shear in skewed multi-beam...

20-7/Task 107 COPY NO. _____

SHEAR IN SKEWED MULTI-BEAM BRIDGES

FINAL REPORT

Prepared forNational Cooperative Highway Research Program

Transportation Research BoardNational Research Council

Modjeski and Masters, Inc.NCHRP Project 20-7/Task 107

March 2002

iii

ACKNOWLEDGMENT OF SPONSORSHIP

This work was sponsored by the American Association of State Highway andTransportation Officials, in cooperation with the Federal Highway Administration, and wasconducted in the National Cooperative Highway Research Program, which is administeredby the Transportation Research Board of the National Research Council.

DISCLAIMER

This is an uncorrected draft as submitted by the research agency. The opinions andconclusions expressed or implied in the report are those of the research agency. They arenot necessarily those of the Transportation Research Board, the National Research Council,the Federal Highway Administration, the American Association of State Highway andTransportation Officials, or the individual states participating in the National CooperativeHighway Research Program.

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TABLE OF CONTENTS

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

CHAPTER 1 INTRODUCTION AND RESEARCH OBJECTIVES . . . . . . . . . . . . 5

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

CHAPTER 2 LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 NCHRP Project 12-26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Ontario Highway Bridge Design Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Additional Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

CHAPTER 3 METHODOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

CHAPTER 4 STUDY FINDINGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.1 Simple Span Beam-Slab Bridge Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.1.1 Live Load Shear Along Exterior Beam Length . . . . . . . . . . . . . . . . . . . . 40

4.1.1.1 Influence of Skew Angle . . . . . . . . . . . . . . . . . . . . . . . . . 404.1.1.2 Influence of Beam Stiffness . . . . . . . . . . . . . . . . . . . . . . . 434.1.1.3 Influence of Span Length . . . . . . . . . . . . . . . . . . . . . . . . . 474.1.1.4 Influence of Intermediate Cross Frames . . . . . . . . . . . . . 504.1.1.5 Influence of Beam Spacing . . . . . . . . . . . . . . . . . . . . . . . 534.1.1.6 Influence of Slab Thickness . . . . . . . . . . . . . . . . . . . . . . . 554.1.1.7 Influence of Bridge Aspect Ratio . . . . . . . . . . . . . . . . . . 57

4.1.2 Live Load Shear Across Bearing Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 604.2 Simple Span Concrete T-Beam Bridge Models . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2.1 Live Load Shear Along Exterior Beam Length . . . . . . . . . . . . . . . . . . . . 694.2.2 Live Load Shear Across Bearing Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 71

TABLE OF CONTENTS (continued)

v

4.3 Simple Span Spread Concrete Box Girder Bridge Models . . . . . . . . . . . . . . . . . 734.4 Two-Span Continuous Beam-Slab Bridge Models . . . . . . . . . . . . . . . . . . . . . . . 78

4.4.1 Simple Span vs. Two-Span Correction Factors at Obtuse Cornersof Abutments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.4.2 Correction Factors at Obtuse Corners of Abutments and Piers . . . . . . . . 814.4.3 Live Load Shear Along Exterior Beam Length . . . . . . . . . . . . . . . . . . . . 84

4.4.3.1 Influence of Skew Angle . . . . . . . . . . . . . . . . . . . . . . . . . 844.4.3.2 Influence of Beam Stiffness . . . . . . . . . . . . . . . . . . . . . . . 884.4.3.3 Influence of Span Length . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.4.4 Live Load Shear Across Abutment Bearing Lines . . . . . . . . . . . . . . . . . 934.4.5 Live Load Shear Across Pier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.4.6 Live Load Reactions at Pier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.5 Skew Correction Factors from LRFD Specifications and Research Results . . . 126

CHAPTER 5 INTERPRETATION AND APPLICATION . . . . . . . . . . . . . . . . . . . 128

5.1 Simple Span Beam-Slab Bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.2 Simple Span Concrete T-Beam Bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.3 Simple Span Spread Concrete Box Girder Bridges . . . . . . . . . . . . . . . . . . . . . . 1335.4 Two-Span Continuous Beam-Slab Bridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.5 Application of Study Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

CHAPTER 6 CONCLUSIONS AND SUGGESTED RESEARCH . . . . . . . . . . . . 145

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

APPENDIX A ANALYSIS MATRICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1

APPENDIX B CROSS SECTIONS AND FRAMING PLANS OFBRIDGE MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1

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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 10. Effect of Girder Stiffness on Skew Corrections Along Exterior Beams . . . . . . . 45




Figure 14. Effect of Span Length on Skew Corrections Along Exterior Beams . . . . . . . . . . 48



Figure 17. Effect of Intermediate Cross Frames on Skew Corrections Along Exterior Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Figure 18. Effect of Intermediate Cross Frames on Skew Corrections Along Exterior Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

vii

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. Effect of Skew Angle on End Shear Skew Corrections . . . . . . . . . . . . . . . . . . . . 62


Figure 24. Effect of Girder Stiffness on End Shear Skew Corrections . . . . . . . . . . . . . . . . . 63




Figure 28. Effect of Span Length on End Shear Skew Corrections . . . . . . . . . . . . . . . . . . . 65



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 38. Comparison of Simple Span and Two-Span Continuous SkewCorrection Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

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Figure 39. Comparison of Skew Correction Factors at Abutments and Pier . . . . . . . . . . . . 83

Figure 40. Nomenclature for Investigation of Correction Factors Along the Lengthof the Exterior Girders of Two-Span Continuous Bridge Models . . . . . . . . . . . . 85

Figure 41. Effect of Skew Angle on Skew Corrections Along Exterior Beamsof Continuous Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Figure 42. Effect of Beam Stiffness on Skew Corrections Along Exterior Beamsof Continuous Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Figure 43. Effect of Beam Stiffness on Skew Corrections Along Exterior Beamsof Continuous Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Figure 44. Effect of Span Length on Skew Corrections Along Exterior Beamsof Continuous Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Figure 45. Effect of Span Length on Skew Corrections Along Exterior Beamsof Continuous Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Figure 46. Nomenclature for Investigation of Correction Factors Across theAbutment 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. Complete Results Set for End Shear Skew Corrections At Abutments . . . . . . . . 99

Figure 53. Average Variation of End Shear Skew Corrections Across Abutmentsof Two-Span Continuous Beam-Slab Bridges . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Figure 54. Nomenclature for Investigation of Correction Factors Across thePier of Two-Span Continuous Bridge Models . . . . . . . . . . . . . . . . . . . . . . . . . . 104

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Figure 55. Effect of Skew Angle on Skew Corrections for Shear Across Pier . . . . . . . . . . 104

Figure 56. Effect of Girder Stiffness on Skew Corrections for Shear Across Pier . . . . . . . 106

Figure 57. Effect of Girder Stiffness on Skew Corrections for Shear Across Pier . . . . . . . 106

Figure 58. Effect of Span Length on Skew Corrections for Shear Across Pier . . . . . . . . . 108

Figure 59. Effect of Span Length on Skew Corrections for Shear Across Pier . . . . . . . . . 108

Figure 60. Complete Results Set for Skew Corrections for Shear Across Pier . . . . . . . . . 110

Figure 61. Average Variation of Skew Corrections for Shear Across Piersof Two-Span Continuous Beam-Slab Bridges . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Figure 62. Nomenclature for Investigation of Correction Factors for Reaction at thePier of Two-Span Continuous Bridge Models . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Figure 63. Comparison of Skew Correction Factors for Shear and Reaction at Pier . . . . . 115





Figure 68. Effect of Skew Angle on Skew Corrections for Reaction Across Pier . . . . . . . 121

Figure 69. Effect of Girder Stiffness on Skew Corrections for Reaction Across Pier . . . . 123

Figure 70. Effect of Girder Stiffness on Skew Corrections for Reaction Across Pier . . . . 123

Figure 71. Effect of Span Length on Skew Corrections for Reaction Across Pier . . . . . . . 125

Figure 72. Effect of Span Length on Skew Corrections for Reaction Across Pier . . . . . . . 125

Figure 73. Results for the Variation of the Skew Correction Along the Lengthof the Exterior Girders of Simple-Span Beam-Slab Bridges . . . . . . . . . . . . . . . 130

x


Figure 74. Average Results for the Variation of the Skew Correction Along theBearing Lines of Simple-Span Beam-Slab Bridges . . . . . . . . . . . . . . . . . . . . . . 132

Figure 75. Results for the Variation of the Skew Correction Along the Lengthof the Exterior Girders of Two-Span Continuous Beam-Slab Bridges . . . . . . . 136

Figure 76. Average Results for the Variation of the Skew Correction AcrossAbutments and Piers of Two-Span Continuous Beam-Slab Bridges . . . . . . . . . 138

Figure 77. 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 . . . . . . . . . . . 141

Figure 78. 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 . . . . . . . . . . . 141

Figure 79. Proposed Variation of the Skew Correction Factors for Shear Acrossthe 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 . . . . . . . . . . . 144

Figure 80. Proposed Variation of the Skew Correction Factors for Shear Acrossthe 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 . . . . . . . . . . . 144

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LIST OF TABLES

Table 1. Correction Factors for Load Distribution Factors for Support Shearof the Obtuse Corner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Table 2. Maximum Shear Forces at Pier Support for Three-Lane Bridge withDifferent Skew Angles Predicted Using Different Methods . . . . . . . . . . . . . . . . 22

Table 3. Base Analysis Matrix for Beam and Slab Bridges . . . . . . . . . . . . . . . . . . . . . . . 27

Table 4. Average NCHRP 12-26 and Base Parameters for Beam-Slab Bridge Models . . 30

Table 5. Average NCHRP 12-26 and Base Parameters for Concrete T-beamBridge Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Table 6. Average NCHRP 12-26 and Base Parameters for Spread Concrete BoxGirder Bridge Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Table 7. Comparison of Maximum Live Load Shears from BSDI and anLRFD Line Girder Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Table 8. Comparison of Skew Correction Factors for End Shear of Exterior Girdersat the Obtuse Corners of Simple Span and Two-Span Bridge Models . . . . . . . . 80

Table 9. Comparison of Skew Correction Factors for Shear of Exterior Girdersat the Obtuse Abutment Corners and Obtuse Pier Corners ofTwo-Span Bridge Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Table 10. Correction Factors for Reaction at the Pier of Two-SpanBeam-Slab Bridge Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Table 11. Comparison of Skew Correction Factors from LRFD Specificationsand Research Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

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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.

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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.

1

SUMMARY

This research focused on an investigation of the skew correction factors for live load

shear defined in Article 4.6.2.2.3c 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 12-262 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 12-262.

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.

2

For the simple span bridge models studied, the research results indicate that:

• Regardless of changes in the aforementioned bridge parameters, areasonable approximation for the variation of the skew correction factoralong the length of exterior girders of simple span beam-slab and concreteT-beam bridges is a linear distribution of the factor from its value at theobtuse corner of the bridge, determined according to the LRFDSpecifications, to a value of 1.0 at girder mid-span.

• Regardless of changes in the aforementioned bridge parameters, areasonable approximation of the skew correction factor for live load shearacross the bearing lines of simple span beam-slab and concrete T-beambridges is a linear distribution of the correction factor from its value at theobtuse corner of the bridge, determined according to the LRFDSpecifications, 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 ofexterior girders in each span and for shear across both the abutments andpiers of two-span continuous beam-slab bridges are identical to thoseproposed for simple span bridges. The correction factor variation alongthe exterior girder may be approximated by a linear distribution of thefactor 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 lineardistribution of the factor at the obtuse corner to a value of 1.0 at the acutecorner.

• The skew correction factor defined by the LRFD Specifications is valid forthe girder shear at the obtuse corners of both the abutments and piers ofthe continuous bridges.

• Skew correction factors for reaction at the piers of continuous bridges arepresent and are unique from those calculated for shear at the piers. Fromthe limited continuous bridge model data of this study, however, accurateempirical equations which define the correction factor or define itsvariation across the pier could not be derived. Therefore, the developmentof such equations for continuous bridges is necessary and is recommendedfor further research.

For application of the research findings, the recommendations are as follows:

3

Skew Correction Factor for Shear, Variation Along Exterior Beam Length

• For superstructure types “Concrete Deck, Filled Grid, or Partially Filled Grid onSteel or Concrete Beams; Concrete T-Beams, T- and Double T Section,” withinthe applicable ranges of skew angle (θ), spacing of beams or webs (S), span ofbeam (L) and number of beams, stringers or girders (Nb) as defined by Table4.6.2.2.3c-1 of the LRFD Specifications, the skew correction factor for shear maybe varied linearly from its value at the obtuse corner of the bridge, determined inaccordance with the empirical equation defined in Table 4.6.2.2.3c-1, to a valueof 1.0 at girder mid-span.

• This approximate variation is applicable for both simple span structures andcontinuous structures. For continuous structures, the skew correction factorcalculated at the obtuse corner of the abutment per Table 4.6.2.2.3c-1 is also validat the obtuse corners of the interior piers. Likewise, the variation of thecorrection factor is applicable from both the obtuse corner of the abutment and theobtuse 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 onSteel or Concrete Beams; Concrete T-Beams, T- and Double T Section,” withinthe applicable ranges of skew angle (θ), spacing of beams or webs (S), span ofbeam (L) and number of beams, stringers or girders (Nb) as defined by Table4.6.2.2.3c-1 of the LRFD Specifications, the skew correction factor for shear maybe varied linearly from its value at the obtuse corner of the bridge, determined inaccordance with Table 4.6.2.2.3c-1, to a value of 1.0 at the acute corner of thebearing line.

• This approximate variation is applicable for both simple span structures andcontinuous structures. For continuous structures, the skew correction factorcalculated at the obtuse corner of the abutment per Table 4.6.2.2.3c-1 is also validat the obtuse corners of the interior piers. Likewise, the variation of thecorrection factor is applicable from both the obtuse corner of the abutment and theobtuse 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

4

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 4.6.2.2.3c-1 of the LRFD

Specifications.

5

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 12-26 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 12-262 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 12-33 was the development of AASHTO Bridge Design

Specifications utilizing the Load and Resistance Factor Design (LRFD) methodology. This

6

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 12-26.

7

SUPPORTINGCOMPONENTS

TYPE OF DECK TYPICAL CROSS-SECTION

Steel Beam Cast-in-place concreteslab, precast concreteslab, steel grid,glued/spiked panels,stressed wood

Precast Concrete I or Bulb-Tee Sections

Cast-in-placeconcrete, precastconcrete

Closed Steel or PrecastConcrete Boxes

Cast-in-place concreteslab

Open Steel or PrecastConcrete Boxes

Cast-in-place concreteslab, precast concretedeck slab

Cast-in-Place Concrete TeeBeam

Monolithic concrete

Figure 1. Typical Beam and Slab Superstructures1.

8

The current design methodology in Section 4 of the LRFD Specifications1 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 4.6.2.2.3c-1 of the LRFD Specifications1 and as shown in Table

1, have been the subject of much discussion following the adoption of the LRFD Specifications

in 1993. As stated in the scope of services provided by the NCHRP for this project,

“Article 4.6.2.2.3c, Skewed Bridges, in the AASHTO LRFD Bridge DesignSpecifications, requires that shear in the exterior beam at the obtuse corner of thebridge shall be adjusted when the line of support is skewed. The Specificationsprovide correction factors for this adjustment and require that the correctionfactors be applied to all beams in the cross-section.

In the development of these correction factors, the variation of the effect of skewon the individual beam reactions was not considered. In addition, theSpecifications provide no guidance on the influence of skew on the shear alongthe length of the beam. The commentary to the Specifications states that theprescribed corrections are conservative. As a consequence of this conservatismsome 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.

9

Table 1. Correction Factors for Load Distribution Factors for Support Shear of theObtuse Corner1.

Type of Superstructure Correction Factor Range ofApplicability

Concrete Deck, Filled Grid, orPartially Filled Grid on Steel orConcrete Beams; Concrete T-Beams, T- and Double TSection

0° # 2 # 60°3.5 # S # 16.020 # L # 240Nb $ 4

Multicell Concrete Box Beams,Box Sections

0° < 2 # 60°6.0 < S # 13.020 # L # 24035 # d # 110Nc $ 3

Concrete Deck on SpreadConcrete Box Beams

0 < 2 # 60°6.0 # S # 11.520 # L # 14018 # d # 65Nb $ 3

Concrete Box Beams Used inMultibeam Decks

0 < 2 # 60°20 # L # 12017 # d # 6035 # b # 605 # Nb # 20

Where: 2 = skew angle (degrees) Nb = number of beams, stringers or girdersS = spacing of beams or webs (ft) Nc = number of cells in a concrete box girderL = span of beam (ft) ts = depth of concrete slab (in)b = width of beam (in) Kg = longitudinal stiffness parameter (in4)d = depth of beam or stringer (in)

10

The development of the skew correction factors for beam and slab bridges in the LRFD

Specifications was part of NCHRP Project 12-26. The report for that project, Distribution of

Wheel Loads on Highway Bridges4, 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 12-33 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 section5, as shown in the typical skewed bridge plan of Figure 2.

The working group for NCHRP 12-33 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 beam5, 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.

11

C Abutment (Typ.)L

Correction Factor Calculated forand Applied to End Shear atObtuse Corner of Exterior Girder(Typ.)

