SHEAR IN SKEWED MULTI-BEAM BRIDGES

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1 20-7/Task 107 COPY NO. SHEAR IN SKEWED MULTI-BEAM BRIDGES FINAL REPORT Prepared for National Cooperative Highway Research Program Transportation Research Board National Research Council Modjeski and Masters, Inc. NCHRP Project 20-7/Task 107 March 2002

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3 ACKNOWLEDGMENT OF SPONSORSHIP This work was sponsored by the American Association of State Highway and Transportation Officials, in cooperation with the Federal Highway Administration, and was conducted in the National Cooperative Highway Research Program, which is administered by the Transportation Research Board of the National Research Council. DISCLAIMER This is an uncorrected draft as submitted by the research agency. The opinions and conclusions expressed or implied in the report are those of the research agency. They are not necessarily those of the Transportation Research Board, the National Research Council, the Federal Highway Administration, the American Association of State Highway and Transportation Officials, or the individual states participating in the National Cooperative Highway Research Program. iii

4 TABLE OF CONTENTS LIST OF FIGURES... vi LIST OF TABLES... xi ACKNOWLEDGMENTS... xii ABSTRACT... xiii SUMMARY...1 CHAPTER 1 INTRODUCTION AND RESEARCH OBJECTIVES Introduction Research Objectives...13 CHAPTER 2 LITERATURE REVIEW Introduction NCHRP Project Ontario Highway Bridge Design Code Additional Work...19 CHAPTER 3 METHODOLOGY...26 CHAPTER 4 STUDY FINDINGS Simple Span Beam-Slab Bridge Models Live Load Shear Along Exterior Beam Length Influence of Skew Angle Influence of Beam Stiffness Influence of Span Length Influence of Intermediate Cross Frames Influence of Beam Spacing Influence of Slab Thickness Influence of Bridge Aspect Ratio Live Load Shear Across Bearing Lines Simple Span Concrete T-Beam Bridge Models Live Load Shear Along Exterior Beam Length Live Load Shear Across Bearing Lines...71 TABLE OF CONTENTS (continued) iv

5 4.3 Simple Span Spread Concrete Box Girder Bridge Models Two-Span Continuous Beam-Slab Bridge Models Simple Span vs. Two-Span Correction Factors at Obtuse Corners of Abutments Correction Factors at Obtuse Corners of Abutments and Piers Live Load Shear Along Exterior Beam Length Influence of Skew Angle Influence of Beam Stiffness Influence of Span Length Live Load Shear Across Abutment Bearing Lines Live Load Shear Across Pier Live Load Reactions at Pier Skew Correction Factors from LRFD Specifications and Research Results CHAPTER 5 INTERPRETATION AND APPLICATION Simple Span Beam-Slab Bridges Simple Span Concrete T-Beam Bridges Simple Span Spread Concrete Box Girder Bridges Two-Span Continuous Beam-Slab Bridges Application of Study Findings CHAPTER 6 CONCLUSIONS AND SUGGESTED RESEARCH REFERENCES APPENDIX A ANALYSIS MATRICES... A-1 APPENDIX B CROSS SECTIONS AND FRAMING PLANS OF BRIDGE MODELS...B-1 v

6 LIST OF FIGURES Figure 1. Typical Beam and Slab Superstructures...7 Figure 2. Plan View of Typical Skewed Superstructure. Current Application of the Skew Correction Factor for Shear per the AASHTO LRFD Bridge Design Specifications...11 Figure 3. General Truck Placement Pattern used in NCHRP 12-26/1 for Maximum Shear...17 Figure 4. Bridge Plan Geometries for Analysis...27 Figure 5. Schematic Diagram of BSDI Finite Element Modeling...33 Figure 6. Transformation of Concrete Section to Steel Section...33 Figure 7. Procedure for Calculation of the Normalized Skew Corrections...37 Figure 8. Effect of Skew Angle on Skew Corrections Along Exterior Beams...42 Figure 9. Effect of Skew Angle on Skew Corrections Along Exterior Beams...42 Figure 10. Effect of Girder Stiffness on Skew Corrections Along Exterior Beams...45 Figure 11. Effect of Girder Stiffness on Skew Corrections Along Exterior Beams...45 Figure 12. Effect of Girder Stiffness on Skew Corrections Along Exterior Beams...46 Figure 13. Effect of Girder Stiffness on Skew Corrections Along Exterior Beams...46 Figure 14. Effect of Span Length on Skew Corrections Along Exterior Beams...48 Figure 15. Effect of Span Length on Skew Corrections Along Exterior Beams...48 Figure 16. Effect of Span Length on Skew Corrections Along Exterior Beams...49 Figure 17. Figure 18. Effect of Intermediate Cross Frames on Skew Corrections Along Exterior Beams...52 Effect of Intermediate Cross Frames on Skew Corrections Along Exterior Beams...52 vi

