EFFECT OF ABUTMENT SKEW AND HORIZONTALLY CURVED ALIGNMENT ON
BRIDGE REACTION FORCES
A Thesis
Presented to the faculty of the Department of Civil Engineering
California State University, Sacramento
Submitted in partial satisfaction of
the requirements for the degree of
MASTER OF SCIENCE
In
Civil Engineering
(Structural Engineering)
by
Lucas Richard Miner
SPRING
2014
© 2014
Lucas Richard Miner
ALL RIGHTS RESERVED
ii
EFFECT OF ABUTMENT SKEW AND HORIZONTALLY CURVED ALIGNMENT ON
BRIDGE REACTION FORCES
A Thesis
by
Lucas Richard Miner
Approved by:
__________________________________, Committee Chair
Dr. Benjamin Fell
__________________________________, Second Reader
Dr. Matthew Salveson
__________________________________, Third Reader
Dr. Toorak Zokaie
____________________________
Date
iii
Student: Lucas Richard Miner
I certify that this student has met the requirements for format contained in the University format
manual, and that this thesis is suitable for shelving in the Library and credit is to be awarded for
the thesis.
__________________________, Graduate Coordinator ___________________
Dr. Matthew Salveson
Date
Department of Civil Engineering
iv
Abstract
Of
EFFECT OF ABUTMENT SKEW AND HORIZONTALLY CURVED ALIGNMENT ON
BRIDGE REACTION FORCES
by
Lucas Richard Miner
This thesis presents analysis results from a parametric study investigating the effects of abutment
skew and horizontal alignment curvature on bridge reaction forces. Over 800 single-span box
girder bridges were modeled using three-dimensional finite element analysis software. The
effects of skew angle, curve angle, and a coupled skew-curve effect on abutment reactions in the
obtuse corners of single-span box girder bridges were found to be significant. These results are
compared with state-of-the-art bridge design practice and LRFD Specification procedures that
provide guidance on designing bridges with skew and horizontal curvature. Interestingly, the
results demonstrate that the obtuse corner reaction forces are greatly influence by the bridge
aspect ratio across a wide variety of skew and curve angles. Furthermore, the horizontal
alignment curvature has a large effect, even at very small alignment central angles. Moreover,
the effect of skew angle is shown to be partially dependent on bridge curvature; although this
coupling of the skew and curve effects is minimal at small skew angles. The bearing stiffness was
also varied, which had a large effect on the reactions of skewed and curved bridges. Empirical
equations for skew and curve correction are proposed and additional research recommended in
the summary chapter.
v
_______________________, Committee Chair
Dr. Benjamin Fell
_______________________
Date
vi
ACKNOWLEDGEMENTS
I am grateful for the formal instruction of Dr. Benjamin Fell, Dr. Matthew Salveson, and
Dr. Toorak Zokaie in my classes at CSUS who clearly expounded the principles of structural
engineering and bridge design in a challenging and stimulating way.
I am also grateful for my bridge design job at Drake Haglan and Associates and my past
internships with the Caltrans Office of Earthquake Engineering and Dr. Toorak Zokaie. Without
these, I would not have received the direction and guidance that was vital to both the scope and
the details of this work.
I am most thankful for the patience and steadfastness of my wife Hannah, the
encouragement and support from my parents Richard and Christina, the prayers of all my friends
and family, and the kindness of my grandmother-in-law Bernice who thought my pursuit of
graduate education in structural engineering important enough to contribute with her own
finances.
vii
TABLE OF CONTENTS
Page
Acknowledgements ................................................................................................................. vii
List of Tables ............................................................................................................................. x
List of Figures ...........................................................................................................................xi
Chapter
1. INTRODUCTION ................................................................................................................. 1
1.1 Motivation ....................................................................................................................... 1
1.2 Objectives and Scope....................................................................................................... 8
1.3 Organization and Outline................................................................................................. 9
1.4 Definitions ..................................................................................................................... 10
2. PREVIOUS RESEARCH AND STATE OF PRACTICE ................................................... 14
2.1 Design Specifications ................................................................................................... 14
2.2 Bridge Design Practice .................................................................................................. 16
3. STRUCTURAL ANALYSIS............................................................................................... 18
3.1 Format............................................................................................................................ 18
3.1.1 Bridge Configurations ............................................................................................ 18
3.1.2 Plots ........................................................................................................................ 19
3.2 Software ........................................................................................................................ 22
3.3 Modeling........................................................................................................................ 22
4. ANALYSIS OF THE DATA ............................................................................................... 27
4.1 Introduction to Results.................................................................................................. 27
4.2 Visualization of Bridge Parametric Variation ............................................................... 28
viii
4.3 Results ........................................................................................................................... 30
4.3.1 Aspect Ratio = 1.0 .................................................................................................. 30
4.3.2 Aspect Ratio = 2.0 .................................................................................................. 33
4.3.3 Aspect Ratio = 4.0 .................................................................................................. 37
4.3.4 Aspect Ratio = 8.0 .................................................................................................. 39
4.3.5 General Observations ............................................................................................. 41
5. FINDINGS AND INTERPRETATIONS ............................................................................ 42
5.1 General .......................................................................................................................... 42
5.2 Discussion...................................................................................................................... 42
6. CONCLUSION .................................................................................................................... 51
6.1 Summary of Findings ................................................................................................... 51
6.2 Design Applications ..................................................................................................... 51
6.2.1 Aspect Ratio Dependency ...................................................................................... 51
6.2.2 Bearing Stiffness .................................................................................................... 52
6.3 Further Research ............................................................................................................ 52
Appendix A: Modeling Details ............................................................................................... 55
Appendix B: Effect of Softer Bearings ................................................................................... 60
Appendix C: Effect Of Span Length ....................................................................................... 62
References ................................................................................................................................ 64
ix
LIST OF TABLES
Page
Table 2.1: Adapted from AASHTO LRFD Table 4.6.2.2.3c-1................................................ 15
Table 2.2: Adapted from Caltrans 2014 Amendments to AASHTO LRFD Table
4.6.2.2.3c-1 ............................................................................................................... 15
Table 3.1- Table showing parametric variation of models analyzed in this study ................... 18
Table 3.2: The variations used in this study and extent of variation considered ..................... 26
Table 4.1: A table of graphics that illustrate part of the spectrum of bridge models
considered in this report. All of the bridges in this table have an aspect ratio of
4.0. ............................................................................................................................ 30
x
LIST OF FIGURES
Page
Figure 1.1: Example of curved bridge alignment and skewed end abutment (left).
