shown, placing a cross-fr ame normal to the girders and at the bearing location of a skewed support, provide s an alternate load path and attracts a
12 significant amount of force in that crossframe. NCHRP Report 725 referred to these cross-frames 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 cross-frame along the skew at the supports. Staggered cross-frame layout configurations appear to induce the least amount of forces within its cross-frames. This is esp ecially true and desira ble for interior crossframes closest to highly skewed supports Staggered cross-frames allow for more flexibility in the system and therefore attract less load. Al so, K-frame type cross-frames tend to be the better choice over X-frame type cross-frames (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 self-weight, 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 mid-span. There are, theoretically, no lateral deflections or twisting. However, as cross-frames 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 s-frames to the girders during erection. This is known as Steel Dead Load Fit (SDLF) and Total Dead Load Fit (TDLF)
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 cross-frames 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 non-composite 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 cross-frames are skewed parall el to the support, which is typically standard practice, these end cross-frames contribute to the rotations and transverse movements described above. The end cross-fram 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 non-skewed 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 mid-span 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
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 fit-up condition assumed during the design of the girders and cross-frames. Otherwise, unintended locked-in 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 cross-frames shall be designed:
15 No-Load 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 cross-frames 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 fit-up issues, produce stability effects and s econd-order 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 cross-frames do not fit-up with the connection work points on the initi ally fabricated girders. During fit-up of cross-frames 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 cross-frame 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 fit-up as seen in Figure III.1.
16 Figure III.1 Typical Fit-up Procedure for Skewed I-Girders (AASHTO/NSBA, 2003)
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 locked-in fo rces during erection when the girders are forced to fit up with the stiffer cross-frames. In many cases especially for strai ght girders on skewed supports, locked-in 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 locked-i n forces from fit-up and forces due to differential deflections between adjacent girder s can be additive. Or in highly skewed straight bridges, if the first intermediate cro ss-frames are too close to the bearing line, the locked-in cross-frame 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 cross-frame 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 time-consuming and with a large number of variables, it leaves a lot of room for error. 1D and 2D simplified analysis are much less time-consuming and therefore preferred by engineers for the design of non-complex 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 2D-grid and 3D-FEA. It is ultimately left up to engineering
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 cross-frames are taken into account; however, most common 2D software, such as DESCUS and MDX, use equivalent beam element properties when modeling the cross-frames. As discussed in NCHRP Report 725, how these common 2D so ftware compute the equivalent beam element properties for the cross-frames 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
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 3D-FEA 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 cross-frames 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 Cross-Frame Modeling Most designers use the methods descri bed in the AASHTO/ NSBA (2011) G13.1 document for finding the equivalent beam sti ffness of cross-frames in 2D analysis models. There are two approaches here:
20 1. Calculate the equivalent mo ment of inertia based on th e flexural analogy method. In a model of the cross-frame, a unit force couple is applied to one end to find the equivalent rotation that is then used to back-calculate the equivalent moment of inertia. 2. Calculate the equivalent mo ment of inertia based on the shear analogy method. In a model of the cross-frame, a unit vertical force is applied to one end to find the equivalent deflection that is then used to back-calculate the equivalent moment of inertia (AASHTO/NSBA, 2011). Both methods use Euler-Bernoulli 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 cross-frame equivalent beam stiffness. This approach includes an equivalent shear area for a shear-deformable beam element repres entation (Timoshenko beam theory) of the cross-frame. 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
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 cross-frame. The cross-frame 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 cross-frame behavior compared to the Euler-Bernoulli beam for all types of cross-frames (incl uding X and K type cross-frames) that are typically used in I-gir der bridges. Not only are the cal culated forces more accurate but the deflections and rotations are more accura te. Predicting deflections and rotations