C Girder (Typ.)L

Correction FactorConservatively Applied toEnd Shear of All Girders(Typ.)

Skew Angle

Figure 2. Plan View of Typical Skewed Superstructure. Current Application of the SkewCorrection Factor for Shear per the AASHTO LRFD Bridge Design Specifications1.

12

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 12-33 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 bridges5. 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 two-

span 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.

13

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.

14

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 12-262, Khaleel and Itani7, Bishara, et al.8 and Ebeido and

Kennedy9 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 12-26

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

12-264. The primary objective of NCHRP Project 12-26 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

15

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 12-26 for the skew correction

factors were developed for simple span bridges utilizing the programs GENDEK5A11 and

FINITE12 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

16

responses. The general loading condition that maximized shear at the obtuse corner of the

skewed bridges is shown in Figure 3.

17

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

18

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 4.6.2.2.3c of the AASHTO LRFD Bridge Design Specifications1.

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 12-26.

2.3 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

19

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 Bakht14 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%.

(Equation 1)ε = ≤S Tan

L1

18( )Ψ

where:S = beam spacingL = span lengthQ = skew angle

2.4 ADDITIONAL WORK

Additional work regarding skewed beam and slab bridges was reported by Ebeido and

Kennedy15,16. 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

20

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 ABAQUS17 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 12-26 and the LRFD Specifications.

21

The comparison of distribution factors was limited to those for shear at interior piers, as

NCHRP 12-26 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 12-26 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 12-26 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.

22

Table 2. Maximum Shear Forces at Pier Support for Three-Lane Bridge with Different SkewAngles Predicted Using Different Methods17

Shear force(kN)(1)

Skewangle

(degrees)(2)

LRFD(1994)

(3)

NCHRP(1988)

(4)

Proposedformulas

(5)

Finite-elementanalysis

(6)

Maximum exterior girder shear force at the pier support

2 = 0 338 440 296 291

2 = 30 376 517 315 308

2 = 45 405 581 353 349

2 = 60 455 657 391 382

Maximum interior girder shear force at the pier support

2 = 0 423 440 314 319

2 = 30 473 517 297 288

2 = 45 508 581 275 268

2 = 60 569 657 253 250

23

The effects of skew angle and intermediate transverse cross frames on load distribution in

skewed, simple span are investigated by Aggour and Aggour18. 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 Bell19 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 ANSYS20. 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 loads21. While most design

24

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 ADINA22 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 shears23. Finite element analyses of the four bridge models,

with skew angles of 0°, 20°, 40°and 60°, were performed using SAP IV24. 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

25

factors from the Standard Specifications. The ratio of the shear values obtained from the finite

element analyses to those calculated according to the Standard Specifications25 varied from 0.45

to approximately 1.

26

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.

27

Table 3. Base Analysis Matrix for Beam and Slab Bridges

Beam and Slab Bridges

SkewAngle,2

(I+Ae2)1 (I+Ae2)2 (I+Ae2)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

28

Also included in the bridge analysis matrices were major parameters, such as I+Ae2

(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 12-26 for development of the current skew correction factors in the

LRFD Specifications, and include skew angle, beam spacing, beam stiffness, span length and

slab thickness4. 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 12-26 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 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, 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.

29

For the beam-slab bridge types, the average properties calculated in NCHRP 12-262, 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 in4 and

1,870,000 in4, 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 12-262, 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 12-26.

The base bridge parameters for the spread box girder bridge models were also developed

from the results of NCHRP 12-262. Table 6 contains the average properties from NCHRP 12-26

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.

30

Average BaseNCHRP 12-26 Model

Parameter ParameterBeam Spacing, ft. 7.8 7.75Beam Stiffness (I+Ae2), in4 339,000 358,000Slab Thickness, in. 7 7Number of Girders in X-Section 5.5 6

Bridge Parameter


Parameter ParameterGirder Spacing, ft. 7.77 7.75Girder Stiffness (I+Ae2), in4 357,000 333,000Slab Thickness, in. 7 7Number of Girders in X-Section 5 6

Bridge Parameter


Parameter ParameterBeam Spacing, ft. 8.83 8.83Box Depth, in. 39 39Box Width, in. 48 48Box Web Thickness, in. 5.5 5Box Top Flange Thickness, in. 3.8 3Box Bottom Flg Thickness, in. 5.8 6Slab Thickness, in. 7.6 7.5Number of Girders in X-Section 6 5

Bridge Parameter

Table 4. Average NCHRP 12-26 and Base Parameters for Beam-Slab Bridge Models

Table 5. Average NCHRP 12-26 and Base Parameters for Concrete T-beam Bridge Models

Table 6. Average NCHRP 12-26 and Base Parameters for Spread Concrete Box Girder BridgeModels

31

Investigation of each of the bridge models identified in the analysis matrices was

performed using finite element analyses. The services of Bridge Software Development

International, Ltd. (BSDI)26 were utilized for the finite element modeling. BSDI allows the user

to define the geometry, members, support conditions and loading conditions necessary for

construction of the finite element model. The model processing and generation of the live load

results was performed by BSDI.

The three-dimensional finite element modeling of the bridges by the BSDI software

allowed for individual modeling of the deck, beams and cross frames and optimization of the

live load placement. The deck slab was modeled with eight-node solid elements, each

possessing three translational degrees of freedom. The deck elements were modeled in their

actual position with respect to the neutral axes of the beams, which allowed the in-plane shear

stiffness of the deck to be considered in the analyses. Composite action between the deck slab

and beams was achieved through the use of rigid links prohibiting rotation of the deck with

respect to the beams. A combination of plate elements for the webs and beam elements for the

flanges were utilized to model the bridge beams. In modeling the flanges as beam elements, the

axial and lateral flange stiffness was incorporated into the models. Cross frames, X or K

configuration, were modeled with truss elements. Diaphragms were modeled with plate

elements for the webs and beam elements for the flanges, similar to the modeling of the girders.

All supports for the analysis models were free to translate laterally and longitudinally, with

restraint provided as required to ensure global stability. A schematic diagram of the bridge

modeling technique for an I-girder bridge is shown in Figure 5.

32

The BSDI software is tailored toward the analysis of steel I-girder and steel box girder

cross sections. The analysis of concrete I-girders and concrete box girders was achieved,

however, by transformation of the concrete sections into equivalent steel sections. The concrete

sections were transformed to produce steel sections which matched both the non-composite and

composite section properties of the concrete sections. The haunch depth above the girders was

modified as required in order to achieve the required composite section properties. Figure 6

displays the transformation of a concrete I-girder into an equivalent steel I-girder. A similar

procedure was utilized for transformation of the concrete box girders into equivalent steel boxes.

Transformation of the concrete T-beams was not required, as the BSDI input processor was

modified to facilitate the analysis of these bridge types

33

Plate Element for Web(Typ.)

Beam Element for Flange(Typ.)

Truss Element for CrossFrame (Typ.)

Rigid Link (Typ.)

Solid Element forDeck

L Beam (Typ.)C

7" Slab

Equivalent SteelBeam

AASHTO 28/63Concrete I-Beam

(EI)conc beam = (EI)steel beam

Haunch

(EI)composite conc beam = (EI)composite steel beam

Figure 5. Schematic Diagram of BSDI Finite Element Modeling

.

34

Figure 6. Transformation of Concrete Section to Steel Section

35

Influence surfaces were generated and utilized by the BSDI software for calculation of

the controlling live load effects for each of the bridge beams. The construction of the influence

surfaces was achieved by individual application of unit loads at each node of the entire deck

surface. From the bridge response under each unit load, influence surfaces were created for each

element of the model for each effect under consideration (moment, shear, lateral flange bending,

etc.). An automated live loader program placed the specified live loads in the position that

created the worst case effects for each of the members.

For all models of this investigation, the applied live load was two 12-foot lanes of

AASHTO HS20 trucks26, without a concurrent uniform load. While the LRFD Specifications

utilize a live load condition that combines the truck loading with a uniform load1, it is assumed

that the omission of the uniform load does not have a significant influence on the analysis

results. The skew correction factors, based upon normalized live load responses, i.e., live load

results based upon one particular live load configuration, are assumed to be relatively insensitive

to the exact configuration of the live load. The simultaneous application of the uniform load

with the truck load, therefore, was not considered. The application of two lanes of live load was

selected based upon previous experience that this configuration typically governs the response of

the bridge types investigated in this study. For the continuous span models, an additional live

load case of two lanes of 90% of two HS20 trucks spaced 50 feet apart was included for

determination of the pier reactions, as stipulated by the LRFD Specifications1.

Processing of the live load shear results from the BSDI bridge models attempted to

recognize the complexity of the bridge analyses of this study and the likelihood that individual

analysts may arrive at unique solutions. Therefore, through consultation with BSDI, it was

36

determined that curve-fitting techniques should be utilized during processing of the BSDI output.

The Least Squares Method of curve-fitting was applied to the live load shear diagram of each

bridge girder, obtained from the “raw” BSDI model output. This analysis approach was

considered to be a prudent method for obtaining results representative of the range of possible

solutions from various analysts and analysis tools. Three-dimensional finite element modeling

of even the simplest of bridge structures is a complex task. The skewed bridges studied in this

project merely added to the level of complexity in the finite element analysis. To arrive at

solutions to these complex bridge models, individual engineers may employ not only different

modeling techniques and philosophies, but also different analysis tools and/or software

packages. Hence, the final solutions obtained by each analyst for the same bridge may differ

slightly, whether it be a result of the modeling philosophy, the technique or the tool.

The BSDI software, as one example, is tailored for use in the design of bridge structures.

The BSDI modeling techniques and analysis methods, therefore, are geared toward producing

accurate solutions, while retaining a high level of confidence that a conservative solution has

been obtained for a structure designed for a service life of 50, 75 or possibly 100 years. Hence,

curve-fitting the results of the BSDI analyses was viewed as a reasonable method for obtaining

results representative of the range of possible solutions.

After obtaining the live load results from BSDI and curve-fitting the shear diagrams of

each bridge girder, the influence of skew angle and other primary geometric bridge parameters

on live load shears along the length of exterior beams of skewed beam and slab bridges was

presented in terms of normalized skew corrections. The live load shear diagrams obtained from

the bridge models were used to calculate the skew correction factors for the exterior beams at

37

each 10th point along the beam length. The skew correction factors are defined as the ratio of the

live load shear at a given location of a skewed bridge to that of a “right” bridge with identical

geometric parameters, VLL,s / VLL,r. The actual skew correction, (VLL,s / VLL,r) -1.0, when

positive, represents an additional fraction of the right bridge shear that is present when the same

bridge is skewed. The variation of this skew correction, (VLL,s / VLL,r) -1.0, is utilized in this

study to depict the variation of the skew correction factor itself. Therefore, the skew correction

at each 10th point along the exterior girders was calculated and then normalized to the skew

correction at the end of the beam at the obtuse corner of the bridge. Figure 7 illustrates this

process for calculating the normalized skew correction at the two-tenth point of an exterior

beam.

38

20k

C Abutment (Typ.)L

Girder 6

Girder 5

Girder 4

Girder 3

Girder 2

Girder 125kVLL 22k

C Abutment (Typ.)L

Girder 6

Girder 5

Girder 4

Girder 3

Girder 2

Girder 130kVLL

Skew Correction Factor 1.101.20

Normalized Skew Correction 0.501.00

RIGHT BRIDGE SKEWED BRIDGE

Skew Correction 0.100.20

Exterior Beam LL End Shear, “Right” Bridge = 25 kipsExterior Beam LL End Shear, Obtuse Corner, Skewed Bridge = 30 kipsSkew Correction Factor ( = 30/25) = 1.20Skew Correction = 0.20

Exterior Beam LL Shear, Two-tenth Point, “Right” Bridge = 20 kipsExterior Beam LL Shear, Two-tenth Point, Skewed Bridge = 22 kipsSkew Correction Factor ( = 22/20) = 1.10Skew Correction = 0.10

Therefore,Normalized Skew Correction at Two-tenth Point (0.10/0.20) = 0.50 (50%)

Thus, the normalized correction indicates that the skew correction at the Two-tenth Point is 50% of the skew correction at the end of the beam.

Figure 7. Procedure for Calculation of the Normalized Skew Corrections

39

This procedure of calculating, and then plotting, the normalized skew corrections enabled

graphic visualization of the variation of the skew correction along the length of the exterior

beams. It also facilitated direct comparison of this variation between bridges with different

geometric parameters, and hence, different magnitudes of skew corrections. A calculated skew

correction factor of 1.0 within the length of a beam produces a normalized skew correction of

0.0, indicating that no correction for skew is necessary. A calculated skew correction factor less

than 1.0 produces a normalized skew correction less than 0.0, indicating that this point has a

negative correction for skew, i.e., the shear in the skewed bridge model is less than the shear in

the “right” bridge model. The normalized skew corrections were plotted at each tenth point

along the exterior girders, defining location 0.0 as the beam end at the obtuse corner of the

bridge, location 1.0 at the acute corner, exterior girder 1 at the “bottom” of the bridge plan

(Girder 1 in Figure 7) and exterior girder 2 at the “top” of the bridge plan (Girder 6 in Figure 7).

This same procedure of plotting normalized skew corrections was utilized for

investigation of both shear across the abutments and piers and reactions across the piers of the

beam and slab bridges. The skew correction factors for shear of each beam across the bearing

line were calculated as the ratio of the live load shear from the skewed bridge model to that of

the corresponding “right” bridge model with identical geometric parameters. The skew

correction of each beam was then normalized to the skew correction for the beam at the obtuse

corner of the bearing line. Thus, the variation of the skew correction across the bearing lines

could be directly compared for bridge models with varying geometric parameters and

magnitudes of correction factors. The data plots of the normalized correction factors were

40

constructed by defining Girder 1 at the obtuse corner of the bearing line and defining the

remaining girders in ascending order to the acute corner.

A separate comparison of the skew correction factors for bearing reactions and those for

end shear of simple span bridges was not performed. That investigation, with the intent of

studying the influence of end cross frames and the effects of various load paths present at

bearings on end shears and reactions, was not possible due to the analysis procedure employed

by BSDI. The influence surfaces for the girder reactions are utilized by BSDI for calculation of

the end shears, thus assuming that the end shear is equal to the end reaction. A study of the load

paths through end cross frames and diaphragms, and their effect on the end shears and bearing

reactions, therefore, was not feasible.

41

CHAPTER 4 STUDY FINDINGS

4.1 SIMPLE SPAN BEAM-SLAB BRIDGE MODELS

4.1.1 Live Load Shear Along Exterior Beam Length

4.1.1.1 Influence of Skew Angle

The influence of skew angle on the variation of the skew correction factor along the

length of exterior beams was investigated in two sets of beam-slab bridge models. Each set of

models was based upon a 42' span length, a six-beam cross section with beam spacings of 7.75-

ft., a 7-in. deck slab and no intermediate cross-frames. The first set of models studied girder

stiffnesses of 44,400 in4 (I + Ae2) and skew angles of 30° and 60°. The second set studied girder

stiffnesses of 333,000 in4 (I + Ae2) and skew angles of 30° and 60°.

The plots of the normalized skew corrections for these two sets of models display a

diminishing influence of the skew correction factor from the end of the exterior beam at the

obtuse corner to the acute corner (see Figures 8 and 9). For the models with girder stiffnesses of

44,400 in4, the skew correction falls from its normalized value of 1.0 to zero or below zero

within the length of the beam span. For both the 30° and 60° skew angles, the correction factor

falls rapidly from its normalized value at the end of the span to zero near the four-tenth point of

the span length. The model with the 30° skew does have a slight skew correction present at mid-

42

span of approximately 30% of the correction at the end of the beam, but the correction falls to

zero by the eight-tenth point of the span length.

For the models with girder stiffnesses of 333,000 in4, the data displays the same general

trend of a diminishing influence of the skew correction factor along the length of the beam;

however, at the end of the beam adjacent to the acute corner, a slight skew correction of

approximately 20-45% the value at the obtuse corner is present. One of the exterior girders of

the 30° skew model also displays a small “spike” in the correction factor at mid-span. These

models, however, were created using an 8-ft. deep beam with a 42-ft. span length. This

geometry produces a span length to beam depth ratio 5.25– a ratio well outside the range of

typical beam-slab bridges.

The occurrence of the correction factor at the acute corner of the bridge and the “spike”

in the correction factor at mid-span is not as prevalent in the models that utilized the girder

stiffness of 44,400 in4. These models possess a span to depth ratio of 21, much more

representative of actual design situations. For development of design guidelines for the variation

of the skew correction factor along the length of the exterior girders, therefore, the results of the

models with 42-ft. spans and girder stiffnesses of 333,000 in4 are not considered to be as

representative of actual design conditions, as are the results of the models with 42-ft. spans and

girder stiffnesses of 44,400 in4.