Retrieved April 25, 2014 from www.abkj.com. Copyright 2014 ABKJ, Inc............. 1
Figure 1.2: Highly skewed railway bridge. Since the railroad above does not cross the
road below at a perpendicular angle, the skew is necessary. Retrieved May 1,
2014 from www.geograph.org.uk .............................................................................. 2
Figure 1.3: Curved skewed bridge (a) Plan view, (b) Spine model from CBridge, and (c)
Full 3D beam-plate model from CSIBridge of Scales Road Bridge over Slate
Creek in Yuba County, California.............................................................................. 3
Figure 1.4: Truck driving over left lane of bridge. The left girders will bear most of the
shear force that resists this truck. ............................................................................... 4
Figure 1.5: Plan view of three dimensional curved box girder bridge analysis model. The
abutment reaction is monitored at the circled node. ................................................... 4
Figure 1.6: Illustrative example showing the effect of skew angle on abutment reactions
at the obtuse corner for the bridge model shown in Figure 1.5. ................................. 5
Figure 1.7: Idealization of a simply supported bridge as a beam with pin and roller
boundary conditions. .................................................................................................. 6
Figure 1.8: Idealization of a simply supported bridge as a plate with edge pin boundary
conditions. In a 3D bridge model, usually the edge pin would be broken up into
multiple conventional pin supports place at the bearing locations. ............................ 6
Figure 1.9: Curved skewed bridge model showing inside, outside, obtuse, and acute
corners ........................................................................................................................ 7
Figure 1.10: A (a) curved box-girder bridge model and a straight box-girder bridge model .. 11
xi
Figure 1.11 A curved skewed bridge model with inside, outside, obtuse, and acute
corners labeled.......................................................................................................... 12
Figure 1.12 Parallelogram with skew angle labeled. .............................................................. 13
Figure 3.1: An example of a typical curve for this study. The aspect ratio and bearing
stiffness are shown on the top. The obtuse abutment reaction percentage (Ro) is
plotted against skew from 0 to 60 degree skew.
A design curve which
represents the spine model approximation with LRFD modification factors
applied is shown with a dotted line and the rest of the curves correspond to
different central angles. The curve that represents a straight bridge (i.e. central
angle 0) contains markers. ........................................................................................ 20
Figure 3.2: 3D view of typical box girder model with supports at the end of each girder. ... 23
Figure 3.3: Cross section view of typical box girder with no overhangs. Bridge models
with overhangs would simply have an extra plate element on either side of the
top of the cross section. ............................................................................................ 24
Figure 3.4: Plan view of typical box girder model with 20 degree skew and no curve. .......... 25
Figure 4.1: A bridge model with obtuse corner circled. This shows the reaction point
under consideration in this section of the report. If this bridge were resting on
bents or abutments, the bearings would push up on the bridge with a force equal
and opposite to the bridge weight. The obtuse corner reaction is the force that
the bearing sitting under the obtuse corner of the bridge must exert to hold the
bridge up................................................................................................................... 27
Figure 4.2: Obtuse Corner Reaction Percentage (Ro) vs. Abutment Skew for central
angles between -36 and 36 degrees.
xii
The curve with the circle markers
represents the non-curved bridge (central angle = 0) and the dashed line
represents the spine model results with Caltrans modification factors applied
(see Section 3.1.2 to clarify plots). ........................................................................... 31
Figure 4.3: Zoomed in screenshot showing the difference between the curve response of
bridges with positive and negative central angle...................................................... 32
Figure 4.4: Obtuse Corner Reaction Percentage (Ro) vs. Abutment Skew for central
angles between -36 and 36 degrees.
The curve with the circle markers
represents the non-curved bridge (central angle = 0) and the dashed line
represents the spine model results with Caltrans modification factors applied
(see Section 3.1.2 to clarify plots). ........................................................................... 34
Figure 4.5: Extreme case with central angle 36 degrees and skew 50 degrees to illustrate
the close proximity of the obtuse corner to the bridge center of gravity. ................. 35
Figure 4.6: Obtuse Corner Reaction Percentage (Ro) vs. Abutment Skew for central
angles between -36 and 36 degrees.
The curve with the circle markers
represents the non-curved bridge (central angle = 0) and the dashed line
represents the spine model results with Caltrans modification factors applied
(see Section 3.1.2 to clarify plots). ........................................................................... 37
Figure 4.7: Obtuse Corner Reaction Percentage (Ro) vs. Abutment Skew for central
angles between -36 and 36 degrees.
The curve with the circle markers
represents the non-curved bridge (central angle = 0) and the dashed line
represents the spine model results with LRFD modification factors applied (see
Section 3.1.2 to clarify plots). .................................................................................. 39
xiii
Figure 5.1: A plot showing the skew effect for multiple aspect ratios (AR in the legend).
