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
174
Embed
SHEAR IN SKEWED MULTI-BEAM BRIDGES107)_FR.pdfSHEAR IN SKEWED MULTI-BEAM BRIDGES FINAL REPORT Prepared for National Cooperative Highway Research Program Transportation Research Board
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
20-7/Task 107 COPY NO. _____
SHEAR IN SKEWED MULTI-BEAM BRIDGES
FINAL REPORT
Prepared forNational Cooperative Highway Research Program
Transportation Research BoardNational Research Council
Modjeski and Masters, Inc.NCHRP Project 20-7/Task 107
March 2002
ii
iii
ACKNOWLEDGMENT OF SPONSORSHIP
This work was sponsored by the American Association of State Highway andTransportation Officials, in cooperation with the Federal Highway Administration, and wasconducted in the National Cooperative Highway Research Program, which is administeredby the Transportation Research Board of the National Research Council.
DISCLAIMER
This is an uncorrected draft as submitted by the research agency. The opinions andconclusions expressed or implied in the report are those of the research agency. They arenot necessarily those of the Transportation Research Board, the National Research Council,the Federal Highway Administration, the American Association of State Highway andTransportation Officials, or the individual states participating in the National CooperativeHighway Research Program.
Figure 62. Nomenclature for Investigation of Correction Factors for Reaction at thePier of Two-Span Continuous Bridge Models . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Figure 63. Comparison of Skew Correction Factors for Shear and Reaction at Pier . . . . . 115
Figure 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 Lengthof 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 theBearing Lines of Simple-Span Beam-Slab Bridges . . . . . . . . . . . . . . . . . . . . . . 132
Figure 75. Results for the Variation of the Skew Correction Along the Lengthof the Exterior Girders of Two-Span Continuous Beam-Slab Bridges . . . . . . . 136
Figure 76. Average Results for the Variation of the Skew Correction AcrossAbutments and Piers of Two-Span Continuous Beam-Slab Bridges . . . . . . . . . 138
Figure 77. Proposed Variation of the Skew Correction Factors for Shear Along the Length of the Exterior Girders in Simple Span Superstructures of Concrete Deck, Filled Grid, or Partially Filled Grid on Steel or Concrete Beams; Concrete T-Beams, T- and Double T Sections . . . . . . . . . . . 141
Figure 78. Proposed Variation of the Skew Correction Factors for Shear Along the Length of the Exterior Girders in Continuous Superstructures of Concrete Deck, Filled Grid, or Partially Filled Grid on Steel or Concrete Beams; Concrete T-Beams, T- and Double T Sections . . . . . . . . . . . 141
Figure 79. Proposed Variation of the Skew Correction Factors for Shear Acrossthe Bearing Lines of Simple Span Superstructures of Concrete Deck, Filled Grid, or Partially Filled Grid on Steel or Concrete Beams; Concrete T-Beams, T- and Double T Sections . . . . . . . . . . . 144
Figure 80. Proposed Variation of the Skew Correction Factors for Shear Acrossthe Abutments and Piers of Continuous Superstructures of Concrete Deck, Filled Grid, or Partially Filled Grid on Steel or Concrete Beams; Concrete T-Beams, T- and Double T Sections . . . . . . . . . . . 144
Table 7. Comparison of Maximum Live Load Shears from BSDI and anLRFD Line Girder Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Table 8. Comparison of Skew Correction Factors for End Shear of Exterior Girdersat the Obtuse Corners of Simple Span and Two-Span Bridge Models . . . . . . . . 80
slab thickness and bridge aspect ratio on the skew correction factor variation was investigated.
2
For the simple span bridge models studied, the research results indicate that:
• Regardless of changes in the aforementioned bridge parameters, areasonable approximation for the variation of the skew correction factoralong the length of exterior girders of simple span beam-slab and concreteT-beam bridges is a linear distribution of the factor from its value at theobtuse corner of the bridge, determined according to the LRFDSpecifications, to a value of 1.0 at girder mid-span.
• Regardless of changes in the aforementioned bridge parameters, areasonable approximation of the skew correction factor for live load shearacross the bearing lines of simple span beam-slab and concrete T-beambridges is a linear distribution of the correction factor from its value at theobtuse corner of the bridge, determined according to the LRFDSpecifications, to a value of 1.0 at the acute corner of the bridge.
For the two-span continuous bridge models studied, the research results indicate that:
• The variations of the skew correction factors for shear along the length ofexterior girders in each span and for shear across both the abutments andpiers of two-span continuous beam-slab bridges are identical to thoseproposed for simple span bridges. The correction factor variation alongthe exterior girder may be approximated by a linear distribution of thefactor at the obtuse corner to a value of 1.0 at girder mid-span. Likewise,the variation across the abutments and piers is approximated by a lineardistribution of the factor at the obtuse corner to a value of 1.0 at the acutecorner.
• The skew correction factor defined by the LRFD Specifications is valid forthe girder shear at the obtuse corners of both the abutments and piers ofthe continuous bridges.
• Skew correction factors for reaction at the piers of continuous bridges arepresent and are unique from those calculated for shear at the piers. Fromthe limited continuous bridge model data of this study, however, accurateempirical equations which define the correction factor or define itsvariation across the pier could not be derived. Therefore, the developmentof such equations for continuous bridges is necessary and is recommendedfor further research.
For application of the research findings, the recommendations are as follows:
3
Skew Correction Factor for Shear, Variation Along Exterior Beam Length
• For superstructure types “Concrete Deck, Filled Grid, or Partially Filled Grid onSteel or Concrete Beams; Concrete T-Beams, T- and Double T Section,” withinthe applicable ranges of skew angle (θ), spacing of beams or webs (S), span ofbeam (L) and number of beams, stringers or girders (Nb) as defined by Table4.6.2.2.3c-1 of the LRFD Specifications, the skew correction factor for shear maybe varied linearly from its value at the obtuse corner of the bridge, determined inaccordance with the empirical equation defined in Table 4.6.2.2.3c-1, to a valueof 1.0 at girder mid-span.
• This approximate variation is applicable for both simple span structures andcontinuous structures. For continuous structures, the skew correction factorcalculated at the obtuse corner of the abutment per Table 4.6.2.2.3c-1 is also validat the obtuse corners of the interior piers. Likewise, the variation of thecorrection factor is applicable from both the obtuse corner of the abutment and theobtuse corners of the interior piers to the girder mid-span.
Skew Correction Factor for Shear, Variation Across Bearing Lines
• For superstructure types “Concrete Deck, Filled Grid, or Partially Filled Grid onSteel or Concrete Beams; Concrete T-Beams, T- and Double T Section,” withinthe applicable ranges of skew angle (θ), spacing of beams or webs (S), span ofbeam (L) and number of beams, stringers or girders (Nb) as defined by Table4.6.2.2.3c-1 of the LRFD Specifications, the skew correction factor for shear maybe varied linearly from its value at the obtuse corner of the bridge, determined inaccordance with Table 4.6.2.2.3c-1, to a value of 1.0 at the acute corner of thebearing line.
• This approximate variation is applicable for both simple span structures andcontinuous structures. For continuous structures, the skew correction factorcalculated at the obtuse corner of the abutment per Table 4.6.2.2.3c-1 is also validat the obtuse corners of the interior piers. Likewise, the variation of thecorrection factor is applicable from both the obtuse corner of the abutment and theobtuse corners of the interior piers to the acute corner of the bearing lines.
Additional suggested research includes an investigation of the effects of torsion on web
shear in spread box girder bridges. The study results indicate that although torsion is typically
4
neglected in “right” bridges, the introduction of skew may increase torsional effects to levels that
are not negligible. Without further research, however, and given the lack of substantial field
documentation indicating problems with torsion and shear in skewed spread box girder bridges,
the current design practices are considered to be acceptable.
Finally, this study investigated only a few types of beam and slab bridges and provides
recommendations regarding only superstructures consisting of concrete decks, filled grids, or
partially filled grids on steel or concrete beams; concrete T-beams; or T- and double T sections.
Additional research is recommended, therefore, to determine the effects of skew on shear in the
remaining beam and slab bridge types included within Table 4.6.2.2.3c-1 of the LRFD
Specifications.
5
CHAPTER 1 INTRODUCTION AND RESEARCH OBJECTIVES
1.1 INTRODUCTION
Beam and slab bridges are basic and common elements of the national system of
roadways and bridges. Examples of typical beam and slab superstructures are shown in Figure 1,
and include structures such as beam-slab (i.e. steel I-beam, concrete I-beam and concrete T-
beam), box girder, multi-box beam and spread box beam bridges. Design procedures for these
structures are well documented and standardized through research, physical testing and
development of design codes, especially for “right” (i.e., non-skewed) bridges. The design of
skewed bridges, however, is often based more upon engineering experience and extrapolation of
limited analyses, rather than upon extensive research. In fact, for many years, little was done to
incorporate the effect of skew on live load distribution, with the result that many skewed bridges
were designed as right bridges. This was often the case for shear design in skewed beam and
slab structures.