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 fit-up method. The cont ractor more accurately understands the deflections and rotations to expect during c onstruction and the forces necessary for the chosen fit-up method. The engineer can more accurately and efficiently design the girders and cross-frames for the expected movements and locked-in forces from fit-up and final condition loads. I-Girder Torsion Modeling Current practice in 2D-grid 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 cross-frame 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 RISA-3D. 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 open-section thin-walled beam
23 associated with warping fixity as each end of a given unbraced length (cross-frame spacing) is: Where Lb is the unbraced length between the cross-frames, J is the St. Venant torsional constant, and p2 is defined as GJ/ECw. Assuming warping fixity at the intermediate cross-frame locations leads to a reasonably accurate characterization of the girder torsional stiffness (NCHRP, 2012). I-Girder Flange Lateral Bending Modeling AASHTO LRFD Bridge Specifications section C126.96.36.199.4b provides a simplified equation to calculate the lateral moment fo r a horizontally curved girder based on the radius, major-axis 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 cross-frame or diaphragm due to the use of discontinuous cross-frame 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 cross-frames or diaphragms, these values need not be c onsidered. Lateral flange bending in the exterior girders is substantially reduced when cross-frames or diaphragms are placed in discontinuous lines over the entire bridge due to the reduced cross-frame 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 major-axis dead and
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 cross-frame, utilizing the forces delivered to the flanges from the cross-frames 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 Fixed-End Conditions (NCHRP, 2012) Calculation of Locked-In Forces Due to Cross-Frame Detailing Regardless the type of analysis used (2D-grid, 2D-frame, or 3D-FEA), the analysis essentially assumes a NLF condition unless the lock ed-in forces are accounted for in the model. Any lock-in forces, due to the lack of fit of the cross-frames 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 locked-in forces
25 tend to be opposite in sign to the internal forc es due to dead loads. Therefore the 2D-grid or 3D-FEA analysis solutions for cross-fram e forces and flange lateral bending stresses are conservative when SDLF or TDLF initial fit-up 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 locked-in cross-frame forces. Figure III.3 shows four configurati ons that visually expl ain how the locked-in 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 cross-frames 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 lack-of-fit 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 s-frame ends. In 3D-FEA, 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
26 multiplied by the cross-frame member areas to determine the axial forces. In 2D-grid models that use equivalent beam elements for the cross-frames, the displacements calculated above are converted into beam end displacements and end rotations. Assuming fixed-end conditions, the end displace ments are used to calculate the fixed-end forces, which are then applied to the e quivalent cross-frame beam element. Figure III.3 Conceptual Configurations A ssociated with Dead Load Fit (TDLF or SDLF) Detailing (NCHRP, 2012)
27 Figure III.3 (Continued) Conceptual Configurations Associated with Dead Load Fit (TDLF or SDLF) Detailing (NCHRP, 2012)
28 The behavior of the end cross-frames at skewed bearing lines is slightly different, however the locked-in forces due to cross-fram e fit-up 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 cross-frames 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 cross-frame. In order for the skewed end cross-frame to fit up with the girders in Configuration 2, the cross-frame 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 no-load, 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 I-gird 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 I-girder 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
29 this point, the stress ra tio of flange lateral bending st ress over major-axis bending stress ( fl/fb) increases above 0.30 where cross-frames 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 major-axis bending responses. Below this level the vertical components of the forces from the cross-frames are too small to noticeably influence the major-axis be nding response (NCHRP Appendix C, 2012). NCHRP Report 725 provides a matrix of gr ades for traditional 2D-grid 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), 2D-grid and 1D-line girder analysis receive really poo r grades. However, it should be noted that the recommended improvements to 2D-grid 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 break-down 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
30 F: over 30% normalized mean error, reflecting unreliable accuracy and inadequate for design (NCHRP, 2012). NCHRP Report 725 also provides recomm endations for cross-frame detailing methods for straight skewed I-girder 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) cross-frame fo rces and girder flange lateral bending stresses will essentially be canceled out by the TDLF locked-i n forces. With a low skew index level, the forces required for crossframe fit-up during steel erection are very manageable. Ensuring that the first intermed iate cross-frames are a minimum distance offset from centerline of bear ing, will help alleviate nuisanc e stiffness effects and reduce fit-up forces by providing enough flexibility at th e end of girder to force the girders into position with the relatively stiff cross-fr 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).