The data used to create these curves is the same data shown in Chapter 4. The
design curve is plotted in a dashed line .................................................................... 43
Figure 5.2: A plot of skew angle vs. percent of abutment reaction at obtuse corner with
dotted lines showing approximation by proposed formula. ..................................... 44
Figure 5.3: Plot showing the curve effect for various aspect ratios. When the curves
show negative y-axis values, uplift is occurring at the obtuse corner. ..................... 45
Figure 5.4: Figure showing the obtuse inside and obtuse outside corners of a bridge
model. For any skew angle, a positive central angle will produce exactly the
same curve effect on the outside corner as a negative central angle will produce
on the inside corner. ................................................................................................. 46
Figure 5.5: Plot comparing recommended formula with analysis results ............................... 47
Figure 5.6: Comparison of plots of percent abutment reaction at obtuse corner vs.
abutment skew for all four aspect ratios. .................................................................. 49
Figure 5.7: Extreme case with central angle 36 degrees and skew 50 degrees to illustrate
the close proximity of the inside obtuse corner to .......... the bridge center of gravity.
xiv
50
1
CHAPTER 1 : INTRODUCTION
1.1 Motivation
With the increasing rate of urbanization and rapid infrastructure growth, the need for
complex transportation systems has also increased; often leading to road and bridge geometries
with unconventional, non-collinear, configurations. According to the bridge database maintained
by the Caltrans maintenance division, approximately 10,600 of 23,800 bridges in California are
skewed such that the end abutments or bents are not perpendicular to the longitudinal direction of
the bridge deck as shown in Figures 1.1 and 1.2. Many bridges also contain horizontally curved
alignments in addition to the skew. At this time, structural analysis capabilities due to software
development have removed many of the limitations that once existed in curved skewed bridge
analysis and such bridges are becoming much easier to design.
This report discusses the
structural analysis issues that arise as a result of skewed and curved bridge geometries, especially
related to the calculation of girder reaction forces.
Figure 1.1: Example of curved bridge alignment and skewed end abutment (left). Retrieved April
25, 2014 from www.abkj.com. Copyright 2014 ABKJ, Inc
2
Figure 1.2: Highly skewed railway bridge. Since the railroad above does not cross the road
below at a perpendicular angle, the skew is necessary. Retrieved May 1, 2014 from
www.geograph.org.uk
Designing the girders and bearings to support a curved skewed bridge can be a complex
problem since the reaction, shear, and moment demands due to gravity loads will vary across the
girders of a skewed bridge deck. Furthermore, it is difficult to predict the response of a curved
and skewed bridge with the analysis techniques commonly used in bridge design. The most
prevalent bridge design software packages such as CTBridge, CONBOX, CONSPAN, and
VBridge rely on a spine model (example shown in Figure 1.3b); utilizing line elements placed
along the center of the bridge. This treatment is accurate for many applications for bridges with
simple alignments but is unable to capture many load distribution effects of curved and skewed
geometries and may generate unconservative results. A more accurate analysis model of a curved
and/or skewed bridge is a 3D beam-plate model as illustrated in Figure 1.3(c).
3
(a)
Node
End
constraint
3D Plate-Beam Model
2D beam element
(b)
(c)
Figure 1.3: Curved skewed bridge (a) Plan view, (b) Spine model from CBridge, and (c) Full 3D
beam-plate model from CSIBridge of Scales Road Bridge over Slate Creek in Yuba County,
California
To highlight the differences between results from a spine model and a full 3D plate-beam
analysis, consider the example illustrated in Figure 1.4 featuring a truck driving in the left lane
over a highway bridge (Figure 1.4). In the more accurate beam-plate model, the left girder
would have a larger shear force as compared to the right side. However, in a spine model, there
are no actual girders modeled, so the bridge is assumed to have an equal lateral distribution of the
shear force imposed by the truck.
4
Figure 1.4: Truck driving over left lane of bridge. The left girders will bear most of the shear
force that resists this truck.
For a skewed bridge, a full 3D model shows that the reactions at the abutments vary
widely across the abutment. A spine model ignores these effects. To illustrate, the curved bridge
(Central Angle = 11 degrees) below in Figure 1.5 was analyzed to find the vertical reaction at the
right side of the abutment.
Figure 1.5: Plan view of three dimensional curved box girder bridge analysis model. The
abutment reaction is monitored at the circled node.
Abutment Reaction at Obtuse
Corner [kips]
5
1200
1000
800
360% error
600
3D Model
400
Spine Model
200
0
0
20
40
Skew Angle [degrees]
60
Figure 1.6: Illustrative example showing the effect of skew angle on abutment reactions at the
obtuse corner for the bridge model shown in Figure 1.5.
As the curves above show, the spine model that meets the LRFD Section 4 requirements
completely ignores the effect of curve and skew and produces very non-conservative results.1
Thinking about a bridge response that varies from girder to girder due to skew is counterintuitive
to someone taking a statics-based approach because they would assume that rigid statically
determinate bridge would behave uniformly no matter how wide or skewed it is (see Figure 1.7).
1
Of course, the LRFD code mandates modifications to the spine model results that attempt to approximate
the true response. These modifications are detailed later in this report. Furthermore, the reason that these
curves do not have a similar reaction for a skew angle of zero is that the 11 degree central angle is ignored
by the spine model that meets the LRFD code requirements. Then, as skew angle increases, the spine
model reactions do not change but the full 3D model reaction increases; capturing the true 3D response.
6
Figure 1.7: Idealization of a simply supported bridge as a beam with pin and roller boundary
conditions.
The problem with this reasoning is that a bridge, in reality, is not statically determinate as long as
it is supported by multiple bearings on each end (transversely along the width). The bridge is
more accurately modeled as a plate with multiple essential boundary conditions at the edge (see
Figure1.8 ). The loads that are borne by each section along the support width can differ greatly.