7 LIST OF FIGURES (continued) Figure 19. Effect of Beam Spacing on Skew Corrections Along Exterior Beams...54 Figure 20. Effect of Slab Thickness on Skew Corrections Along Exterior Beams...56 Figure 21. Effect of Bridge Aspect Ratio on Exterior Girder Shear...59 Figure 22. Figure 23. Figure 24. Figure 25. Figure 26. Figure 27. Effect of Skew Angle on End Shear Skew Corrections...62 Effect of Skew Angle on End Shear Skew Corrections...62 Effect of Girder Stiffness on End Shear Skew Corrections...63 Effect of Girder Stiffness on End Shear Skew Corrections...63 Effect of Girder Stiffness on End Shear Skew Corrections...64 Effect of Girder Stiffness on End Shear Skew Corrections...64 Figure 28. Effect of Span Length on End Shear Skew Corrections...65 Figure 29. Effect of Span Length on End Shear Skew Corrections...65 Figure 30. Effect of Span Length on End Shear Skew Corrections...66 Figure 31. Effect of Intermediate Cross Frames on End Shear Skew Corrections...66 Figure 32. Effect of Intermediate Cross Frames on End Shear Skew Corrections...67 Figure 33. Effect of Slab Thickness on End Shear Skew Corrections...67 Figure 34. Complete Results for End Shear Skew Corrections...68 Figure 35. Average Variation of End Shear Skew Corrections for Simple Span Beam-Slab Bridges...68 Figure 36. Effect of Skew Angle on Skew Corrections Along Exterior Beams...70 Figure 37. Figure 38. Effect of Skew Angle on End Shear Skew Corrections...72 Comparison of Simple Span and Two-Span Continuous Skew Correction Factors...80 vii

8 LIST OF FIGURES (continued) Figure 39. Comparison of Skew Correction Factors at Abutments and Pier...83 Figure 40. Figure 41. Figure 42. Figure 43. Figure 44. Figure 45. Figure 46. Nomenclature for Investigation of Correction Factors Along the Length of the Exterior Girders of Two-Span Continuous Bridge Models...85 Effect of Skew Angle on Skew Corrections Along Exterior Beams of Continuous Models...87 Effect of Beam Stiffness on Skew Corrections Along Exterior Beams of Continuous Models...90 Effect of Beam Stiffness on Skew Corrections Along Exterior Beams of Continuous Models...90 Effect of Span Length on Skew Corrections Along Exterior Beams of Continuous Models...92 Effect of Span Length on Skew Corrections Along Exterior Beams of Continuous Models...92 Nomenclature for Investigation of Correction Factors Across the Abutment Bearing Lines of Two-Span Continuous Bridge Models...94 Figure 47. Effect of Skew Angle on End Shear Skew Corrections At Abutments...95 Figure 48. Effect of Girder Stiffness on End Shear Skew Corrections At Abutments...95 Figure 49. Effect of Girder Stiffness on End Shear Skew Corrections At Abutments...96 Figure 50. Effect of Span Length on End Shear Skew Corrections At Abutments...96 Figure 51. Effect of Span Length on End Shear Skew Corrections At Abutments...97 Figure 52. Figure 53. Figure 54. Complete Results Set for End Shear Skew Corrections At Abutments...99 Average Variation of End Shear Skew Corrections Across Abutments of Two-Span Continuous Beam-Slab Bridges Nomenclature for Investigation of Correction Factors Across the Pier of Two-Span Continuous Bridge Models viii

9 LIST OF FIGURES (continued) Figure 55. Effect of Skew Angle on Skew Corrections for Shear Across Pier Figure 56. Effect of Girder Stiffness on Skew Corrections for Shear Across Pier Figure 57. Effect of Girder Stiffness on Skew Corrections for Shear Across Pier Figure 58. Effect of Span Length on Skew Corrections for Shear Across Pier Figure 59. Effect of Span Length on Skew Corrections for Shear Across Pier Figure 60. Complete Results Set for Skew Corrections for Shear Across Pier Figure 61. Figure 62. Average Variation of Skew Corrections for Shear Across Piers of Two-Span Continuous Beam-Slab Bridges Nomenclature for Investigation of Correction Factors for Reaction at the Pier of Two-Span Continuous Bridge Models Figure 63. Comparison of Skew Correction Factors for Shear and Reaction at Pier Figure 64. Comparison of Skew Correction Factors for Shear and Reaction at Pier Figure 65. Comparison of Skew Correction Factors for Shear and Reaction at Pier Figure 66. Comparison of Skew Correction Factors for Shear and Reaction at Pier Figure 67. Comparison of Skew Correction Factors for Shear and Reaction at Pier Figure 68. Effect of Skew Angle on Skew Corrections for Reaction Across Pier Figure 69. Effect of Girder Stiffness on Skew Corrections for Reaction Across Pier Figure 70. Effect of Girder Stiffness on Skew Corrections for Reaction Across Pier Figure 71. Effect of Span Length on Skew Corrections for Reaction Across Pier Figure 72. Effect of Span Length on Skew Corrections for Reaction Across Pier Figure 73. Results for the Variation of the Skew Correction Along the Length of the Exterior Girders of Simple-Span Beam-Slab Bridges ix

10 LIST OF FIGURES (continued) Figure 74. Figure 75. Figure 76. Figure 77. Figure 78. Figure 79. Figure 80. Average Results for the Variation of the Skew Correction Along the Bearing Lines of Simple-Span Beam-Slab Bridges Results for the Variation of the Skew Correction Along the Length of the Exterior Girders of Two-Span Continuous Beam-Slab Bridges Average Results for the Variation of the Skew Correction Across Abutments and Piers of Two-Span Continuous Beam-Slab Bridges Proposed Variation of the Skew Correction Factors for Shear Along the Length of the Exterior Girders in Simple Span Superstructures of Concrete Deck, Filled Grid, or Partially Filled Grid on Steel or Concrete Beams; Concrete T-Beams, T- and Double T Sections Proposed Variation of the Skew Correction Factors for Shear Along the Length of the Exterior Girders in Continuous Superstructures of Concrete Deck, Filled Grid, or Partially Filled Grid on Steel or Concrete Beams; Concrete T-Beams, T- and Double T Sections Proposed Variation of the Skew Correction Factors for Shear Across the Bearing Lines of Simple Span Superstructures of Concrete Deck, Filled Grid, or Partially Filled Grid on Steel or Concrete Beams; Concrete T-Beams, T- and Double T Sections Proposed Variation of the Skew Correction Factors for Shear Across the Abutments and Piers of Continuous Superstructures of Concrete Deck, Filled Grid, or Partially Filled Grid on Steel or Concrete Beams; Concrete T-Beams, T- and Double T Sections x