43

EFFEC T O F SKEW AN G LE O N SKEW C O R R EC TIO NS ALO N G EXTE RIO R BEAM S42' S im ple S pan, Be am-S lab Bridge s, I+Ae 2 = 44,400 in 4, w/o In te rm e d. C ross

Fram e s

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Tenth Point Along S pan

Nor

mal

ized

Ske

w C

orre

ctio

ns

Ext. Girder 1, 30 deg . Skew Ext. Girder 2, 30 deg . SkewExt. Girder 1, 60 deg . Skew Ext. Girder 2, 60 deg . Skew

E FFE C T O F SKE W AN G LE O N SKE W C O R R E C T IO N S ALO N G E XT E R IO R B E AM S

42' S im ple S pan, Be am -S lab Bridge s, I+Ae 2 = 333,000 in 4, w/o Inte rm e d. C ross Fram e s

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Nor

mal

ized

Ske

w C

orre

ctio

ns

Ext. Gird er 1, 30 deg . Skew Ext. Girder 2, 30 d eg . SkewExt. Gird er 1, 60 deg . Skew Ext. Girder 2, 60 d eg . Skew

Figure 8. Effect of Skew Angle on Skew Corrections Along Exterior Beams

44


4.1.1.2 Influence of Beam Stiffness

The influence of beam stiffness on the variation of the skew correction factor along the

length of exterior beams was investigated in four sets of beam-slab bridge models. Each set of

models was based upon a six-beam cross section with beam spacings of 7.75-ft., a 7-in. deck slab

and no intermediate cross-frames. The first set of models studied girder stiffnesses of 44,400 in4

and 333,000 in4 at a span length of 42-ft. and a skew angle of 30°. The second set was the same

as the first, except that a skew angle of 60° was used. The third set investigated girder

stiffnesses of 333,000 in4 and 1,870,000 in4 at a span length of 105-ft. and a skew angle of 60°.

The fourth set studied girder stiffnesses of 44,400 in4, 333,000 in4 and 1,870,000 in4 at a span

length of 168-ft. and a skew angle of 60°.

Each of the models displays that the variation of the skew correction factor along the

length of the exterior beams is essentially the same among the varying beam stiffnesses at each

span length, with the exception of a few anomalies (see Figures 10, 11, 12 and 13). For the

majority of the model results, the skew correction quickly drops from its value at the end of the

beam adjacent to the obtuse corner to zero near the three- or four-tenth point of the span length.

A change in beam stiffness does not have an appreciable effect on the length along the exterior

girder over which a skew correction factor applies.

An anomaly in the results occurs, however, in the 168-ft. span models. These models

indicate that a substantial percentage of the skew correction at the end of the beams may also be

effective near mid-span. The most evident case of this occurs on the 168-ft. spans with beam

45

stiffnesses of 333,000 in4. At mid-span, the skew correction is approximately equal to the end

correction. The significance of this data point, however, is amplified by the relatively small

magnitude of the shears at mid-span. In this case, for example, the shears in Exterior Girder 2 at

mid-span are 22.7 kips and 25.5 kips for the right and skewed bridges, respectively. At the end

of the beam (obtuse end for the skewed model), the live load shears are 50.9 kips and 56.7 kips

for the right and skewed bridges, respectively. Therefore, the skew correction factors at mid-

span and at the obtuse end of the beam are 1.12 and 1.11, respectively. Given that shears at mid-

span are less than one-half of the end shears, and therefore, will not control for design purposes,

the presence of this anomaly in these few models will not be considered to have a great impact

on the study conclusions. It is recognized that the mid-span shears may be utilized for

determination of reinforcing steel and beam stiffener spacing; however, significant correction

factors at mid-span occur in a very limited number of study models. Therefore, incorporation of

these corrections in a design approximation will not be pursued in detail.

The occurrence of a skew correction factor at the acute corner of the models with a span

length of 42-ft. and a beam stiffness of 333,000 in4, as evident in Figures 10 and 11, was

discussed in the previous section. The geometry of these models is well outside of the typical

bridge geometry, and therefore, the anomaly in these few models also will not be considered to

have a great impact on the study conclusions.

46

EFFECT O F G IR DER STIFFNESS O N SKEW C O RR ECTIO N S ALO NG EXTERIO R BEAM

42' S imple S pan, Be am-S lab Bridge s, 30 de g. S ke w, w/o Inte rme d. C ross Frame s

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Nor

mal

ized

Ske

w C

orre

ctio

ns

Ext. Girder 1, I+A e2 = 333,000 in4 Ext. Girder 2, I+A e2 = 333,000 in4Ext. Girder 1, I+A e2 = 44,400 in4 Ext. Girder 2, I+A e2 = 44,400 in4

E FF E C T O F G IR D E R ST IFFN E SS O N SKE W C O R R E C T IO N S ALO N G E XT E R IO R B E AM42' S im ple S pan , Be am -S lab Bridge s, 60 de g. S k e w, w/o In te rm e d. C ross

Fram e s

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Nor

mal

ized

Ske

w C

orre

ctio

ns

Ext. Gird er 1, I+A e2 = 333,000 in 4 Ext. Gird er 2, I+A e2 = 333,000 in 4Ext. Gird er 1, I+A e2 = 44,400 in 4 Ext. Gird er 2, I+A e2 = 44,400 in 4

Figure 10. Effect of Girder Stiffness on Skew Corrections Along Exterior Beams

47


48

EFFECT OF GIRDER STIFFNESS ON SKEW CORRECTIONS ALONG EXTERIOR BEAMS

105' S imple S pan, Beam-S lab Bridges, 60 deg. S kew, w/o Intermed. Cross Frames

-1.60-1.40-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Nor

mal

ized

Ske

w C

orre

ctio

ns

Ext. Girder 1, I+A e2 = 333,000 in4 Ext. Girder 2, I+A e2 = 333,000 in4Ext. Girder 1, I+A e2 = 1,870,000 in4 Ext. Girder 2, I+A e2 = 1,870,000 in4

E F F E C T O F G IR D E R S T IF F N E SS O N S KE W C O R R E C T IO N S ALO N G E XT E R IO R B E AM S

168' S im ple S pan , Be am -S lab Bridge s, 60 de g. S ke w, w/o In te rm e d. C ross Fram e s

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Nor

mal

ized

Ske

w C

orre

ctio

ns

Ext. Gird er 1, I+A e2 = 333,000 in 4 Ext. Gird er 2, I+A e2 = 333,000 in 4Ext. Gird er 1, I+A e2 = 44,400 in 4 Ext. Gird er 2, I+A e2 = 44,400 in 4Ext. Gird er 1, I+A e2 = 1,870,000 in 4 Ext. Gird er 2, I+A e2 = 1,870,000 in 4


49


4.1.1.3 Influence of Span Length

The influence of span length on the variation of the skew correction factor along the

length of exterior beams was investigated in three sets of models. Each set of models was based

upon a six-beam cross section with beam spacings of 7.75-ft., a 7-in. deck slab, no intermediate

cross-frames and a skew angle of 60°. The first model set included bridges with beam stiffnesses

of 44,400 in4 and span lengths of 42-ft. and 168-ft. The second set investigated models with

beam stiffnesses of 333,000 in4 and span lengths of 105-ft. and 168-ft. Similarly, the third set

investigated beam stiffnesses of 1,870,000 in4 with span lengths of 105-ft. and 168-ft.

The variation of the skew correction factor along the length of the exterior beams are

essentially the same between the models of each set investigated (see Figures 14, 15 and 16).

The skew correction quickly drops from its value at the end of the beam to zero near the three- or

four-tenth point of the span length. The longer spans may tend to slightly increase the length

along the beam over which the correction factor is effective, but in all cases the correction factor

disappears between the three- and four-tenth point of the span length. As discussed in the

previous section, a correction factor approximately equal in magnitude to the end correction is

present near mid-span of the 168-ft. model with the 333,000 in4 beam stiffness, but the shear

values in this region will not govern for design purposes.

50

E F F E C T O F SP AN LE N G T H O N SKE W C O R R E C T IO N S ALO N G E XT E R IO R B E AM S

S im ple S pan , Be am -S lab Bridge s, I+Ae 2 = 44,400 in 4, 60 de g. S ke w, w/o In te rm e d. C ross Fram e s

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Te nth Point Along Span

Nor

mal

ized

Ske

w C

orre

ctio

ns

Ext. Gird er 1, 42' Span Ext. Gird er 2, 42' Sp anExt. Gird er 1, 168' Span Ext. Gird er 2, 168' Sp an


S im ple S pan , Be am -S lab Bridge s , I+Ae 2 = 333,000 in 4, 60 de g. S k e w, w/o In te rm e d. C ross Fram e s

-1.60-1.40-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Nor

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ized

Ske

w C

orre

ctio

ns

Ext. Gird er 1, 105' Sp an Ext. Gird er 2, 105' Sp anExt. Gird er 1, 168' Sp an Ext. Gird er 2, 168' Sp an

Figure 14. Effect of Span Length on Skew Corrections Along Exterior Beams

51


52


S im ple S pan , Be am -S lab Bridge s, I+Ae 2 = 1,870,000 in 4, 60 de g. S k e w, w/o In te rm e d. C ross Fram e s

-1.60-1.40-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Nor

mal

ized

Ske

w C

orre

ctio

ns

Ext. Gird er 1, 105' Span Ext. Gird er 2, 105' Sp anExt. Gird er 1, 168' Span Ext. Gird er 2, 168' Sp an


53

4.1.1.4 Influence of Intermediate Cross Frames

The influence of intermediate cross frames on the variation of the skew correction factor

along the length of exterior beams was investigated in two sets of models. Models were

generated for cases both with and without intermediate cross frames. Each model possessed a

six-beam cross section with beam spacings of 7.75-ft., beam stiffnesses of 333,000 in4, a skew

angle of 60°, 7-in. slab thickness and span lengths of 105-ft. or 168-ft.. The cross frame spacing

was set at 21-ft. for the right bridges and at approximately 13-ft. to 25-ft. for the skewed bridges,

contingent upon the model geometry. All intermediate cross frames were contiguous and

modeled as K-type cross frames, constructed from single angle members.

While the presence of intermediate cross frames produced much more uniform load

distribution between the two exterior beams of each model, as depicted by the similarity of the

data points for each exterior beam of the models with intermediate cross frames, the variation of

the skew correction factor along the length of the exterior beams are essentially the same for

models with and without intermediate cross frames (see Figures 17 and 18). The skew

correction quickly drops from its value at the end of the beam to zero near the three-tenth point

of the span. Additionally, the correction factor spike near mid-span of the 168-ft. models is

occurs regardless of the presence of intermediate cross frames. The magnitude of the spike,

however, is much smaller when intermediate cross frames are present.

Although the presence of intermediate cross frames did not effect the variation of the

skew correction factor along the length of the exterior beams, the magnitudes of the skew

correction were in the order of three times greater for models that possessed cross frames than

54

for models without cross frames. Figures 17 and 18 do not display these differences due to the

use of normalized data. The magnitude of the skew corrections may not be purely a function of

the presence of cross frames, but also of the articulation of the cross frames. The models

investigated possessed contiguous cross frames that framed directly into the girder bearings; the

effects of staggered cross frames were not investigated.

55

E F F E C T O F IN T E R M E D IAT E C R O S S F R AM E S O N S KE W C O R R E C T IO N S ALO N G

E XT E R IO R B E AM S105' S i m pl e S pan , B e am -S lab B ridge s , I+ Ae 2 = 333,000 i n 4, 60 de g. S k e w

-1 .60-1.40-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

0.0 0.1 0 .2 0.3 0 .4 0 .5 0 .6 0.7 0.8 0 .9 1 .0

Te n th Poin t A lon g S pan

Nor

mal

ized

Ske

w C

orre

ctio

ns

Ext. Gird e r 1, No X-Frames Ext . Gird e r 2, No X-FramesExt. Gird e r 1, X-Frames Ext . Gird e r 2, X-Frames

Figure 17. Effect of Intermediate Cross Frames on Skew Corrections Along Exterior Beams

56

E F F E C T O F IN T E R M E D IAT E C R O S S F R AM E S O N S KE W C O R R E C T IO N S ALO N G E XT E R IO R

B E AM S168' S im ple S pan , B e am -S lab B ridge s , I+Ae 2 = 333,000 in 4, 60 de g. S k e w

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Te n th Poin t A long S pan

Nor

mal

ized

Ske

w C

orre

ctio

ns

Ext. G irder 1, N o X -Fram es Ext. G irder 2, N o X -F ram esExt. G irder 1, X -F ram es Ext. G irder 2, X -F ram es

Figure 18. Effect of Intermediate Cross Frames on Skew Corrections Along Exterior Beams4.1.1.5 Influence of Beam Spacing

The influence of beam spacing on the variation of the skew correction factor along the

length of exterior beams was investigated in one set of models. The models were constructed

with a 42-ft. span length, beam stiffnesses of 44,400 in4, a skew angle of 60°, 7-in. slab thickness

and six beams spaced at 7.75-ft. or nine beams at 4.84-ft. Intermediate cross frames were not

included in the models. The variation of the skew correction factor along the length of the

exterior beams is essentially the same for the models with the two different beam spacings (see

Figure 19). The skew correction quickly drops from its value at the end of the beam to zero near

the three-tenth point along the span length. The spacing of the beams does not significantly

57

affect this variation.

58

E F F E C T O F B E AM SP AC IN G O N SKE W C O R R E C T IO N S ALO N G E XT E R IO R B E AM S42' S im ple S pan , Be am -S lab Bridge s, I+Ae 2 = 44,400 in 4, 60 de g. S ke w, w/o

In te rm e d. C ross Fram e s

-1.40-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Nor

mal

ized

Ske

w C

orre

ctio

ns

Ext. Gird er 1, 7.75' Beam Spa. Ext. Gird er 2, 7.75' Beam Sp a.Ext. Gird er 1, 4.84' Beam Spa. Ext. Gird er 2, 4.84' Beam Sp a.

Figure 19. Effect of Beam Spacing on Skew Corrections Along Exterior Beams

59

4.1.1.6 Influence of Slab Thickness

The influence of slab thickness on the variation of the skew correction factor along the

length of exterior beams was investigated in one set of models. The models were investigated

for slab thicknesses of 7-in. and 9-in. with a 42-ft. span length, a six-beam cross section with

beam spacings of 7.75-ft., beam stiffnesses of 44,400 in4 and a skew angle of 60°. Intermediate

cross frames were not included in the models. The variation of the skew correction factor along

the length of the exterior beams are nearly identical for the models with the two different slab

thicknesses (see Figure 20). The skew correction quickly drops from its value at the end of the

beam to zero near the three-tenth point along the span length. The thickness of the slab does not

significantly affect this variation.

60

E F F E C T O F SLAB T HIC KN E SS O N SKE W C O R R E C T IO N S ALO N G E XT E R IO R B E AM S42' S im ple S pan , Be am -S lab Bridge s, I+Ae 2 = 44,400 in 4, 60 de g. S k e w, w/o

In te rm e d. C ross Fram e s

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Nor

mal

ized

Ske

w C

orre

ctio

ns

Ext. Gird er 1, 7" Slab Ext. Gird er 2, 7" SlabExt. Gird er 1, 9" Slab Ext. Gird er 2, 9" Slab

Figure 20. Effect of Slab Thickness on Skew Corrections Along Exterior Beams

61

4.1.1.7 Influence of Bridge Aspect Ratio

A cursory investigation of the influence of bridge aspect ratio on the variation of shear

along the length of exterior girders was completed using a set of two models with 60° skew.

Different bridge aspect ratios were obtained by holding a constant span length and varying the

number of girders at a fixed spacing, and hence, the bridge width. While the bridge aspect ratio

also changes when span length is varied for a constant bridge width, these effects were addressed

in the span length investigation.

The two bridge models studied had a 42-ft. span length, 44,400 in4 beam stiffness, beam

spacing of 7.75-ft., 7-in. slab thickness, no intermediate cross frames and a skew angle of 60°.

Six girders were included in the first model (Model A) and ten in the second (Model B), yielding

bridge aspect ratios of 1.0 and 1.74, respectively, calculated as the ratio of the curb-to-curb width

to the span length.

Comparison of the live load shear diagrams of the exterior girders of these skewed bridge

models, shown in Figure 21, displays nearly identical results. For both Models A and B, the

shear diagrams of Girder 1 are nearly identical. Similarly, the shear diagrams of Girder 6 of

Model A and Girder 10 of Model B are nearly identical.

While the accompanying right bridge model was not constructed for each of these

skewed bridge models, it is inferred that the live load shear diagrams of the right bridge girders

would also be very similar regardless of the number of girders in each model. It is deduced,

therefore, that the skew correction factors along the length of the exterior girders would be very

62

similar and are not greatly affected by a change in the number of girders, and hence, a change in

the bridge aspect ratio.

63

EFFECT OF BRIDGE ASPECT RATIO ON LIVE LOAD SHEAR OF EXTERIOR GIRDERSAspect Ratios = 1.0 (Model A) vs. 1.73 (Model B)

-60-50-40-30-20-10

0102030405060

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Tenth Point Along Span Length

Liv

e L

oad

Shea

r (k

)

Model A, Girder 1 Model A, Girder 6

Model B, Girder 1 Model B, Girder 10

Figure 21. Effect of Bridge Aspect Ratio on Exterior Girder Shear

64

4.1.2 Live Load Shear Across Bearing Lines

For each of the simple span, six girder beam-slab bridge models discussed in Section

4.1.1, plots of the skew correction variation for end shear across each bearing line were

constructed (see Figures 22 thru 33). These plots were constructed to investigate the influence

of skew angle (Figures 22 and 23), girder stiffness (Figures 24 through 27), span length (Figures

28 through 30), intermediate cross frames (Figures 31 and 32) and slab thickness (Figure 33) on

the variation of the end shear correction factor across the bearing lines. Comparison of the skew

correction plots does not reveal consistent, distinct relationships between changes in any of these

parameters and the variation of the end shear correction factors. Figures 25 through 27 may

indicate a slight increase in the presence of a skew correction factor across the bearing lines with

an increase in girder stiffness, but the trend is not consistent for all cases. All of the data plots

depict, however, a general decline in the influence of the skew correction factor from the obtuse

corner to the acute corner of the bearing line. In most cases, the normalized skew correction

falls from its initial value at the obtuse corner of the bearing line to a negative value at the acute

corner, indicating that the end shear at the acute corner is greater in the right bridge than in the

skewed bridge.