Figure 1.8: Idealization of a simply supported bridge as a plate with edge pin boundary
conditions. In a 3D bridge model, usually the edge pin would be broken up into multiple
conventional pin supports place at the bearing locations.
As abutment skew increases, the spine beam idealization becomes less accurate because the
reactions of the bridge will vary across the width.
7
A bridge’s skew angle is not the only aspect of its configuration that affects support
reactions. The curved bridge shown in Figure 1.9 will yield a different response than its
equivalent spine model approximation. The abutment bearings supporting the outside exterior
girder will support a higher percentage of the total dead load as compared to the abutment
bearings supporting the inside exterior girder. The torsion produced by the eccentric bridge
center of gravity is not accurately captured in a spine model. Furthermore, the spine model
ignores the shear due to torsion since there is no out-of-plane dimension to allow the torsion to be
transferred into the girders (i.e. resisted by a shear force couple).2
Outside Obtuse Corner
Outside Acute Corner
Inside Acute Corner
Inside Obtuse Corner
Figure 1.9: Curved skewed bridge model showing inside, outside, obtuse, and acute corners
Although a full 3D modeling procedure is necessary for accurately modeling skewed and curved
bridges, full 3D bridge modeling is inefficient and undesirable for most design purposes because
of the lack of available software, and the difficulty of utilizing full 3D analysis results in the
LRFD Specifications. There are currently no bridge design software packages that contain the
option to use a full 3D model.
2
Another major issue with the spine model approximation is that it may not give accurate results for bridge
bents and columns. Therefore, in bridge design practice, an entirely separate 2D bent model is developed
in a separate software program such as VBent or LEAP RCPier.
8
If a bridge designer wishes to investigate a bridge’s full 3D response, he or she must
either model it from scratch in an FE program such as SAP2000 or use a bridge modeler such as
CSIBridge; which does not have the capability to produce all of the results needed for bridge
design. The latter is usually the bridge designers’ most efficient option. The problem with this
option is that the bridge designer must also know how to interpret the 3D results and apply the
correct LRFD Specifications to these results.
Therefore, AASHTO LRFD Specifications have produced modification factors and
approximate design procedures that allow the designer to continue using a spine approximation
for bridges that are similar to the ideal straight case. The Live Load Distribution Factors (LRFD
4.6.2), Skew Shear Correction Factors (LRFD 4.6.2), Horizontal Curve Limitations (LRFD
4.6.1), and many other modifiers are an attempt to give a reasonable approximation of the true 3D
behavior of a bridge without performing a full 3D analysis.
1.2 Objectives and Scope
In many cases, the bridge designer is faced with a choice between spine model
idealization and the longer process of full 3D analysis. In the vast majority of cases, a spine
idealization is used. When a spine idealization is used, and the full 3D response is not
considered, many effects of skew and curve may slip through the design and analysis process
completely unnoticed. The bridge configurations studied below mimic common bridge design
situations, and purpose to provide information to the designer concerning the consequences of
using the analysis approximations at his or her disposal.
The purpose of this study is to provide structural analysis results for vertical abutment
reaction forces for skewed curved bridges. These results shed light on some of the significant
differences between 3D spine idealization and full 3D bridge modeling. In design, this data is
9
directly applicable to the calculation of bridge bearing forces and girder shear forces. The results
shown in this study were compiled from the finite element analysis of over 800 bridge models.
The report results specifically focus on bridge obtuse corner abutment reactions as they vary with
curve and skew of single span box girder bridges.
1.3 Organization and Outline
Chapter 1 of this report provides an explanation of the motivation for this study. The
beginning of this chapter provides a general discussion of the problem being studied. It also
contains the scope and organization of the report. Relevant terms are defined at the end of the
chapter.
The second chapter discusses the AASHTO LRFD Specification procedure for correcting
the design shear forces in girders of skewed bridges and explains the California Amendments to
the LRFD Specifications. It also presents the LRFD Specification limits for spine model
idealization for horizontally curved bridges and some of the research considered in the
development of these specifications. This chapter also contains a breakdown of some of the main
approximations in current bridge design practice on which this study will comment.
Chapter 3 is a discussion of the details of the structural modeling and analysis that
underlies the conclusions of this report. An explanation, both of the format for the datapresentation and of the plots used to display the analysis information, is found at the beginning of
the chapter. At the end of this chapter, the software and structural modeling assumptions are
detailed.
The results from this study are presented in Chapter 4. These results consist of plots of
the parametric variation of skew and curve for bridges with differing aspect ratios. There are four
individual plots that show the entire spectrum of skew and curve variation for each of the four
10
aspect ratios analyzed by this report. Important comments and observations are found underneath
each of the result plots. These comments explain what the results indicate about the skew effect,
curve effect, coupled skew-curve effect, and the design curve.
A discussion of the analysis findings and an interpretation of the results are found in
Chapter 5. The comments from the previous chapter are discussed in four categories. The skew
effect category contains a discussion of how the obtuse abutment reaction varies with skew angle.
A formula is proposed which takes the aspect ratio into account for determining skew correction.
The curve effect category also proposes a formula that approximates the variation of obtuse
reaction with aspect ratio. The coupled skew-curve effect category discusses how the curve
effect varies with abutment skew and how the skew effect varies with horizontal curve. Lastly,
this chapter contains a discussion of the limitations of the LRFD Specification procedure for
determining skew correction for shear and reactions at the obtuse corner of skewed abutments.
Chapter 6 contains a summary of the conclusions in this report. A section of this chapter
is devoted to summarizing the major points of this research that are directly relevant to bridge
design. At the end of this chapter, additional research is recommended.
