207/Task 107 COPY NO. _____
SHEAR IN SKEWED MULTIBEAM BRIDGES
FINAL REPORT
Prepared for
National Cooperative Highway Research Program
Transportation Research Board
National Research Council
Modjeski and Masters, Inc.
NCHRP Project 207/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 1226.............................................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 BeamSlab 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 TBeam 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 TwoSpan Continuous BeamSlab Bridge Models.......................78
4.4.1 Simple Span vs. TwoSpan 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 BeamSlab Bridges....................................128
5.2 Simple Span Concrete TBeam Bridges..............................133
5.3 Simple Span Spread Concrete Box Girder Bridges......................133
5.4 TwoSpan Continuous BeamSlab Bridges............................134
5.5 Application of Study Findings......................................139
CHAPTER 6 CONCLUSIONS AND SUGGESTED RESEARCH............145
REFERENCES.............................................................149
APPENDIX A ANALYSIS MATRICES..................................A1
APPENDIX B CROSS SECTIONS AND FRAMING PLANS OF
BRIDGE MODELS.......................................B1
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 1226/1
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
BeamSlab 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 TwoSpan 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 TwoSpan 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 TwoSpan 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 TwoSpan Continuous BeamSlab Bridges..........................102
Figure 54.Nomenclature for Investigation of Correction Factors Across the
Pier of TwoSpan 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 TwoSpan Continuous BeamSlab Bridges..........................110
Figure 62.Nomenclature for Investigation of Correction Factors for Reaction at the
Pier of TwoSpan 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 SimpleSpan BeamSlab Bridges...............130
x
LIST OF FIGURES (continued)
Figure 74.Average Results for the Variation of the Skew Correction Along the
Bearing Lines of SimpleSpan BeamSlab Bridges......................132
Figure 75.Results for the Variation of the Skew Correction Along the Length
of the Exterior Girders of TwoSpan Continuous BeamSlab Bridges.......136
Figure 76.Average Results for the Variation of the Skew Correction Across
Abutments and Piers of TwoSpan Continuous BeamSlab 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 TBeams, 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 TBeams, 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 TBeams, 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 TBeams, 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 ThreeLane 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 1226 and Base Parameters for BeamSlab Bridge Models..30
Table 5.Average NCHRP 1226 and Base Parameters for Concrete Tbeam
Bridge Models...................................................30
Table 6.Average NCHRP 1226 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 TwoSpan 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
TwoSpan Bridge Models..........................................83
Table 10.Correction Factors for Reaction at the Pier of TwoSpan
BeamSlab 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 twospan
continuous beam and slab bridges. The report also documents an investigation of skew
correction factors for live load reactions at the piers of twospan 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 Tbeams, 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 midspan. 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 1226
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 1226
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 twospan 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 beamslab models, 3 simple span concrete Tbeam models, 4 simple
span spread concrete box girder models and 9 twospan continuous beamslab 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 beamslab and concrete
Tbeam 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 midspan.
• 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 beamslab and concrete Tbeam
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 twospan 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 twospan continuous beamslab 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 midspan. 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 TBeams, 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
4.6.2.2.3c1 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 4.6.2.2.3c1, to a value
of 1.0 at girder midspan.
• 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 4.6.2.2.3c1 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 midspan.
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 TBeams, 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
4.6.2.2.3c1 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 4.6.2.2.3c1, 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 4.6.2.2.3c1 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 Tbeams; 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 4.6.2.2.3c1 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 beamslab (i.e. steel Ibeam, concrete Ibeam and concrete T
beam), box girder, multibox 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., nonskewed) 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 1226 and Project 1233, 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 1226
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 1233 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 1226.
