20-7/Task 107 COPY NO. _____

SHEAR IN SKEWED MULTI-BEAM BRIDGES

FINAL REPORT

Prepared for

National Cooperative Highway Research Program

Transportation Research Board

National Research Council

Modjeski and Masters, Inc.

NCHRP Project 20-7/Task 107

March 2002

ii

iii

ACKNOWLEDGMENT OF SPONSORSHIP

This work was sponsored by the American Association of State Highway and

Transportation Officials, in cooperation with the Federal Highway Administration, and was

conducted in the National Cooperative Highway Research Program, which is administered

by the Transportation Research Board of the National Research Council.

DISCLAIMER

This is an uncorrected draft as submitted by the research agency. The opinions and

conclusions expressed or implied in the report are those of the research agency. They are

not necessarily those of the Transportation Research Board, the National Research Council,

the Federal Highway Administration, the American Association of State Highway and

Transportation Officials, or the individual states participating in the National Cooperative

Highway Research Program.

iv

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......................................................5

1.2 Research Objectives...............................................13

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

2.1 Introduction.....................................................14

2.2 NCHRP Project 12-26.............................................14

2.3 Ontario Highway Bridge Design Code................................18

2.4 Additional Work.................................................19

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

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

4.1 Simple Span Beam-Slab Bridge Models...............................40

4.1.1 Live Load Shear Along Exterior Beam Length....................40

4.1.1.1 Influence of Skew Angle.........................40

4.1.1.2 Influence of Beam Stiffness.......................43

4.1.1.3 Influence of Span Length.........................47

4.1.1.4 Influence of Intermediate Cross Frames.............50

4.1.1.5 Influence of Beam Spacing.......................53

4.1.1.6 Influence of Slab Thickness.......................55

4.1.1.7 Influence of Bridge Aspect Ratio..................57

4.1.2 Live Load Shear Across Bearing Lines..........................60

4.2 Simple Span Concrete T-Beam Bridge Models..........................69

4.2.1 Live Load Shear Along Exterior Beam Length....................69

4.2.2 Live Load Shear Across Bearing Lines..........................71

TABLE OF CONTENTS (continued)

v

4.3 Simple Span Spread Concrete Box Girder Bridge Models.................73

4.4 Two-Span Continuous Beam-Slab Bridge Models.......................78

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

of Abutments..............................................78

4.4.2 Correction Factors at Obtuse Corners of Abutments and Piers........81

4.4.3 Live Load Shear Along Exterior Beam Length....................84

4.4.3.1 Influence of Skew Angle.........................84

4.4.3.2 Influence of Beam Stiffness.......................88

4.4.3.3 Influence of Span Length.........................91

4.4.4 Live Load Shear Across Abutment Bearing Lines.................93

4.4.5 Live Load Shear Across Pier.................................103

4.4.6 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....................................128

5.2 Simple Span Concrete T-Beam Bridges..............................133

5.3 Simple Span Spread Concrete Box Girder Bridges......................133

5.4 Two-Span Continuous Beam-Slab Bridges............................134

5.5 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 OF

BRIDGE MODELS.......................................B-1

vi

LIST OF FIGURES

Figure 1.Typical Beam and Slab Superstructures................................7

Figure 2.Plan View of Typical Skewed Superstructure. Current Application

of the Skew Correction Factor for Shear per the AASHTO LRFD

Bridge Design Specifications........................................11

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

for Maximum Shear...............................................17

Figure 4.Bridge Plan Geometries for Analysis.................................27

Figure 5.Schematic Diagram of BSDI Finite Element Modeling...................33

Figure 6.Transformation of Concrete Section to Steel Section.....................33

Figure 7.Procedure for Calculation of the Normalized Skew Corrections............37

Figure 8.Effect of Skew Angle on Skew Corrections Along Exterior Beams..........42

Figure 9.Effect of Skew Angle on Skew Corrections Along Exterior Beams..........42

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

Figure 11.Effect of Girder Stiffness on Skew Corrections Along Exterior Beams.......45

Figure 12.Effect of Girder Stiffness on Skew Corrections Along Exterior Beams.......46

Figure 13.Effect of Girder Stiffness on Skew Corrections Along Exterior Beams.......46

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

Figure 15.Effect of Span Length on Skew Corrections Along Exterior Beams..........48

Figure 16.Effect of Span Length on Skew Corrections Along Exterior Beams..........49

Figure 17.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 23.Effect of Skew Angle on End Shear Skew Corrections....................62

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

Figure 25.Effect of Girder Stiffness on End Shear Skew Corrections.................63

Figure 26.Effect of Girder Stiffness on End Shear Skew Corrections.................64