Two recent NCHRP research projects, Project 12-26 and Project 12-33, focused on
updating and refining the AASHTO Bridge Design Specifications, and in doing so, refined the
shear design procedures for skewed beam and slab bridges. NCHRP Project 12-262 focused on
investigating the live load distribution in beam and slab bridges and on developing refined live
load distribution formulas to be incorporated in an updated AASHTO Bridge Design
Specification. The objective of Project 12-33 was the development of AASHTO Bridge Design
Specifications utilizing the Load and Resistance Factor Design (LRFD) methodology. This
6
project culminated with the publication of the first edition of the AASHTO LRFD Bridge Design
Specifications (LRFD Specifications)3 in 1994 and incorporated the refined shear design
procedures for skewed beam and slab bridges developed in NCHRP 12-26.
The current design methodology in Section 4 of the LRFD Specifications1 for typical,
right beam and slab bridges permits the use of empirical distribution factors for determination of
the live load effects in bridge beams. For the mid-span bending moment and end shear of
exterior beams in skewed beam and slab bridges, the LRFD Specifications provide correction
factors that are to be applied to the moment and shear distribution factors, calculated for the
corresponding right bridge. These empirical skew correction factors for end shear in beam and
slab bridges, as defined in Table 4.6.2.2.3c-1 of the LRFD Specifications1 and as shown in Table
1, have been the subject of much discussion following the adoption of the LRFD Specifications
in 1993. As stated in the scope of services provided by the NCHRP for this project,
“Article 4.6.2.2.3c, Skewed Bridges, in the AASHTO LRFD Bridge DesignSpecifications, requires that shear in the exterior beam at the obtuse corner of thebridge shall be adjusted when the line of support is skewed. The Specificationsprovide correction factors for this adjustment and require that the correctionfactors be applied to all beams in the cross-section.
In the development of these correction factors, the variation of the effect of skewon the individual beam reactions was not considered. In addition, theSpecifications provide no guidance on the influence of skew on the shear alongthe length of the beam. The commentary to the Specifications states that theprescribed corrections are conservative. As a consequence of this conservatismsome beams in the bridge are overdesigned.”
It is not only this conservatism that has been the topic of discussions surrounding the skew
correction factors, but also the extent to which the correction factors apply to the shear along the
length of the exterior girder.
9
Table 1. Correction Factors for Load Distribution Factors for Support Shear of theObtuse Corner1.
Type of Superstructure Correction Factor Range ofApplicability
Concrete Deck, Filled Grid, orPartially Filled Grid on Steel orConcrete Beams; Concrete T-Beams, T- and Double TSection
0° # 2 # 60°3.5 # S # 16.020 # L # 240Nb $ 4
Multicell Concrete Box Beams,Box Sections
0° < 2 # 60°6.0 < S # 13.020 # L # 24035 # d # 110Nc $ 3
Concrete Deck on SpreadConcrete Box Beams
0 < 2 # 60°6.0 # S # 11.520 # L # 14018 # d # 65Nb $ 3
Concrete Box Beams Used inMultibeam Decks
0 < 2 # 60°20 # L # 12017 # d # 6035 # b # 605 # Nb # 20
Where: 2 = skew angle (degrees) Nb = number of beams, stringers or girdersS = spacing of beams or webs (ft) Nc = number of cells in a concrete box girderL = span of beam (ft) ts = depth of concrete slab (in)b = width of beam (in) Kg = longitudinal stiffness parameter (in4)d = depth of beam or stringer (in)
10
The development of the skew correction factors for beam and slab bridges in the LRFD
Specifications was part of NCHRP Project 12-26. The report for that project, Distribution of
Wheel Loads on Highway Bridges4, indicated that the skew correction factors were derived for
only the end shears of the exterior girders at the obtuse corners of simple span bridges. In
general, the end shear tends to increase as the skew angle of the supports increases beyond
approximately 15° to 20°. For the LRFD Specifications, however, the working group for
NCHRP 12-33 conservatively extended the applicability of the correction factor to include not
only the end shear at the obtuse corner of the exterior beams, but also the end shear of each beam
in the bridge cross section5, as shown in the typical skewed bridge plan of Figure 2.
The working group for NCHRP 12-33 also assumed that it may be reasonable to extend
the correction factors for end shear of the exterior beam to the shear along the length of the
exterior beam5, but made no provisions in the LRFD Specifications to do so. During the
development of the skew correction factors in NCHRP 12-26, the effect of skew on the shear
along the length of the exterior beams was not investigated, and the current LRFD Specifications
do not address this issue.
11
C Abutment (Typ.)L
Correction Factor Calculated forand Applied to End Shear atObtuse Corner of Exterior Girder(Typ.)
C Girder (Typ.)L
Correction FactorConservatively Applied toEnd Shear of All Girders(Typ.)
Skew Angle
Figure 2. Plan View of Typical Skewed Superstructure. Current Application of the SkewCorrection Factor for Shear per the AASHTO LRFD Bridge Design Specifications1.
12
An additional topic of discussion regarding the design of skewed bridges is the treatment
of reactions at interior supports of continuous spans. Based upon the NCHRP 12-33 working
group’s previous experience with curved and simple-span skewed structures, it was speculated
that skew effects also account for the reduced reaction at interior supports, and, in some cases,
the uplift at the acute corner of skewed bridges5. Intuition may suggest, therefore, that at the
interior supports of continuous spans, where both an obtuse and acute corner exist opposite each
other, the skew effects on shear may cancel out for determination of the total reaction. This
hypothesis, however, has not yet been investigated and is not addressed in the LRFD
Specifications.
As a result of these outstanding issues regarding the skew correction factors for shear,
this project focuses on investigating and more accurately assessing the effect of skew on end
shear across bearing lines and on shear along the length of exterior beams of beam and slab
bridges. This research concentrates on simple span bridges, with a cursory evaluation of two-
span continuous beam-slab bridges. The importance of this topic lies in the fact that while
research has been performed to determine the shear correction factor for end shears at the obtuse
corners of skewed bridges, these factors also have been conservatively applied to the end shear
of all beams in the cross section and, in some cases, to the shear along the length of the exterior
girder, without supporting research. The possibility exists, therefore, that some beams in beam
and slab bridges are over-designed for shear. Further research on this topic may enable the use
of more precise skew correction factors, and hence, may result in more economical structures.
13
1.2 RESEARCH OBJECTIVES
The main objective of this study is to develop practical and reasonably accurate design
guidelines for estimating the variation of the skew correction factor for live load shear along the
length of exterior beams and across the beam supports of simple-span beam and slab bridges.
This study also investigates a limited number of two-span continuous bridge models to address
the variation of the skew correction factor along the length of the exterior beams and across the
abutments and piers of these bridge types. Additionally, the continuous models are studied to
address the need for skew correction factors for live load reactions at piers. The proposed
guidelines for the skew correction factors of both simple-span and two-span continuous bridges
are intended to be developed in a manner suitable for incorporation into the current AASHTO
LRFD Bridge Design Specifications.
14
CHAPTER 2 LITERATURE REVIEW
2.1 INTRODUCTION
Extensive research has been performed by bridge engineers in an attempt to accurately
predict the path of loads through bridges and to present the predictions in reasonably accurate,
yet practical load distribution formulas for designers. Specific to beam and slab bridges, much
research has been performed to develop approximate, algebraic equations for the distribution of
moment and shear in right bridges. A further extension of that work is the area of research
devoted to the distribution of moment in skewed beam and slab bridges. Research by Marx, et
al.6, Nutt, et al. for the NCHRP Project 12-262, Khaleel and Itani7, Bishara, et al.8 and Ebeido and
Kennedy9 has concentrated on moment distributions in skewed, simply-supported and
continuous beam and slab bridges. The research devoted to the distribution of shear and bearing
reactions in skewed bridges, however, is confined to a rather limited set of sources.
2.2 NCHRP PROJECT 12-26
One of the major comprehensive studies aimed at predicting the effect of skew on the
distribution of shear in beam and slab bridges was the work by Zokaie, et. al. for NCHRP Project
12-264. The primary objective of NCHRP Project 12-26 was to investigate the live load
distribution in beam and slab bridges and develop, where necessary, more accurate live load
distribution formulas to replace those specified in the AASHTO Standard Specifications for
Highway Bridges (Standard Specifications)10. While experiencing only minor revisions since
15
incorporation into the Standard Specifications in 1931, the “S/over” equations (i.e., S/5.5 or
similar equations) for live load distribution provide little guidance on the treatment of skewed
bridges. One goal of NCHRP Project 12-26, therefore, was aimed at developing distribution
factors that would account for skew effects.
The analysis of load distribution and, ultimately, the development of the new load
distribution factor formulas for “right” beam and slab bridges in NCHRP Project 12-26, was
initiated by construction of a database of 850 existing beam and slab bridges from a nationwide
survey of state transportation officials. From the database, the “average” beam and slab bridge
parameters were defined for five different bridge types: beam-slab (i.e., steel I-beam, concrete I-
beam and concrete T-beam), box girder, slab, multi-box beam and spread box beam. Parametric
analyses were performed by varying a single parameter at a time to determine each parameter’s
effect on the distribution of HS20 truck live load. The parametric studies utilized both finite
element analyses and grillage analyses with a number of different software packages. From the
results, new live load distribution equations for right bridges were derived to incorporate the
effects of each parameter that had a significant effect on load distribution.