11 LIST OF TABLES Table 1. Table 2. Correction Factors for Load Distribution Factors for Support Shear of the Obtuse Corner...9 Maximum Shear Forces at Pier Support for Three-Lane Bridge with Different Skew Angles Predicted Using Different Methods...22 Table 3. Base Analysis Matrix for Beam and Slab Bridges...27 Table 4. Average NCHRP and Base Parameters for Beam-Slab Bridge Models.. 30 Table 5. Table 6. Table 7. Table 8. Table 9. Table 10. Table 11. Average NCHRP and Base Parameters for Concrete T-beam Bridge Models...30 Average NCHRP and Base Parameters for Spread Concrete Box Girder Bridge Models...30 Comparison of Maximum Live Load Shears from BSDI and an LRFD Line Girder Analysis...76 Comparison of Skew Correction Factors for End Shear of Exterior Girders at the Obtuse Corners of Simple Span and Two-Span Bridge Models...80 Comparison of Skew Correction Factors for Shear of Exterior Girders at the Obtuse Abutment Corners and Obtuse Pier Corners of Two-Span Bridge Models...83 Correction Factors for Reaction at the Pier of Two-Span Beam-Slab Bridge Models Comparison of Skew Correction Factors from LRFD Specifications and Research Results xi

12 ACKNOWLEDGMENTS The authors acknowledge and appreciate the assistance provided by Wagdy G. Wassef, Ph.D., during the analysis and interpretation of the finite element models of this research. Additionally, Chris W. Smith assisted in the development of the bridge models and Adnan Kurtovic assisted in the post processing of the bridge models. Their efforts are greatly appreciated. The authors also appreciate the efforts of Dann Hall and Rich Lawin of Bridge Software Development International, LTD., who performed the finite element analysis of the bridge models in this study. xii

13 ABSTRACT This report documents an investigation of the skew correction factors for live load shear and the development of design guidelines for the variation of the skew correction factors along the exterior beam length and across the end bearing lines of simple span and two-span continuous beam and slab bridges. The report also documents an investigation of skew correction factors for live load reactions at the piers of two-span continuous bridges. The research was performed through finite element analysis of 41 bridge models. The study findings suggest that a reasonable approximation for the variation of the skew correction factor along the length of exterior girders of superstructures consisting of concrete decks, filled grids, or partially filled grids on steel or concrete beams; concrete T-beams, T- and double T sections is a linear distribution of the factor from its value at the obtuse corner of the bridge, determined according to the AASHTO LRFD Bridge Design Specifications (LRFD Specifications), to a value of 1.0 at girder mid-span. Similarly, the skew correction factor variation across the bearing lines of those bridges may be approximated by a linear distribution of the correction factor from its value at the obtuse corner of the bridge, determined according to the LRFD Specifications, to a value of 1.0 at the acute corner of the bridge. The variations of the skew correction factors for shear along the length of exterior girders and for shear across both the abutments and piers of continuous bridges are identical to those proposed for simple span bridges. Skew correction factors for reaction at the piers of continuous bridges are present and are unique from those calculated for shear at the piers. From the limited data, however, accurate empirical equations for the correction factor or its variation across the pier could not be derived. Therefore, the development of such equations for continuous bridges is necessary. xiii

14 SUMMARY This research focused on an investigation of the skew correction factors for live load shear defined in Article c of the AASHTO LRFD Bridge Design Specifications (LRFD Specifications) 1. The LRFD Specifications stipulate that the skew correction factors for shear, derived in NCHRP Project for exterior beams at obtuse corners of skewed, simple span bridges, be applied not only to the end shear of the exterior beams, but also to the end shear of each beam in the bridge cross section. During the development of the skew correction factors, however, variation of the effect of skew on the end shear of interior beams was not investigated. Additionally, the effect of skew on shear along the length of exterior beams of beam and slab bridges was not investigated in NCHRP Project The objective of this research, therefore, was the development of design guidelines for the variation of the skew correction factor for shear along the exterior beam length and across the end bearing lines of simple span beam and slab bridges. This study also investigated a limited number of two-span continuous bridge models and the variation of the skew correction factor for shear in these bridge types. Additionally, the need for skew correction factors for live load reactions at the piers of continuous bridges was investigated. The research was performed through finite element analysis of 41 bridge models, including 25 simple span beam-slab models, 3 simple span concrete T-beam models, 4 simple span spread concrete box girder models and 9 two-span continuous beam-slab models. The influence of skew angle, beam stiffness, span length, intermediate cross frames, beam spacing, slab thickness and bridge aspect ratio on the skew correction factor variation was investigated. 1