Figure 27.Effect of Girder Stiffness on End Shear Skew Corrections.................64

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

Figure 29.Effect of Span Length on End Shear Skew Corrections...................65

Figure 30.Effect of Span Length on End Shear Skew Corrections...................66

Figure 31.Effect of Intermediate Cross Frames on End Shear Skew Corrections........66

Figure 32.Effect of Intermediate Cross Frames on End Shear Skew Corrections........67

Figure 33.Effect of Slab Thickness on End Shear Skew Corrections.................67

Figure 34.Complete Results for End Shear Skew Corrections......................68

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

Beam-Slab Bridges...............................................68

Figure 36.Effect of Skew Angle on Skew Corrections Along Exterior Beams..........70

Figure 37.Effect of Skew Angle on End Shear Skew Corrections....................72

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

Correction Factors................................................80

viii

LIST OF FIGURES (continued)

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

Figure 40.Nomenclature for Investigation of Correction Factors Along the Length

of the Exterior Girders of Two-Span Continuous Bridge Models............85

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

of Continuous Models.............................................87

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

of Continuous Models.............................................90

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

of Continuous Models.............................................90

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

of Continuous Models.............................................92

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

of Continuous Models.............................................92

Figure 46.Nomenclature for Investigation of Correction Factors Across the

Abutment Bearing Lines of Two-Span Continuous Bridge Models..........94

Figure 47.Effect of Skew Angle on End Shear Skew Corrections At Abutments........95

Figure 48.Effect of Girder Stiffness on End Shear Skew Corrections At Abutments.....95

Figure 49.Effect of Girder Stiffness on End Shear Skew Corrections At Abutments.....96

Figure 50.Effect of Span Length on End Shear Skew Corrections At Abutments.......96

Figure 51.Effect of Span Length on End Shear Skew Corrections At Abutments.......97

Figure 52.Complete Results Set for End Shear Skew Corrections At Abutments........99

Figure 53.Average Variation of End Shear Skew Corrections Across Abutments

of Two-Span Continuous Beam-Slab Bridges..........................102

Figure 54.Nomenclature for Investigation of Correction Factors Across the

Pier of Two-Span Continuous Bridge Models..........................104

ix

LIST OF FIGURES (continued)

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 Piers

of Two-Span Continuous Beam-Slab Bridges..........................110

Figure 62.Nomenclature for Investigation of Correction Factors for Reaction at the

Pier of Two-Span Continuous Bridge Models..........................113

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

Figure 64.Comparison of Skew Correction Factors for Shear and Reaction at Pier.....116

Figure 65.Comparison of Skew Correction Factors for Shear and Reaction at Pier.....117

Figure 66.Comparison of Skew Correction Factors for Shear and Reaction at Pier.....118

Figure 67.Comparison of Skew Correction Factors for Shear and Reaction at Pier.....119

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 Length

of the Exterior Girders of Simple-Span Beam-Slab Bridges...............130

x

LIST OF FIGURES (continued)

Figure 74.Average Results for the Variation of the Skew Correction Along the

Bearing Lines of Simple-Span Beam-Slab Bridges......................132

Figure 75.Results for the Variation of the Skew Correction Along the Length

of the Exterior Girders of Two-Span Continuous Beam-Slab Bridges.......136

Figure 76.Average Results for the Variation of the Skew Correction Across

Abutments 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 Across

the Bearing Lines of Simple Span Superstructures of

Concrete Deck, Filled Grid, or Partially Filled Grid on Steel or

Concrete Beams; Concrete T-Beams, T- and Double T Sections...........144

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

the Abutments and Piers of Continuous Superstructures of

Concrete Deck, Filled Grid, or Partially Filled Grid on Steel or

Concrete Beams; Concrete T-Beams, T- and Double T Sections...........144

xi

LIST OF TABLES

Table 1.Correction Factors for Load Distribution Factors for Support Shear

of the Obtuse Corner...............................................9

Table 2.Maximum Shear Forces at Pier Support for Three-Lane Bridge with

Different Skew Angles Predicted Using Different Methods................22

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

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

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

Bridge Models...................................................30

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

Girder Bridge Models.............................................30

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

LRFD Line Girder Analysis........................................76

Table 8.Comparison of Skew Correction Factors for End Shear of Exterior Girders

at the Obtuse Corners of Simple Span and Two-Span Bridge Models........80

Table 9.Comparison of Skew Correction Factors for Shear of Exterior Girders

at the Obtuse Abutment Corners and Obtuse Pier Corners of

Two-Span Bridge Models..........................................83

Table 10.Correction Factors for Reaction at the Pier of Two-Span

Beam-Slab Bridge Models.........................................113

Table 11. Comparison of Skew Correction Factors from LRFD Specifications

and Research Results.............................................127

xii

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.

xiii

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 12-26

2

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 12-26

2

.