The approximate equations developed in NCHRP Project 12-26 for the skew correction
factors were developed for simple span bridges utilizing the programs GENDEK5A11 and
FINITE12 for finite element analysis. The skew correction factors were developed such that they
could be applied to the newly derived distribution factors of a right bridge with the same
geometric parameters as the skewed bridge under investigation. In order to incorporate the
effects of each bridge parameter that had a significant impact on the load distribution of right
bridges, parametric studies of skewed bridges were completed, similar to those performed for the
right bridges. The live load used in the parametric studies consisted of two trucks placed
transversely on the bridge cross section to maximize the girder responses. Test models of
different live load placements confirmed that two trucks typically produced the governing girder
16
responses. The general loading condition that maximized shear at the obtuse corner of the
skewed bridges is shown in Figure 3.
17
Figure 3. General Truck Placement Pattern used in NCHRP 12-26/1 for Maximum Shear.
18
From the parametric analyses, the equations for the skew correction factors for shear
were derived from the ratio of the maximum exterior girder shear of a skewed bridge to that of a
right bridge, each with the same geometric parameters and live load positioning. These
equations, developed for the end shear of exterior beams at obtuse corners of beam and slab
bridges, are presented in Article 4.6.2.2.3c of the AASHTO LRFD Bridge Design Specifications1.
As discussed in Section 1.1, the LRFD Specifications require that the correction factors be
applied not only to the end shear of the exterior beams, but also to the end shear of each beam in
the bridge cross section. During the development of the skew correction factors, however,
variation of the effect of skew on the end shear of interior beams was not investigated. The
application of the skew correction factors to all beams of a cross section is considered to be
conservative; therefore, it is suspected that certain beams may be over-designed. Additionally,
the effect of skew on shear along the length of exterior beams of beam and slab bridges was not
investigated in NCHRP Project 12-26.
2.3 ONTARIO HIGHWAY BRIDGE DESIGN CODE
The treatment of skew and its effects on load distribution are handled differently in the
third edition of the Ontario Highway Bridge Design Code (OHBDC)13 than the method utilized
in the LRFD Specifications. Rather than modify the load distribution factors developed for
“right” bridges, the OHBDC defines a limit for the “skewness” of a bridge, beyond which
refined methods of analysis must be used. Prior to the third edition of the OHBDC, the Ontario
19
code implied that the measure of a bridge’s skewness was only its skew angle, as the “skewness”
limitation was defined by a skew angle of 20E (measured from centerline of bearings to a line
normal to the bridge centerline). The third edition of the OHBDC, however, incorporated the
work of Jaeger and Bakht14 which indicated that the measure of bridge “skewness” is also a
function of span length, bridge width and girder spacing. Hence, the skew limitation, ε, was
redefined in the third edition to incorporate these effects, as shown in Equation 1. Bridges
beyond the skewness limit of 1/18 must be analyzed using a refined method such as grillage
analysis, orthotropic plate theory or finite element analysis. Skewed bridges within this limit
may be analyzed using the load distribution factors developed for right bridges, with the
associated error of this procedure estimated at less than 5%.
beam-slab bridges. The expanded analysis matrices for each bridge type are provided in
Appendix A and the typical framing plans and cross sections of the bridge models are provided
in Appendix B.
The basic cross section parameters (i.e. number of beams, beam spacing, beam
inertia/beam depth, slab thickness) for the beam and slab bridges were selected using the results
of NCHRP 12-26 as a guide. The analysis of load distribution, and ultimately, the development
of the new load distribution factor formulas for “right” beam and slab bridges, in NCHRP 12-26
was initiated by construction of a database of 850 existing beam and slab bridges from a
nationwide survey of state transportation officials. From the database, the “average” beam and
slab bridge parameters were defined for five different bridge types: beam-slab, box girder, slab,
multi-box beam and spread box beam. These average bridge properties were used as a guide in
setting the base parameters of the models to be investigated in this project.
29
For the beam-slab bridge types, the average properties calculated in NCHRP 12-262, and
the base bridge model parameters used in this study are shown in Table 4. Additional beam-slab
bridge parameters, specifically, girder spacings of 4.84 ft., girder stiffnesses of 44,400 in4 and
1,870,000 in4, a slab thickness of 9 in. and a 10-girder cross section, were also selected for
additional investigations. The two-span continuous beam-slab bridge models were based upon
the same base parameters, with the addition of a second, equal span.
For the concrete T-beam models, the base bridge parameters utilized in this research
were again established using the average properties from NCHRP 12-262, as shown in Table 5.
The analysis matrix for the T-beam bridges was developed using typical span lengths for this
bridge type, determined from NCHRP 12-26, rather than the base case span lengths defined
previously. The matrix also includes a second beam with a stiffness typical of those identified in
NCHRP 12-26.
The base bridge parameters for the spread box girder bridge models were also developed
from the results of NCHRP 12-262. Table 6 contains the average properties from NCHRP 12-26
and the base parameters utilized in this study. The analysis matrix for this bridge type, found in
Appendix A, was created by selecting a two additional, typical box girders, one shallower and
one deeper than the base case girder.
30
Average BaseNCHRP 12-26 Model
Parameter ParameterBeam Spacing, ft. 7.8 7.75Beam Stiffness (I+Ae2), in4 339,000 358,000Slab Thickness, in. 7 7Number of Girders in X-Section 5.5 6
Bridge Parameter
Average BaseNCHRP 12-26 Model
Parameter ParameterGirder Spacing, ft. 7.77 7.75Girder Stiffness (I+Ae2), in4 357,000 333,000Slab Thickness, in. 7 7Number of Girders in X-Section 5 6
Bridge Parameter
Average BaseNCHRP 12-26 Model
Parameter ParameterBeam Spacing, ft. 8.83 8.83Box Depth, in. 39 39Box Width, in. 48 48Box Web Thickness, in. 5.5 5Box Top Flange Thickness, in. 3.8 3Box Bottom Flg Thickness, in. 5.8 6Slab Thickness, in. 7.6 7.5Number of Girders in X-Section 6 5
Bridge Parameter
Table 4. Average NCHRP 12-26 and Base Parameters for Beam-Slab Bridge Models
Table 5. Average NCHRP 12-26 and Base Parameters for Concrete T-beam Bridge Models
Table 6. Average NCHRP 12-26 and Base Parameters for Spread Concrete Box Girder BridgeModels
31
Investigation of each of the bridge models identified in the analysis matrices was
performed using finite element analyses. The services of Bridge Software Development
International, Ltd. (BSDI)26 were utilized for the finite element modeling. BSDI allows the user
to define the geometry, members, support conditions and loading conditions necessary for
construction of the finite element model. The model processing and generation of the live load
results was performed by BSDI.
The three-dimensional finite element modeling of the bridges by the BSDI software
allowed for individual modeling of the deck, beams and cross frames and optimization of the
live load placement. The deck slab was modeled with eight-node solid elements, each
possessing three translational degrees of freedom. The deck elements were modeled in their
actual position with respect to the neutral axes of the beams, which allowed the in-plane shear
stiffness of the deck to be considered in the analyses. Composite action between the deck slab
and beams was achieved through the use of rigid links prohibiting rotation of the deck with
respect to the beams. A combination of plate elements for the webs and beam elements for the
flanges were utilized to model the bridge beams. In modeling the flanges as beam elements, the
axial and lateral flange stiffness was incorporated into the models. Cross frames, X or K
configuration, were modeled with truss elements. Diaphragms were modeled with plate
elements for the webs and beam elements for the flanges, similar to the modeling of the girders.
All supports for the analysis models were free to translate laterally and longitudinally, with
restraint provided as required to ensure global stability. A schematic diagram of the bridge
modeling technique for an I-girder bridge is shown in Figure 5.
32
The BSDI software is tailored toward the analysis of steel I-girder and steel box girder
cross sections. The analysis of concrete I-girders and concrete box girders was achieved,
however, by transformation of the concrete sections into equivalent steel sections. The concrete
sections were transformed to produce steel sections which matched both the non-composite and
composite section properties of the concrete sections. The haunch depth above the girders was
modified as required in order to achieve the required composite section properties. Figure 6
displays the transformation of a concrete I-girder into an equivalent steel I-girder. A similar
procedure was utilized for transformation of the concrete box girders into equivalent steel boxes.
Transformation of the concrete T-beams was not required, as the BSDI input processor was
modified to facilitate the analysis of these bridge types
Therefore,Normalized Skew Correction at Two-tenth Point (0.10/0.20) = 0.50 (50%)
Thus, the normalized correction indicates that the skew correction at the Two-tenth Point is 50% of the skew correction at the end of the beam.
Figure 7. Procedure for Calculation of the Normalized Skew Corrections
39
This procedure of calculating, and then plotting, the normalized skew corrections enabled
graphic visualization of the variation of the skew correction along the length of the exterior
beams. It also facilitated direct comparison of this variation between bridges with different
geometric parameters, and hence, different magnitudes of skew corrections. A calculated skew
correction factor of 1.0 within the length of a beam produces a normalized skew correction of
0.0, indicating that no correction for skew is necessary. A calculated skew correction factor less
than 1.0 produces a normalized skew correction less than 0.0, indicating that this point has a
negative correction for skew, i.e., the shear in the skewed bridge model is less than the shear in
the “right” bridge model. The normalized skew corrections were plotted at each tenth point
along the exterior girders, defining location 0.0 as the beam end at the obtuse corner of the
bridge, location 1.0 at the acute corner, exterior girder 1 at the “bottom” of the bridge plan
(Girder 1 in Figure 7) and exterior girder 2 at the “top” of the bridge plan (Girder 6 in Figure 7).