1.4 Definitions
Below are the theoretical definitions of key terms as they function in this thesis.
Aspect Ratio (AR) – The ratio between the length of a given bridge and its width.
𝐿
𝐴𝑅 = 𝑊
(1.1)
For example, if a bridge is twice as long as it is wide, it will have an aspect ratio of 2.0.
Central Angle - The angle included between two points along the centerline of a curved
bridge measured from the center of the curve as shown in Figure 4.6.1.2.3-1 of the LRFD
Specifications.
11
Coupled Skew-Curve Effect – The change in the skew effect as the curve is varied or the
change in the curve effect as the abutment skew is varied. Since the variation of bridge
skew is dependent on both the skew angle and on the bridge curvature, a coupled skewcurve effect exists. For example, the reaction at the obtuse corner of a highly curved
bridge (Figure 1.10a) will not increase with skew angle at the same rate as the reaction at
the obtuse corner of a straight bridge (Figure 1.10b).
(a)
(b)
Figure 1.10: A (a) curved box-girder bridge model and a straight box-girder bridge model
Curve Effect – The bridge response variation due to varying the horizontal bridge
curvature. In this study, the bridge central angle was varied from -48 degrees to 48
degrees in order to observe the curve effect on obtuse corner abutment reactions.
Full 3D Analysis – The utilization of the beam-plate model that is detailed in this report.
More generally, the utilization of a modeling technique that captures 3D effects that
cause response variation transversely along the bridge width.
Girder – A structural component whose primary function is to resist loads in flexure and
shear. The webs of box girder bridges are also called girders when combined with the
deck and soffit slabs, which are referred to as the girder flanges.
12
Obtuse Corner – The corner of a skewed bridge where the angle between the
superstructure alignment and the abutment direction is greater than 90 degrees. A bridge
with one skewed abutment will have one obtuse corner, one acute corner, and two
orthogonal corners. A bridge with two skewed abutments will have two obtuse corners
and two acute corners. A curved bridge with two skewed abutments will have two
outside corners and two inside corners.
Outside Obtuse Corner
Outside Acute Corner
Inside Acute Corner
Inside Obtuse Corner
Figure 1.11 A curved skewed bridge model with inside, outside, obtuse, and acute corners
labeled.
Skew Angle– The off-normal angle between a line along the length of an abutment or bent
and the bridge alignment at the point of interest.
13
Skew Angle

Figure 1.12 Parallelogram with skew angle labeled.
Skew Effect – The bridge response variation due to changing the abutment skew angle. In
this study, the skew angle was varied between 0 and 60 degrees in order to observe the
skew effect on the obtuse corner abutment reactions.
14
CHAPTER 2 : PREVIOUS RESEARCH AND STATE OF PRACTICE
2.1 Design Specifications
Currently, the AASHTO LRFD Bridge Design Specifications have adopted some simple
guidelines for the use of spine model idealization in bridge analysis. This provides bridge
designers an alternative to modeling and analyzing every bridge as a full 3D finite element model.
In summary, the LRFD Specification states that bridges with central angles below 12 degrees
may be idealized as straight spine model bridges, and that bridges with central angles less than 34
degrees may be modeled with a curved spine model. For skewed bridges up to 60 degree skew
(45 degrees for curved bridges in California), the skew shear correction factors found in LRFD
Section 4.6.2.2.3c may be applied to shear responses obtained from the bridge analysis model to
approximate the actual skew response. Caltrans has also amended the requirements for boxgirder bridges as explained below. The LRFD Specifications lack guidance on many issues
pertaining to skewed and curved bridges including: abutment reaction calculations, bent reaction
calculations, bearing design, and other related issues.
The section relevant to horizontal bridge curvature states “Horizontally curved concrete
box girders may be designed with straight segments, for central angles up to 12 degrees within
one span unless concerns about other force effects dictate otherwise (4.6.1.2.3)” In other words, a
straight spine model may be used to approximate demand for bridges with central angles below
12 degrees. Furthermore, it says “Horizontally curved nonsegmental concrete box girder bridge
superstructures may be analyzed and designed for global force effects as single-spine beams with
straight segments for central angles up to 34 degrees within one span … unless concerns about
local force effects dictate otherwise.” This conclusion is based on research in NCHRP 620
Development of Design Specifications and Commentary for Horizontally Curved Concrete Box-
15
Girder Bridges by NRV, David Evans and Assoc, and Zocon Consulting Engineers (National
Research Council), and on Live Load Distribution Factors for Concrete Box-Girder Bridges by
Song, Shin-Tai, Y. H. Chai and Hida (Song).
The section of the LRFD Specification that pertains to skew and is relevant to this report
is Section 4.6.2.2.3c, which gives skew correction factors for shear, and the California
Amendments to this section. Below are modified tables for shear correction on skewed bridges
between 0 and 60 degree skew.
Type of Superstructure
Cast-In-Place Concrete Multicell Box
Correction Factor
1.0 + (0.25 +
12 ∗ 𝐿
) ∗ tan⁡(𝜃)
70 ∗ 𝑑
Table 2.1: Adapted from AASHTO LRFD Table 4.6.2.2.3c-1
The variable L is the span length, d is the span depth, and  is the skew angle in degrees. Since
this specification is based on a study from 1975 (Wallace), Caltrans has amended the AASHTO
specification. The Caltrans amendment to AASHTO uses the factors in Table 2.2 below.