15 For the simple span bridge models studied, the research results indicate that: Regardless of changes in the aforementioned bridge parameters, a reasonable approximation for the variation of the skew correction factor along the length of exterior girders of simple span beam-slab and concrete T-beam bridges is a linear distribution of the factor from its value at the obtuse corner of the bridge, determined according to the LRFD Specifications, to a value of 1.0 at girder mid-span. Regardless of changes in the aforementioned bridge parameters, a reasonable approximation of the skew correction factor for live load shear across the bearing lines of simple span beam-slab and concrete T-beam bridges is a linear distribution of the correction factor from its value at the obtuse corner of the bridge, determined according to the LRFD Specifications, to a value of 1.0 at the acute corner of the bridge. For the two-span continuous bridge models studied, the research results indicate that: The variations of the skew correction factors for shear along the length of exterior girders in each span and for shear across both the abutments and piers of two-span continuous beam-slab bridges are identical to those proposed for simple span bridges. The correction factor variation along the exterior girder may be approximated by a linear distribution of the factor at the obtuse corner to a value of 1.0 at girder mid-span. Likewise, the variation across the abutments and piers is approximated by a linear distribution of the factor at the obtuse corner to a value of 1.0 at the acute corner. The skew correction factor defined by the LRFD Specifications is valid for the girder shear at the obtuse corners of both the abutments and piers of the continuous bridges. Skew correction factors for reaction at the piers of continuous bridges are present and are unique from those calculated for shear at the piers. From the limited continuous bridge model data of this study, however, accurate empirical equations which define the correction factor or define its variation across the pier could not be derived. Therefore, the development of such equations for continuous bridges is necessary and is recommended for further research. For application of the research findings, the recommendations are as follows: 2

16 Skew Correction Factor for Shear, Variation Along Exterior Beam Length For superstructure types Concrete Deck, Filled Grid, or Partially Filled Grid on Steel or Concrete Beams; Concrete T-Beams, T- and Double T Section, within the applicable ranges of skew angle (θ), spacing of beams or webs (S), span of beam (L) and number of beams, stringers or girders (N b ) as defined by Table c-1 of the LRFD Specifications, the skew correction factor for shear may be varied linearly from its value at the obtuse corner of the bridge, determined in accordance with the empirical equation defined in Table c-1, to a value of 1.0 at girder mid-span. This approximate variation is applicable for both simple span structures and continuous structures. For continuous structures, the skew correction factor calculated at the obtuse corner of the abutment per Table c-1 is also valid at the obtuse corners of the interior piers. Likewise, the variation of the correction factor is applicable from both the obtuse corner of the abutment and the obtuse corners of the interior piers to the girder mid-span. Skew Correction Factor for Shear, Variation Across Bearing Lines For superstructure types Concrete Deck, Filled Grid, or Partially Filled Grid on Steel or Concrete Beams; Concrete T-Beams, T- and Double T Section, within the applicable ranges of skew angle (θ), spacing of beams or webs (S), span of beam (L) and number of beams, stringers or girders (N b ) as defined by Table c-1 of the LRFD Specifications, the skew correction factor for shear may be varied linearly from its value at the obtuse corner of the bridge, determined in accordance with Table c-1, to a value of 1.0 at the acute corner of the bearing line. This approximate variation is applicable for both simple span structures and continuous structures. For continuous structures, the skew correction factor calculated at the obtuse corner of the abutment per Table c-1 is also valid at the obtuse corners of the interior piers. Likewise, the variation of the correction factor is applicable from both the obtuse corner of the abutment and the obtuse corners of the interior piers to the acute corner of the bearing lines. Additional suggested research includes an investigation of the effects of torsion on web shear in spread box girder bridges. The study results indicate that although torsion is typically 3

17 neglected in right bridges, the introduction of skew may increase torsional effects to levels that are not negligible. Without further research, however, and given the lack of substantial field documentation indicating problems with torsion and shear in skewed spread box girder bridges, the current design practices are considered to be acceptable. Finally, this study investigated only a few types of beam and slab bridges and provides recommendations regarding only superstructures consisting of concrete decks, filled grids, or partially filled grids on steel or concrete beams; concrete T-beams; or T- and double T sections. Additional research is recommended, therefore, to determine the effects of skew on shear in the remaining beam and slab bridge types included within Table c-1 of the LRFD Specifications. 4

18 CHAPTER 1 INTRODUCTION AND RESEARCH OBJECTIVES 1.1 INTRODUCTION Beam and slab bridges are basic and common elements of the national system of roadways and bridges. Examples of typical beam and slab superstructures are shown in Figure 1, and include structures such as beam-slab (i.e. steel I-beam, concrete I-beam and concrete T- beam), box girder, multi-box beam and spread box beam bridges. Design procedures for these structures are well documented and standardized through research, physical testing and development of design codes, especially for right (i.e., non-skewed) bridges. The design of skewed bridges, however, is often based more upon engineering experience and extrapolation of limited analyses, rather than upon extensive research. In fact, for many years, little was done to incorporate the effect of skew on live load distribution, with the result that many skewed bridges were designed as right bridges. This was often the case for shear design in skewed beam and slab structures. Two recent NCHRP research projects, Project and Project 12-33, focused on updating and refining the AASHTO Bridge Design Specifications, and in doing so, refined the shear design procedures for skewed beam and slab bridges. NCHRP Project focused on investigating the live load distribution in beam and slab bridges and on developing refined live load distribution formulas to be incorporated in an updated AASHTO Bridge Design Specification. The objective of Project was the development of AASHTO Bridge Design Specifications utilizing the Load and Resistance Factor Design (LRFD) methodology. This 5

19 project culminated with the publication of the first edition of the AASHTO LRFD Bridge Design Specifications (LRFD Specifications) 3 in 1994 and incorporated the refined shear design procedures for skewed beam and slab bridges developed in NCHRP