31 Figure III.4 Matrix of Grades for Reco mmended Level of Analysis for I-Girder Bridges (NCHRP, 2012)
32 For straight skewed I-girder 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 s-frame fit-up 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 cross-frame 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).
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 2D-grid base model, an improved 2D-grid model, a 2D -frame base model, an improved 2D-frame model, a 3D-FEA NLF-detailing model, a nd a 3D-FEA TDLF-detailing model. The results for major-axis 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 Design-Build 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 SR-114 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
34 category as per NCHRP Report 725 and as seen in Figure III.4. There are nine steel plate I-girders spaced at 11Â’-0 Â” on center and 7Â’-5Â” overhangs. A ll structural steel conforms to AASHTO M 270-50W, 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)
Figure IV.2 Plan Layout of the Geneva Road Bridge (Parsons, 2011)
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 I-girders. 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 cross-frames are K-type cross-frames with WT members for the bottom, top, and diagonal chords. The interior cross-fram 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 cross-frame near each obtuse corner of the framing plan was removed. Those cross-frames 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 cross-frames at
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 cross-frame at the obtuse corners came from the article, Â“Des ign and Construction of the Curved and Severely Skewed Steel I-Girder East-West Connector Bridges over I-88.Â” The article explains how non-skewed 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 cross-frames that are in-l ine with the removed cross-frames 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 cross-frames. 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 oss-frames 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 cross-frame in orde r to reduce the cross-frame loads and mitigate nuisance stiffness effects. However, arrangi ng the cross-frames 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 2D-grid 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.
Figure IV.4 Girder Elevation of the Geneva Road Bridge (Parsons, 2011)
39 Figure IV.5 Framing Plan of the Geneva Road Bridge (Parsons, 2011)
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 cross-frames 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 cast-in-place 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 2D-grid models and the 3D structural analysis software,
41 LARSA 4D with the Steel Bridge Module, is us ed for the other four models that include 2D-frame and 3D-FEA. 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 cross-frames, 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 locked-in cross-frame forces are included in the anal ysis. The final improvement includes adding the locked-in cross-frame forces due to TDLF detailing for the 3D-FEA model. The theory behind calculating more accura te lateral flange bending stresses is based on assuming a staggered cross-frame layout is used. Since the cross-frames are continuous, the lateral flange bending stresses will not be calculated as per the outlined 2D-grid improvements in the analysis of th e example bridge. This improvement would have been a post-processing step and will co ntinue to be one unless 2D-grid software companies choose to re-write their code a nd implement it directly into the software. 2D-Grid 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:
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. 2D-Grid Improved Model Â– MDX This model includes any possible reco mmended improvements to a 2D-grid 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 cross-frame 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 locked-in cross-frame forces for TDLF or SDLF detailing.
43 That leaves adjusting the equivalent beam stiffness assumed for cross-frames as the only improvement that can be implemented in the 2D-grid model. The user has the option of inputting the cross-frame type (K-type, X-type, or diaphragm) and the associated member sizes or manually input the equivalent cross-frame properties. If the first option is chosen, the software automa tically converts the cross-frame 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. 2D-Frame 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 2D-grid softwa re. There are two methods of modeling a steel girder structure: 2D-frame and 3D-FEA 2D-frame 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
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. 2D-Frame 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 cross-frames are modeled in 3D, therefore computing the equiva lent beam stiffness is unnecessary. The flange lateral bending stresses are automatically computed. Locked-in crossframe forces due to TDLF detailing are only analyzed in the 3D-FEA TDLF model for ease of comparison with the 3D-FEA NLF model. 3D-FEA NLF Model Â– LARSA 4D 3D-FEA 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 cross-frames 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 cross-frames are modeled in 3D
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 locked-in cross-frame 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 full-size test bridge. However, this is out of the scope of this thesis and the 3D-FEA NLF model is used as the benchmark and assumed to be the most accurate. The major-axis 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 major-axis 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 cross-frame 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. 3D-FEA TDLF Model Â– LARSA 4D The only improvement needed for this m odel is including the initial locked-in cross-frame forces due to TDLF detaili ng. The locked-in cross-frame forces are calculated by determining the axial strain of the truss type members of the cross-frames due to the camber differences for total dead load differential deflections. These initial
46 strains are inputted into the model as an equi valent thermal strain load. See Appendix A for example calculations.