This same procedure of plotting normalized skew corrections was utilized for
investigation of both shear across the abutments and piers and reactions across the piers of the
beam and slab bridges. The skew correction factors for shear of each beam across the bearing
line were calculated as the ratio of the live load shear from the skewed bridge model to that of
the corresponding “right” bridge model with identical geometric parameters. The skew
correction of each beam was then normalized to the skew correction for the beam at the obtuse
corner of the bearing line. Thus, the variation of the skew correction across the bearing lines
could be directly compared for bridge models with varying geometric parameters and
magnitudes of correction factors. The data plots of the normalized correction factors were
40
constructed by defining Girder 1 at the obtuse corner of the bearing line and defining the
remaining girders in ascending order to the acute corner.
A separate comparison of the skew correction factors for bearing reactions and those for
end shear of simple span bridges was not performed. That investigation, with the intent of
studying the influence of end cross frames and the effects of various load paths present at
bearings on end shears and reactions, was not possible due to the analysis procedure employed
by BSDI. The influence surfaces for the girder reactions are utilized by BSDI for calculation of
the end shears, thus assuming that the end shear is equal to the end reaction. A study of the load
paths through end cross frames and diaphragms, and their effect on the end shears and bearing
reactions, therefore, was not feasible.
41
CHAPTER 4 STUDY FINDINGS
4.1 SIMPLE SPAN BEAM-SLAB BRIDGE MODELS
4.1.1 Live Load Shear Along Exterior Beam Length
4.1.1.1 Influence of Skew Angle
The influence of skew angle on the variation of the skew correction factor along the
length of exterior beams was investigated in two sets of beam-slab bridge models. Each set of
models was based upon a 42' span length, a six-beam cross section with beam spacings of 7.75-
ft., a 7-in. deck slab and no intermediate cross-frames. The first set of models studied girder
stiffnesses of 44,400 in4 (I + Ae2) and skew angles of 30° and 60°. The second set studied girder
stiffnesses of 333,000 in4 (I + Ae2) and skew angles of 30° and 60°.
The plots of the normalized skew corrections for these two sets of models display a
diminishing influence of the skew correction factor from the end of the exterior beam at the
obtuse corner to the acute corner (see Figures 8 and 9). For the models with girder stiffnesses of
44,400 in4, the skew correction falls from its normalized value of 1.0 to zero or below zero
within the length of the beam span. For both the 30° and 60° skew angles, the correction factor
falls rapidly from its normalized value at the end of the span to zero near the four-tenth point of
the span length. The model with the 30° skew does have a slight skew correction present at mid-
42
span of approximately 30% of the correction at the end of the beam, but the correction falls to
zero by the eight-tenth point of the span length.
For the models with girder stiffnesses of 333,000 in4, the data displays the same general
trend of a diminishing influence of the skew correction factor along the length of the beam;
however, at the end of the beam adjacent to the acute corner, a slight skew correction of
approximately 20-45% the value at the obtuse corner is present. One of the exterior girders of
the 30° skew model also displays a small “spike” in the correction factor at mid-span. These
models, however, were created using an 8-ft. deep beam with a 42-ft. span length. This
geometry produces a span length to beam depth ratio 5.25– a ratio well outside the range of
typical beam-slab bridges.
The occurrence of the correction factor at the acute corner of the bridge and the “spike”
in the correction factor at mid-span is not as prevalent in the models that utilized the girder
stiffness of 44,400 in4. These models possess a span to depth ratio of 21, much more
representative of actual design situations. For development of design guidelines for the variation
of the skew correction factor along the length of the exterior girders, therefore, the results of the
models with 42-ft. spans and girder stiffnesses of 333,000 in4 are not considered to be as
representative of actual design conditions, as are the results of the models with 42-ft. spans and
girder stiffnesses of 44,400 in4.
43
EFFEC T O F SKEW AN G LE O N SKEW C O R R EC TIO NS ALO N G EXTE RIO R BEAM S42' S im ple S pan, Be am-S lab Bridge s, I+Ae 2 = 44,400 in 4, w/o In te rm e d. C ross
E FF E C T O F G IR D E R ST IFFN E SS O N SKE W C O R R E C T IO N S ALO N G E XT E R IO R B E AM42' S im ple S pan , Be am -S lab Bridge s, 60 de g. S k e w, w/o In te rm e d. C ross
Ext. Gird er 1, I+A e2 = 333,000 in 4 Ext. Gird er 2, I+A e2 = 333,000 in 4Ext. Gird er 1, I+A e2 = 44,400 in 4 Ext. Gird er 2, I+A e2 = 44,400 in 4Ext. Gird er 1, I+A e2 = 1,870,000 in 4 Ext. Gird er 2, I+A e2 = 1,870,000 in 4
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 in4 and span lengths of 42-ft. and 168-ft. The second set investigated models with
beam stiffnesses of 333,000 in4 and span lengths of 105-ft. and 168-ft. Similarly, the third set
investigated beam stiffnesses of 1,870,000 in4 with span lengths of 105-ft. and 168-ft.
The variation of the skew correction factor along the length of the exterior beams are
essentially the same between the models of each set investigated (see Figures 14, 15 and 16).
The skew correction quickly drops from its value at the end of the beam to zero near the three- or
four-tenth point of the span length. The longer spans may tend to slightly increase the length
along the beam over which the correction factor is effective, but in all cases the correction factor
disappears between the three- and four-tenth point of the span length. As discussed in the
previous section, a correction factor approximately equal in magnitude to the end correction is
present near mid-span of the 168-ft. model with the 333,000 in4 beam stiffness, but the shear
values in this region will not govern for design purposes.
50
E F F E C T O F SP AN LE N G T H O N SKE W C O R R E C T IO N S ALO N G E XT E R IO R B E AM S
S im ple S pan , Be am -S lab Bridge s, I+Ae 2 = 44,400 in 4, 60 de g. S ke w, w/o In te rm e d. C ross Fram e s
Ext. G irder 1, N o X -Fram es Ext. G irder 2, N o X -F ram esExt. G irder 1, X -F ram es Ext. G irder 2, X -F ram es
Figure 18. Effect of Intermediate Cross Frames on Skew Corrections Along Exterior Beams4.1.1.5 Influence of Beam Spacing
The influence of beam spacing on the variation of the skew correction factor along the
length of exterior beams was investigated in one set of models. The models were constructed
with a 42-ft. span length, beam stiffnesses of 44,400 in4, a skew angle of 60°, 7-in. slab thickness
and six beams spaced at 7.75-ft. or nine beams at 4.84-ft. Intermediate cross frames were not
included in the models. The variation of the skew correction factor along the length of the
exterior beams is essentially the same for the models with the two different beam spacings (see
Figure 19). The skew correction quickly drops from its value at the end of the beam to zero near
the three-tenth point along the span length. The spacing of the beams does not significantly
57
affect this variation.
58
E F F E C T O F B E AM SP AC IN G O N SKE W C O R R E C T IO N S ALO N G E XT E R IO R B E AM S42' S im ple S pan , Be am -S lab Bridge s, I+Ae 2 = 44,400 in 4, 60 de g. S ke w, w/o
Ext. Gird er 1, 7.75' Beam Spa. Ext. Gird er 2, 7.75' Beam Sp a.Ext. Gird er 1, 4.84' Beam Spa. Ext. Gird er 2, 4.84' Beam Sp a.
Figure 19. Effect of Beam Spacing on Skew Corrections Along Exterior Beams
59
4.1.1.6 Influence of Slab Thickness
The influence of slab thickness on the variation of the skew correction factor along the
length of exterior beams was investigated in one set of models. The models were investigated
for slab thicknesses of 7-in. and 9-in. with a 42-ft. span length, a six-beam cross section with
beam spacings of 7.75-ft., beam stiffnesses of 44,400 in4 and a skew angle of 60°. Intermediate
cross frames were not included in the models. The variation of the skew correction factor along
the length of the exterior beams are nearly identical for the models with the two different slab
thicknesses (see Figure 20). The skew correction quickly drops from its value at the end of the
beam to zero near the three-tenth point along the span length. The thickness of the slab does not
significantly affect this variation.
60
E F F E C T O F SLAB T HIC KN E SS O N SKE W C O R R E C T IO N S ALO N G E XT E R IO R B E AM S42' S im ple S pan , Be am -S lab Bridge s, I+Ae 2 = 44,400 in 4, 60 de g. S k e w, w/o
Figure 38. Comparison of Simple Span and Two-Span Continuous Skew Correction Factors
Table 8. Comparison of Skew Correction Factors for End Shear of Exterior Girders at theObtuse Abutment Corners of Simple Span and Two-Span Bridge Models.
87
88
4.4.2 Correction Factors at Obtuse Corners of Abutments and Piers
The examination of the two-span beam-slab bridge results also included a comparison of
the skew correction factors for shear in the exterior girders at the obtuse corners of the abutments
and the pier, as shown in Figure 39.
As discussed in Section 4.2.1, NCHRP Project 12-26 did not provide explicit guidance
regarding application of the skew correction factors at the piers of continuous bridges.