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, a

reasonable approximation for the variation of the skew correction factor

along the length of exterior girders of simple span beam-slab and concrete

T-beam bridges is a linear distribution of the factor from its value at the

obtuse corner of the bridge, determined according to the LRFD

Specifications, to a value of 1.0 at girder mid-span.

• Regardless of changes in the aforementioned bridge parameters, a

reasonable approximation of the skew correction factor for live load shear

across the bearing lines of simple span beam-slab and concrete T-beam

bridges is a linear distribution of the correction factor from its value at the

obtuse corner of the bridge, determined according to the LRFD

Specifications, to a value of 1.0 at the acute corner of the bridge.

For the two-span continuous bridge models studied, the research results indicate that:

• The variations of the skew correction factors for shear along the length of

exterior girders in each span and for shear across both the abutments and

piers of two-span continuous beam-slab bridges are identical to those

proposed for simple span bridges. The correction factor variation along

the exterior girder may be approximated by a linear distribution of the

factor at the obtuse corner to a value of 1.0 at girder mid-span. Likewise,

the variation across the abutments and piers is approximated by a linear

distribution of the factor at the obtuse corner to a value of 1.0 at the acute

corner.

• The skew correction factor defined by the LRFD Specifications is valid for

the girder shear at the obtuse corners of both the abutments and piers of

the continuous bridges.

• Skew correction factors for reaction at the piers of continuous bridges are

present and are unique from those calculated for shear at the piers. From

the limited continuous bridge model data of this study, however, accurate

empirical equations which define the correction factor or define its

variation across the pier could not be derived. Therefore, the development

of such equations for continuous bridges is necessary and is recommended

for further research.

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

3

Skew Correction Factor for Shear, Variation Along Exterior Beam Length

• For superstructure types “Concrete Deck, Filled Grid, or Partially Filled Grid on

Steel or Concrete Beams; Concrete T-Beams, T- and Double T Section,” within

the applicable ranges of skew angle (θ), spacing of beams or webs (S), span of

beam (L) and number of beams, stringers or girders (N

b

) as defined by Table

4.6.2.2.3c-1 of the LRFD Specifications, the skew correction factor for shear may

be varied linearly from its value at the obtuse corner of the bridge, determined in

accordance with the empirical equation defined in Table 4.6.2.2.3c-1, to a value

of 1.0 at girder mid-span.

• This approximate variation is applicable for both simple span structures and

continuous structures. For continuous structures, the skew correction factor

calculated at the obtuse corner of the abutment per Table 4.6.2.2.3c-1 is also valid

at the obtuse corners of the interior piers. Likewise, the variation of the

correction factor is applicable from both the obtuse corner of the abutment and the

obtuse corners of the interior piers to the girder mid-span.

Skew Correction Factor for Shear, Variation Across Bearing Lines

• For superstructure types “Concrete Deck, Filled Grid, or Partially Filled Grid on

Steel or Concrete Beams; Concrete T-Beams, T- and Double T Section,” within

the applicable ranges of skew angle (θ), spacing of beams or webs (S), span of

beam (L) and number of beams, stringers or girders (N

b

) as defined by Table

4.6.2.2.3c-1 of the LRFD Specifications, the skew correction factor for shear may

be varied linearly from its value at the obtuse corner of the bridge, determined in

accordance with Table 4.6.2.2.3c-1, to a value of 1.0 at the acute corner of the

bearing line.

• This approximate variation is applicable for both simple span structures and

continuous structures. For continuous structures, the skew correction factor

calculated at the obtuse corner of the abutment per Table 4.6.2.2.3c-1 is also valid

at the obtuse corners of the interior piers. Likewise, the variation of the

correction factor is applicable from both the obtuse corner of the abutment and the

obtuse corners of the interior piers to the acute corner of the bearing lines.

Additional suggested research includes an investigation of the effects of torsion on web

shear in spread box girder bridges. The study results indicate that although torsion is typically

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 12-26

2

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

SUPPORTING

COMPONENTS

TYPE OF DECK TYPICAL CROSS-SECTION

Steel Beam Cast-in-place concrete

slab, precast concrete

slab, steel grid,

glued/spiked panels,

stressed wood

Precast Concrete I or Bulb-

Tee Sections

Cast-in-place

concrete, precast

concrete

Closed Steel or Precast

Concrete Boxes

Cast-in-place concrete

slab

Open Steel or Precast

Concrete Boxes

Cast-in-place concrete

slab, precast concrete

deck slab

Cast-in-Place Concrete Tee

Beam

Monolithic concrete

Figure 1. Typical Beam and Slab Superstructures

1

.