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 I-girder bridges. Six models were created, analyzed, and result s compared. The results include: Major-axis bending stresses Vertical displacements Cross-frame 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 2D-grid and 1D-linear 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 3D-FEA NLF LARSA 4D model is a ssumed to be the most accurate and therefore used as the benchmark against wh ich all other 2D-grid and 2D-frame models are compared. The 2D-grid MDX models (bas e and improved) are first compared to the 3D-FEA NLF LARSA 4D model. Next, th e 2D-frame LARSA 4D models (base and improved) are compared to the 3D-FEA NL F LARSA 4D model. Finally, the 3D-FEA NLF LARSA 4D model is compared to th e 3D-FEA TDLF LARS A 4D model.
48 2D-Grid Models Major-axis bending stresses. The average grade of traditional 2D-grid analyses for major-axis girder bending stresses as re ported by NCHRP is a C and the worst-case 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 major-axis 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 2D-grid MDX models (base and improved) are compared to the 3D-FEA 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 3D-FEA 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 cross-frame 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 cross-frames. 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 cross-frames 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
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 Major-Axis Bending Stress of Girder 1 Top Flange Figure V.2 Major-Axis 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
50 Vertical Displacements. The average grade of traditional 2D-grid analysis for girder vertical displacements as reported by NCHRP is a C and the worst-case grade is a D. Figure V.3 compares the vertical disp lacements due to noncomposite dead loads among the 2D-grid MDX models (base and im proved) and the 3D-FEA 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
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. Cross-Frame Forces. The average and worst-case grade of traditional 2D-grid analysis models for cross-frame 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 cross-frame forces due to noncomposite dead loads for the 2D -grid MDX base model and 3D-FEA LARSA 4D model for each member of the cross-fr ames. The cross-frames along bay 2 between girders 2 and 3 are shown. Figure V.4 Cross-Frame 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
52 Figure V.4 (Continued) Cross-Frame 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
53 Figure V.4 (Continued) Cross-Frame 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
54 The pattern of cross-frame me mber forces along bay 2 is similar for both analysis models. A spike in axial load in all member s at cross-frames 11 and 12 can clearly been seen. This illustrates the increased loads th at occur at the obtuse corners. However, a significant difference in cross-frame forces can clearly be seen between the two models from the graphs. The normalized mean error in comparison to the 3D-FEA 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 cross-frames with the highest fo rces. With percenta ges of error well over 30% for these critical cross-frames, the gr ade should be an F and the method deemed unacceptable for calculating cross-frame 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 Cross-Frame Forces in the 2D-Grid MDX Model Flange Lateral Bending Stresses. The average and worst-case grade of traditional 2D-grid 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 post-processing using the cross-frame forces would need to be comp leted. Since the cross-frame forces results received a grade of F for the 2D-grid model, calculation results for flange lateral bending stresses would be inaccurate as well. Theref ore, these calculations were not performed.
55 Girder Layover at Bearings. The average grade of traditional 2D-grid analysis models for girder layover at bearings as reported by NCHRP is a C and the worst-case 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. 2D-Frame Models Major-axis bending stresses. Figure V.5 compares the unfactored major-axis 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 2D-frame LARSA 4D models (base and improved) are compared to the 3D-FEA 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 2D-frame 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 2D-frame LARSA 4D models (base and improved) and the 3D-FEA NLF LARSA 4D mode l along Girder 1 and Girder 3.