Furthermore, the current LRFD Specifications are silent on this issue. Hence, it was necessary to
determine whether the skew correction factors for shear, developed in Project NCHRP 12-26 for
exterior girders of simple span bridges, and found in this study to be valid at the obtuse corners
of abutments of continuous bridges, are also valid for exterior girders at the obtuse corners of
piers of continuous bridges.
For each of the two-span continuous models investigated in this study, the skew
correction factors for the exterior girders were calculated at the obtuse corners of both abutments
and at the girder location adjacent to the obtuse corner of the pier. Comparison of the results,
shown in Table 9, indicates that the correction factors at the pier are typically greater than those
at the abutments. Additionally, increases in the skew angle and the girder stiffness tend to
increase the differences between the correction factors at the pier and abutments. With the
limited number of data sets, however, it is difficult to accurately predict a trend in the results.
Most of the bridge model results yield differences between the abutment and pier correction
factors of less than 5%. Given that the LRFD Specifications have regarded corrections of this
magnitude to imply misleading accuracy in approximate methods, the skew correction factors of
89
the exterior girders at the obtuse corner of the abutments are considered to be representative of
those that occur at the piers. Therefore, the skew correction factors developed in Project
NCHRP 12-26 for exterior girders of simple span bridges, found to be valid at the obtuse
corners of abutments of continuous bridges, are also considered to be applicable to the exterior
girders at the obtuse corners of piers of continuous bridges.
90
C PierL
Span 1 Span 2
Girder (Typ.)
Are Skew Correction Factorssimilar at these locations (Typ.)?
Figure 39. Comparison of Skew Correction Factors at Abutments and Pier
Table 9. Comparison of Skew Correction Factors for Shear of Exterior Girders at the ObtuseAbutment Corners and Obtuse Pier Corners of Two-Span Bridge Models.
91
C Abutment (Typ.)L
C PierL
Span 1 Span 2ExteriorGirder 1
ExteriorGirder 2
4.4.3 Live Load Shear Along Exterior Beam Length
4.4.3.1 Influence of Skew Angle
The influence of skew angle on the variation of the skew correction factor along the
length of exterior beams of two span continuous bridges was investigated in one set of beam-slab
bridge models. The bridge models were based upon two equal spans, 168-ft. in length, a six-
beam cross section with beam spacings of 7.75-ft., girder stiffnesses of 333,000 in4 (I + Ae2), a
7-in. deck slab and no intermediate cross-frames. Skew angles of 30° and 60° were studied.
The variation of the skew correction was investigated along the length of each of the
“four” exterior girders: exterior girders 1 and 2 in each of span 1 and span 2. This nomenclature
is shown in Figure 40. The plot of the variation of the skew correction for each girder was
created by defining location 0.0 as the end of the girder at the obtuse corner created with its
support. The skew corrections at each tenth point along each girder were normalized against the
correction at the girder’s obtuse corner. Plotting each girder simultaneously enabled direct
comparison of the variation of the skew correction along the length of each girder.
92
Figure 40. Nomenclature for Investigation of Correction Factors Along the Length of theExterior Girders of Two-Span Continuous Bridge Models
93
The plots of the normalized skew corrections for these models display a diminishing
influence of the skew correction from the end of the exterior beams at the obtuse corners to the
acute corners (see Figure 41). The results indicate that the variation of the skew correction along
the girder length is similar regardless of whether the obtuse corner is located at the abutments or
pier. Additionally, the variation of the skew correction factor is not sensitive to changes in the
skew angle, as 30° and 60° skew angles both produce variations in which the correction factor
falls rapidly from its normalized value at the obtuse corner to zero near the three-tenth point of
the span length.
The results from the model with 30° skew display the presence of a correction factor at
mid-span that exceeds the correction factor at the obtuse corner. Further investigation of the
girder shears from this model, however, reveal that the plotted data greatly amplifies the actual
analysis results. The shears at the obtuse corner of this exterior girder are 50.2 kips and 51.6
kips, for the right and skewed bridge, respectively. This produces a correction factor of
approximately 1.03 at the end of the girder. At mid-span, the shears are 28.95 kips and 30.0
kips, for the right and skewed bridge, respectively, producing a correction factor at this location
of 1.04. When normalized to the skew correction of 0.03 at the end of the girder, the correction
of 0.04 at mid-span produces a normalized value of 1.32. Incorporation of this isolated
correction factor into a design approximation for the variation of the skew correction factor is
not considered to be necessary. The shears at this location of the girders do not control for
design purposes and the minimal differences between the skewed bridge and right bridge model
results are amplified by the manner in which the results are presented. As a result, the mid-span
skew correction will be neglected in the design approximation for the variation of the skew
94
correction factor.
95
EFFEC T O F SKEW AN G LE O N SKE W C O R R E C T IO N S ALO N G E XT E R IO R BE AM S
Two-S pan C ontinu ous, Be am -S lab Bridge s, 168' S pans,I+Ae 2 = 333,000 in 4, w/o Inte rm e d. C ross Fram e s
Sp an 1, Ext. Gird er 1, 105' Sp an s Sp an 1, Ext. Gird er 2, 105' Sp an sSp an 2, Ext. Gird er 1, 105' Sp an s Sp an 2, Ext. Gird er 2, 105' Sp an sSp an 1, Ext. Gird er 1, 168' Sp an s Sp an 1, Ext. Gird er 2, 168' Sp an sSp an 2, Ext. Gird er 1, 168' Sp an s Sp an 2, Ext. Gird er 2, 168' Sp an s
E F F E C T O F SP AN LE N G T H O N SKE W C O R R E C T IO N S ALO N G E XT E R IO R B E AM S
Two-S pan C ontin u ou s, Be am -S lab Bridge s,I+Ae 2 = 1,870,000 in 4, 60 de g. S ke w, w/o In te rm e d. C ross Fram e s
Figure 56. Effect of Girder Stiffness on Skew Corrections for Shear Across Pier
115
Figure 57. Effect of Girder Stiffness on Skew Corrections for Shear Across PierThe influence of span length on the variation of the correction factor across the pier is
shown in Figures 58 and 59. The results indicate that an increase in span length may tend to
increase the influence of the skew correction across the pier. As the span length increases, the
skew correction of girders 2 through 5 become a larger percentage of the corrections of girder 1.
When studying these results in conjunction with those from the investigation of girder stiffness,
it appears that an increase in the flexibility of the structure, caused by either a decrease in beam
stiffness or an increase in span length, results in a greater presence of a skew correction across
the piers of continuous bridges. All of the results for the investigation of span length display,
however, a general decrease in the magnitude of the skew correction across the pier. Again, the
pronounced skew correction at girder 5 in Figure 55, is not present in these models (Figures 58
and 59).
116
EFFECT OF SPAN LENGTH ON SKEW CORRECTIONS FOR SHEAR
ACROSS PIERTwo-Span Continuous, Beam-Slab Bridges, I+Ae2 = 333,000 in4,
Figure 58. Effect of Span Length on Skew Corrections for Shear Across Pier
117
Figure 59. Effect of Span Length on Skew Corrections for Shear Across PierFigure 60 superimposes the results from each of the models of Figures 55 through 59.
Except for a few isolated data points, the general decline in the correction factor across the
abutment bearing lines is evident. Similar to the results for the end shear correction factors
across the abutments of the continuous bridges, certain models produce correction factor at
Girder 2, the first interior girder adjacent to the obtuse corner, equal to or greater than the
correction factor at the obtuse corner of the pier.
Figure 61 condenses all of the results into a single plot of the average variation of the
skew correction for shear across the pier. Superimposed on the results is a linear variation of the
correction factor from its value at the obtuse corner to a value of zero at the acute corner. Except
for the data point at girder 5, all of the average results fall within this approximation. The
average data point of 0.55 at girder 5 includes the normalized value of approximately 2.75 from
Figure 55. When this value is not included in the pool of results, an average value of 0.12 is
produced at girder 5. This value falls well within the linear approximation. The data point of
2.75 is considered to be an isolate value, not representative of the results anticipated from typical
beam-slab bridges. As a result, it is suggested that the variation of the correction factor across
the pier can be reasonably approximated by a linear distribution from its value at the obtuse
corner to zero at the acute corner, identical to the variation suggested across the abutments.
Figure 69. Effect of Girder Stiffness on Skew Corrections for Reaction Across Pier
132
Figure 70. Effect of Girder Stiffness on Skew Corrections for Reaction Across PierFinally, the influence of span length on the variation of the skew correction for reaction
is displayed in Figures 71 and 72. Again, the variation of the correction factor is symmetrical
across the pier for each of the data sets. These results, however, do not yield a correlation
between the change in span length and the variation of the skew correction across the pier.
Figure 71 displays that an increase in span length tends to decrease the magnitude of the
correction at the interior girders, with respect to the correction at girder 1. Furthermore, each
data set of Figure 71 produces a greater skew correction at the interior girders than at the exterior
girders. Figure 72, however, indicates that the corrections at the interior girders are less than
those at the exterior girders. Additionally, the increase in span length increases the magnitude of
the correction at the interior girders, with respect to the correction at girder 1. Figures 71 and 72,
therefore, do not reveal a definite trend between the change in span length and the variation of
the skew correction for reaction at the pier.