Figure 34 superimposes the results from each of the models of Figures 22 through 33.

Again, the general decline in the skew correction across the bearing line is evident.

Additionally, this graph displays that the deviation of the data at each girder bearing location

tends to decrease as the girders nearest the acute corner of the bearing line are reached (girders 5

and 6). The deviation in the data points is greatest at the first interior girder adjacent to the

65

obtuse corner. In some isolated cases, this location may have a skew correction greater than that

found at the obtuse corner. It is also evident from Figure 34 that girder 4 possesses a number of

fairly significant skew corrections; normalized values as great as 1.4 were obtained at this girder.

Additionally, girder 6, at the acute corner of the bearing line, possesses a number of positive

normalized skew corrections, indicating that a skew correction factor is present at this location.

However, the four greatest normalized skew corrections at girder 4 and all of the positive

normalized corrections at girder 6 were obtained from the bridge models utilizing 8-ft. deep

beams with a 42-ft. span length. As discussed in Section 4.1.1.1, this span to depth ratio is well

outside the range of typical beam-slab bridges.

Figure 35, nevertheless, condenses all of the results into a single plot of the average

variation of the skew correction for end shear across the bearing lines and superimposes on the

results a linear variation of the skew correction from its value at the obtuse corner to zero at the

acute corner. This figure reveals the conservative nature of applying the skew correction factor

at the obtuse corner to the end shear of each girder in the cross section, the current practice

defined in the LRFD Specifications. It is suggested, from the data results, that the variation of

the correction factor across the bearing lines could approximated by a linear distribution from its

value at the obtuse corner to zero at the acute corner. While select data points fall outside of this

distribution, the average results fall well within this linear variation.

66

EFFECT OF SKEW ANGLE ON SKEW CORRECTIONS FOR END SHEAR ACROSS BEARING LINES

42' Simple Span, Beam-Slab Bridges, I+Ae2 = 44,400 in4, w/o Intermed. Cross Frames

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Brg. Line 1, 30 deg. Skew Brg. Line 2, 30 deg. SkewBrg. Line 1, 60 deg. Skew Brg. Line 2, 60 deg. Skew

EFFECT OF SKEW ANGLE ON SKEW CORRECTIONS FOR END SHEAR ACROSS BEARING LINES

42' Simple Span, Beam-Slab Bridges, I+Ae2 = 333,000 in4, w/o Intermed. Cross Frames

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.201.401.60

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Brg. Line 1, 30 deg. Skew Brg. Line 2, 30 deg. SkewBrg. Line 1, 60 deg. Skew Brg. Line 2, 60 deg. Skew

Figure 22. Effect of Skew Angle on End Shear Skew Corrections

67


68

EFFECT OF GIRDER STIFFNESS ON SKEW CORRECTIONS FOR END SHEAR ACROSS

BEARING LINES42' Simple Span, Beam-Slab Bridges, 60 deg. Skew, w/o Intermed. Cross Frames

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Brg. Line 1, I+Ae2 = 44,400 in4 Brg. Line 2, I+Ae2 = 44,400 in4Brg. Line 1, I+Ae2 = 333,000 in4 Brg. Line 2, I+Ae2 = 333,000 in4


BEARING LINES42' Simple Span, Beam-Slab Bridges, 30 deg. Skew, w/o Intermed. Cross Frames

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.201.401.60

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Brg. Line 1, I+Ae2 = 44,400 in4 Brg. Line 2, I+Ae2 = 44,400 in4Brg. Line 1, I+Ae2 = 333,000 in4 Brg. Line 2, I+Ae2 = 333,000 in4

Figure 24. Effect of Girder Stiffness on End Shear Skew Corrections


69

EFFECT OF GIRDER STIFFNESS ON SKEW CORRECTIONS FOR END SHEAR ACROSS BEARING

LINES168' Simple Span, Beam-Slab Bridges, 60 deg. Skew, w/o Intermed. Cross Frames

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.201.40

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Brg. Line 1, I+Ae2 = 44,400 in4 Brg. Line 2, I+Ae2 = 44,400 in4Brg. Line 1, I+Ae2 = 333,000 in4 Brg. Line 2, I+Ae2 = 333,000 in4Brg. Line 1, I+Ae2 = 1,870,000 in4 Brg. Line 2, I+Ae2 = 1,870,000 in4

EFFECT OF GIRDER STIFFNESS ON SKEW CORRECTIONS FOR END SHEAR ACROSS BEARING

LINES105' Simple Span, Beam-Slab Bridges, 60 deg. Skew, w/o Intermed. Cross Frames

-1.40-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.201.401.601.80

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

nS

Brg. Line 1, I+Ae2 = 333,000 in4 Brg. Line 2, I+Ae2 = 333,000 in4Brg. Line 1, I+Ae2 = 1,870,000 in4 Brg. Line 2, I+Ae2 = 1,870,000 in4



70

EFFECT OF SPAN LENGTH ON SKEW CORRECTIONS FOR END SHEAR ACROSS

BEARING LINESSimple Span, Beam-Slab Bridges, I+Ae2 = 44,400 in4, 60 deg. Skew, w/o Intermed.

Cross Frames

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Brg. Line 1, 42' Span Brg. Line 2, 42' SpanBrg. Line 1, 168' Span Brg. Line 2, 168' Span


BEARING LINESSimple Span, Beam-Slab Bridges, I+Ae2 = 333,000 in4, 60 deg. Skew, w/o

Intermed. Cross Frames

-1.40-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Brg. Line 1, 42' Span Brg. Line 2, 42' Span Brg. Line 1, 105' SpanBrg. Line 2, 105' Span Brg. Line 1, 168' Span Brg. Line 2, 168' Span

Figure 28. Effect of Span Length on End Shear Skew CorrectionsFigure 29. Effect of Span Length on End Shear Skew Corrections

71


BEARING LINESS imple Span, Beam-Slab Bridges , I+Ae2 = 1,870,000 in4, 60 deg. S k ew, w/o


-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.201.401.601.80

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Brg. Line 1, 105' Span Brg. Line 2, 105' SpanBrg. Line 1, 168' Span Brg. Line 2, 168' Span

EFFECT OF INTERMEDIATE CROSS FRAMES ON SKEW CORRECTIONS FOR END SHEAR

ACROSS BEARING LINES105' Simple Span, Beam-Slab Bridges , I+Ae2 = 333,000 in4, 60 deg. Skew

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Brg. Line 1, No X-Frames Brg. Line 2, No X-FramesBrg. Line 1, X-Frames Brg. Line 2, X-Frames

Figure 30. Effect of Span Length on End Shear Skew CorrectionsFigure 31. Effect of Intermediate Cross Frames on End Shear Skew Corrections

72

EFFECT OF SLAB THICKNESS ON SKEW CORRECTIONS FOR END SHEAR ACROSS

BEARING LINES42' S imple Span, Beam-Slab Bridges , I+Ae2 = 44,400 in4, 60 deg. Skew, w/o

Intermediate Cross Frames

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Brg. Line 1, 7" Slab Brg. Line 2, 7" Slab Brg. Line 1, 9" Slab Brg. Line 2, 9" Slab

EFFECT OF INTERMEDIATE CROSS FRAMES ON SKEW CORRECTIONS FOR END SHEAR

ACROSS BEARING LINES168' S imple Span, Beam-Slab Bridges, I+Ae2 = 333,000 in4, 60 deg. Skew

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Brg. Line 1, No X-Frames Brg. Line 2, No X-FramesBrg. Line 1, X-Frames Brg. Line 2, X-Frames

Figure 32. Effect of Intermediate Cross Frames on End Shear Skew CorrectionsFigure 33. Effect of Slab Thickness on End Shear Skew Corrections

73

NORMALIZED SKEW CORRECTIONS

FOR END SHEAR ACROSS BEARING LINESSimple Span Beam-Slab Bridges

-1.40-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.201.401.60

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

AVERAGE NORMALIZED SKEW CORRECTIONS FOR END SHEAR ACROSS BEARING LINES

Simple Span Beam-Slab Bridges

1.00

0.49

0.08

0.25

-0.23

-0.54

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Figure 34. Complete Results Set for End Shear Skew Corrections

74

Figure 35. Average Variation of End Shear Skew Corrections for Simple Span Beam-SlabBridges

75

4.2 SIMPLE SPAN CONCRETE T-BEAM BRIDGE MODELS


A cursory investigation of simple span, monolithic, concrete T-beam bridges was made

to determine whether the effects of skew on shear in this bridge type differs significantly from

those found in the simple span beam-slab bridge models. The T-beam bridges investigated were

based upon a 42' span length, a six-beam cross section with beam spacings of 7.75-ft., a 7-in.

deck slab and no intermediate diaphragms. The T-beams utilized were 14 in. wide and 39 in.

deep, producing a stiffness of 358,000 in4 (I + Ae2). Two models were analyzed, one with a 30°

skew angle and one with a 60° skew angle.

The plot of the normalized skew corrections along the length of the exterior girders,

shown in Figure 36, are very similar to those created for the beam-slab bridges. The skew

correction falls quickly from its initial value at the obtuse corner to zero by the four-tenth point

of the span length, regardless of skew angle. Similar to some of the beam-slab model results,

there is an isolated spike in the plot of the skew corrections at mid-span. This data point,

however, is produced by differences in relatively small shear values at mid-span of the skew and

right bridge models. As discussed in section 4.1, the live load shears at mid-span are less than

one-half of the end shears, and therefore, will not control for design purposes. The isolated spike

in the skew correction at this location, therefore, will not be considered to have a great impact on

the study conclusions.

76

EF F ECT O F SKEW ANG LE O N SKEW C O RRECTIO NS ALO NG EXTER IO R B EAM LENG TH

4 2 ' S i m pl e S pa n , C o n cre te T-B e a m B ri dg e s ,I+A e 2 = 3 5 8 ,0 0 0 i n 4

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

T e n th Poin t Alon g S pan

Nor

mal

ized

Ske

w C

orre

ctio

ns

Ext . G irder 1 , 30 deg. Skew Ext . G irder 2, 30 deg. Skew

Ext . G irder 1 , 60 deg. Skew Ext . G irder 2, 60 deg. Skew


77

4.2.2 Live Load Shear Across Bearing Lines

For the two models described previously in section 4.2.1, the variation of the skew

correction for end shear across the bearing lines was investigated to determine whether is varied

significantly from the results of the beam-slab models. Similar to those results, Figure 37

indicates that the change in skew angle does create a consistent trend in the variation of the skew

correction across the bearings of the T-beam models. The diminishing influence of the skew

correction from the obtuse corner to the acute corner is again apparent in Figure 37.

78

E F F E C T O F S KE W AN G LE O N S KE W C O R R E C T IO N S F O R E N D SHE AR AC R O SS

B E AR IN G LIN E S42' S im ple S pan , C on cre te T -B e am Bridge s ,

I+Ae 2 = 358,000 in 4

-1.40-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Bearin g Lin e 1, 30 d eg . Skew Bearin g Lin e 2, 30 d eg . SkewBearin g Lin e 1, 60 d eg . Skew Bearin g Lin e 2, 60 d eg . Skew


79

4.3 SIMPLE SPAN SPREAD CONCRETE BOX GIRDER BRIDGE MODELS

The investigation of the skew correction factors for shear in spread concrete box girder

bridges was initiated with the analysis of four 105-ft. simple span spread concrete box girder

bridges. Each bridge model possessed a cross section of five prestressed concrete spread box

girders, spaced at 8'-10" on centers, composite with a 7½" concrete deck. Two models utilized

48" wide x 39" deep boxes, while the remaining two models utilized 48" wide x 66" deep boxes.

For each box girder type, one model had 0°skew and one had 60° skew. For modeling using the

BSDI software, the prestressed concrete box girders were transformed into equivalent steel box

girders. The bridge models incorporated plate diaphragms both inside and between the boxes at

the supports; however, intermediate diaphragms were not included.

The live load shear results provided by BSDI for the box girders were the maximum

shears for one of the two webs of the box girder, incorporating the effects of torque. The shear

values are tabulated for only the controlling web of the box girder; the shear in the second web

was not included in the output. Additionally, the torque that contributed to the controlling web

shear was not tabulated, only the maximum torque was provided. Separation of the maximum

shear into its vertical bending and torsion components, therefore, was not possible.

To ensure that the live load shear distribution factors and skew correction factors of the

LRFD Specifications were developed using similar shear results (i.e., combined vertical bending

shear and shear flow due to torsion), the methodology utilized in NCHRP 12-26 was

investigated. Through this investigation, it was discovered that the shear distribution factors and

skew correction factors were derived neglecting the effects of box girder torsion. These factors

80

were developed through finite element modeling using the program FINITE. The program

MUPDI was utilized for analysis of box girder moments of simple span, right bridges. The

program FINITE was used for the investigation of box girder shear, as well as the analysis of

skewed and continuous box girder bridges. The box girder shear obtained from the FINITE

output was considered to be the shear force in the total box girder “system,” i.e., the integration

and summation of the shear stresses across the effective deck width of the box girder, the box

girder flanges and both box girder webs. Thus, a total box girder shear is calculated, assuming

that each web is subject to one-half of the total shear, neglecting the effects of torsion.

In an effort to quantify the influence of torsion on box girder shear, the controlling live

load shears obtained from each of the models analyzed by BSDI were compared to results

calculated from a line girder analysis, using the distribution factors and skew correction factors

defined in the current LRFD Specifications. The comparison results, shown in Table 7, indicate

that the controlling web shears from the BSDI results, incorporating the effects of torsion, may

be 4% to 29% more conservative than the LRFD results for the right bridge with the 39" deep

boxes. For the right, 66" deep boxes, the BSDI results are closer to the LRFD results, but may

still be as much as 20% more conservative. For both bridge models, the differences between the

BSDI and LRFD results are greater for the interior girder than for the exterior girder.

The disparity between the BSDI and LRFD results increases, however, when the bridges

are skewed. For the skewed, 39" deep boxes, the BSDI results are as much as 59% more

conservative than the LRFD results; for the 66" deep boxes, the BSDI results are as much as

52% more conservative. Both the interior beams and the exterior beams exhibit differences of

81

these magnitudes. These results indicate that for skewed bridges, the effects of torsion on box

girder shear may not be negligible.

82

Tenth Pt. BSDI LRFD BSDI LRFD BSDI LRFD BSDI LRFDMax V Max V ∆V Max V Max V ∆V Max V Max V ∆V Max V Max V ∆V

0.0 38 31 -18% 46 33 -29% 121 * 50 -59% 85 * 52 -39%0.1 31 28 -10% 38 29 -23% 38 28 -27% 34 29 -14%0.2 27 25 -9% 33 26 -22% 41 25 -40% 32 26 -19%0.3 23 21 -7% 28 22 -20% 39 21 -45% 36 22 -38%0.4 18 18 1% 22 19 -14% 34 18 -47% 39 19 -51%0.5 16 15 -8% 16 15 -3% 35 15 -58% 34 15 -55%0.6 -18 -18 1% -22 -19 -14% -36 -18 -50% -34 -19 -44%0.7 -23 -21 -7% -27 -22 -17% -40 -21 -47% -35 -22 -36%0.8 -27 -25 -9% -33 -26 -22% -44 -25 -44% -33 -26 -22%0.9 -29 -28 -4% -38 -29 -23% -45 -28 -38% -30 -29 -3%1.0 -38 -31 -18% -46 -33 -29% -107 * -50 -53% -74 * -52 -30%

Tenth Pt. BSDI LRFD BSDI LRFD BSDI LRFD BSDI LRFDMax V Max V ∆V Max V Max V ∆V Max V Max V ∆V Max V Max V ∆V

0.0 36 33 -9% 41 34 -16% 103 * 58 -43% 85 * 61 -28%0.1 29 29 1% 33 31 -7% 44 * 29 -33% 38 * 31 -19%0.2 26 26 0% 28 27 -3% 39 26 -33% 35 27 -22%0.3 23 23 -2% 30 24 -21% 39 23 -42% 36 24 -34%0.4 18 19 6% 48 * 20 -58% 34 19 -44% 35 20 -43%0.5 15 16 4% 19 16 -14% 32 16 -51% 32 16 -49%0.6 -18 -19 6% -21 -20 -5% -36 -19 -47% -35 -20 -43%0.7 -22 -23 3% -25 -24 -5% -39 -23 -42% -46 -24 -49%0.8 -26 -26 0% -28 -27 -3% -42 -26 -38% -38 -27 -28%0.9 -29 -29 1% -32 -31 -4% -43 -29 -32% -34 -31 -10%1.0 -36 -33 -9% -41 -34 -16% -86 * -58 -32% -70 * -61 -13%

* Denotes that actual shear has this magnitude, but the opposite sign.