7
SUPPORTING
COMPONENTS
TYPE OF DECK TYPICAL CROSSSECTION
Steel Beam Castinplace concrete
slab, precast concrete
slab, steel grid,
glued/spiked panels,
stressed wood
Precast Concrete I or Bulb
Tee Sections
Castinplace
concrete, precast
concrete
Closed Steel or Precast
Concrete Boxes
Castinplace concrete
slab
Open Steel or Precast
Concrete Boxes
Castinplace concrete
slab, precast concrete
deck slab
CastinPlace 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 midspan 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 4.6.2.2.3c1 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 crosssection.
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 1226. 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 1233 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 1233 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 1226, 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 1233 working
group’s previous experience with curved and simplespan 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 beamslab 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 overdesigned 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 simplespan beam and slab bridges.
This study also investigates a limited number of twospan 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 simplespan and twospan 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 1226
2
, Khaleel and Itani
7
, Bishara, et al.
8
and Ebeido and
Kennedy
9
has concentrated on moment distributions in skewed, simplysupported 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 1226
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
1226
4
. The primary objective of NCHRP Project 1226 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 1226, 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 1226, 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: beamslab (i.e., steel Ibeam, concrete I
beam and concrete Tbeam), box girder, slab, multibox 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 1226 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 1226/1 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 overdesigned. Additionally,
the effect of skew on shear along the length of exterior beams of beam and slab bridges was not
investigated in NCHRP Project 1226.
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 steelconcrete 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 twospan 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 HS2044 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 1226 and the LRFD Specifications.
21
The comparison of distribution factors was limited to those for shear at interior piers, as
NCHRP 1226 and the LRFD Specifications do not address the distribution of pier reactions in
skewed bridges. Ebeido and Kennedy used a threelane continuous bridge with skew angles of
0°, 30°, 45° and 60° to compare the maximum shear force determined from the LRFD
Specifications, NCHRP 1226, 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 1226 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 1226 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 ThreeLane 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
HS2044 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 multistringer bridges to evaluate the reaction components at the
rocker and bolster bearings under both dead load and HS2044 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.
ElAli investigated the internal forces in four 137foot simplysupported, 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 HS2044 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 twospan continuous beam and slab bridge models.
Analysis matrices were developed based on key parameters of simple span and twospan
continuous beam and slab bridges. These analysis matrices served to guide the study, to allow
for assessment of nonlinear 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 curbtocurb 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 1226 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 beamslab bridges, 3 simple span concrete
Tbeam bridges, 4 simple span spread concrete box girder bridges and 9 twospan continuous
beamslab 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 1226 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 1226
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: beamslab, box girder, slab,
multibox 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 beamslab bridge types, the average properties calculated in NCHRP 1226
2
, and
the base bridge model parameters used in this study are shown in Table 4. Additional beamslab
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 10girder cross section, were also selected for
additional investigations. The twospan continuous beamslab bridge models were based upon
the same base parameters, with the addition of a second, equal span.
For the concrete Tbeam models, the base bridge parameters utilized in this research
were again established using the average properties from NCHRP 1226
2
, as shown in Table 5.
The analysis matrix for the Tbeam bridges was developed using typical span lengths for this
bridge type, determined from NCHRP 1226, 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 1226.
The base bridge parameters for the spread box girder bridge models were also developed
from the results of NCHRP 1226
2
. Table 6 contains the average properties from NCHRP 1226
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 1226 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 XSection 5.5 6
Bridge Parameter
Average Base
NCHRP 1226 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 XSection 5 6
Bridge Parameter
Average Base
NCHRP 1226 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 XSection 6 5
Bridge Parameter
Table 4. Average NCHRP 1226 and Base Parameters for BeamSlab Bridge Models
Table 5. Average NCHRP 1226 and Base Parameters for Concrete Tbeam Bridge Models
Table 6. Average NCHRP 1226 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 threedimensional 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 eightnode 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 inplane 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 Igirder bridge is shown in Figure 5.
32
The BSDI software is tailored toward the analysis of steel Igirder and steel box girder
cross sections. The analysis of concrete Igirders 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 noncomposite 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 Igirder into an equivalent steel Igirder. A similar
procedure was utilized for transformation of the concrete box girders into equivalent steel boxes.