56 Figure V.5 Major-Axis 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
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 mid-span 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
58 Cross-Frame Forces. Figure V.7 compares the unfact ored cross-frame forces due to noncomposite dead loads for the 2D-frame LARSA 4D models (base and improved) and 3D-FEA LARSA 4D model for each member of the cross-frames. The cross-frames along bay 2 (between girders 2 and 3) are shown. The pattern of cross-frame me mber forces in the graph along bay 2 is similar for all models, except for the end cross-frames in the 2D-frame models. The spike in axial load in all members at cross-frames 11 and 12 illustrates the increased loads that occur at the obtuse corners and is much more accura tely represented in both 2D-frame LARSA 4D models versus the 2D-grid MDX models. Figure V.7 Cross-Frame 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
59 Figure V.7 (Continued) Cross-Frame 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
60 Figure V.7 (Continued) Cross-Frame 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
61 A significant difference in cross-frame for ces can clearly be seen for the end and first interior cross-frames between the im proved 2D-frame LARSA 4D model and the 3D benchmark. The Â“improvementsÂ” to the gird er torsional stiffne ss appears to provide inaccurate results for cross-frame forces near the girder ends and provide no noticeable improvement over the 2D-frame base model. The normalized mean error ranges from 3.1% to 6.8% for the 2D-frame base model and from 6.0% to 12.0% for the 2D-frame 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 oss-Frame Forces in the 2D-Frame LARSA 4D Models Flange Lateral Bending Stresses. Figure V.8 compares the unfactored flange lateral bending stresses due to noncomposite dead loads for the 2D-frame LARSA 4D base model and 3D-FEA 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.
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
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
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 cross-frame 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 cross-frame or diaphragm due to the use of discontinuous cr oss-frame 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 cross-frames 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 cross-frames 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 2D-frame models (base and improved) and the 3D-FEA 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.
65 The normalized mean error for the 2D-frame 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 2D-frame 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 fit-up 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
66 3D-FEA Model with TDLF Detailing The 3D-FEA TDLF model incl udes the initial locked-in cr oss-frame forces due to TDLF detailing. Results are compared to the 3D-FEA NLF model that by default assumes no initial cross-frame forces and the gi rder webs are plumb in the no-load case. Results at different constructi on stages are also compared. Major-axis bending stresses. As seen in Figure V.10, the major-axis 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 Major-Axis 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
67 Vertical Displacements. The vertical displacements due to dead loads on noncomposite girder sections are also very similar between the 3D-FEA 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
68 Cross-Frame Forces. Figure V.12 compares the unfactored cross-frame forces due to the girder, cross-frame, and deck noncomposite dead loads for the 3D-FEA NLF and TDLF models, and the forces due to only the steel gird er and cross-frame noncomposite dead loads for the 3D-FEA TDLF model. For most of the cross-frame members with TDLF detailing, the maximum fo rce in the cross-frames occurs during the fit-up of the cross-frames with the girders. The girders are twisted and forced into an out of plumb orientation and as dead loads are applied, the cross-frame forces are offset or relieved as the girders twist back into th e vertically plumb pos ition under total dead loads. Figure V.12 Cross-Frame 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
69 Figure V.12 (Continued) Cross-Frame 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
70 Figure V.12 (Continued) Cross-Frame 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
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 3D-FEA NLF and 3D-FEA TDLF models. The TDLF model results incl ude two construction stages Â– steel girder and cross-frame 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
72 As expected the flange lateral bending stress is minimal at the total dead load case when using TDLF detailing. The results for the cross-frame 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 3D-FEA NLF and TDLF models with girders, cross-frames, and d eck dead loads and th e 3D-FEA TDLF model with the steel girders and cross-frame 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 locked-in cross-frame 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 3D-FEA 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 3D-FEA 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
73 intent of this figure is to validate the beha vior found in the 3D-FEA 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
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