Type of Superstructure
Cast-In Place Concrete Multicell Box
Correction Factor
1.0 +
𝜃
50
Table 2.2: Adapted from Caltrans 2014 Amendments to AASHTO LRFD Table 4.6.2.2.3c-1
It is also noteworthy that there are no skew correction factors for slab bridges. A straight
skewed slab bridge is modeled without skew for design purposes. The reason for this is that slab
bridges are assumed to have ample shear capacity to resist extra shear from the skew and there is
16
no shear check for a slab bridge. Since there is no direct guidance on skewed bent or abutment
reactions, sometimes the skew correction factors are also applied to reactions to calculate bearing
forces. This assumption is neither confirmed or denied by the LRFD Specification, since it
simply recommends that skew be considered in the design of bearings. The LRFD Specification
gives no definite guidance on how skew should be considered in bearing design.
Much study has been conducted pertaining to Skewed Multicell Box Girder Bridges and
this has been utilized in the LRFD Specifications. Much study has also been conducted
pertaining to Curved Multicell Box Girder Bridges. However, to the knowledge of the author, no
study has specifically focused on the relationship between skew and curve for box girder bridges.
This study overlaps the previous curve and skew studies in some respects, but approaches the
variation of curve and skew based on bridge configurations that are purposefully relevant to real
design situations.
2.2 Bridge Design Practice
In a typical bridge design situation, a curved skewed bridge will most likely be analyzed
only as a spine model. Current bridge design practice relies on the LRFD modification factors to
approximate the skew or curved response. Some of the major approximations that this introduces
are:
1. A curve less than a central angle of 12 degrees is ignored
2. No skew correction will be applied for skewed slab bridges
3. Skew correction for all bridge types is completely independent of bridge aspect
ratio.
17
4. The skew correction factors are usually assumed to also apply to the reactions at
the bearings since the reactions are usually equal and opposite to the shear at the
end of a girder.
The results presented in this report specifically focus on some of the effects that are missed by
spine idealization.
18
CHAPTER 3 : STRUCTURAL ANALYSIS
3.1 Format
3.1.1 Bridge Configurations
This research looks at the combination between curve and skew in four different major
bridge configurations, i.e. with four different aspect ratios, 1.0, 2.0, 4.0, and 8.0, because the
skew and curve effects were found to vary widely with aspect ratio. First, bridge models were
developed to represent each of these aspect ratios. Then, they were parametrically varied by
changing only the central angle and skew angle.
The bridge configuration variations that were considered are:
Variation
Range
Discretization
Aspect Ratio
1.0 to 8.0
1.0, 2.0, 4.0, and 8.0
Skew Angle
0o to 60o
Increments of 10o
Central Angle
-48o to 48o
Increments of 6o
Table 3.1- Table showing parametric variation of models analyzed in this study
Single-span 4-cell box girder bridges were analyzed for all three of these variations. The bridge
models with different aspect ratios differed slightly in girder spacing, overhang width, slab
thickness and soffit thickness. For bridge models within a specific aspect ratio, there are no
differences except for the two varied parameters (skew and curve angles) listed above. The most
noteworthy variation between the models of different aspect ratios is the difference in the
overhang length. This is because a larger overhang will distribute a higher percentage of the
reactions to the exterior supports. For this study, it should be expected that approximately 20% of
the reaction will go to each support in the non-curved, non-skewed case in each plot. This is
19
because the overhang lengths were chosen to produce this result. The details of each of these
bridge configurations are found in Appendix A.
3.1.2 Plots
Figure 3.1 is a typical plot for this study. The plot is the percent of the abutment reaction
concentrated at the obtuse Corner (Ro) vs. abutment skew for a range of curve central angles (see
Figure 3.1 for an example curve). The abutment skew is on the x-axis, the percent of abutment
reaction is on the y-axis, and the curve legend on the right shows the central angles that each
curve represents in degrees.
20
Aspect Ratio 4.0, Kbearing = 500,000 k/in
140
120
Legend (central angle values in degrees)
o
48 central angle
100
Ro
80
60
40
Design Curve
20
Non-curved case
(i.e. 0o central angle
0
-20 0
-40
20
40
-48o central angle
60
Design
48
42
36
30
24
18
12
6
0
-6
-12
-18
-24
-30
-36
-42
-48
Abutment Skew [deg]
Figure 3.1: An example of a typical curve for this study. The aspect ratio and bearing stiffness
are shown on the top. The obtuse abutment reaction percentage (Ro) is plotted against skew from
0 to 60 degree skew. A design curve which represents the spine model approximation with
LRFD modification factors applied is shown with a dotted line and the rest of the curves
correspond to different central angles. The curve that represents a straight bridge (i.e. central
angle 0) contains markers.
The y-axis value will start near 20% for the non-curved bridge since all models are 4-cell box
girders and are supported in five locations. According to the tributary area method, a straight
non-skewed bridge will distribute approximately 20% of the force to each of the five girders if the
bridge overhang is equal to half the girder spacing.
21
The y-axis value (Ro) is obtained by dividing the reaction force at the obtuse corner by
the sum of all of the reaction forces. These reaction forces are obtained from the bridge model
analysis results. The equation below shows the calculation.
𝑅𝑜 = 100 ∗
𝑅𝑒𝑎𝑐𝑡𝑖𝑜𝑛⁡𝑎𝑡⁡𝑂𝑏𝑡𝑢𝑠𝑒⁡𝐶𝑜𝑟𝑛𝑒𝑟⁡𝑜𝑓⁡𝐴𝑏𝑢𝑡𝑚𝑒𝑛𝑡⁡1
∑ 𝐴𝑙𝑙⁡𝑅𝑒𝑎𝑐𝑡𝑖𝑜𝑛𝑠⁡𝐴𝑡⁡𝐴𝑏𝑢𝑡𝑚𝑒𝑛𝑡⁡1
(3.1)
Ro is the percent of the total abutment reaction that is concentrated at the obtuse corner. The
shape of this curve is the same as the shape of the absolute reaction curve and is similar to the
shape of the skew correction factor curve since they are all related by a constant factor. In order
to compare the y-axis value to the LRFD skew correction factors, one must multiply the y-axis
value by the number of support locations (i.e. the number of bearing pads or the number of
reaction points) and divide by 100.