20 SUPPORTING COMPONENTS Steel Beam Precast Concrete I or Bulb- Tee Sections TYPE OF DECK Cast-in-place concrete slab, precast concrete slab, steel grid, glued/spiked panels, stressed wood Cast-in-place concrete, precast concrete TYPICAL CROSS-SECTION Closed Steel or Precast Concrete Boxes Cast-in-place concrete slab Open Steel or Precast Concrete Boxes Cast-in-place concrete slab, precast concrete deck slab Cast-in-Place Concrete Tee Beam Monolithic concrete Figure 1. Typical Beam and Slab Superstructures 1. 7

21 The current design methodology in Section 4 of the LRFD Specifications 1 for typical, right beam and slab bridges permits the use of empirical distribution factors for determination of the live load effects in bridge beams. For the mid-span bending moment and end shear of exterior beams in skewed beam and slab bridges, the LRFD Specifications provide correction factors that are to be applied to the moment and shear distribution factors, calculated for the corresponding right bridge. These empirical skew correction factors for end shear in beam and slab bridges, as defined in Table c-1 of the LRFD Specifications 1 and as shown in Table 1, have been the subject of much discussion following the adoption of the LRFD Specifications in As stated in the scope of services provided by the NCHRP for this project, Article c, Skewed Bridges, in the AASHTO LRFD Bridge Design Specifications, requires that shear in the exterior beam at the obtuse corner of the bridge shall be adjusted when the line of support is skewed. The Specifications provide correction factors for this adjustment and require that the correction factors be applied to all beams in the cross-section. In the development of these correction factors, the variation of the effect of skew on the individual beam reactions was not considered. In addition, the Specifications provide no guidance on the influence of skew on the shear along the length of the beam. The commentary to the Specifications states that the prescribed corrections are conservative. As a consequence of this conservatism some beams in the bridge are overdesigned. It is not only this conservatism that has been the topic of discussions surrounding the skew correction factors, but also the extent to which the correction factors apply to the shear along the length of the exterior girder. 8

22 Table 1. Correction Factors for Load Distribution Factors for Support Shear of the Obtuse Corner 1. Type of Superstructure Correction Factor Range of Applicability Concrete Deck, Filled Grid, or Partially Filled Grid on Steel or Concrete Beams; Concrete T- Beams, T- and Double T Section Multicell Concrete Box Beams, Box Sections Concrete Deck on Spread Concrete Box Beams Concrete Box Beams Used in Multibeam Decks 0 # 2 # # S # # L # 240 N b $ 4 0 < 2 # < S # # L # # d # 110 N c $ 3 0 < 2 # # S # # L # # d # 65 N b $ 3 0 < 2 # # L # # d # # b # 60 5 # N b # 20 Where: 2 = skew angle (degrees) N b = number of beams, stringers or girders S = spacing of beams or webs (ft) N c = number of cells in a concrete box girder L = span of beam (ft) t s = depth of concrete slab (in) b = width of beam (in) K g = longitudinal stiffness parameter (in 4 ) d = depth of beam or stringer (in) 9

23 The development of the skew correction factors for beam and slab bridges in the LRFD Specifications was part of NCHRP Project The report for that project, Distribution of Wheel Loads on Highway Bridges 4, indicated that the skew correction factors were derived for only the end shears of the exterior girders at the obtuse corners of simple span bridges. In general, the end shear tends to increase as the skew angle of the supports increases beyond approximately 15 to 20. For the LRFD Specifications, however, the working group for NCHRP conservatively extended the applicability of the correction factor to include not only the end shear at the obtuse corner of the exterior beams, but also the end shear of each beam in the bridge cross section 5, as shown in the typical skewed bridge plan of Figure 2. The working group for NCHRP also assumed that it may be reasonable to extend the correction factors for end shear of the exterior beam to the shear along the length of the exterior beam 5, but made no provisions in the LRFD Specifications to do so. During the development of the skew correction factors in NCHRP 12-26, the effect of skew on the shear along the length of the exterior beams was not investigated, and the current LRFD Specifications do not address this issue. 10

24 C Girder (Typ.) L Correction Factor Conservatively Applied to End Shear of All Girders (Typ.) C Abutment (Typ.) L Skew Angle Correction Factor Calculated for and Applied to End Shear at Obtuse Corner of Exterior Girder (Typ.) Figure 2. Plan View of Typical Skewed Superstructure. Current Application of the Skew Correction Factor for Shear per the AASHTO LRFD Bridge Design Specifications 1. 11

25 An additional topic of discussion regarding the design of skewed bridges is the treatment of reactions at interior supports of continuous spans. Based upon the NCHRP working group s previous experience with curved and simple-span skewed structures, it was speculated that skew effects also account for the reduced reaction at interior supports, and, in some cases, the uplift at the acute corner of skewed bridges 5. Intuition may suggest, therefore, that at the interior supports of continuous spans, where both an obtuse and acute corner exist opposite each other, the skew effects on shear may cancel out for determination of the total reaction. This hypothesis, however, has not yet been investigated and is not addressed in the LRFD Specifications. As a result of these outstanding issues regarding the skew correction factors for shear, this project focuses on investigating and more accurately assessing the effect of skew on end shear across bearing lines and on shear along the length of exterior beams of beam and slab bridges. This research concentrates on simple span bridges, with a cursory evaluation of twospan continuous beam-slab bridges. The importance of this topic lies in the fact that while research has been performed to determine the shear correction factor for end shears at the obtuse corners of skewed bridges, these factors also have been conservatively applied to the end shear of all beams in the cross section and, in some cases, to the shear along the length of the exterior girder, without supporting research. The possibility exists, therefore, that some beams in beam and slab bridges are over-designed for shear. Further research on this topic may enable the use of more precise skew correction factors, and hence, may result in more economical structures. 12