8

The current design methodology in Section 4 of the LRFD Specifications

1

for typical,

right beam and slab bridges permits the use of empirical distribution factors for determination of

the live load effects in bridge beams. For the mid-span bending moment and end shear of

exterior beams in skewed beam and slab bridges, the LRFD Specifications provide correction

factors that are to be applied to the moment and shear distribution factors, calculated for the

corresponding right bridge. These empirical skew correction factors for end shear in beam and

slab bridges, as defined in Table 4.6.2.2.3c-1 of the LRFD Specifications

1

and as shown in Table

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

in 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 Design

Specifications, requires that shear in the exterior beam at the obtuse corner of the

bridge shall be adjusted when the line of support is skewed. The Specifications

provide correction factors for this adjustment and require that the correction

factors be applied to all beams in the cross-section.

In the development of these correction factors, the variation of the effect of skew

on the individual beam reactions was not considered. In addition, the

Specifications provide no guidance on the influence of skew on the shear along

the length of the beam. The commentary to the Specifications states that the

prescribed corrections are conservative. As a consequence of this conservatism

some beams in the bridge are overdesigned.”

It is not only this conservatism that has been the topic of discussions surrounding the skew

correction factors, but also the extent to which the correction factors apply to the shear along the

length of the exterior girder.

9

Table 1. Correction Factors for Load Distribution Factors for Support Shear of the

Obtuse Corner

1

.

Type of Superstructure Correction Factor Range of

Applicability

Concrete Deck, Filled Grid, or

Partially Filled Grid on Steel or

Concrete Beams; Concrete T-

Beams, T- and Double T

Section

0°

#

2

#

60°

3.5

#

S

#

16.0

20

#

L

#

240

N

b

$

4

Multicell Concrete Box Beams,

Box Sections

0° <

2

#

60°

6.0 < S

#

13.0

20

#

L

#

240

35

#

d

#

110

N

c

$

3

Concrete Deck on Spread

Concrete Box Beams

0 <

2

#

60°

6.0

#

S

#

11.5

20

#

L

#

140

18

#

d

#

65

N

b

$

3

Concrete Box Beams Used in

Multibeam Decks

0 <

2

#

60°

20

#

L

#

120

17

#

d

#

60

35

#

b

#

60

5

#

N

b

#

20

Where:

2

= skew angle (degrees) N

b

= number of beams, stringers or girders

S = spacing of beams or webs (ft) N

c

= number of cells in a concrete box girder

L = span of beam (ft) t

s

= depth of concrete slab (in)

b = width of beam (in) K

g

= longitudinal stiffness parameter (in

4

)

d = depth of beam or stringer (in)

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 Bridges

4

, indicated that the skew correction factors were derived for

only the end shears of the exterior girders at the obtuse corners of simple span bridges. In

general, the end shear tends to increase as the skew angle of the supports increases beyond

approximately 15° to 20°. For the LRFD Specifications, however, the working group for

NCHRP 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 section

5

, 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 beam

5

, but made no provisions in the LRFD Specifications to do so. During the

development of the skew correction factors in NCHRP 12-26, the effect of skew on the shear

along the length of the exterior beams was not investigated, and the current LRFD Specifications

do not address this issue.

11

C Abutment (Typ.)

L

Correction Factor Calculated for

and Applied to End Shear at

Obtuse Corner of Exterior Girder

(Typ.)

C Girder (Typ.)

L

Correction Factor

Conservatively Applied to

End Shear of All Girders

(Typ.)

Skew Angle

Figure 2. Plan View of Typical Skewed Superstructure. Current Application of the Skew

Correction Factor for Shear per the AASHTO LRFD Bridge Design Specifications

1

.

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 bridges

5

. Intuition may suggest, therefore, that at the

interior supports of continuous spans, where both an obtuse and acute corner exist opposite each

other, the skew effects on shear may cancel out for determination of the total reaction. This

hypothesis, however, has not yet been investigated and is not addressed in the LRFD

Specifications.

As a result of these outstanding issues regarding the skew correction factors for shear,

this project focuses on investigating and more accurately assessing the effect of skew on end

shear across bearing lines and on shear along the length of exterior beams of beam and slab

bridges. This research concentrates on simple span bridges, with a cursory evaluation of 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 12-26

2

, Khaleel and Itani

7

, Bishara, et al.