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 I-girder 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 locked-in for ces, which will counterbalance to a certain extent the cross-frame loads, lateral deflectio ns, and rotations caused by the dead loads. If these locked-in forces are not included, the designer is assuming NLF detailing, which can lead to overly conservative cross-frames forces and lateral flange bending stresses in cases of very large skews and with the pres ence of nuisance stiffness cross-frames close to the bearing line. Correctly modeling the behavior for the chosen detailing method is important to develop an accurate and efficient design. The 3D-FEA 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 3D-FEA model. Cr eating the model, running through the analysis, and sorting through the massive amount of output data can be very time-
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 3D-FEA 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 I-girder example bridge in the 3D-FEA 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 full-sized test bridge with stress gauges and with similar geometry is needed to truly validat e the assumptions made in the creating and analyzing the 3D-FEA m odel. This was considered outside the scope of this thesis. The 2D-grid method of analysis for highly skewed I-girder bridges appears to be the least accurate for calculating cross-fram e forces for nuisance stiffness cross-frames and lateral flange bending stresses. 2D-g 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 cross-frames, designers need to do a significant amount of post-processing calculations to check for additional cross-frame forces and later flange bending stresses that may not have been captured in the 2D-grid software analysis. NCHRP Report 725 repeat edly encouraged the bridge software industry to implement these improvements into the software.
77 The 2D-frame method using the LARSA 4D software or something similar, currently appears to be the most efficient me thod to use for highly skewed steel I-girder bridge design. LARSA 4D c ontains a steel bridge module that makes creating the model much easier, similar to the 2D-grid 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 highly-skewed steel I-girder bridges. With this flexibility, the necessity of pos t-processing can be eliminated or greatly reduced. Compared to 3D-FEA models, 2D-frame model output is much more manageable and therefore less time-cons uming and efficient for the designer. General Recommendations and Future Work Current Recommendations Recommendations for designers currently analyzing highly sk ewed steel I-girder bridges include: Use the 2D-frame method with software th at includes a bridge module for easily creating the geometry. Include improvements to the 2D-frame 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 cross-frame layouts. When contiguous cross-frames are used throughout each span, a more detailed analysis should be used to analyze th e cross-frame forces and lateral flange bending stresses near the girder ends.
78 Include initial locked-in cross-frame forces when TDLF or SDLF detailing is used to get a more efficient design. The 3D-FEA method should be used on a limited basis to verify behavior and check localized stresses. It is a good tool to use to check cross-frame forces and lateral flange bending stresses; however it is too cumbersome to use as an allencompassing analysis. The 2D-grid method is not recommended for steel I-girder bridges with a high skew index until improvements are made to the software or the designer decides to accompany this analysis with extensive post-processing. Future Work Considerations Future work considerations and recommendations include: Encourage the bridge software indus try to implement the recommended improvements to 2D-grid software. Encourage the 3D-capable bridge softwa re industry to implement improvements to steel bridge design modules to accu rately and easily include appropriate locked-in cross-frame 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 cross-frame layouts. Specifically, analyze the girders and cross-frames with contiguous crossframe layouts and nuisance stiffness crossframe near bearing supports. Adjust the 2D analysis method improvements accordingly.
79 Research fit-up practices typically used in highly-skewed steel I-girder erection. Determine at what point the fit-up forces for TDLF detailing become too large. Research and analyze more innovative cross-frame configurations, including partially skewed cross-frames, lean-on bracing, temporary bracing, and different connection and bearing plate detailing.
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), 361-374. 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 I-Gi rder East-West Connector Bridges over I88. 27th Annual International Bri dge Conference 2010. IBC-10-24. 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 12-79. Parsons Corporation (2011). Design Plans for SR-114 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 Highly-Skewed and Curved Steel I-Girder Bridges. Thesis submitted to the Univer sity of Colorado Denver. ProQuest LLC.
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.
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
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
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
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
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
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
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