𝑅
𝑜
𝑆𝑘𝑒𝑤⁡𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛⁡𝐹𝑎𝑐𝑡𝑜𝑟 = #⁡𝑜𝑓⁡𝑆𝑢𝑝𝑝𝑜𝑟𝑡𝑠 ∗ 100
(3.2)
For this study, the number of supports at Abutment 1 is always five since all models are 4-cell
box-girder bridges.
The plot will also contain a dashed curve for the design response. The design response
represents the results from using the spine model approximation with LRFD modifications
applied. The design curve is calculated by the following equation:
𝐷𝑒𝑠𝑖𝑔𝑛⁡𝐶𝑢𝑟𝑣𝑒⁡(𝜃) = 𝑅𝑒𝑎𝑐𝑡𝑖𝑜𝑛⁡𝑎𝑡⁡0𝑜 ⁡𝑆𝑘𝑒𝑤⁡ ∗ ⁡𝐿𝑅𝐹𝐷⁡𝑆𝑘𝑒𝑤⁡𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛⁡𝐹𝑎𝑐𝑡𝑜𝑟(𝜃)
(3.3)
Which, considering the Caltrans equation, is the same as
𝜃
𝐷𝑒𝑠𝑖𝑔𝑛⁡𝐶𝑢𝑟𝑣𝑒(𝜃) = ⁡𝑅𝑒𝑎𝑐𝑡𝑖𝑜𝑛⁡𝑎𝑡⁡0𝑜 𝑆𝑘𝑒𝑤 ∗ (1.0 + 50)
(3.4)
This enables the reader to compare the skew effect with the LRFD design equation prediction.
22
3.2 Software
The modeling for this study was conducted using private software developed by the
author of this report and Zocon Consulting Engineers (Zocon) and owned by Zocon. This
software uses matrix structural analysis based on the stiffness method. Frame elements and plate
elements were used in the 3D models for this study. The frame element is an Euler beam that
accounts for shear deformation. A diagonal boundary spring element was used to model the
boundary conditions. The plate elements defined at the input are broken into constant stress
triangles with membrane and bending degrees of freedom. The only load case that was used was
a self-weight load case. The model results were verified in SAP2000.
3.3 Modeling
The typical skewed single span box girder for this study is shown below.
23
Figure 3.2: 3D view of typical box girder model with supports at the end of each girder.
24
Figure 3.3: Cross section view of typical box girder with no overhangs. Bridge models with
overhangs would simply have an extra plate element on either side of the top of the cross section.
25
Figure 3.4: Plan view of typical box girder model with 20 degree skew and no curve.
This box girder is pinned (bearing element with 500,000 kip/in vertical stiffness) at the end of
each girder as if it were sitting on a bearing upon an abutment seat. The results in the body of this
report are for bridges that were modeled with stiff bearings (500,000 kips/in) to emulate PTFE
bearings since stiffer bearings are growing rapidly in modern usage and they produce a more
extreme skew response. Fabric bearings are softer and, therefore, produce different curve and
skew responses. Appendix B contains a more detailed discussion of the ramifications of bearing
stiffness as they pertain to this study.
No sloping of the exterior girders was used in this study.
26
Diaphragms (3 ft thick) were modeled at both abutments. These were modeled as beam
elements connecting the span end nodes together along the width of the abutment. In order to
obtain uniformity in the analysis results, the diaphragm material was modeled to be the same as
the rest of the superstructure except for the unit weight. The unit weight was chosen to be 0
pounds per cubic foot since diaphragm length varies with skew. Therefore, the increased weight
of the diaphragm as skew increases has no effect on the results.
The box girder was modeled with longitudinal beam elements for webs and plate
elements at the top and bottom of the bridge to model the flanges (see Figure 3.3). These were
joined by rigid links as shown in the cross section view in Figure 3.3. Realistic lengths, widths,
and girder spacings were chosen for all box girder models in order to make them comparable to
design situations (see Appendix A for complete bridge model input data).
There are four major bridge configurations used in this study that differ primarily in
aspect ratio. Each of these four bridge configurations are varied parametrically as is show by
Table 3.2.
Parameter
Range
Increment Size
Abutment Skew
0 – 60 degrees
10 degrees
Span Central Angle
-48 – 48 degrees
6 degrees
Table 3.2: The variations used in this study and extent of variation considered
The exact modeling specifications of each of the 4 bridge configurations are listed in the tables in
Appendix A.
27
CHAPTER 4 : ANALYSIS OF THE DATA
4.1 Introduction to Results
This section provides vertical reaction results for the variation of curve and skew for four
different bridge aspect ratios. The data reported is vertical reaction in the obtuse corner of
skewed curved single-span box-girder bridges as shown in Figure 4.1. After a bridge is
constructed, its ends will sit on a load-bearing device designed to bear the reaction forces of the
bridge. The response on the y-axis of the plots presented below captures the bearing force of the
bearing supporting the obtuse corner of the bridge.
Figure 4.1: A bridge model with obtuse corner circled. This shows the reaction point under
consideration in this section of the report. If this bridge were resting on bents or abutments, the
bearings would push up on the bridge with a force equal and opposite to the bridge weight. The
obtuse corner reaction is the force that the bearing sitting under the obtuse corner of the bridge
must exert to hold the bridge up.