26 1.2 RESEARCH OBJECTIVES The main objective of this study is to develop practical and reasonably accurate design guidelines for estimating the variation of the skew correction factor for live load shear along the length of exterior beams and across the beam supports of simple-span beam and slab bridges. This study also investigates a limited number of two-span continuous bridge models to address the variation of the skew correction factor along the length of the exterior beams and across the abutments and piers of these bridge types. Additionally, the continuous models are studied to address the need for skew correction factors for live load reactions at piers. The proposed guidelines for the skew correction factors of both simple-span and two-span continuous bridges are intended to be developed in a manner suitable for incorporation into the current AASHTO LRFD Bridge Design Specifications. 13

27 CHAPTER 2 LITERATURE REVIEW 2.1 INTRODUCTION Extensive research has been performed by bridge engineers in an attempt to accurately predict the path of loads through bridges and to present the predictions in reasonably accurate, yet practical load distribution formulas for designers. Specific to beam and slab bridges, much research has been performed to develop approximate, algebraic equations for the distribution of moment and shear in right bridges. A further extension of that work is the area of research devoted to the distribution of moment in skewed beam and slab bridges. Research by Marx, et al. 6, Nutt, et al. for the NCHRP Project , Khaleel and Itani 7, Bishara, et al. 8 and Ebeido and Kennedy 9 has concentrated on moment distributions in skewed, simply-supported and continuous beam and slab bridges. The research devoted to the distribution of shear and bearing reactions in skewed bridges, however, is confined to a rather limited set of sources. 2.2 NCHRP PROJECT One of the major comprehensive studies aimed at predicting the effect of skew on the distribution of shear in beam and slab bridges was the work by Zokaie, et. al. for NCHRP Project The primary objective of NCHRP Project was to investigate the live load distribution in beam and slab bridges and develop, where necessary, more accurate live load distribution formulas to replace those specified in the AASHTO Standard Specifications for Highway Bridges (Standard Specifications) 10. While experiencing only minor revisions since 14

28 incorporation into the Standard Specifications in 1931, the S/over equations (i.e., S/5.5 or similar equations) for live load distribution provide little guidance on the treatment of skewed bridges. One goal of NCHRP Project 12-26, therefore, was aimed at developing distribution factors that would account for skew effects. The analysis of load distribution and, ultimately, the development of the new load distribution factor formulas for right beam and slab bridges in NCHRP Project 12-26, was initiated by construction of a database of 850 existing beam and slab bridges from a nationwide survey of state transportation officials. From the database, the average beam and slab bridge parameters were defined for five different bridge types: beam-slab (i.e., steel I-beam, concrete I- beam and concrete T-beam), box girder, slab, multi-box beam and spread box beam. Parametric analyses were performed by varying a single parameter at a time to determine each parameter s effect on the distribution of HS20 truck live load. The parametric studies utilized both finite element analyses and grillage analyses with a number of different software packages. From the results, new live load distribution equations for right bridges were derived to incorporate the effects of each parameter that had a significant effect on load distribution. The approximate equations developed in NCHRP Project for the skew correction factors were developed for simple span bridges utilizing the programs GENDEK5A 11 and FINITE 12 for finite element analysis. The skew correction factors were developed such that they could be applied to the newly derived distribution factors of a right bridge with the same geometric parameters as the skewed bridge under investigation. In order to incorporate the effects of each bridge parameter that had a significant impact on the load distribution of right bridges, parametric studies of skewed bridges were completed, similar to those performed for the right bridges. The live load used in the parametric studies consisted of two trucks placed transversely on the bridge cross section to maximize the girder responses. Test models of different live load placements confirmed that two trucks typically produced the governing girder 15

29 responses. The general loading condition that maximized shear at the obtuse corner of the skewed bridges is shown in Figure 3. 16

30 Figure 3. General Truck Placement Pattern used in NCHRP 12-26/1 for Maximum Shear. 17

31 From the parametric analyses, the equations for the skew correction factors for shear were derived from the ratio of the maximum exterior girder shear of a skewed bridge to that of a right bridge, each with the same geometric parameters and live load positioning. These equations, developed for the end shear of exterior beams at obtuse corners of beam and slab bridges, are presented in Article c of the AASHTO LRFD Bridge Design Specifications 1. As discussed in Section 1.1, the LRFD Specifications require that the correction factors be applied not only to the end shear of the exterior beams, but also to the end shear of each beam in the bridge cross section. During the development of the skew correction factors, however, variation of the effect of skew on the end shear of interior beams was not investigated. The application of the skew correction factors to all beams of a cross section is considered to be conservative; therefore, it is suspected that certain beams may be over-designed. Additionally, the effect of skew on shear along the length of exterior beams of beam and slab bridges was not investigated in NCHRP Project ONTARIO HIGHWAY BRIDGE DESIGN CODE The treatment of skew and its effects on load distribution are handled differently in the third edition of the Ontario Highway Bridge Design Code (OHBDC) 13 than the method utilized in the LRFD Specifications. Rather than modify the load distribution factors developed for right bridges, the OHBDC defines a limit for the skewness of a bridge, beyond which refined methods of analysis must be used. Prior to the third edition of the OHBDC, the Ontario 18