EXTERIOR GIRDER INTERIOR GIRDER EXTERIOR GIRDER INTERIOR GIRDER

48/39 SPREAD BOX BEAMS

48/66 SPREAD BOX BEAMS0 DEGREE SKEW 60 DEGREE SKEW

0 DEGREE SKEW 60 DEGREE SKEWEXTERIOR GIRDER INTERIOR GIRDER EXTERIOR GIRDER INTERIOR GIRDER

Table 7. Comparison of Maximum Live Load Shears from BSDI and an LRFD Line GirderAnalysis

83

One final investigation regarding the box girder torsion was performed to compare the

torque calculated by BSDI against the threshold limit of 25% of the torsional cracking moment,

as specified in Article 5.8.2.1 of the LRFD Specifications. When the factored torsional moment

is below this limit, only a small reduction in shear capacity results and the effects of torsion are

neglected. Recognizing that the live load used in the BSDI models was two lanes of HS20

trucks, rather than the HL93 loading, and therefore, that this investigation does not produce

completely accurate results, it still provides a measure of the torques under investigation. The

results indicate that the box girder torsion in the two right bridge models tends to be less than the

threshold limit. The torsion in the skewed boxes, however, exceeds the limit at numerous

locations along the span of the girders.

These results raise an issue regarding the design methodology to be followed in

development of the skew correction factors for box girder bridges. While the influence of torque

may be negligible in right bridges, skew tends to amplify box girder torsion to levels that may

need to be considered in design. Although these bridge models were constructed without

intermediate diaphragms, the presence of which may reduce torsional effects by maintaining

relatively equal deflections among the boxes, the premise of equal deflections among girders of

skewed bridges may not be valid. Regardless of whether torsion is or is not included in the

determination of the box girder shear, the data obtained from BSDI is not conducive to the

development of skew correction factors for the box girders. It is incongruous to apply a skew

correction factor, derived from analyses that incorporate the effects of torsion, to distribution

factors derived from models that do not incorporate the effects of torsion.

84

4.4 TWO-SPAN CONTINUOUS BEAM-SLAB BRIDGE MODELS

4.4.1 Simple Span vs. Two-Span Correction Factors at Obtuse Corners of Abutments

One of the first tasks undertaken in the analysis of the two-span continuous beam-slab

bridges was a comparison of the skew correction factors calculated for the exterior girders at the

obtuse corner of the abutments versus those calculated from similar simple span bridge models,

as shown in Figure 38.

The skew correction factors developed in NCHRP Project 12-26 were determined from

simple span bridge models, but no guidance was provided regarding application of the skew

correction factors to continuous bridges. The report for Project 12-26, however, did propose that

when dealing with shear at the piers of right, continuous bridges, correction factors should be

applied to the empirical shear distribution factors developed for right, simple span bridges. The

LRFD Specifications, however, do not incorporate these continuity corrections. The corrections

suggested in Project 12-26 were in the range of 5%, i.e., 1.05, and commentary article C4.6.2.2.1

of the 1998 LRFD Specifications indicates that corrections of this magnitude may misrepresent

the level of accuracy in the approximate, empirical distribution factors1. Although it was not

anticipated in this study that the skew correction factors at the abutments of simple span and

continuous bridges would differ greatly, the influence lines for shear in simple span and two-

span beams are not identical. As a result, it was necessary to confirm that the skew correction

factors calculated at the obtuse corner of the abutments were similar in the simple span and two-

span models.

85

The skew correction factors from four simple span and four corresponding two-span

bridge models were compared. Each of the four pairs of simple span and continuous span

models were identical except for the addition of a second, equal span in the continuous models.

As shown in Table 8, the skew correction factors calculated at the obtuse corner of the abutments

of the simple span and two-span models were within approximately 4%. These results indicate

that the skew correction factors for shear, developed in Project NCHRP 12-26 for exterior

girders of simple span bridges, are also valid at the obtuse corner of abutments of continuous

bridges.

86

Girder (Typ.)

C PierL

Span 1 Span 2

SIMPLE SPAN BRIDGE

Girder (Typ.)

TWO-SPAN CONTINUOUS BRIDGE

Are Skew Correction Factorssimilar at these locations (Typ.)?

Abutment 2

Abutment 2

Abutment 1

Abutment 1

Span Beam Skew

Length Stiffness Angle

(ft) (in4) (deg) Abutment 1 Abutment 2 Abutment 1 Abutment 2 Abut. 1 Abut. 2

105 333,000 60 1.09 1.07 1.08 1.07 -0.9% 0%105 1,870,000 60 1.10 1.10 1.14 1.13 3.6% 2.7%168 333,000 60 1.12 1.11 1.10 1.09 -1.8% -1.8%168 1,870,000 60 1.16 1.16 1.17 1.15 0.9% -0.9%

Skew Correction Factors Percentage

Simple Span Models Two-Span Cont. Models Difference

Figure 38. Comparison of Simple Span and Two-Span Continuous Skew Correction Factors

Table 8. Comparison of Skew Correction Factors for End Shear of Exterior Girders at theObtuse Abutment Corners of Simple Span and Two-Span Bridge Models.

88

4.4.2 Correction Factors at Obtuse Corners of Abutments and Piers

The examination of the two-span beam-slab bridge results also included a comparison of

the skew correction factors for shear in the exterior girders at the obtuse corners of the abutments

and the pier, as shown in Figure 39.

As discussed in Section 4.2.1, NCHRP Project 12-26 did not provide explicit guidance

regarding application of the skew correction factors at the piers of continuous bridges.

Furthermore, the current LRFD Specifications are silent on this issue. Hence, it was necessary to

determine whether the skew correction factors for shear, developed in Project NCHRP 12-26 for

exterior girders of simple span bridges, and found in this study to be valid at the obtuse corners

of abutments of continuous bridges, are also valid for exterior girders at the obtuse corners of

piers of continuous bridges.

For each of the two-span continuous models investigated in this study, the skew

correction factors for the exterior girders were calculated at the obtuse corners of both abutments

and at the girder location adjacent to the obtuse corner of the pier. Comparison of the results,

shown in Table 9, indicates that the correction factors at the pier are typically greater than those

at the abutments. Additionally, increases in the skew angle and the girder stiffness tend to

increase the differences between the correction factors at the pier and abutments. With the

limited number of data sets, however, it is difficult to accurately predict a trend in the results.

Most of the bridge model results yield differences between the abutment and pier correction

factors of less than 5%. Given that the LRFD Specifications have regarded corrections of this

magnitude to imply misleading accuracy in approximate methods, the skew correction factors of

89

the exterior girders at the obtuse corner of the abutments are considered to be representative of

those that occur at the piers. Therefore, the skew correction factors developed in Project

NCHRP 12-26 for exterior girders of simple span bridges, found to be valid at the obtuse

corners of abutments of continuous bridges, are also considered to be applicable to the exterior

girders at the obtuse corners of piers of continuous bridges.

90

C PierL

Span 1 Span 2

Girder (Typ.)

Are Skew Correction Factorssimilar at these locations (Typ.)?

Abutment 2

Abutment 1

Exterior Girder 1

Exterior Girder 2

Girder locationadjacent to Pier (Typ.)

Span Beam SkewLength Stiffness Angle Abutment 1 Abutment 2 Pier Pier

(ft) (in4) (deg) Ext. Girder 1 Ext. Girder 2 Ext. Girder 1 Ext. Girder 2 Ext. Girder 1 Ext. Girder 2105 333,000 60 1.08 1.07 1.07 1.09 -0.9% 1.9%105 1,870,000 60 1.14 1.13 1.19 1.25 4.4% 10.6%168 333,000 30 1.03 1.03 1.02 1.03 -1.0% 0.0%168 333,000 60 1.10 1.09 1.13 1.12 2.7% 2.8%168 1,870,000 60 1.17 1.15 1.24 1.24 6.0% 7.8%

Skew Correction Factors PercentageDifference

Figure 39. Comparison of Skew Correction Factors at Abutments and Pier

Table 9. Comparison of Skew Correction Factors for Shear of Exterior Girders at the ObtuseAbutment Corners and Obtuse Pier Corners of Two-Span Bridge Models.

91

C Abutment (Typ.)L

C PierL

Span 1 Span 2ExteriorGirder 1

ExteriorGirder 2


4.4.3.1 Influence of Skew Angle

The influence of skew angle on the variation of the skew correction factor along the

length of exterior beams of two span continuous bridges was investigated in one set of beam-slab

bridge models. The bridge models were based upon two equal spans, 168-ft. in length, a six-

beam cross section with beam spacings of 7.75-ft., girder stiffnesses of 333,000 in4 (I + Ae2), a

7-in. deck slab and no intermediate cross-frames. Skew angles of 30° and 60° were studied.

The variation of the skew correction was investigated along the length of each of the

“four” exterior girders: exterior girders 1 and 2 in each of span 1 and span 2. This nomenclature

is shown in Figure 40. The plot of the variation of the skew correction for each girder was

created by defining location 0.0 as the end of the girder at the obtuse corner created with its

support. The skew corrections at each tenth point along each girder were normalized against the

correction at the girder’s obtuse corner. Plotting each girder simultaneously enabled direct

comparison of the variation of the skew correction along the length of each girder.

92

Figure 40. Nomenclature for Investigation of Correction Factors Along the Length of theExterior Girders of Two-Span Continuous Bridge Models

93

The plots of the normalized skew corrections for these models display a diminishing

influence of the skew correction from the end of the exterior beams at the obtuse corners to the

acute corners (see Figure 41). The results indicate that the variation of the skew correction along

the girder length is similar regardless of whether the obtuse corner is located at the abutments or

pier. Additionally, the variation of the skew correction factor is not sensitive to changes in the

skew angle, as 30° and 60° skew angles both produce variations in which the correction factor

falls rapidly from its normalized value at the obtuse corner to zero near the three-tenth point of

the span length.

The results from the model with 30° skew display the presence of a correction factor at

mid-span that exceeds the correction factor at the obtuse corner. Further investigation of the

girder shears from this model, however, reveal that the plotted data greatly amplifies the actual

analysis results. The shears at the obtuse corner of this exterior girder are 50.2 kips and 51.6

kips, for the right and skewed bridge, respectively. This produces a correction factor of

approximately 1.03 at the end of the girder. At mid-span, the shears are 28.95 kips and 30.0

kips, for the right and skewed bridge, respectively, producing a correction factor at this location

of 1.04. When normalized to the skew correction of 0.03 at the end of the girder, the correction

of 0.04 at mid-span produces a normalized value of 1.32. Incorporation of this isolated

correction factor into a design approximation for the variation of the skew correction factor is

not considered to be necessary. The shears at this location of the girders do not control for

design purposes and the minimal differences between the skewed bridge and right bridge model

results are amplified by the manner in which the results are presented. As a result, the mid-span

skew correction will be neglected in the design approximation for the variation of the skew

94

correction factor.

95

EFFEC T O F SKEW AN G LE O N SKE W C O R R E C T IO N S ALO N G E XT E R IO R BE AM S

Two-S pan C ontinu ous, Be am -S lab Bridge s, 168' S pans,I+Ae 2 = 333,000 in 4, w/o Inte rm e d. C ross Fram e s

-2.00-1.75-1.50-1.25-1.00-0.75-0.50-0.250.000.250.500.751.001.251.50

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Nor

mal

ized

Ske

wC

orre

ctio

ns

Span 1, Ext. Girder 1, 30 deg . Skew Span 1, Ext. Girder 2, 30 deg . SkewSpan 2, Ext. Girder 1, 30 deg . Skew Span 2, Ext. Girder 2, 30 deg . SkewSpan 1, Ext. Girder 1, 60 deg . Skew Span 1, Ext. Girder 2, 60 deg . SkewSpan 2, Ext. Girder 1, 60 deg . Skew Span 2, Ext. Girder 2, 60 deg . Skew

Figure 41. Effect of Skew Angle on Skew Corrections Along Exterior Beams of ContinuousModels

96

4.4.3.2 Influence of Beam Stiffness

The influence of beam stiffness on the variation of the skew correction factor along the

length of exterior beams was investigated in two sets of two-span continuous beam-slab bridge

models. Each set of models was based upon a six-beam cross section with beam spacings of

7.75-ft., a 7-in. deck slab and no intermediate cross-frames. The first set of continuous models

studied girder stiffnesses of 333,000 in4 and 1,870,000 in4 with two equal spans of 105-ft. and a

skew angle of 60°. The second set of models was similar to the first; however, two equal spans

of 168-ft. were used.

Each set of results, shown in Figures 42 and 43, displays that the variation of the skew

correction along the length of the exterior beams is similar between the varying beam stiffnesses,

with the exception of a few anomalies. For the majority of the model results, the correction

factor quickly drops from its value at the end of the beams at the obtuse corner to zero near the

three- or four-tenth point of the span length. For the models with 105-ft. spans, the increase in

beam stiffness may tend to increase the length along the girder over which the correction factor

is effective, but in most cases, the correction factor falls to zero near the four-tenth point of the

span length.

Some of the model results again produce a skew correction factor mid-span of the

girders. Investigation of the largest skew correction of Figures 42 and 43 leads to the same

conclusions discussed in section 4.2.3.1. The corrections at mid-span are created by

amplification of relatively small differences between the right and skewed bridge mid-span

girder shears. The amplification occurs when these small differences are normalized against a

97

small skew correction at the end of the girder. While the mid-span shears in the right and

skewed bridge models do differ, the magnitude of the shears at this location and the magnitude

of the difference do not warrant special consideration in a design approximation. Therefore, the

mid-span skew corrections will be neglected in the development of the design approximation for

the variation of the skew correction factor.

98

E F F E C T O F G IR D E R S T IF F N E S S O N S KE W C O R R E C T IO N S ALO N G E XT E R IO R B E AM S

Two-S pan C on tin u ou s, Be am -S lab B ridge s , 105' S pan s,60 de g. S k e w, w/o In te rm e d. C ross Fram e s

-2.50-2.25-2.00-1.75-1.50-1.25-1.00-0.75-0.50-0.250.000.250.500.751.001.25

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Tenth Point Along S pan

Nor

mal

ized

Ske

w C

orre

ctio

ns

Sp an 1, Ext. Gird . 1, I+A e2 = 333,000 in 4 Sp an 1, Ext. Gird . 2, I+A e2 = 333,000 in 4Sp an 2, Ext. Gird . 1, I+A e2 = 333,000 in 4 Sp an 2, Ext. Gird . 2, I+A e2 = 333,000 in 4Sp an 1, Ext. Gird . 1, I+A e2 = 1,870,000 in 4 Sp an 1, Ext. Gird . 2, I+A e2 = 1,870,000 in 4Sp an 2, Ext. Gird . 1, I+A e2 = 1,870,000 in 4 Sp an 2, Ext. Gird . 2, I+A e2 = 1,870,000 in 4

EFFECT OF GIRDER STIFFNESS ON SKEW CORRECTIONS ALONG EXTERIOR BEAMS

Two-S pan Continuous, Beam-S lab Bridges, 168' S pans,60 deg. S kew, w/o Intermed. Cross Frames

-1.50-1.25-1.00-0.75-0.50-0.250.000.250.500.751.001.25

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Tenth Point Along Span

Nor

mal

ized

Ske

w C

orre

ctio

ns

Span 1, Ext. Gird. 1, I+A e2 = 333,000 in4 Span 1, Ext. Gird. 2, I+A e2 = 333,000 in4Span 2, Ext. Gird. 1, I+A e2 = 333,000 in4 Span 2, Ext. Gird. 2, I+A e2 = 333,000 in4Span 1, Ext. Gird. 1, I+A e2 = 1,870,000 in4 Span 1, Ext. Gird. 2, I+A e2 = 1,870,000 in4Span 2, Ext. Gird. 1, I+A e2 = 1,870,000 in4 Span 2, Ext. Gird. 2, I+A e2 = 1,870,000 in4

Figure 42. Effect of Beam Stiffness on Skew Corrections Along Exterior Beams of Continuous Models

Figure 43. Effect of Beam Stiffness on Skew Corrections Along Exterior Beams of Continuous Models

99

4.4.3.3 Influence of Span Length

The influence of span length on the variation of the skew correction factor along the

length of exterior beams was investigated in two sets of two-span continuous beam-slab bridge

models. Each set of models was based upon a six-beam cross section with beam spacings of

7.75-ft., a 7-in. deck slab and no intermediate cross-frames. The first set of continuous models

compared span lengths of 105-ft. and 168-ft. with girder stiffnesses of 333,000 in4 and a skew

angle of 60°. The second set of models was similar to the first, except that the girder stiffness

was increased to 1,870,000 in4.

Similar to the findings in Sections 4.4.3.1 and 4.4.3.2, the results indicate that the

variation of the skew correction along the length of the exterior girders is not significantly

affected by changes in span length. As shown on Figures 44 and 45, regardless of span length,

the skew corrections drop from their value at the obtuse corner of the girder to zero near the

three- or four-tenth point of the span length. The mid-span skew corrections were discussed

previously and will not be considered in the development of the design approximation for the

variation of the skew correction factor.