8

and Ebeido and

Kennedy

9

has concentrated on moment distributions in skewed, simply-supported and

continuous beam and slab bridges. The research devoted to the distribution of shear and bearing

reactions in skewed bridges, however, is confined to a rather limited set of sources.

2.2 NCHRP PROJECT 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

12-26

4

. 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 GENDEK5A

11

and

FINITE

12

for finite element analysis. The skew correction factors were developed such that they

could be applied to the newly derived distribution factors of a right bridge with the same

geometric parameters as the skewed bridge under investigation. In order to incorporate the

effects of each bridge parameter that had a significant impact on the load distribution of right

bridges, parametric studies of skewed bridges were completed, similar to those performed for the

right bridges. The live load used in the parametric studies consisted of two trucks placed

transversely on the bridge cross section to maximize the girder responses. Test models of

different live load placements confirmed that two trucks typically produced the governing girder

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 Specifications

1

.

As discussed in Section 1.1, the LRFD Specifications require that the correction factors be

applied not only to the end shear of the exterior beams, but also to the end shear of each beam in

the bridge cross section. During the development of the skew correction factors, however,

variation of the effect of skew on the end shear of interior beams was not investigated. The

application of the skew correction factors to all beams of a cross section is considered to be

conservative; therefore, it is suspected that certain beams may be over-designed. Additionally,

the effect of skew on shear along the length of exterior beams of beam and slab bridges was not

investigated in NCHRP Project 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 Bakht

14

which indicated that the measure of bridge “skewness” is also a

function of span length, bridge width and girder spacing. Hence, the skew limitation, ε, was

redefined in the third edition to incorporate these effects, as shown in Equation 1. Bridges

beyond the skewness limit of 1/18 must be analyzed using a refined method such as grillage

analysis, orthotropic plate theory or finite element analysis. Skewed bridges within this limit

may be analyzed using the load distribution factors developed for right bridges, with the

associated error of this procedure estimated at less than 5%.

(Equation 1)

ε = ≤

S Tan

L

1

18

( )Ψ

where:

S = beam spacing

L = span length

Q = skew angle

2.4 ADDITIONAL WORK

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

Kennedy

15,16

. Their research focused on load distribution in skewed composite bridges, both

simple span and continuous, and included studies of moment, shear and reactions. Two separate

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 ABAQUS

17

for the finite

element analysis of the bridge models. The results of the computer analyses were verified

through physical testing of six scale bridge models. Empirical formulas were developed for end

shear distribution factors of both dead load and OHBDC truck live load. The empirical

formulas were derived separately for exterior girders at the acute corner of the bridge, exterior

girders at the obtuse corner and interior girders. The effect of skew on shear along the length of

the girders was not addressed.

The second study by Ebeido and Kennedy focused on continuous skewed composite

bridges and the distribution of both shear and reactions at interior piers. Similar to the study for

simple spans, this research incorporated over 600 two-span continuous bridges with

investigation of the aforementioned parameters, as well as the ratio of adjacent span lengths.

ABAQUS was again used for the finite element modeling, and verified through physical testing

of three scale models of continuous bridges. The live load used in this research, however, was

the AASHTO HS20-44 truck. This facilitated comparison of the empirical formulas for

distribution of shear at pier supports developed in Ebeido and Kennedy’s research with those

from NCHRP 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 Skew

Angles Predicted Using Different Methods

17

Shear force

(kN)

(1)

Skew

angle

(degrees)

(2)

LRFD

(1994)

(3)

NCHRP

(1988)

(4)

Proposed

formulas

(5)

Finite-

element

analysis

(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 Aggour

18

. Their analysis of 12 single track

railway bridges, with superstructures consisting of two steel plate girders, focused on the

distribution of bending moments. The authors’ findings, however, indicate that the variation in

number of intermediate cross frames had little impact on the magnitude of reactions at the acute

and obtuse corners of the bridges. The girder reactions for models with varying numbers of

intermediate cross frames did not differ from those of a model possessing only end cross frames.

The research performed by Bell

19

in 1998 focused on evaluating the shear and moment

distribution factors currently specified in the Standard Specifications and the LRFD

Specifications. Bell investigated straight, skewed, simple span and continuous beam and slab

bridges, both with and without intermediate diaphragms, using both field test data and finite

element analysis with ANSYS

20

. The research objective was to develop empirical equations for

load distribution in continuous bridges, if it was determined that modifications to the existing

equations were required to provide more accurate distribution results. Using the AASHTO

HS20-44 truck for live load, parametric studies were performed, investigating the effects of the

number of spans, span length, span length ratio, skew angle and girder spacing. The results

indicated that the distribution factors provided in the LRFD Specifications accurately assess the

effect of skew on the distribution of shear, and therefore, no modifications to the current

equations for shear distribution were recommended.