Transformation of the concrete Tbeams 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 IBeam
(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 12foot 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 curvefitting techniques should be utilized during processing of the BSDI output.
The Least Squares Method of curvefitting 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. Threedimensional 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,
curvefitting 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 curvefitting 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 twotenth 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, Twotenth Point, “Right” Bridge = 20 kips
Exterior Beam LL Shear, Twotenth Point, Skewed Bridge = 22 kips
Skew Correction Factor ( = 22/20) = 1.10
Skew Correction = 0.10
Therefore,
Normalized Skew Correction at Twotenth Point (0.10/0.20) = 0.50 (50%)
Thus, the normalized correction indicates that the skew correction at the
Twotenth 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 BEAMSLAB 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 beamslab bridge models. Each set of
models was based upon a 42' span length, a sixbeam cross section with beam spacings of 7.75
ft., a 7in. deck slab and no intermediate crossframes. 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 fourtenth 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 eighttenth 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 2045% 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 midspan. These
models, however, were created using an 8ft. deep beam with a 42ft. span length. This
geometry produces a span length to beam depth ratio 5.25– a ratio well outside the range of
typical beamslab bridges.
The occurrence of the correction factor at the acute corner of the bridge and the “spike”
in the correction factor at midspan 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 42ft. 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 42ft. 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, BeamS 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, BeamS 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 beamslab bridge models. Each set of
models was based upon a sixbeam cross section with beam spacings of 7.75ft., a 7in. deck slab
and no intermediate crossframes. The first set of models studied girder stiffnesses of 44,400 in
4
and 333,000 in
4
at a span length of 42ft. 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 105ft. 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 168ft. 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 fourtenth 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 168ft. span models. These models
indicate that a substantial percentage of the skew correction at the end of the beams may also be
effective near midspan. The most evident case of this occurs on the 168ft. spans with beam
45
stiffnesses of 333,000 in
4
. At midspan, 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 midspan. In this case, for example, the shears in Exterior Girder 2 at
midspan 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 onehalf 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 midspan shears may be utilized for
determination of reinforcing steel and beam stiffener spacing; however, significant correction
factors at midspan 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 42ft. 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, BeamS 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, BeamS 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, BeamS 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, BeamS 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 sixbeam cross section with beam spacings of 7.75ft., a 7in. deck slab, no intermediate
crossframes and a skew angle of 60°. The first model set included bridges with beam stiffnesses
of 44,400 in
4
and span lengths of 42ft. and 168ft. The second set investigated models with
beam stiffnesses of 333,000 in
4
and span lengths of 105ft. and 168ft. Similarly, the third set
investigated beam stiffnesses of 1,870,000 in
4
with span lengths of 105ft. and 168ft.
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
fourtenth 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 fourtenth 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 midspan of the 168ft. 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, BeamS 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, BeamS 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, BeamS 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
sixbeam cross section with beam spacings of 7.75ft., beam stiffnesses of 333,000 in
4
, a skew
angle of 60°, 7in. slab thickness and span lengths of 105ft. or 168ft.. The cross frame spacing
was set at 21ft. for the right bridges and at approximately 13ft. to 25ft. for the skewed bridges,
contingent upon the model geometry. All intermediate cross frames were contiguous and
modeled as Ktype 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 threetenth point
of the span. Additionally, the correction factor spike near midspan of the 168ft. 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 amS 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 XFrames
Ext. Girder 2, No XFrames
Ext. Girder 1, XFrames
Ext. Girder 2, XFrames
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, BeamS 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 XFrames
Ext. Girder 2, No XFrames
Ext. Girder 1, XFrames
Ext. Girder 2, XFrames
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 42ft. span length, beam stiffnesses of 44,400 in
4
, a skew angle of 60°, 7in. slab thickness
and six beams spaced at 7.75ft. or nine beams at 4.84ft. 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 threetenth point along the span length. The spacing of the beams does not significantly
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