32 code implied that the measure of a bridge s skewness was only its skew angle, as the skewness limitation was defined by a skew angle of 20E (measured from centerline of bearings to a line normal to the bridge centerline). The third edition of the OHBDC, however, incorporated the work of Jaeger and Bakht 14 which indicated that the measure of bridge skewness is also a function of span length, bridge width and girder spacing. Hence, the skew limitation, ε, was redefined in the third edition to incorporate these effects, as shown in Equation 1. Bridges beyond the skewness limit of 1/18 must be analyzed using a refined method such as grillage analysis, orthotropic plate theory or finite element analysis. Skewed bridges within this limit may be analyzed using the load distribution factors developed for right bridges, with the associated error of this procedure estimated at less than 5%. STan ( Ψ) 1 ε = L 18 (Equation 1) where: S = beam spacing L = span length Q = skew angle 2.4 ADDITIONAL WORK Additional work regarding skewed beam and slab bridges was reported by Ebeido and Kennedy 15,16. Their research focused on load distribution in skewed composite bridges, both simple span and continuous, and included studies of moment, shear and reactions. Two separate 19

33 studies were performed regarding the distribution of shear and reactions in skewed bridges: (i) Simply supported composite bridges, and (ii) Continuous composite bridges. The first study analyzed the influence of skew and other bridge geometric parameters on the distribution of shear in simply supported composite steel-concrete bridges. The parameters investigated included: skew angle, beam number and spacing, bridge aspect ratio, number of loaded lanes, number of intermediate diaphragms and the presence of end diaphragms. A parametric study of over 400 bridge cases was completed using ABAQUS 17 for the finite element analysis of the bridge models. The results of the computer analyses were verified through physical testing of six scale bridge models. Empirical formulas were developed for end shear distribution factors of both dead load and OHBDC truck live load. The empirical formulas were derived separately for exterior girders at the acute corner of the bridge, exterior girders at the obtuse corner and interior girders. The effect of skew on shear along the length of the girders was not addressed. The second study by Ebeido and Kennedy focused on continuous skewed composite bridges and the distribution of both shear and reactions at interior piers. Similar to the study for simple spans, this research incorporated over 600 two-span continuous bridges with investigation of the aforementioned parameters, as well as the ratio of adjacent span lengths. ABAQUS was again used for the finite element modeling, and verified through physical testing of three scale models of continuous bridges. The live load used in this research, however, was the AASHTO HS20-44 truck. This facilitated comparison of the empirical formulas for distribution of shear at pier supports developed in Ebeido and Kennedy s research with those from NCHRP and the LRFD Specifications. 20

34 The comparison of distribution factors was limited to those for shear at interior piers, as NCHRP and the LRFD Specifications do not address the distribution of pier reactions in skewed bridges. Ebeido and Kennedy used a three-lane continuous bridge with skew angles of 0, 30, 45 and 60 to compare the maximum shear force determined from the LRFD Specifications, NCHRP 12-26, their empirical formulas and their finite element analyses. The comparison results, shown in Table 2, indicated that the distribution factors developed by the authors result in less conservative shear forces at the piers. These results, the authors state, are due to the fact that NCHRP and the LRFD Specifications do not account for intermediate diaphragms and apply the same skew correction factors to both the interior and exterior girders. Additionally, the factors developed by Ebeido and Kennedy account for the effect of skew on the distribution of dead load, an effect not considered in NCHRP and the LRFD Specifications. Similar to the first study, however, the effect of skew on shear along the lengths of the girders was not addressed. 21

35 Table 2. Maximum Shear Forces at Pier Support for Three-Lane Bridge with Different Skew Angles Predicted Using Different Methods 17 Shear force (kn) (1) Maximum exterior girder shear force at the pier support Skew angle (degrees) (2) LRFD (1994) (3) NCHRP (1988) (4) Proposed formulas (5) Finiteelement analysis (6) 2 = = = = Maximum interior girder shear force at the pier support 2 = = = =

36 The effects of skew angle and intermediate transverse cross frames on load distribution in skewed, simple span are investigated by Aggour and Aggour 18. Their analysis of 12 single track railway bridges, with superstructures consisting of two steel plate girders, focused on the distribution of bending moments. The authors findings, however, indicate that the variation in number of intermediate cross frames had little impact on the magnitude of reactions at the acute and obtuse corners of the bridges. The girder reactions for models with varying numbers of intermediate cross frames did not differ from those of a model possessing only end cross frames. The research performed by Bell 19 in 1998 focused on evaluating the shear and moment distribution factors currently specified in the Standard Specifications and the LRFD Specifications. Bell investigated straight, skewed, simple span and continuous beam and slab bridges, both with and without intermediate diaphragms, using both field test data and finite element analysis with ANSYS 20. The research objective was to develop empirical equations for load distribution in continuous bridges, if it was determined that modifications to the existing equations were required to provide more accurate distribution results. Using the AASHTO HS20-44 truck for live load, parametric studies were performed, investigating the effects of the number of spans, span length, span length ratio, skew angle and girder spacing. The results indicated that the distribution factors provided in the LRFD Specifications accurately assess the effect of skew on the distribution of shear, and therefore, no modifications to the current equations for shear distribution were recommended. In his research project Forces At Bearings Of Skewed Bridges, Bishara investigated 36 simply supported composite multi-stringer bridges to evaluate the reaction components at the rocker and bolster bearings under both dead load and HS20-44 live loads 21. While most design 23