100

E F F E C T O F SP AN LE N G T H O N S KE W C O R R E C T IO N S ALO N G E XT E R IO R B E AM S

Two-S pan C on tin u ou s, Be am -S lab Bridge s ,I+Ae 2 = 333,000 in 4, 60 de g. S k e w, w/o In te rm e d. C ross Fram e s

-2.50-2.25-2.00-1.75-1.50-1.25-1.00-0.75-0.50-0.250.000.250.500.751.001.25

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Nor

mal

ized

Ske

w C

orre

ctio

ns

Sp an 1, Ext. Gird er 1, 105' Sp an s Sp an 1, Ext. Gird er 2, 105' Sp an sSp an 2, Ext. Gird er 1, 105' Sp an s Sp an 2, Ext. Gird er 2, 105' Sp an sSp an 1, Ext. Gird er 1, 168' Sp an s Sp an 1, Ext. Gird er 2, 168' Sp an sSp an 2, Ext. Gird er 1, 168' Sp an s Sp an 2, Ext. Gird er 2, 168' Sp an s


Two-S pan C ontin u ou s, Be am -S lab Bridge s,I+Ae 2 = 1,870,000 in 4, 60 de g. S ke w, w/o In te rm e d. C ross Fram e s

-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.001.20

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Nor

mal

ized

Ske

w C

orre

ctio

ns

Span 1, Ext. Girder 1, 105' Sp ans Span 1, Ext. Girder 2, 105' Sp ansSpan 2, Ext. Girder 1, 105' Sp ans Span 2, Ext. Girder 2, 105' Sp ansSpan 1, Ext. Girder 1, 168' Sp ans Span 1, Ext. Girder 2, 168' Sp ansSpan 2, Ext. Girder 1, 168' Sp ans Span 2, Ext. Girder 2, 168' Sp ans

Figure 44. Effect of Span Length on Skew Corrections Along Exterior Beams of Continuous Models

Figure 45. Effect of Span Length on Skew Corrections Along Exterior Beams of Continuous Models

101

4.4.4 Live Load Shear Across Abutment Bearing Lines

For each of the two-span continuous beam-slab bridge models discussed in Section 4.4.3,

the skew correction factors were calculated for the end shear of each girder across each of the

abutment bearing lines. Similar to the investigation of the simple span models, data plots were

then created to investigate the variation of the end shear skew corrections across the abutments

and the effects of skew angle, girder stiffness and span length on this variation. For generation

of these plots and for direct comparison of the girders at each abutment, the girders at the obtuse

corner of the bearing lines were defined as girder 1, as shown in Figure 46.

The influence of skew angle on the variation of the end shear correction factor is shown

in Figure 47, the influence of girder stiffness is shown in Figures 48 and 49, and the influence of

span length is shown in Figure 50. Investigation of these figures does not reveal distinct

relationships between changes in any of these parameters and the variation of the end shear skew

correction. Similar to the results from the simple span models, each data plots depicts a general

decline in the end shear skew correction from the obtuse corner to the acute corner of the

abutment bearing line. In all cases, the normalized skew correction falls from its initial value at

the obtuse corner to a negative value at the acute corner, indicating that the end shear at the acute

corner is greater in the right bridge than in the skewed bridge.

102

C Abutment (Typ.)L

C PierL

Span 1 Span 2

Girder 6

Girder 5

Girder 4

Girder 3

Girder 2

Girder 1 Girder 6

Girder 5

Girder 4

Girder 3

Girder 2

Girder 1

Figure 46. Nomenclature for Investigation of Correction Factors Across the AbutmentBearing Lines of Two-Span Continuous Bridge Models

103

EFFECT OF SKEW ANGLE ON SKEW CORRECTIONS FOR END SHEAR ACROSS

ABUTMENT BEARING LINESTwo-Span Continuous, Beam-Slab Bridges, 168' Spans,

I + Ae2 = 333,000 in4, w/o Intermed. Cross Frames

-1.50-1.25-1.00-0.75-0.50-0.250.000.250.500.751.001.25

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Abut. 1, 30 deg. Skew Abut. 2, 30 deg. SkewAbut. 1, 60 deg. Skew Abut. 2, 60 deg. Skew



60 deg. Skew, w/o Intermed. Cross Frames

-1.50-1.25-1.00-0.75-0.50-0.250.000.250.500.751.001.25

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Abut. 1, I+Ae2 = 333,000 in4 Abut. 2, I+Ae2 = 333,000 in4Abut. 1, I+Ae2 = 1,870,000 in4 Abut. 2, I+Ae2 = 1,870,000 in4

Figure 47. Effect of Skew Angle on End Shear Skew Corrections At Abutments

Figure 48. Effect of Girder Stiffness on End Shear Skew Corrections At Abutments

104




-1.50-1.25-1.00-0.75-0.50-0.250.000.250.500.751.001.25

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Abut. 1, I+Ae2 = 333,000 in4 Abut. 2, I+Ae2 = 333,000 in4Abut. 1, I+Ae2 = 1,870,000 in4 Abut. 2, I+Ae2 = 1,870,000 in4


ABUTMENT BEARING LINESTwo-Span Continuous, Beam-Slab Bridges, I+Ae2 = 333,000 in4, 60 deg. Skew, w/o


-1.50-1.25-1.00-0.75-0.50-0.250.000.250.500.751.001.25

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Abut. 1, 105' Spans Abut. 2, 105' SpansAbut. 1, 168' Spans Abut. 2, 168' Spans

Figure 49. Effect of Girder Stiffness on End Shear Skew Corrections At Abutments

Figure 50. Effect of Span Length on End Shear Skew Corrections At Abutments

105


ABUTMENT BEARING LINESTwo-Span Continuous, Beam-Slab Bridges, I+Ae2 = 1,870,000 in4, 60 deg. Skew,

w/o Intermed. Cross Frames

-1.00-0.75-0.50-0.250.000.250.500.751.001.25

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Abut. 1, 105' Spans Abut. 2, 105' SpansAbut. 1, 168' Spans Abut. 2, 168' Spans

Figure 51. Effect of Span Length on End Shear Skew Corrections At Abutments

106

Figure 52 superimposes the results from each of the models of Figures 47 through 51.

Again, the general decline in the skew correction across the abutment bearing lines is evident.

This figure also displays that in some cases, the skew correction at Girder 2, the first interior

girder adjacent to the obtuse corner, is equal to or greater than the skew correction at the obtuse

corner of the abutment bearing line.

107

NORMALIZED SKEW CORRECTIONSFOR END SHEAR ACROSS

ABUTMENT BEARING LINESTwo-Span Continuous Beam-Slab Bridges

-1.50

-1.25

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

1.25

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Figure 52. Complete Results Set for End Shear Skew Corrections at Abutments

108

Figure 53 condenses all of the results into a single plot of the average variation of the

skew correction for end shear across the abutment bearing lines. Superimposed on the results is

a linear variation of the skew correction from its value at the obtuse corner to a value of zero at

the acute corner. While the study results reveal that some bridge models have skew corrections

at Girder 2 in excess of that predicted by the linear variation from the obtuse corner, a close

inspection of the data indicates that the linear variation is a reasonable design approximation.

For example, conservatively assume that the skew correction factor for end shear at the

obtuse corner of an abutment is 1.75, a correction factor exceeding any determined in this study.

According to the proposed linear approximation of the correction factor variation, the skew

correction at the first interior girder adjacent to the obtuse corner is 80% of the obtuse corner

correction, for a six girder cross section. This produces a skew correction of 0.60 (0.80*0.75 =

0.60) at the first interior girder. Using the average results from this study, shown in Figure 53,

the skew correction at this same girder was found to be 89% of the obtuse corner correction.

Thus, from the study results, the skew correction at the first interior girder is 0.6675 (0.89*0.75

= 0.6675).

If the skew correction factor of 1.60 from the linear approximation was used rather that

the correction factor of 1.6675 from the average results, a 4% difference is produced in the

calculated girder shear. This difference is within expected levels of accuracy of a design

approximation. Furthermore, as the magnitude of the skew correction at the obtuse corner

decreases to more typical values found in this study, the percentage difference between the

approximated skew correction at the first interior girder and that predicted by the study results

decreases. As a result, it is suggested that the variation of the correction factor across the

109

abutment bearing lines can be reasonably approximated by a linear distribution from its value at

the obtuse corner to zero at the acute corner, an approximation identical to that proposed for the

simple span bridges.

110

AVERAGE NORMALIZED SKEW CORRECTIONS FOR END SHEAR ACROSS

ABUTMENT BEARING LINESTwo-Span Continuous Beam-Slab Bridges

1.000.89

0.29

0.00

-0.44

-1.06

-1.20

-1.00-0.80

-0.60

-0.40

-0.200.00

0.20

0.40

0.600.80

1.00

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Figure 53. Average Variation of End Shear Skew Corrections Across Abutments of Two-Span

Continuous Beam-Slab Bridges

111

4.4.5 Live Load Shear Across Pier

Similar to the investigation regarding the variation of the end shear skew correction

factor across each of the abutment bearing lines, the variation of the correction factor for shear

across the piers of the two-span continuous models was investigated. The girder shear directly

adjacent to the pier was investigated in both span 1 and span 2. Again, data plots were created to

study the effects of skew angle, girder stiffness and span length on the variation of the skew

correction across the pier. For generation of these plots and for direct comparison of the results

on either side of the pier, girder 1 was defined in each span as the exterior girder that created an

obtuse corner with the pier, as shown in Figure 54.

The influence of skew angle on the variation of the skew correction across the pier is

shown in Figure 55. Investigation of this figure does not reveal a distinct relationship between a

change in skew angle and the variation of the skew correction across the pier. This figure does,

however, display the presence of a significant skew correction, with respect to girder 1, at girder

5 in the model with 30 degree skew. In span 2 of this model, the shear in girder 1 is 56.2 kips

and 57.5 kips in the right and skewed bridge models, respectively. This yields a skew correction

factor of 1.02 for girder 1, span 2. For girder 5, the shear is 57.4 kips and 61.0 kips in the right

and skewed bridge models, respectively. This yields a correction factor of 1.06. When the skew

correction of 0.06 is normalized against 0.02 at girder 1, a value of approximately 2.75 is

produced. In comparison to the results of all of the models studied, this data point appears to be

an anomaly. The large normalized skew correction is produced by the small skew correction at

the obtuse corner.

112

C Abutment (Typ.)L

C PierL

Span 1 Span 2Girder 6

Girder 5

Girder 4

Girder 3

Girder 2

Girder 1 Girder 6

Girder 5

Girder 4

Girder 3

Girder 2

Girder 1

EFFECT OF SKEW ANGLE ON SKEWCORRECTIONS FOR SHEAR

ACROSS PIER Two-Span Continuous, Beam-Slab Bridges, 168' Spans,

I + Ae2 = 333,000 in4, w/o Intermed. Cross Frames

-1.00-0.75-0.50-0.250.000.250.500.751.001.251.501.752.002.252.502.75

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Span 1 @ Pier, 30 deg. Skew Span 2 @ Pier, 30 deg. SkewSpan 1 @ Pier, 60 deg. Skew Span 2 @ Pier, 60 deg. Skew

Figure 54. Nomenclature for Investigation of Correction Factors Across the Pier of Two-Span Continuous Bridge Models

Figure 55. Effect of Skew Angle on Skew Corrections for Shear Across Pier

113

The influence of girder stiffness on the variation of the correction factor across the pier is

shown in Figures 56 and 57. The results indicate that an increase in girder stiffness may tend to

diminish the influence of the skew correction across the pier. As the girder stiffness increases,

the skew correction of girders 2 through 5 become a smaller percentage of the skew correction of

girder 1. All of the results display, however, a general decrease in the magnitude of the skew

correction across the pier. The pronounced correction found at girder 5 in Figure 55, is not

present in Figures 56 and 57.

114

EFFECT OF GIRDER STIFFNESS ON SKEW CORRECTIONS FOR SHEAR

ACROSS PIERTwo-Span Continuous, Beam-Slab Bridges, 105' Spans,


-1.75-1.50-1.25-1.00-0.75-0.50-0.250.000.250.500.751.001.251.50

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Span 1 @ Pier, I+Ae2 = 333,000 in4 Span 2 @ Pier, I+Ae2 = 333,000 in4Span 1 @ Pier, I+Ae2 = 1,870,000 in4 Span 2 @ Pier, I+Ae2 = 1,870,000 in4

EFFECT OF GIRDER STIFFNESS ON SKEW CORRECTIONS FOR SHEAR

ACROSS PIERTwo-Span Continuous, Beam-Slab Bridges, 168' Spans,


-1.00-0.75-0.50-0.250.000.250.500.751.001.251.50

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Span 1 @ Pier, I+Ae2 = 333,000 in4 Span 2 @ Pier, I+Ae2 = 333,000 in4Span 1 @ Pier, I+Ae2 = 1,870,000 in4 Span 2 @ Pier, I+Ae2 = 1,870,000 in4

Figure 56. Effect of Girder Stiffness on Skew Corrections for Shear Across Pier

115

Figure 57. Effect of Girder Stiffness on Skew Corrections for Shear Across PierThe influence of span length on the variation of the correction factor across the pier is

shown in Figures 58 and 59. The results indicate that an increase in span length may tend to

increase the influence of the skew correction across the pier. As the span length increases, the

skew correction of girders 2 through 5 become a larger percentage of the corrections of girder 1.

When studying these results in conjunction with those from the investigation of girder stiffness,

it appears that an increase in the flexibility of the structure, caused by either a decrease in beam

stiffness or an increase in span length, results in a greater presence of a skew correction across

the piers of continuous bridges. All of the results for the investigation of span length display,

however, a general decrease in the magnitude of the skew correction across the pier. Again, the

pronounced skew correction at girder 5 in Figure 55, is not present in these models (Figures 58

and 59).

116

EFFECT OF SPAN LENGTH ON SKEW CORRECTIONS FOR SHEAR

ACROSS PIERTwo-Span Continuous, Beam-Slab Bridges, I+Ae2 = 333,000 in4,


-1.75-1.50-1.25-1.00-0.75-0.50-0.250.000.250.500.751.001.251.50

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Span 1 @ Pier, 105' Spans Span 2 @ Pier, 105' SpansSpan 1 @ Pier, 168' Spans Span 2 @ Pier, 168' Spans

EFFECT OF SPAN LENGTH ON SKEW CORRECTIONS FOR SHEAR

ACROSS PIERTwo-Span Continuous, Beam-Slab Bridges, I+Ae2 = 1,870,000 in4, 60 deg. Skew,

w/o Intermed. Cross Frames

-1.00-0.75-0.50-0.250.000.250.500.751.001.25

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Span 1 @ Pier, 105' Spans Span 2 @ Pier, 105' SpansSpan 1 @ Pier, 168' Spans Span 2 @ Pier, 168' Spans

Figure 58. Effect of Span Length on Skew Corrections for Shear Across Pier

117

Figure 59. Effect of Span Length on Skew Corrections for Shear Across PierFigure 60 superimposes the results from each of the models of Figures 55 through 59.

Except for a few isolated data points, the general decline in the correction factor across the

abutment bearing lines is evident. Similar to the results for the end shear correction factors

across the abutments of the continuous bridges, certain models produce correction factor at

Girder 2, the first interior girder adjacent to the obtuse corner, equal to or greater than the

correction factor at the obtuse corner of the pier.

Figure 61 condenses all of the results into a single plot of the average variation of the

skew correction for shear across the pier. Superimposed on the results is a linear variation of the

correction factor from its value at the obtuse corner to a value of zero at the acute corner. Except

for the data point at girder 5, all of the average results fall within this approximation. The

average data point of 0.55 at girder 5 includes the normalized value of approximately 2.75 from

Figure 55. When this value is not included in the pool of results, an average value of 0.12 is

produced at girder 5. This value falls well within the linear approximation. The data point of

2.75 is considered to be an isolate value, not representative of the results anticipated from typical

beam-slab bridges. As a result, it is suggested that the variation of the correction factor across

the pier can be reasonably approximated by a linear distribution from its value at the obtuse

corner to zero at the acute corner, identical to the variation suggested across the abutments.

118

NORMALIZED SKEW CORRECTIONS FORSHEAR ACROSS PIER

Two-Span Continuous Beam-Slab Bridges

-1.75-1.50-1.25-1.00-0.75-0.50-0.250.000.250.500.751.001.251.501.752.002.252.502.75

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

AVERAGE NORMALIZED SKEW CORRECTIONS FOR SHEAR ACROSS PIER

Two-Span Continuous Beam-Slab Bridges

1.00

0.76

0.04

-0.14

0.55

-0.61

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Figure 60. Complete Results Set for Skew Corrections for Shear Across Pier

Figure 61. Average Variation of Skew Corrections for Shear Across Piers of Two-SpanContinuous Beam-Slab Bridges

119

4.4.6 Live Load Reactions at Pier

Intuition suggests that at the piers of skewed, continuous bridges, where an obtuse corner

and acute corner adjoin on opposite sides of a girder’s bearing, a skew correction for reaction

may not be necessary. It is speculated that the effects of the obtuse and acute corners on the

girder shear on either side of the bearing may tend to negate each other and eliminate the skew

correction. The current LRFD Specifications, however, do not address skew correction factors

for reactions at the piers of continuous bridges. As a result, a number of questions were

investigated as part of this study: Is there a skew correction factor for girder reaction at piers,

i.e., do the effects of acute and obtuse corners negate each other? If a correction factor for

reaction does exist, it is the same as the correction factor for shear at the pier? If the correction

factor exists, how does it vary across the pier, and what bridge parameters influence the

variation?