133
EFFECT OF SPAN LENGTH ONSKEW CORRECTIONS FORREACTION ACROSS PIER
Definition of Variables:Kg = Longitudinal Stiffness Parameter (in4)
ts = Depth of Concrete Slab (in)θ = Skew Angle measured from a line normal to the CL of Bridge
LRFD Empirical Skew Correction Factor = 1.0 + 0.20 * ( 12.0 L ts3 / Kg )
0.3 tan θ
BRIDGE MODEL DATA
Table 11. Comparison of Skew Correction Factors from LRFD Specifications and ResearchResults
137
CHAPTER 5 INTERPRETATION AND APPLICATION
5.1 SIMPLE SPAN BEAM-SLAB BRIDGES
With respect to simple span beam-slab bridges, the results of this study indicate that:
• The variation of the skew correction factors for shear along the length ofthe exterior girders of simple span beam-slab bridges is not significantlyinfluenced by changes in skew angle, beam stiffness, span length, beamspacing, slab thickness, bridge aspect ratio, or by the presence ofintermediate cross frames.
• While the magnitude of the correction factors may change in concert withchanges of these parameters, the variation of the correction factor alongthe exterior beam length is not significantly altered.
• The skew correction factor typically falls quickly from its initial value at
the end of the exterior girder at the obtuse bridge corner to zero by thefour-tenth point of the span, independent of changes in the aforementionedbridge parameters.
• An appropriate, conservative design approximation for the variation of theskew correction factor for shear along the length of the exterior girder is alinear variation from its initial value at the obtuse corner of the bridge planto zero at some point within the girder span.
Regarding the last conclusion, Figure 73, a plot of the results from all models that
represent realistic bridge designs (span-to-depth ratios range from 13 to 21), displays that a
reasonable approximation for the skew correction factor variation is a straight line from its value
at the obtuse corner to a correction factor of 1.0 at mid-span. In isolated cases, a correction
factor is present near mid-span of the exterior girders, but the magnitudes of the shears near mid-
span are still much less that the magnitudes of the end shears. It is suggested, therefore, that
these corrections near mid-span may be neglected as the shear values at these locations will not
138
govern for typical design applications. It is recognized that design cases in which the web depth,
thickness, yield strength, stiffener spacing, etc. vary along the beam length, the controlling
region for shear design may not be at the typical end of beam location. The correction factor
spikes found at mid-span in a limited number of the data sets, however, were amplified by the
manner of data reduction. Given that the magnitude of the mid-span shear is small, any slight
deviations between the right bridge and skewed bridge mid-span shears may produce appreciable
correction factors in terms of percentages. To illustrate, the data displays some mid-span skew
correction factors between 1.03 to 1.13. The actual difference between the mid-span shears in
the right and skewed bridge models for the correction factor of 1.13 is only 2.8 kips (22.7 k right
bridge; 25.5 k skewed bridge). Attempts to pinpoint design corrections to this level of accuracy
may be misleading in terms of the accuracy of the approximation itself. Furthermore, the bridge
models which possess these mid-span spikes are predominately models with a 168’ span length
and a 60 degree skew. Given the small number of occurrences of the mid-span spikes and the
extreme skew angle of the models in which it did occur, the inclusion of the spikes in a general
design approximation was not considered necessary.
Additionally, select model results yielded a correction factor greater than 1.0 at the acute
corner of the exterior girders. These results, however, were obtained from bridge models with a
span-to-depth ratio of 5.25, well outside ratios of practical design applications. It is suggested,
therefore, that these corrections at the acute corner may be neglected given the fact that these
results did not occur in models that are more representative of actual design cases.
139
NORMALIZED SKEW CORRECTIONS FOR SHEAR IN EXTERIOR GIRDERS of SIMPLE SPAN BEAM-SLAB
BRIDGES
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Tenth Point Along Span
Nor
mal
ized
Ske
w C
orre
ctio
ns
Proposed Design Approximation for the Variation of the Skew Correction Factor
Figure 73. Results for the Variation of the Skew Correction Along the Length of the ExteriorGirders of Simple-Span Beam-Slab Bridges.
140
The study results also indicate that:
• The variation of the skew correction factors for end shear across thebearing lines of simple span beam-slab bridges is not significantly alteredby changes in skew angle, beam stiffness, span length, beam spacing, slabthickness, or by the presence of intermediate cross frames.
• A reasonable, conservative approximation of this variation across the
bearing lines is a linear distribution of the correction factor from its valueat the obtuse corner to a correction factor of 1.0 at the acute corner.
The average variation of the skew correction for end shear across the bearing lines of the
models studied is depicted in Figure 74, indicating that the linear distribution across the bearing
lines is conservative and encompasses the average results.
141
AVERAGE NORMALIZED SKEW CORRECTIONS FOR END SHEAR ACROSS BEARING LINES of
SIMPLE SPAN BEAM-SLAB BRIDGES1.00
0.49
0.080.25
-0.23
-0.54
-1.00
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
1 2 3 4 5 6
Girder
Nor
mal
ized
Ske
w C
orre
ctio
ns
Proposed Design Approximation for the Variation of the Skew Correction Factor
Figure 74. Average Results for the Variation of the Skew Correction Along the BearingLines of Simple Span Beam-Slab Bridges.
142
5.2 SIMPLE SPAN CONCRETE T-BEAM BRIDGES
The cursory investigation of simple span, monolithic, concrete T-beam bridge models
indicated that:
• The variation of the skew correction factors along both the length of theexterior beams and across the beam supports is very similar to thatobtained from the simple span beam-slab bridge models.
• Regardless bridge model skew angle, the variation of the skew correctionfactors along the length of the beams can be approximated reasonably by alinear distribution of the correction factor at the obtuse corner of theexterior girder to a value of 1.0 at mid-span of the exterior girder.
• Across the bearing lines, the skew correction factor variation can beapproximated by a linear distribution of the correction factor at the obtusecorner of the bearing line to a value of 1.0 at the acute corner.
Although only the effect of skew angle on the correction factor variation was
investigated, it is presumed that changes in other bridge parameters will produce similar results.
Therefore, the design approximations developed for the beam-slab bridges are considered to be
The analysis of the spread concrete box girder bridge models raised an issue regarding
the influence of torsion on the maximum shear in box girders and the design methodology to be
followed in the development of skew correction factors for this bridge type. Although typically
neglected in right bridges due to the premise of equal deflections of the bridge girders and
143
negligible differential shears between the webs of the box girders, the effects of torsion on box
girder shear may be amplified due to the introduction of skew. The box girder shear data
obtained in this study, which incorporates the effects of torsion, indicates that torsion may not be
negligible in skewed bridges. Without further conclusive research, however, and given the lack
of substantial field documentation indicating problems with torsion and shear in skewed spread
box girder bridges, the current design practices of neglecting torsion are considered to be
acceptable. It is recommended, however, that further studies of box girder shear in skewed
bridges be performed to investigate the influence of torsion. Such studies may help determine
whether torsion should be included in approximations for the variation of the skew correction
factors for shear.
5.4 TWO-SPAN CONTINUOUS BEAM-SLAB BRIDGES
The investigation of the two-span continuous beam-slab bridge models reveals that:
• The skew correction factors for shear at the obtuse corner of skewed,simple span beam-slab bridges are valid for the shear at the obtuse cornerof the abutments of skewed, continuous beam-slab bridges.
• The skew correction factors for shear at the obtuse corner of skewed,simple span beam-slab bridges are also valid for shear in exterior girdersof continuous bridges at the obtuse corner created by the girders and thepiers.
• The variation of the skew correction factors for shear along the length ofthe exterior girders is not significantly influenced by changes in skewangle, beam stiffness and span length.
• The variation of the correction factor along the length of the exterior
144
girder is similar when the obtuse corner is located at the abutment andwhen it is located at the pier. As a result, each span of the continuousexterior girders possesses the same correction factor at its obtuse corner,as well as the same variation of the correction factor along the span length.
• While the magnitude of the correction factors may change in concert withchanges of skew angle, beam stiffness and span length, the variation of thecorrection factor along the span length is not significantly altered.
• The skew correction factor typically falls quickly from its initial value atthe end of the exterior girder at the obtuse corner to zero by the four-tenthpoint of the span, independent of changes in the aforementioned bridgeparameters.
• An appropriate, conservative design approximation for the variation of theskew correction factor for shear along the length of each span of theexterior girder is a linear variation from its initial value at the girder’sobtuse corner to 1.0 at some point within the girder span, very similar tothe results from the simple span models.
Regarding the last conclusion, Figure 75 displays that a reasonable approximation for the
skew correction factor variation is a straight line from its value at the obtuse corner to a
correction factor of 1.0 at mid-span. Similar to the simple span model results, a correction factor
is present near mid-span of the exterior girders in isolated cases. However, the manner in which
the data is presented tends to amplify the results and the magnitudes of the shears near mid-span
are still much less that the magnitudes of the end shears. It is suggested, therefore, that these
corrections near mid-span may be neglected as the shear values at these locations will not govern
for typical design applications.