In his research project “Forces At Bearings Of Skewed Bridges”, Bishara investigated 36

simply supported composite multi-stringer bridges to evaluate the reaction components at the

rocker and bolster bearings under both dead load and HS20-44 live loads

21

. While most design

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 ADINA

22

for

the finite element analysis, a parametric study was performed to determine the effects span

length, deck width and skew angle on the girder reactions. Two field tests were performed, one

on a simple span bridge and one on a two span continuous bridge, to validate the results of the

finite element analysis.

The research conclusions that addressed the live load vertical reactions were: (i) Bearing

forces differ substantially between the interior and exterior girders and between the obtuse and

acute corners; (ii) The maximum live load reaction for the exterior girder is obtained when the

trucks are placed at the obtuse corner; (iii) The maximum live load reaction for the interior

girders was about 98% of the value computed per the Standard Specifications; therefore, the

design approximations in the Standard Specifications are suitable for design, and; (iv) The

maximum live load reaction for the exterior girder was less than that obtained from the

AASHTO procedures.

El-Ali investigated the internal forces in four 137-foot simply-supported, welded steel

plate girder bridges with various skew angles to determine the effect of skew on girder bending

moments, torsional moments and shears

23

. Finite element analyses of the four bridge models,

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

24

. The girder spacing of

each bridge model was constant and intermediate and end cross frames were included. Four

lanes of HS20-44 live load were applied in six different configurations in order to obtain the

maximum results. The research conclusions indicated that the live load shears obtained from the

finite element models did not have a definite correlation to those calculated using the distribution

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 Specifications

25

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

Skew

Angle,

2

(I+Ae

2

)

1

(I+Ae

2

)

2

(I+Ae

2

)

3

0 L 2.5L 4L L 2.5L 4L L 2.5L 4L

30 L 2.5L 4L L 2.5L 4L L 2.5L 4L

60 L 2.5L 4L L 2.5L 4L L 2.5L 4L

Figure 4. Bridge Plan Geometries for Analysis

28

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

2

(where I = girder stiffness, A = the beam cross sectional area and e = the distance between the

centers of the deck and the girder), that have a significant influence on the load distribution of

beam and slab bridges. These parameters were identified during the skewed bridge sensitivity

studies performed in NCHRP 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 thickness

4

. As a result, those same parameters, as well as bridge aspect ratio and the

presence of intermediate cross frames, were investigated in a total of 41 bridge models. This

group of 41 models was comprised of 25 simple span beam-slab bridges, 3 simple span concrete

T-beam bridges, 4 simple span spread concrete box girder bridges and 9 two-span continuous

beam-slab bridges. The expanded analysis matrices for each bridge type are provided in

Appendix A and the typical framing plans and cross sections of the bridge models are provided

in Appendix B.

The basic cross section parameters (i.e. number of beams, beam spacing, beam

inertia/beam depth, slab thickness) for the beam and slab bridges were selected using the results

of NCHRP 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 12-26

2

, and

the base bridge model parameters used in this study are shown in Table 4. Additional beam-slab

bridge parameters, specifically, girder spacings of 4.84 ft., girder stiffnesses of 44,400 in

4

and

1,870,000 in

4

, a slab thickness of 9 in. and a 10-girder cross section, were also selected for

additional investigations. The two-span continuous beam-slab bridge models were based upon

the same base parameters, with the addition of a second, equal span.

For the concrete T-beam models, the base bridge parameters utilized in this research

were again established using the average properties from NCHRP 12-26

2

, 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 12-26

2

. 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 Base

NCHRP 12-26 Model

Parameter Parameter

Beam Spacing, ft.7.8 7.75

Beam Stiffness (I+Ae

2

), in

4

339,000 358,000

Slab Thickness, in.7 7

Number of Girders in X-Section 5.5 6

Bridge Parameter

Average Base

NCHRP 12-26 Model

Parameter Parameter

Girder Spacing, ft.7.77 7.75

Girder Stiffness (I+Ae

2

), in

4

357,000 333,000

Slab Thickness, in.7 7

Number of Girders in X-Section 5 6

Bridge Parameter

Average Base

NCHRP 12-26 Model

Parameter Parameter

Beam Spacing, ft.8.83 8.83

Box Depth, in.39 39

Box Width, in.48 48

Box Web Thickness, in.5.5 5

Box Top Flange Thickness, in.3.8 3

Box Bottom Flg Thickness, in.5.8 6

Slab Thickness, in.7.6 7.5

Number 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 Bridge

Models

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 Cross

Frame (Typ.)