For investigation of these questions, the girders of the bridge models were labeled as

shown in Figure 62, with girder 1 located at the “bottom” of the plan view. The same two-span

continuous beam-slab bridge models used for the investigation of the skew correction factors for

shear were used for the investigation of correction factors for pier reactions.

As shown by the results in Table 10, correction factors for reaction at the interior pier of

the continuous bridge models were present, and were greater than 1.0, for almost every girder in

each model. This indicates that the effects of the girders’ obtuse and acute corners on either side

of the bearings do not offset each other for determination of the girder reaction. In fact, the data

suggests that any reduction in shear due to the acute corner is less than the increase in shear due

120

to the obtuse corner. Therefore, the development of skew correction factors for reaction at the

piers of continuous beam-slab bridges is necessary.

121

C PierL

Span 1 Span 2

Abutment (Typ.)Girder 6

Girder 5

Girder 4

Girder 3

Girder 2

Girder 1

Girder Model 1 Model 2 Model 3 Model 4 Model 51 1.01 1.02 1.09 1.11 1.272 1.06 1.08 1.20 1.00 1.073 1.20 1.25 1.22 1.05 1.214 1.19 1.22 1.22 1.06 1.205 1.06 1.08 1.20 1.00 1.076 1.01 1.04 1.09 1.13 1.27

Skew Correction Factors for Reaction at Pier

Figure 62. Nomenclature for Investigation of Correction Factors for Reaction at the Pier ofTwo-Span Continuous Bridge Models

Table 10. Correction Factors for Reaction at the Pier of Two-Span Beam-Slab Bridge Models

122

The presence of correction factors for reaction at the pier leads to a comparison to the

correction factors calculated for shear at the pier. As shown by Figures 63 through 67, the

correction factors for reaction are not identical to the correction factors for shear. In three of the

five sets of results (Figures 63, 64 and 67), the correction factors for reaction at the exterior

girders, girders 1 and 6, are similar to the correction factor for shear at the obtuse corner of the

pier. For the remaining two sets of results (Figures 65 and 66), the correction factors for reaction

at the exterior girders are appreciably less than the correction factor for shear at the obtuse

corner. In all cases, the correction factors for reaction are essentially symmetrical about the

centerline of the bridge.

At the interior girders, girders 2 through 5, the correction factors for reaction are

typically greater than those for shear at the same girder. Additionally, the correction factors for

reaction at these girders may be greater than those for reaction at the exterior girders, and also

greater than those for shear at the obtuse corner of the pier. An attempt, therefore, to simply use

the correction factor for shear at the obtuse corner also for reaction at each girder of the pier is

not conservative for each model of this study. These results indicate that there is no well-defined

relationship between the correction factors for reaction and for shear at the piers of continuous

bridges. It may be necessary, therefore, to develop a set of skew correction factors specific to

the reaction at the piers of continuous bridges.

123

Girder Span 1 Shear Span 2 Shear Reaction1 0.99 1.02 1.022 1.05 1.01 1.083 0.98 1.02 1.254 1.03 1.00 1.225 1.02 1.06 1.086 1.03 0.99 1.04

SKEW CORRECTION FACTORS FOR SHEAR AND REACTIONS AT PIER

SKEW CORRECTION FACTORS FOR SHEAR AND REACTIONS AT THE PIER

Two Span Continuous Beam-Slab Bridge, 168' Spans, w/o Intermed. X-Frames, I+Ae2 = 333,000 in4, 30 deg. Skew

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1 2 3 4 5 6

Girder

Skew

Cor

rect

ion

Fact

ors

Span 1 Shear Span 2 Shear Reaction

Figure 63. Comparison of Skew Correction Factors for Shear and Reaction at Pier

124





0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1 2 3 4 5 6

Girder

Skew

Cor

rect

ion

Fact

ors



125





0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1 2 3 4 5 6

Girder

Skew

Cor

rect

ion

Fact

ors



126




Two Span Continuous Beam-Slab Bridge, 105' Spans, w/o Intermed. X-Frames, I+Ae2 = 1,870,000 in4, 60 deg. Skew

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1 2 3 4 5 6

Girder

Skew

Cor

rect

ion

Fact

ors



127




Two Span Continuous Beam-Slab Bridge, 168' Spans, w/o Intermed. X-Frames, I+Ae2 = 1,870,000 in4, 60 deg. Skew

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1 2 3 4 5 6

Girder

Skew

Cor

rect

ion

Fact

ors



128

Using the limited number of continuous model data sets of this study, the effects of skew

angle, girder stiffness and span length on the variation of the correction factor for reaction across

the pier were investigated. By normalizing the skew correction of each girder against the

correction at girder 1, and plotting the results, the effects of skew angle, girder stiffness and span

length were investigated.

Figure 68 displays the effects of a 30° and 60° skew angle on the variation of the skew

correction for reaction at the pier. An increase in skew angle produces much more uniform skew

corrections across the pier. The 30° skew angle produced skew corrections at the interior girders

that are substantially greater than those at the exterior girders. The data indicates that the

interior girder corrections may be over 10 times greater than those at the exterior girder. For

example, a normalized value of approximately 10.7 is produced when normalizing the correction

of 0.247 at girder 3 against the correction of 0.023 at girder 1. With both the 30° and 60° skew

angle, however, the variation of the skew correction is essentially symmetrical.

129

EFFECT OF SKEW ANGLE ONSKEW CORRECTIONS FORREACTION ACROSS PIER

Two-Span Continuous, Beam-Slab Bridges, 168' Spans,I + Ae2 = 333,000 in4, w/o Intermed. Cross Frames

0.0

2.0

4.0

6.0

8.0

10.0

12.0

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

30 deg. Skew 60 deg. Skew

Figure 68. Effect of Skew Angle on Skew Corrections for Reaction Across Pier

130

The influence of girder stiffness on the variation of the skew correction for reaction is

displayed in Figures 69 and 70. Again, the variation of the correction factor is symmetrical

across the pier for each of the data sets. The results indicate that an increase in girder stiffness

decreases the magnitude of the correction at the interior girders, with respect to the correction at

girder 1. In fact, for the models with greater girder stiffness, the corrections at the interior

girders are less than those for the exterior girders.

131

EFFECT OF GIRDER STIFFNESS ONSKEW CORRECTIONS FORREACTION ACROSS PIER

Two-Span Continuous, Beam-Slab Bridges, 105' Spans,60 deg. Skew, w/o Intermed. Cross Frames

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

I+Ae2 = 333,000 in4 I+Ae2 = 1,870,000 in4

EFFECT OF GIRDER STIFFNESS ONSKEW CORRECTIONS FORREACTION ACROSS PIER

Two-Span Continuous, Beam-Slab Bridges, 168' Spans,60 deg. Skew, w/o Intermed. Cross Frames

0.00

0.50

1.00

1.50

2.00

2.50

3.00

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

I+Ae2 = 333,000 in4 I+Ae2 = 1,870,000 in4

Figure 69. Effect of Girder Stiffness on Skew Corrections for Reaction Across Pier

132

Figure 70. Effect of Girder Stiffness on Skew Corrections for Reaction Across PierFinally, the influence of span length on the variation of the skew correction for reaction

is displayed in Figures 71 and 72. Again, the variation of the correction factor is symmetrical

across the pier for each of the data sets. These results, however, do not yield a correlation

between the change in span length and the variation of the skew correction across the pier.

Figure 71 displays that an increase in span length tends to decrease the magnitude of the

correction at the interior girders, with respect to the correction at girder 1. Furthermore, each

data set of Figure 71 produces a greater skew correction at the interior girders than at the exterior

girders. Figure 72, however, indicates that the corrections at the interior girders are less than

those at the exterior girders. Additionally, the increase in span length increases the magnitude of

the correction at the interior girders, with respect to the correction at girder 1. Figures 71 and 72,

therefore, do not reveal a definite trend between the change in span length and the variation of

the skew correction for reaction at the pier.

133

EFFECT OF SPAN LENGTH ONSKEW CORRECTIONS FORREACTION ACROSS PIER

Two-Span Continuous, Beam-Slab Bridges, I + Ae2 = 333,000 in4,60 deg. Skew, w/o Intermed. Cross Frames

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

105' Spans 168' Spans

EFFECT OF SPAN LENGTH ONSKEW CORRECTIONS FORREACTION ACROSS PIER

Two-Span Continuous, Beam-Slab Bridges, I + Ae2 = 1,870,000 in4,60 deg. Skew, w/o Intermed. Cross Frames

0.00.20.40.60.81.01.21.41.61.82.0

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

105' Spans 168' Spans

Figure 71. Effect of Span Length on Skew Corrections for Reaction Across Pier

134

Figure 72. Effect of Span Length on Skew Corrections for Reaction Across Pier

135

4.5 SKEW CORRECTION FACTORS FROM LRFD SPECIFICATIONS ANDRESEARCH RESULTS

Although not included in the objectives of this research, a cursory evaluation of the skew

correction factors calculated according to Article 4.6.2.2.3c of the LRFD Specifications and

those determined from the analysis models was performed. Table 11 indicates that the empirical

equations of the LRFD Specifications typically corresponded well with the research results. The

research results are within 14% of the empirical equations, with the empirical equations typically

producing more conservative correction factors. In cases where the research models produced

skew correction factors greater than the empirical equations, the greatest differences were found

in bridge models with the extreme 60° skew or with intermediate cross frames. Nevertheless, the

bridge models of this study produced skew correction factors that compare reasonably well to

those calculated according to the LRFD Specifications.

136

LRFD ResearchModel Span Intermed. Span (L) # of Kg ts θ Correction Correction ∆

Type Type X-Frames? (ft) Beams (in4) (in) (deg) Factor FactorBeam-Slab Simple No 42 6 336152 7 30 1.09 1.13 3%Beam-Slab Simple No 42 6 336152 7 60 1.28 1.15 -10%Beam-Slab Simple No 42 9 336152 7 60 1.28 1.11 -14%Beam-Slab Simple No 42 6 357639 9 60 1.35 1.19 -12%Beam-Slab Simple No 168 6 336152 7 60 1.43 1.57 10%

Beam-Slab Simple No 42 6 2522195 7 30 1.05 1.11 6%Beam-Slab Simple No 42 6 2522195 7 60 1.16 1.12 -3%Beam-Slab Simple No 105 6 2522195 7 60 1.20 1.09 -9%Beam-Slab Simple No 168 6 2522195 7 60 1.23 1.12 -9%

Beam-Slab Simple No 105 6 2649798 7 60 1.20 1.10 -8%Beam-Slab Simple No 168 6 2649798 7 60 1.23 1.16 -6%

Beam-Slab Simple Yes 105 6 2522195 7 60 1.20 1.27 5%Beam-Slab Simple Yes 168 6 2522195 7 60 1.23 1.35 9%

T-Beam Simple No 42 6 358040 7 30 1.09 1.09 0%T-Beam Simple No 42 6 358040 7 60 1.28 1.45 13%

Beam-Slab 2-Span Cont. No 168 6 2522195 7 30 1.08 1.03 -4%Beam-Slab 2-Span Cont. No 105 6 2522195 7 60 1.20 1.08 -10%Beam-Slab 2-Span Cont. No 168 6 2522195 7 60 1.23 1.10 -11%Beam-Slab 2-Span Cont. No 105 6 2649798 7 60 1.20 1.14 -5%Beam-Slab 2-Span Cont. No 168 6 2649798 7 60 1.23 1.17 -5%

Definition of Variables:Kg = Longitudinal Stiffness Parameter (in4)

ts = Depth of Concrete Slab (in)θ = Skew Angle measured from a line normal to the CL of Bridge

LRFD Empirical Skew Correction Factor = 1.0 + 0.20 * ( 12.0 L ts3 / Kg )

0.3 tan θ

BRIDGE MODEL DATA

Table 11. Comparison of Skew Correction Factors from LRFD Specifications and ResearchResults

137

CHAPTER 5 INTERPRETATION AND APPLICATION

5.1 SIMPLE SPAN BEAM-SLAB BRIDGES

With respect to simple span beam-slab bridges, the results of this study indicate that:

• The variation of the skew correction factors for shear along the length ofthe exterior girders of simple span beam-slab bridges is not significantlyinfluenced by changes in skew angle, beam stiffness, span length, beamspacing, slab thickness, bridge aspect ratio, or by the presence ofintermediate cross frames.

• While the magnitude of the correction factors may change in concert withchanges of these parameters, the variation of the correction factor alongthe exterior beam length is not significantly altered.

• The skew correction factor typically falls quickly from its initial value at

the end of the exterior girder at the obtuse bridge corner to zero by thefour-tenth point of the span, independent of changes in the aforementionedbridge parameters.

• An appropriate, conservative design approximation for the variation of theskew correction factor for shear along the length of the exterior girder is alinear variation from its initial value at the obtuse corner of the bridge planto zero at some point within the girder span.

Regarding the last conclusion, Figure 73, a plot of the results from all models that

represent realistic bridge designs (span-to-depth ratios range from 13 to 21), displays that a

reasonable approximation for the skew correction factor variation is a straight line from its value

at the obtuse corner to a correction factor of 1.0 at mid-span. In isolated cases, a correction

factor is present near mid-span of the exterior girders, but the magnitudes of the shears near mid-

span are still much less that the magnitudes of the end shears. It is suggested, therefore, that

these corrections near mid-span may be neglected as the shear values at these locations will not

138

govern for typical design applications. It is recognized that design cases in which the web depth,

thickness, yield strength, stiffener spacing, etc. vary along the beam length, the controlling

region for shear design may not be at the typical end of beam location. The correction factor

spikes found at mid-span in a limited number of the data sets, however, were amplified by the

manner of data reduction. Given that the magnitude of the mid-span shear is small, any slight

deviations between the right bridge and skewed bridge mid-span shears may produce appreciable

correction factors in terms of percentages. To illustrate, the data displays some mid-span skew

correction factors between 1.03 to 1.13. The actual difference between the mid-span shears in

the right and skewed bridge models for the correction factor of 1.13 is only 2.8 kips (22.7 k right

bridge; 25.5 k skewed bridge). Attempts to pinpoint design corrections to this level of accuracy

may be misleading in terms of the accuracy of the approximation itself. Furthermore, the bridge

models which possess these mid-span spikes are predominately models with a 168’ span length

and a 60 degree skew. Given the small number of occurrences of the mid-span spikes and the

extreme skew angle of the models in which it did occur, the inclusion of the spikes in a general

design approximation was not considered necessary.

Additionally, select model results yielded a correction factor greater than 1.0 at the acute

corner of the exterior girders. These results, however, were obtained from bridge models with a

span-to-depth ratio of 5.25, well outside ratios of practical design applications. It is suggested,

therefore, that these corrections at the acute corner may be neglected given the fact that these

results did not occur in models that are more representative of actual design cases.

139

NORMALIZED SKEW CORRECTIONS FOR SHEAR IN EXTERIOR GIRDERS of SIMPLE SPAN BEAM-SLAB

BRIDGES

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Tenth Point Along Span

Nor

mal

ized

Ske

w C

orre

ctio

ns

Proposed Design Approximation for the Variation of the Skew Correction Factor

Figure 73. Results for the Variation of the Skew Correction Along the Length of the ExteriorGirders of Simple-Span Beam-Slab Bridges.

140

The study results also indicate that:

• The variation of the skew correction factors for end shear across thebearing lines of simple span beam-slab bridges is not significantly alteredby changes in skew angle, beam stiffness, span length, beam spacing, slabthickness, or by the presence of intermediate cross frames.

• A reasonable, conservative approximation of this variation across the

bearing lines is a linear distribution of the correction factor from its valueat the obtuse corner to a correction factor of 1.0 at the acute corner.

The average variation of the skew correction for end shear across the bearing lines of the

models studied is depicted in Figure 74, indicating that the linear distribution across the bearing

lines is conservative and encompasses the average results.

141

AVERAGE NORMALIZED SKEW CORRECTIONS FOR END SHEAR ACROSS BEARING LINES of

SIMPLE SPAN BEAM-SLAB BRIDGES1.00

0.49

0.080.25

-0.23

-0.54

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns


Figure 74. Average Results for the Variation of the Skew Correction Along the BearingLines of Simple Span Beam-Slab Bridges.

142

5.2 SIMPLE SPAN CONCRETE T-BEAM BRIDGES

The cursory investigation of simple span, monolithic, concrete T-beam bridge models

indicated that:

• The variation of the skew correction factors along both the length of theexterior beams and across the beam supports is very similar to thatobtained from the simple span beam-slab bridge models.

• Regardless bridge model skew angle, the variation of the skew correctionfactors along the length of the beams can be approximated reasonably by alinear distribution of the correction factor at the obtuse corner of theexterior girder to a value of 1.0 at mid-span of the exterior girder.

• Across the bearing lines, the skew correction factor variation can beapproximated by a linear distribution of the correction factor at the obtusecorner of the bearing line to a value of 1.0 at the acute corner.