145
NORMALIZED SKEW CORRECTIONS FOR SHEAR IN EXTERIOR GIRDERS of TWO-SPAN CONTINUOUS BEAM-SLAB BRIDGES
-2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Tenth Point Along Span
Nor
mal
ized
Ske
w C
orre
ctio
nsProposed Design Approximation for the Variation of the Skew Correction Factor
Figure 75. Results for the Variation of the Skew Correction Along the Length of the ExteriorGirders of Two-Span Continuous Beam-Slab Bridges.
146
The study results also indicate that:
• The variation of the skew correction factors for shear across the abutmentsand pier of two-span continuous beam-slab bridges is not significantlyaltered by changes in skew angle, beam stiffness and span length.
• A reasonable approximation of this variation across both the abutmentsand pier is a linear distribution of the correction factor from its value atthe obtuse corner of the abutments and pier to a correction factor of 1.0 atthe acute corner.
The average variation of the skew correction factor for shear across the abutments and
the piers of the models studied is depicted in Figure 76, including the proposed design office
approximation.
147
AVERAGE NORMALIZED SKEW CORRECTIONSFOR SHEAR ACROSS ABUTMENTS
Proposed Design Approximation for the Variation of the Skew Correction Factor
Figure 76. Average Results for the Variation of the Skew Correction Across Abutments andPiers of Two-Span Beam-Slab Bridges.
148
The investigation of the two-span models does reveal that skew correction factors for
reaction at piers of continuous bridges are present at each girder, and that these correction factor
are unique from those required for the shear at the pier. However, the variation of the correction
factor across the pier with changes in the bridge parameters is not clearly understood from the
limited number of data sets in this study. Although changes of skew angle and girder stiffness in
the study models did provide insight into possible effects on the correction factors, it is difficult
to define that variation from the small pool of data sets. The study results do indicate, however,
that further research is required to develop empirical equations for both the skew correction
factors for reactions at piers and the variation of those correction factors across the pier.
5.5 APPLICATION OF STUDY FINDINGS
From the study findings, it was determined that regardless of bridge parameters, a
reasonable design approximation for the variation of the skew correction factor for shear along
the length of the exterior girder of simple span beam-slab and monolithic concrete T-beam
bridges is a linear variation from its initial value at the obtuse corner to a correction factor of 1.0
at mid-span. Similarly, regardless of bridge parameters, the variation of the skew correction
factor for shear along the length of the exterior girders in each span of two-span continuous
beam-slab bridges may be reasonably approximated with a linear variation from its initial value
at the obtuse corner of the girder to a correction factor of 1.0 at mid-span. Therefore, for
149
application of the research findings regarding the variation of the skew correction factor for
shear along the length of exterior girders, the recommendations are as follows:
• For superstructure types “Concrete Deck, Filled Grid, or Partially Filled Grid onSteel or Concrete Beams; Concrete T-Beams, T- and Double T Section,” withinthe applicable ranges of skew angle (θ), spacing of beams or webs (S), span ofbeam (L) and number of beams, stringers or girders (Nb) as defined by Table4.6.2.2.3c-1 of the LRFD Specifications, the skew correction factor for shear maybe varied linearly from its value at the obtuse corner of the bridge, determined inaccordance with the empirical equation defined in Table 4.6.2.2.3c-1, to a valueof 1.0 at girder mid-span, as shown in Figure 77.
• This approximate variation is applicable for both simple span structures andcontinuous structures. For continuous structures, the skew correction factorcalculated at the obtuse corner of the abutment per Table 4.6.2.2.3c-1 is also validat the obtuse corners of the interior piers. Likewise, the variation of thecorrection factor is applicable from both the obtuse corner of the abutment and theobtuse corners of the interior piers to the girder mid-span, as shown in Figure 78.
Although this study did not investigate each superstructure type within the group
“Concrete Deck, Filled Grid, or Partially Filled Grid on Steel or Concrete Beams; Concrete T-
Beams, T- and Double T Section,” the bridge models studied are representative of this class of
superstructure, with the bridge parameters as defined by Table 4.6.2.2.3c-1 of the LRFD
Specifications. The study findings and proposed design approximations, therefore, are
considered to be valid for each type of structure within this class.
150
C Abutment (Typ.)L
Calculated Skew CorrectionFactor at Obtuse Corner (Typ.)
C Girder (Typ.)L
Skew Angle
1.0
1.0
Linear Variation of theCorrection Factor (Typ.)
Mid-Point of GirderSpan (Typ.)
C Abutment (Typ.)L
Calculated Skew CorrectionFactor at Obtuse Corner ofAbutment (Typ.)
C Girder (Typ.)L
Skew Angle
1.0
1.0
Linear Variation of theCorrection Factor (Typ.)
Mid-Point of GirderSpan (Typ.)
C PierL
1.0
1.0
Skew Correction Factor alsoApplied at Obtuse Corner of Pier(Typ.)
Figure 77. Proposed Variation of the Skew Correction Factors for Shear Along the Length ofthe Exterior Girders in Simple Span Superstructures of Concrete Deck, FilledGrid, or Partially Filled Grid on Steel or Concrete Beams; Concrete T-Beams, T-
and Double T Sections
Figure 78. Proposed Variation of the Skew Correction Factors for Shear Along the Length ofthe Exterior Girders in Continuous Superstructures of Concrete Deck, Filled Grid,
151
or Partially Filled Grid on Steel or Concrete Beams; Concrete T-Beams, T- andDouble T Sections
152
The study findings also reveal that regardless of bridge parameters, a reasonable design
approximation for the variation of the skew correction factors for end shear of each girder across
bearing lines of simple span beam-slab bridges and monolithic concrete T-beam bridges is a
linear variation from its initial value at the obtuse corner of the bearing line to a correction factor
of 1.0 at the acute corner of the bearing line. Similarly, regardless of bridge parameters, the
variation of the skew correction factor for shear of each girder across the abutments and piers of
two-span continuous beam-slab bridges may be reasonably approximated with a linear variation
from its initial value at the obtuse corner of the bearing line to a correction factor of 1.0 at the
acute corner of the bearing line. Therefore, for application of the research findings regarding the
variation of the skew correction factor for shear across bearing lines, the recommendations are as
follows:
• For superstructure types “Concrete Deck, Filled Grid, or Partially Filled Grid onSteel or Concrete Beams; Concrete T-Beams, T- and Double T Section,” withinthe applicable ranges of skew angle (θ), spacing of beams or webs (S), span ofbeam (L) and number of beams, stringers or girders (Nb) as defined by Table4.6.2.2.3c-1 of the LRFD Specifications, the skew correction factor for shear maybe varied linearly from its value at the obtuse corner of the bridge, determined inaccordance with Table 4.6.2.2.3c-1, to a value of 1.0 at the acute corner of thebearing line, as shown in Figure 79.
• This approximate variation is applicable for both simple span structures andcontinuous structures. For continuous structures, the skew correction factorcalculated at the obtuse corner of the abutment per Table 4.6.2.2.3c-1 is also validat the obtuse corners of the interior piers. Likewise, the variation of thecorrection factor is applicable from both the obtuse corner of the abutment and theobtuse corners of the interior piers to the acute corner of the bearing lines, asshown in Figure 80.
153
As discussed previously, this study did not investigate each superstructure type within the
group “Concrete Deck, Filled Grid, or Partially Filled Grid on Steel or Concrete Beams;
Concrete T-Beams, T- and Double T Section.” The bridge models studied, however, are
representative of this class of superstructure, with the bridge parameters as defined by Table
4.6.2.2.3c-1 of the LRFD Specifications. The study findings and proposed design
approximation, therefore, are considered to be valid for each type of structure within this class.
154
C Abutment (Typ.)L
Calculated Skew CorrectionFactor at Obtuse Corner (Typ.)
C Girder (Typ.)L
Skew Angle
1.0
1.0
Linear Variation of theCorrection Factor (Typ.)
C Abutment (Typ.)L
Calculated Skew CorrectionFactor at Obtuse Corner ofAbutment (Typ.)
C Girder (Typ.)L
Skew Angle
Linear Variation of theCorrection Factor (Typ.)
C PierL
Skew Correction Factor alsoApplied at Obtuse Corner of Pier(Typ.)
1.0
1.01.0
1.0
Figure 79. Proposed Variation of the Skew Correction Factors for Shear Across the BearingLines of Simple Span Superstructures of Concrete Deck, Filled Grid, or PartiallyFilled Grid on Steel or Concrete Beams; Concrete T-Beams, T- and Double TSections
Figure 80. Proposed Variation of the Skew Correction Factors for Shear Across theAbutments and Piers of Continuous Superstructures of Concrete Deck, FilledGrid, or Partially Filled Grid on Steel or Concrete Beams; Concrete T-Beams, T-and Double T Sections
155
CHAPTER 6 CONCLUSIONS AND SUGGESTED RESEARCH
The variation of the skew correction factors for shear along the length of the exterior
girders of simple span beam-slab bridges is not significantly influenced by changes in skew
angle, beam stiffness, span length, beam spacing, slab thickness, bridge aspect ratio, or by the
presence of intermediate cross frames. It is recommended that a reasonable design
approximation for the variation of the skew correction factor for shear along the length of the
exterior girder of simple span beam-slab bridges is a linear variation from its initial value at the
obtuse corner to a correction factor of 1.0 at mid-span, as shown in Figure 77. Regardless of the
aforementioned bridge parameters, therefore, the skew correction factor may be calculated for
the obtuse corner as defined in the LRFD Specifications and varied linearly to a value of 1.0 at
the mid-point of the girder span. This approximation is recommended for simple span
superstructures of concrete deck, filled grid, or partially filled grid on steel or concrete beams;
concrete T-beams, T- and double T sections, within the geometric limitations defined in Table
4.6.2.2.3c-1 of the LRFD Specifications.