Rigid Link (Typ.)

Solid Element for

Deck

L Beam (Typ.)

C

7" Slab

Equivalent Steel

Beam

AASHTO 28/63

Concrete 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 trucks

26

, without a concurrent uniform load. While the LRFD Specifications

utilize a live load condition that combines the truck loading with a uniform load

1

, 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 Specifications

1

.

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 10

th

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, V

LL,s

/ V

LL,r

. The actual skew correction, (V

LL,s

/ V

LL,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, (V

LL,s

/ V

LL,r

) -1.0, is utilized in this

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

at each 10

th

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

20

k

C Abutment (Typ.)

L

Girder 6

Girder 5

Girder 4

Girder 3

Girder 2

Girder 1

25

k

V

LL

22

k

C Abutment (Typ.)

L

Girder 6

Girder 5

Girder 4

Girder 3

Girder 2

Girder 1

30

k

V

LL

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 kips

Exterior Beam LL End Shear, Obtuse Corner, Skewed Bridge = 30 kips

Skew Correction Factor ( = 30/25) = 1.20

Skew Correction = 0.20

Exterior Beam LL Shear, Two-tenth Point, “Right” Bridge = 20 kips

Exterior Beam LL Shear, Two-tenth Point, Skewed Bridge = 22 kips

Skew Correction Factor ( = 22/20) = 1.10

Skew 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 in

4

(I + Ae

2

) and skew angles of 30° and 60°. The second set studied girder

stiffnesses of 333,000 in

4

(I + Ae

2

) 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 in

4

, 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 in

4

, 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 in

4

. 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 in

4

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 in

4

.

43

EFFECT OF SKEW ANGLE ON SKEW

CORRECTIONS ALONG EXTERIOR BEAMS

42' S i mpl e S pan, Beam-S l ab Bri dges, I+Ae

2

= 44,400 i n

4

, w/o Intermed. Cross

Frames

-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

Tenth Point Along Span

Normalized Skew Corrections

Ext. Girder 1, 30 deg. Skew

Ext. Girder 2, 30 deg. Skew

Ext. Girder 1, 60 deg. Skew

Ext. Girder 2, 60 deg. Skew

EFFECT OF SKEW ANGLE ON SKEW

CORRECTIONS ALONG EXTERIOR BEAMS

42' S i mpl e S pan, Beam-S l ab Bri dges, I+Ae

2

= 333,000 i n

4

, w/o Intermed. Cross

Frames

-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

Tenth Point Along Span

Normalized Skew Corrections

Ext. Girder 1, 30 deg. Skew

Ext. Girder 2, 30 deg. Skew

Ext. Girder 1, 60 deg. Skew

Ext. Girder 2, 60 deg. Skew

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

44

Figure 9. Effect of Skew Angle on Skew Corrections Along Exterior Beams

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 in

4

and 333,000 in

4

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 in

4

and 1,870,000 in

4

at a span length of 105-ft. and a skew angle of 60°.

The fourth set studied girder stiffnesses of 44,400 in

4

, 333,000 in

4

and 1,870,000 in

4

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 in

4

. 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 in

4

, 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 OF GIRDER STIFFNESS ON SKEW

CORRECTIONS ALONG EXTERIOR BEAM

42' S i mpl e S pan, Beam-S l ab Bri dges, 30 deg. S kew, w/o Intermed. Cross Frames

-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

Tenth Point Along Span

Normalized Skew Corrections

Ext. Girder 1, I+Ae2 = 333,000 in4

Ext. Girder 2, I+Ae2 = 333,000 in4

Ext. Girder 1, I+Ae2 = 44,400 in4

Ext. Girder 2, I+Ae2 = 44,400 in4

EFFECT OF GIRDER STIFFNESS ON SKEW

CORRECTIONS ALONG EXTERIOR BEAM

42' S i mpl e S pan, Beam-S l ab Bri dges, 60 deg. S kew, w/o Intermed. Cross

Frames

-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

Tenth Point Along Span

Normalized Skew Corrections

Ext. Girder 1, I+Ae2 = 333,000 in4

Ext. Girder 2, I+Ae2 = 333,000 in4

Ext. Girder 1, I+Ae2 = 44,400 in4

Ext. Girder 2, I+Ae2 = 44,400 in4

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

47

Figure 11. Effect of Girder Stiffness on Skew Corrections Along Exterior Beams

48

EFFECT OF GIRDER STIFFNESS ON SKEW

CORRECTIONS ALONG EXTERIOR BEAMS

105' S i mpl e S pan, Beam-S l ab Bri dges, 60 deg. S kew, w/o Intermed. Cross Frames