Although only the effect of skew angle on the correction factor variation was

investigated, it is presumed that changes in other bridge parameters will produce similar results.

Therefore, the design approximations developed for the beam-slab bridges are considered to be

valid for concrete T-beam bridges.

5.3 SIMPLE SPAN SPREAD CONCRETE BOX GIRDER BRIDGES

The analysis of the spread concrete box girder bridge models raised an issue regarding

the influence of torsion on the maximum shear in box girders and the design methodology to be

followed in the development of skew correction factors for this bridge type. Although typically

neglected in right bridges due to the premise of equal deflections of the bridge girders and

143

negligible differential shears between the webs of the box girders, the effects of torsion on box

girder shear may be amplified due to the introduction of skew. The box girder shear data

obtained in this study, which incorporates the effects of torsion, indicates that torsion may not be

negligible in skewed bridges. Without further conclusive 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 of neglecting torsion are considered to be

acceptable. It is recommended, however, that further studies of box girder shear in skewed

bridges be performed to investigate the influence of torsion. Such studies may help determine

whether torsion should be included in approximations for the variation of the skew correction

factors for shear.

5.4 TWO-SPAN CONTINUOUS BEAM-SLAB BRIDGES

The investigation of the two-span continuous beam-slab bridge models reveals that:

• The skew correction factors for shear at the obtuse corner of skewed,simple span beam-slab bridges are valid for the shear at the obtuse cornerof the abutments of skewed, continuous beam-slab bridges.

• The skew correction factors for shear at the obtuse corner of skewed,simple span beam-slab bridges are also valid for shear in exterior girdersof continuous bridges at the obtuse corner created by the girders and thepiers.

• The variation of the skew correction factors for shear along the length ofthe exterior girders is not significantly influenced by changes in skewangle, beam stiffness and span length.

• The variation of the correction factor along the length of the exterior

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girder is similar when the obtuse corner is located at the abutment andwhen it is located at the pier. As a result, each span of the continuousexterior girders possesses the same correction factor at its obtuse corner,as well as the same variation of the correction factor along the span length.

• While the magnitude of the correction factors may change in concert withchanges of skew angle, beam stiffness and span length, the variation of thecorrection factor along the span length is not significantly altered.

• The skew correction factor typically falls quickly from its initial value atthe end of the exterior girder at the obtuse corner to zero by the four-tenthpoint of the span, independent of changes in the aforementioned bridgeparameters.

• An appropriate, conservative design approximation for the variation of theskew correction factor for shear along the length of each span of theexterior girder is a linear variation from its initial value at the girder’sobtuse corner to 1.0 at some point within the girder span, very similar tothe results from the simple span models.

Regarding the last conclusion, Figure 75 displays that a reasonable approximation for the

skew correction factor variation is a straight line from its value at the obtuse corner to a

correction factor of 1.0 at mid-span. Similar to the simple span model results, a correction factor

is present near mid-span of the exterior girders in isolated cases. However, the manner in which

the data is presented tends to amplify the results and the magnitudes of the shears near mid-span

are still much less that the magnitudes of the end shears. It is suggested, therefore, that these

corrections near mid-span may be neglected as the shear values at these locations will not govern

for typical design applications.

145

NORMALIZED SKEW CORRECTIONS FOR SHEAR IN EXTERIOR GIRDERS of TWO-SPAN CONTINUOUS BEAM-SLAB BRIDGES

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Tenth Point Along Span

Nor

mal

ized

Ske

w C

orre

ctio

nsProposed Design Approximation for the Variation of the Skew Correction Factor

Figure 75. Results for the Variation of the Skew Correction Along the Length of the ExteriorGirders of Two-Span Continuous Beam-Slab Bridges.

146

The study results also indicate that:

• The variation of the skew correction factors for shear across the abutmentsand pier of two-span continuous beam-slab bridges is not significantlyaltered by changes in skew angle, beam stiffness and span length.

• A reasonable approximation of this variation across both the abutmentsand pier is a linear distribution of the correction factor from its value atthe obtuse corner of the abutments and pier to a correction factor of 1.0 atthe acute corner.

The average variation of the skew correction factor for shear across the abutments and

the piers of the models studied is depicted in Figure 76, including the proposed design office

approximation.

147

AVERAGE NORMALIZED SKEW CORRECTIONSFOR SHEAR ACROSS ABUTMENTS

AND PIERS of TWO-SPAN CONTINUOUSBEAM-SLAB BRIDGES

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.801.00

1 2 3 4 5 6

Girder

Nor

mal

ized

Ske

w C

orre

ctio

ns

Abutment Pier


Figure 76. Average Results for the Variation of the Skew Correction Across Abutments andPiers of Two-Span Beam-Slab Bridges.

148

The investigation of the two-span models does reveal that skew correction factors for

reaction at piers of continuous bridges are present at each girder, and that these correction factor

are unique from those required for the shear at the pier. However, the variation of the correction

factor across the pier with changes in the bridge parameters is not clearly understood from the

limited number of data sets in this study. Although changes of skew angle and girder stiffness in

the study models did provide insight into possible effects on the correction factors, it is difficult

to define that variation from the small pool of data sets. The study results do indicate, however,

that further research is required to develop empirical equations for both the skew correction

factors for reactions at piers and the variation of those correction factors across the pier.

5.5 APPLICATION OF STUDY FINDINGS

From the study findings, it was determined that regardless of bridge parameters, a

reasonable design approximation for the variation of the skew correction factor for shear along

the length of the exterior girder of simple span beam-slab and monolithic concrete T-beam

bridges is a linear variation from its initial value at the obtuse corner to a correction factor of 1.0

at mid-span. Similarly, regardless of bridge parameters, the variation of the skew correction

factor for shear along the length of the exterior girders in each span of two-span continuous

beam-slab bridges may be reasonably approximated with a linear variation from its initial value

at the obtuse corner of the girder to a correction factor of 1.0 at mid-span. Therefore, for

149

application of the research findings regarding the variation of the skew correction factor for

shear along the length of exterior girders, the recommendations are as follows:

• For superstructure types “Concrete Deck, Filled Grid, or Partially Filled Grid onSteel or Concrete Beams; Concrete T-Beams, T- and Double T Section,” withinthe applicable ranges of skew angle (θ), spacing of beams or webs (S), span ofbeam (L) and number of beams, stringers or girders (Nb) as defined by Table4.6.2.2.3c-1 of the LRFD Specifications, the skew correction factor for shear maybe varied linearly from its value at the obtuse corner of the bridge, determined inaccordance with the empirical equation defined in Table 4.6.2.2.3c-1, to a valueof 1.0 at girder mid-span, as shown in Figure 77.

• This approximate variation is applicable for both simple span structures andcontinuous structures. For continuous structures, the skew correction factorcalculated at the obtuse corner of the abutment per Table 4.6.2.2.3c-1 is also validat the obtuse corners of the interior piers. Likewise, the variation of thecorrection factor is applicable from both the obtuse corner of the abutment and theobtuse corners of the interior piers to the girder mid-span, as shown in Figure 78.

Although this study did not investigate each superstructure type within the group

“Concrete Deck, Filled Grid, or Partially Filled Grid on Steel or Concrete Beams; Concrete T-

Beams, T- and Double T Section,” the bridge models studied are representative of this class of

superstructure, with the bridge parameters as defined by Table 4.6.2.2.3c-1 of the LRFD

Specifications. The study findings and proposed design approximations, therefore, are

considered to be valid for each type of structure within this class.

150

C Abutment (Typ.)L

Calculated Skew CorrectionFactor at Obtuse Corner (Typ.)

C Girder (Typ.)L

Skew Angle

1.0

1.0

Linear Variation of theCorrection Factor (Typ.)

Mid-Point of GirderSpan (Typ.)

C Abutment (Typ.)L

Calculated Skew CorrectionFactor at Obtuse Corner ofAbutment (Typ.)

C Girder (Typ.)L

Skew Angle

1.0

1.0


Mid-Point of GirderSpan (Typ.)

C PierL

1.0

1.0

Skew Correction Factor alsoApplied at Obtuse Corner of Pier(Typ.)

Figure 77. Proposed Variation of the Skew Correction Factors for Shear Along the Length ofthe Exterior Girders in Simple Span Superstructures of Concrete Deck, FilledGrid, or Partially Filled Grid on Steel or Concrete Beams; Concrete T-Beams, T-

and Double T Sections

Figure 78. Proposed Variation of the Skew Correction Factors for Shear Along the Length ofthe Exterior Girders in Continuous Superstructures of Concrete Deck, Filled Grid,

151

or Partially Filled Grid on Steel or Concrete Beams; Concrete T-Beams, T- andDouble T Sections

152

The study findings also reveal that regardless of bridge parameters, a reasonable design

approximation for the variation of the skew correction factors for end shear of each girder across

bearing lines of simple span beam-slab bridges and monolithic concrete T-beam bridges is a

linear variation from its initial value at the obtuse corner of the bearing line to a correction factor

of 1.0 at the acute corner of the bearing line. Similarly, regardless of bridge parameters, the

variation of the skew correction factor for shear of each girder across the abutments and piers of

two-span continuous beam-slab bridges may be reasonably approximated with a linear variation

from its initial value at the obtuse corner of the bearing line to a correction factor of 1.0 at the

acute corner of the bearing line. Therefore, for application of the research findings regarding the

variation of the skew correction factor for shear across bearing lines, the recommendations are as

follows:

• For superstructure types “Concrete Deck, Filled Grid, or Partially Filled Grid onSteel or Concrete Beams; Concrete T-Beams, T- and Double T Section,” withinthe applicable ranges of skew angle (θ), spacing of beams or webs (S), span ofbeam (L) and number of beams, stringers or girders (Nb) as defined by Table4.6.2.2.3c-1 of the LRFD Specifications, the skew correction factor for shear maybe varied linearly from its value at the obtuse corner of the bridge, determined inaccordance with Table 4.6.2.2.3c-1, to a value of 1.0 at the acute corner of thebearing line, as shown in Figure 79.

• This approximate variation is applicable for both simple span structures andcontinuous structures. For continuous structures, the skew correction factorcalculated at the obtuse corner of the abutment per Table 4.6.2.2.3c-1 is also validat the obtuse corners of the interior piers. Likewise, the variation of thecorrection factor is applicable from both the obtuse corner of the abutment and theobtuse corners of the interior piers to the acute corner of the bearing lines, asshown in Figure 80.

153

As discussed previously, this study did not investigate each superstructure type within the

group “Concrete Deck, Filled Grid, or Partially Filled Grid on Steel or Concrete Beams;

Concrete T-Beams, T- and Double T Section.” The bridge models studied, however, are

representative of this class of superstructure, with the bridge parameters as defined by Table

4.6.2.2.3c-1 of the LRFD Specifications. The study findings and proposed design

approximation, therefore, are considered to be valid for each type of structure within this class.

154

C Abutment (Typ.)L

Calculated Skew CorrectionFactor at Obtuse Corner (Typ.)

C Girder (Typ.)L

Skew Angle

1.0

1.0


C Abutment (Typ.)L

Calculated Skew CorrectionFactor at Obtuse Corner ofAbutment (Typ.)

C Girder (Typ.)L

Skew Angle


C PierL

Skew Correction Factor alsoApplied at Obtuse Corner of Pier(Typ.)

1.0

1.01.0

1.0

Figure 79. Proposed Variation of the Skew Correction Factors for Shear Across the BearingLines of Simple Span Superstructures of Concrete Deck, Filled Grid, or PartiallyFilled Grid on Steel or Concrete Beams; Concrete T-Beams, T- and Double TSections

Figure 80. Proposed Variation of the Skew Correction Factors for Shear Across theAbutments and Piers of Continuous Superstructures of Concrete Deck, FilledGrid, or Partially Filled Grid on Steel or Concrete Beams; Concrete T-Beams, T-and Double T Sections

155

CHAPTER 6 CONCLUSIONS AND SUGGESTED RESEARCH

The variation of the skew correction factors for shear along the length of the exterior

girders of simple span beam-slab bridges is not significantly influenced by changes in skew

angle, beam stiffness, span length, beam spacing, slab thickness, bridge aspect ratio, or by the

presence of intermediate cross frames. It is recommended that a reasonable design

approximation for the variation of the skew correction factor for shear along the length of the

exterior girder of simple span beam-slab bridges is a linear variation from its initial value at the

obtuse corner to a correction factor of 1.0 at mid-span, as shown in Figure 77. Regardless of the

aforementioned bridge parameters, therefore, the skew correction factor may be calculated for

the obtuse corner as defined in the LRFD Specifications and varied linearly to a value of 1.0 at

the mid-point of the girder span. This approximation is recommended for 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, within the geometric limitations defined in Table

4.6.2.2.3c-1 of the LRFD Specifications.

Additionally, the variation of the skew correction factors for end shear of each girder

across bearing lines of simple span beam-slab bridges is not significantly altered by changes in

span length, beam spacing, slab thickness, skew angle and beam stiffness, or by the presence of

intermediate cross frames. Therefore, it is recommended that the skew correction factor be

calculated for the end shear of the exterior girder at the obtuse corner of the bridge, as defined in

the LRFD Specifications, and varied linearly across the bearing lines to a value of 1.0 at the

acute corner, as shown in Figure 79. Again, this approximation is reasonable for simple span

156

superstructures of concrete deck, filled grid, or partially filled grid on steel or concrete beams;

concrete T-beams, T- and double T sections, within the geometric limitations defined in Table

4.6.2.2.3c-1 of the LRFD Specifications.

The skew correction factors for shear defined in the current LRFD Specifications at the

obtuse corner of skewed, simple span beam-slab bridges are also valid for the shear at the obtuse

corner of the abutments of skewed, continuous beam-slab bridges. Furthermore, these same

skew correction factors are also valid for shear in exterior girders of continuous bridges at the

obtuse corner created by the girders and the piers.

Similar to the conclusions regarding the variation of the skew correction factors for shear

along the length of the exterior girders of simple span bridges, the variation of the correction

factors for two-span continuous beam-slab bridges is not significantly influenced by changes in

skew angle, beam stiffness and span length. The same design approximation proposed for the

correction factor variation in the simple span bridges is also recommended for each span of the

continuous bridges. A reasonable design approximation for the variation of the skew correction

factor for shear along the length of the exterior girders in each span of continuous beam-slab

bridges is a linear variation from its initial value at the obtuse corner of the girder to a correction

factor of 1.0 at mid-span, as shown in Figure 78. Regardless of the aforementioned bridge

parameters, therefore, the skew correction factor may be calculated for the end shear of the

exterior girder as defined in the LRFD Specifications, applied at the obtuse corners of both the

abutments and piers, and varied linearly to a value of 1.0 at the mid-point of the girder span.

This approximation is appropriate for continuous superstructures of concrete deck, filled grid, or

157

partially filled grid on steel or concrete beams; concrete T-beams, T- and double T sections,

within the geometric limitations defined in Table 4.6.2.2.3c-1 of the LRFD Specifications.

Additionally, the variation of the skew correction factors for shear of each girder across

the abutments and piers of continuous beam-slab bridges is not significantly altered by changes

in skew angle, girder stiffness and span length. The recommended design approximation for the

simple span bridges is also valid across the abutments and piers of continuous bridges.

Therefore, the skew correction factor can be calculated for the end shear of the exterior girder, as

defined in the LRFD Specifications, applied at the obtuse corners of both the abutments and

piers, and varied linearly across the abutments and piers to a value of 1.0 at the acute corner, as

shown in Figure 80. Again, this approximation is appropriate for continuous superstructures of

concrete deck, filled grid, or partially filled grid on steel or concrete beams; concrete T-beams,

T- and double T sections, within the geometric limitations defined in Table 4.6.2.2.3c-1 of the

LRFD Specifications.

The results of this study lead to the following recommendations for further research:

• Skew Correction Factors for Reaction at the Piers of Continuous Bridges:

It was determined that these correction factors are present and are uniquefrom those calculated for shear at the pier. The effects of the obtuse andacute corners on the girder shear on opposite sides of the bearings do noteliminate a correction factor for reaction. From the limited continuousbridge model data pool of this study, however, it was not feasible todevelop empirical equations that define the correction factor or define itsvariation across the pier. Therefore, the development of such equationsfor continuous bridges is suggested as further research.

• Skew Correction Factors of Other Beam and Slab Bridge Types:

This research focused primarily on the skew correction factors for shear insimple-span and two-span continuous beam-slab bridges and providesrecommendations for only superstructures of concrete deck, filled grid, or

158

partially filled grid on steel or concrete beams; concrete T-beams, T- anddouble T sections. These bridges, however, comprise only one type of thelarger genre of beam and slab bridges. The LRFD Specifications provideempirical equations for skew correction factors of multi-cell concrete boxbeam, spread concrete box beam and multi-beam bridges. It isrecommended that further research be performed to investigate thebehavior of the skew correction factors for shear in these additional bridgetypes.

• Torsion in Skewed Box Beam Bridges:

Additional work could be performed to investigate torsion in skewed,spread box beam bridges and the magnitude of its effect on shear in boxgirder webs. Given the lack of substantial field documentation indicatingproblems with torsion and shear in skewed spread box girder bridges,however, the current design practices which neglect torsion are consideredto be acceptable.

159

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