Additionally, the variation of the skew correction factors for end shear of each girder
across bearing lines of simple span beam-slab bridges is not significantly altered by changes in
span length, beam spacing, slab thickness, skew angle and beam stiffness, or by the presence of
intermediate cross frames. Therefore, it is recommended that the skew correction factor be
calculated for the end shear of the exterior girder at the obtuse corner of the bridge, as defined in
the LRFD Specifications, and varied linearly across the bearing lines to a value of 1.0 at the
acute corner, as shown in Figure 79. Again, this approximation is reasonable for simple span
156
superstructures of concrete deck, filled grid, or partially filled grid on steel or concrete beams;
concrete T-beams, T- and double T sections, within the geometric limitations defined in Table
4.6.2.2.3c-1 of the LRFD Specifications.
The skew correction factors for shear defined in the current LRFD Specifications at the
obtuse corner of skewed, simple span beam-slab bridges are also valid for the shear at the obtuse
corner of the abutments of skewed, continuous beam-slab bridges. Furthermore, these same
skew correction factors are also valid for shear in exterior girders of continuous bridges at the
obtuse corner created by the girders and the piers.
Similar to the conclusions regarding the variation of the skew correction factors for shear
along the length of the exterior girders of simple span bridges, the variation of the correction
factors for two-span continuous beam-slab bridges is not significantly influenced by changes in
skew angle, beam stiffness and span length. The same design approximation proposed for the
correction factor variation in the simple span bridges is also recommended for each span of the
continuous bridges. A reasonable design approximation for the variation of the skew correction
factor for shear along the length of the exterior girders in each span of continuous beam-slab
bridges is a linear variation from its initial value at the obtuse corner of the girder to a correction
factor of 1.0 at mid-span, as shown in Figure 78. Regardless of the aforementioned bridge
parameters, therefore, the skew correction factor may be calculated for the end shear of the
exterior girder as defined in the LRFD Specifications, applied at the obtuse corners of both the
abutments and piers, and varied linearly to a value of 1.0 at the mid-point of the girder span.
This approximation is appropriate for continuous superstructures of concrete deck, filled grid, or
157
partially filled grid on steel or concrete beams; concrete T-beams, T- and double T sections,
within the geometric limitations defined in Table 4.6.2.2.3c-1 of the LRFD Specifications.
Additionally, the variation of the skew correction factors for shear of each girder across
the abutments and piers of continuous beam-slab bridges is not significantly altered by changes
in skew angle, girder stiffness and span length. The recommended design approximation for the
simple span bridges is also valid across the abutments and piers of continuous bridges.
Therefore, the skew correction factor can be calculated for the end shear of the exterior girder, as
defined in the LRFD Specifications, applied at the obtuse corners of both the abutments and
piers, and varied linearly across the abutments and piers to a value of 1.0 at the acute corner, as
shown in Figure 80. Again, this approximation is appropriate for continuous superstructures of
concrete deck, filled grid, or partially filled grid on steel or concrete beams; concrete T-beams,
T- and double T sections, within the geometric limitations defined in Table 4.6.2.2.3c-1 of the
LRFD Specifications.
The results of this study lead to the following recommendations for further research:
• Skew Correction Factors for Reaction at the Piers of Continuous Bridges:
It was determined that these correction factors are present and are uniquefrom those calculated for shear at the pier. The effects of the obtuse andacute corners on the girder shear on opposite sides of the bearings do noteliminate a correction factor for reaction. From the limited continuousbridge model data pool of this study, however, it was not feasible todevelop empirical equations that define the correction factor or define itsvariation across the pier. Therefore, the development of such equationsfor continuous bridges is suggested as further research.
• Skew Correction Factors of Other Beam and Slab Bridge Types:
This research focused primarily on the skew correction factors for shear insimple-span and two-span continuous beam-slab bridges and providesrecommendations for only superstructures of concrete deck, filled grid, or
158
partially filled grid on steel or concrete beams; concrete T-beams, T- anddouble T sections. These bridges, however, comprise only one type of thelarger genre of beam and slab bridges. The LRFD Specifications provideempirical equations for skew correction factors of multi-cell concrete boxbeam, spread concrete box beam and multi-beam bridges. It isrecommended that further research be performed to investigate thebehavior of the skew correction factors for shear in these additional bridgetypes.
• Torsion in Skewed Box Beam Bridges:
Additional work could be performed to investigate torsion in skewed,spread box beam bridges and the magnitude of its effect on shear in boxgirder webs. Given the lack of substantial field documentation indicatingproblems with torsion and shear in skewed spread box girder bridges,however, the current design practices which neglect torsion are consideredto be acceptable.
159
1. American Association of State Highway and Transportation Officials, “LRFDBridge Design Specifications.” Second Edition, (1996).
2. Nutt, R. V., Schamber R. A. and Zokaie T., “Distribution of wheel loads onhighway bridges.” Project No. 12-26, National Cooperative Highway ResearchProgram, Transportation Research Board, National Research Council,Washington, D.C., (1988).
3. American Association of State Highway and Transportation Officials, “LRFDBridge Design Specifications.” First Edition, (1994).
4. Zokaie, T., Osterkamp T. A. and Imbsen R. A., “Distribution of wheel loads onhighway bridges.” Project No. 12-26/1, National Cooperative Highway ResearchProgram, Transportation Research Board, National Research Council,Washington, D.C., (1991).
5. Kulicki, J. M., “Shear in Skewed Multi-Beam Bridges.” Proposal for NCHRP 20-7/Task 107, Modjeski and Masters, Inc., Mechanicsburg, PA, (1999).
6. Marx, H. J., Kachaturion N. and Gamble W. L., “Development of design criteriafor simply supported skew slab-and-girder bridges.” Report No. 522, CivilEngineering Studies, Struct. Res. Ser., University of Illinois, Champaign, IL,(1986).
7. Khaleel, M. A., and Itani, R. Y., “Live-load moments for continuous skewbridges.” Journal of Structural Engineering, ASCE, 116(9), (1990) pp. 2361-2373.
8. Bishara, A. G., Liu, M. C. and El-Ali, N. D., “Wheel Load Distribution on SimplySupported Skew I-beam Composite Bridges.” Structural Engineering, ASCE,119(2), (1993) pp.399-419.
9. Ebeido, T. and Kennedy, J.B., “Girder moments in continuous skew compositebridges.” Journal of Bridge Engineering, ASCE, 1(1), (1996) pp. 37-45.
10. American Association of State Highway and Transportation Officials, “StandardSpecifications for Highway Bridges.” Fifteenth Edition, (1989).
11. Powell, G. H. and Buckle, I. G., “Computer Programs for Bridge Deck Analysis.”Report No. UC SESM 70-6, Division of Structural Engineering and
Structural Mechanics, University of California, Berkeley, CA, (1970).
REFERENCES
160
12. Dodds, R. H. and Lopez, L. A., “A Generalized Software System for NonlinearAnalysis.” International Journal for Advances in Engineering Software, 2(4),(1980), pp. 161-168.
13. Ontario Ministry of Transportation and Communications, “Ontario highwaybridge design code.” Third Edition, Highway Engineering Division, Downsview,Ontario, (1992).
14. Jaeger, L. G. and Bakht B., “Bridge Analysis by Microcomputer.” McGraw-Hill,New York, NY, (1989).
15. Ebeido, T. and Kennedy, J.B., “Shear distribution in simply supported skewcomposite bridges.” Canadian Journal of Civil Engineering, Ottawa, Canada,22(6), (1995) pp. 1143-1154.
16. Ebeido, T. and Kennedy, J.B., “Shear and reaction distributions in continuousskew composite bridges.” Journal of Bridge Engineering, ASCE, 1(4), (1996) pp.155-165.
18. Aggour, M. Sherif and Aggour, M. Shafik, “Skewed bridges with intermediatetransverse bracings.” Journal of the Structural Division, ASCE, 105(8), (1979),pp.1621-1636.
19. Bell, N. B., “Load distribution of continuous bridges using field data and finiteelement analysis.” Master’s Thesis, Florida Atlantic University, Boca Raton,Florida, (1998).
21. Bishara, A. G., “Forces at bearings of skewed bridges.” Report No. FHWA/OH-91/002, Federal Highway Administration, Washington, D.C., (1990).
22. ADINA Engineering, “ADINA, Theory and Modeling Guide.” Report AE 84-6,(1984).
23. El-Ali, N. D., “Evaluation of internal forces in skew multi-stringer simplysupported steel bridge.”, PhD thesis, Ohio State University, Columbus, OH,(1986).
24. “SAP IV, A Structural Analysis Program for Static and Dynamic Response ofLinear Systems.” EERC Report No. 73-11, University of California, Berkeley,CA, (1974).
161
25. American Association of State Highway and Transportation Officials, “StandardSpecifications for Highway Bridges.” Sixteenth Edition, (1996).
26. Bridge Software Development International, Ltd., “Bridge-SystemSM.”Coopersburg, PA, (1988).