-1.60

-1.40

-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

Tenth Point Along Span

Normalized Skew Corrections

Ext. Girder 1, I+Ae2 = 333,000 in4

Ext. Girder 2, I+Ae2 = 333,000 in4

Ext. Girder 1, I+Ae2 = 1,870,000 in4

Ext. Girder 2, I+Ae2 = 1,870,000 in4

EFFECT OF GIRDER STIFFNESS ON SKEW

CORRECTIONS ALONG EXTERIOR BEAMS

168' S i mpl e S pan, Beam-S l ab Bri dges, 60 deg. S kew, w/o Intermed. Cross

Frames

-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

Tenth Point Along Span

Normalized Skew Corrections

Ext. Girder 1, I+Ae2 = 333,000 in4

Ext. Girder 2, I+Ae2 = 333,000 in4

Ext. Girder 1, I+Ae2 = 44,400 in4

Ext. Girder 2, I+Ae2 = 44,400 in4

Ext. Girder 1, I+Ae2 = 1,870,000 in4

Ext. Girder 2, I+Ae2 = 1,870,000 in4

Figure 12. Effect of Girder Stiffness on Skew Corrections Along Exterior Beams

49

Figure 13. Effect of Girder Stiffness on Skew Corrections Along Exterior Beams

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 in

4

and span lengths of 42-ft. and 168-ft. The second set investigated models with

beam stiffnesses of 333,000 in

4

and span lengths of 105-ft. and 168-ft. Similarly, the third set

investigated beam stiffnesses of 1,870,000 in

4

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 in

4

beam stiffness, but the shear

values in this region will not govern for design purposes.

50

EFFECT OF SPAN LENGTH ON SKEW

CORRECTIONS ALONG EXTERIOR BEAMS

S i mpl e S pan, Beam-S l ab Bri dges, I+Ae

2

= 44,400 i n

4

, 60 deg. S kew, w/o

Intermed. Cross Frames

-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

Tenth Point Along Span

Normalized Skew Corrections

Ext. Girder 1, 42' Span

Ext. Girder 2, 42' Span

Ext. Girder 1, 168' Span

Ext. Girder 2, 168' Span

EFFECT OF SPAN LENGTH ON SKEW

CORRECTIONS ALONG EXTERIOR BEAMS

S i mpl e S pan, Beam-S l ab Bri dges, I+Ae

2

= 333,000 i n

4

, 60 deg. S kew, w/o

Inte rmed. Cross Frames

-1.60

-1.40

-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

Tenth Point Along Span

Normalized Skew Corrections

Ext. Girder 1, 105' Span

Ext. Girder 2, 105' Span

Ext. Girder 1, 168' Span

Ext. Girder 2, 168' Span

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

51

Figure 15. Effect of Span Length on Skew Corrections Along Exterior Beams

52

EFFECT OF SPAN LENGTH ON SKEW

CORRECTIONS ALONG EXTERIOR BEAMS

S i mpl e S pan, Beam-S l ab Bri dges, I+Ae

2

= 1,870,000 i n

4

, 60 deg. S kew, w/o

Intermed. Cross Frames

-1.60

-1.40

-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

Tenth Point Along Span

Normalized Skew Corrections

Ext. Girder 1, 105' Span

Ext. Girder 2, 105' Span

Ext. Girder 1, 168' Span

Ext. Girder 2, 168' Span

Figure 16. Effect of Span Length on Skew Corrections Along Exterior Beams

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 in

4

, 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

EFFECT OF INTERMEDIATE CROSS

FRAMES ON SKEW CORRECTIONS ALONG

EXTERIOR BEAMS

105' S i mpl e S pan, Be am-S l ab Bri dge s, I+Ae

2

= 333,000 i n

4

, 60 de g. S ke w

-1.60

-1.40

-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

Te nth Point Along Span

Normalized Skew Corrections

Ext. Girder 1, No X-Frames

Ext. Girder 2, No X-Frames

Ext. Girder 1, X-Frames

Ext. Girder 2, X-Frames

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

56

EFFECT OF INTERMEDIATE CROSS FRAMES

ON SKEW CORRECTIONS ALONG EXTERIOR

BEAMS

168' S i mpl e S pan, Beam-S l ab Bri dge s, I+Ae

2

= 333,000 i n

4

, 60 de g. S ke w

-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

Te nth Point Along Span

Normalized Skew Corrections

Ext. Girder 1, No X-Frames

Ext. Girder 2, No X-Frames

Ext. Girder 1, X-Frames

Ext. Girder 2, X-Frames

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

4.1.1.5 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 in

4

, 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

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