37 codes address the vertical and horizontal reaction components at these bearings, Bishara also addressed the remaining three rotational degrees of freedom at the bearings. Using ADINA 22 for the finite element analysis, a parametric study was performed to determine the effects span length, deck width and skew angle on the girder reactions. Two field tests were performed, one on a simple span bridge and one on a two span continuous bridge, to validate the results of the finite element analysis. The research conclusions that addressed the live load vertical reactions were: (i) Bearing forces differ substantially between the interior and exterior girders and between the obtuse and acute corners; (ii) The maximum live load reaction for the exterior girder is obtained when the trucks are placed at the obtuse corner; (iii) The maximum live load reaction for the interior girders was about 98% of the value computed per the Standard Specifications; therefore, the design approximations in the Standard Specifications are suitable for design, and; (iv) The maximum live load reaction for the exterior girder was less than that obtained from the AASHTO procedures. El-Ali investigated the internal forces in four 137-foot simply-supported, welded steel plate girder bridges with various skew angles to determine the effect of skew on girder bending moments, torsional moments and shears 23. Finite element analyses of the four bridge models, with skew angles of 0, 20, 40 and 60, were performed using SAP IV 24. The girder spacing of each bridge model was constant and intermediate and end cross frames were included. Four lanes of HS20-44 live load were applied in six different configurations in order to obtain the maximum results. The research conclusions indicated that the live load shears obtained from the finite element models did not have a definite correlation to those calculated using the distribution 24

38 factors from the Standard Specifications. The ratio of the shear values obtained from the finite element analyses to those calculated according to the Standard Specifications 25 varied from 0.45 to approximately 1. 25

39 CHAPTER 3 METHODOLOGY The evaluation of the effect of skew on shear along the length of exterior beams and on shear across bearing lines of beam and slab bridges was performed through a parametric study of a selective group of simple span and two-span continuous beam and slab bridge models. Analysis matrices were developed based on key parameters of simple span and two-span continuous beam and slab bridges. These analysis matrices served to guide the study, to allow for assessment of non-linear variation in the results and to identify the major parameters that have a significant effect on the variation of the skew correction factors. The matrices were constructed based upon bridge plans with span lengths of 42 feet (L), 105 feet (2.5L) and 168 feet (4L), a typical curb-to-curb width of 42 feet and skew angles, θ, of 30 and 60. The base case analysis matrix and bridge plan geometries are shown in Table 3 and Figure 4, respectively. 26

40 Table 3. Base Analysis Matrix for Beam and Slab Bridges Beam and Slab Bridges Skew Angle, 2 (I+Ae 2 ) 1 (I+Ae 2 ) 2 (I+Ae 2 ) 3 0 L 2.5L 4L L 2.5L 4L L 2.5L 4L 30 L 2.5L 4L L 2.5L 4L L 2.5L 4L 60 L 2.5L 4L L 2.5L 4L L 2.5L 4L Figure 4. Bridge Plan Geometries for Analysis 27

41 Also included in the bridge analysis matrices were major parameters, such as I+Ae 2 (where I = girder stiffness, A = the beam cross sectional area and e = the distance between the centers of the deck and the girder), that have a significant influence on the load distribution of beam and slab bridges. These parameters were identified during the skewed bridge sensitivity studies performed in NCHRP for development of the current skew correction factors in the LRFD Specifications, and include skew angle, beam spacing, beam stiffness, span length and slab thickness 4. As a result, those same parameters, as well as bridge aspect ratio and the presence of intermediate cross frames, were investigated in a total of 41 bridge models. This group of 41 models was comprised of 25 simple span beam-slab bridges, 3 simple span concrete T-beam bridges, 4 simple span spread concrete box girder bridges and 9 two-span continuous beam-slab bridges. The expanded analysis matrices for each bridge type are provided in Appendix A and the typical framing plans and cross sections of the bridge models are provided in Appendix B. The basic cross section parameters (i.e. number of beams, beam spacing, beam inertia/beam depth, slab thickness) for the beam and slab bridges were selected using the results of NCHRP as a guide. The analysis of load distribution, and ultimately, the development of the new load distribution factor formulas for right beam and slab bridges, in NCHRP was initiated by construction of a database of 850 existing beam and slab bridges from a nationwide survey of state transportation officials. From the database, the average beam and slab bridge parameters were defined for five different bridge types: beam-slab, box girder, slab, multi-box beam and spread box beam. These average bridge properties were used as a guide in setting the base parameters of the models to be investigated in this project. 28

42 For the beam-slab bridge types, the average properties calculated in NCHRP , and the base bridge model parameters used in this study are shown in Table 4. Additional beam-slab bridge parameters, specifically, girder spacings of 4.84 ft., girder stiffnesses of 44,400 in 4 and 1,870,000 in 4, a slab thickness of 9 in. and a 10-girder cross section, were also selected for additional investigations. The two-span continuous beam-slab bridge models were based upon the same base parameters, with the addition of a second, equal span. For the concrete T-beam models, the base bridge parameters utilized in this research were again established using the average properties from NCHRP , as shown in Table 5. The analysis matrix for the T-beam bridges was developed using typical span lengths for this bridge type, determined from NCHRP 12-26, rather than the base case span lengths defined previously. The matrix also includes a second beam with a stiffness typical of those identified in NCHRP The base bridge parameters for the spread box girder bridge models were also developed from the results of NCHRP Table 6 contains the average properties from NCHRP and the base parameters utilized in this study. The analysis matrix for this bridge type, found in Appendix A, was created by selecting a two additional, typical box girders, one shallower and one deeper than the base case girder. 29

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