The angle between the bridge alignment and the abutment line is greater than 90 degrees in the
obtuse corner of a bridge. The reaction at the support in the obtuse corner usually increases as the
skew angle increases because this support gets nearer to the bridge center of gravity while the
support at the acute corner is getting further away from the bridge corner. This section presents
28
the responses of bridge models with four different aspect ratios as they are varied in skew angle
and central angle. These results are compared to spine model results that are modified by LRFD
skew correction factors; thus emulating current bridge design practice. The line labeled “Design”
represents this data. For an explanation of how the values on the plots are calculated, see section
3.1.2 of this report; which is titled “Plots”. For interpreting the results correctly, it is helpful to
note that all of the curves below are for 4-cell box girder bridges. Therefore, all of the supports
of a straight bridge will bear about 20% of the reaction force. This will depend on the length of
the bridge overhang.
4.2 Visualization of Bridge Parametric Variation
In order to provide the reader with some aid in visualizing the meaning of the plots
presented, the following graphics are provided for the third considered bridge aspect ratio. The
third bridge aspect ratio is the model with aspect ratio 4.0. The graphics below illustrate the
parametric variation of curve and skew.
29
Bridge Model – Plan View
Description
Aspect Ratio = 4.0
Central Angle = 0o
Skew Angle = 0o
Aspect Ratio = 4.0
Central Angle = -48o
Skew Angle = 0o
Aspect Ratio = 4.0
Central Angle = 0o
Skew Angle = 30o
30
Aspect Ratio = 4.0
Central Angle = 48o
Skew Angle = 30o
Table 4.1: A table of graphics that illustrate part of the spectrum of bridge models considered in
this report. All of the bridges in this table have an aspect ratio of 4.0.
4.3 Results
The results presented in this section provide graphs of obtuse corner reactions for aspect
ratios 1.0, 2.0, 4.0, and 8.0 from the full 3D analysis and from the 3D spine model modified with
LRFD correction factors.
4.3.1 Aspect Ratio = 1.0
A model of a single-span box girder with a length of 60 feet, a width of 60 feet, a girder
spacing of 12.5 feet and an overhang of 5 feet was parametrically varied between skew angles 0
to 40 degrees and central angles -36 to 36 degrees. The curve below shows the reaction vs. skew
relationship for multiple central angles. The reaction value plotted is the percent of the abutment
reaction that is borne by the obtuse corner. This value is obtained by dividing the obtuse corner
reaction by the sum of the reactions at the abutment.
31
Ro
Aspect Ratio 1.0, Kbearing = 500,000 k/in
50
45
40
35
30
25
20
15
10
5
0
Design
36
30
24
18
12
6
0
-6
-12
-18
-24
-30
-36
0
10
20
30
Abutment Skew [deg]
40
Figure 4.2: Obtuse Corner Reaction Percentage (Ro) vs. Abutment Skew for central angles
between -36 and 36 degrees. The curve with the circle markers represents the non-curved bridge
(central angle = 0) and the dashed line represents the spine model results with Caltrans
modification factors applied (see Section 3.1.2 to clarify plots).
Skew Effect
The skew effect is well approximated by the skew correction. This is evidenced by the
fact that the design curve (dashed line) closely follows the curve that represents the non-curved
bridge (circle markers).
32
Curve Effect
As one would expect, as central angle increases, obtuse corner reaction increases. This
results from torsion due to the change in the center of gravity along the width of the bridge. The
curve effect is not negligible for small curvatures (below 12 degrees). The curve effect is more
pronounced for negative central angles than positive central angles. This can be observed by
noticing that the curves below the 0 degree central angle curve are more spread out than the
curves above the 0 degree central angle curve as is shown below.
The Curve Effect is more
pronounced for negative
central angles than for
positive central angles.
The top curve (36o) starts
at 28%, the middle curve
(0o) starts at 18%, and the
bottom curve (-36o) starts
at 6%.
The distance between the
lower curves (18 – 6 =
12) is greater than the
distance between the
upper curves (28 – 18 =
10).
Figure 4.3: Zoomed in screenshot showing the difference between the curve response of bridges
with positive and negative central angle.
Coupled Skew-Curve Effect
The coupled curve-skew effect appears to be negligible since the slope of the curves does
not change much as the central angle changes.
33
Design Curve
The design curve approximates the skew response very well since it follows the skew
response of the non-curved bridge very closely.
4.3.2 Aspect Ratio = 2.0
A model of a single-span box girder with a length of 80 feet, a width of 40 feet, a girder
spacing of 8 feet, and an overhang of 4 feet was parametrically varied between skew angles 0 to
50 degrees and central angles -48 to 48 degrees. The curve below shows the reaction vs. skew
relationship for multiple central angles. The reaction value plotted is the percent of the abutment
reaction that is borne by the obtuse corner. This value is obtained by dividing the obtuse corner
reaction by the sum of the reactions at the abutment.
34
Aspect Ratio 2.0, Kbearing = 500,000 k/in
80
Design
48
42
36
30
24
18
12
6
0
-6
-12
-18
-24
-30
-36
-42
-48
70
60
Ro
50
40
30
20
10
0
-10 0
20
40
Abutment Skew [deg]
60
Figure 4.4: Obtuse Corner Reaction Percentage (Ro) vs. Abutment Skew for central angles
between -36 and 36 degrees. The curve with the circle markers represents the non-curved bridge
(central angle = 0) and the dashed line represents the spine model results with Caltrans
modification factors applied (see Section 3.1.2 to clarify plots).
Skew Effect
The skew effect is more severe than the bridges with 1.0 aspect ratio. This is shown by
the difference in slope between the plots of aspect ratio 1.0 bridges and aspect ratio 2.0. The
obtuse corner reaction of the non-curved model reaches 40% at about 25o skew angle whereas the
non-curved bridge with aspect ratio 1.0 never reached 40% obtuse corner reaction percent. The
skew effect also develops a peak around 40o skew angle
35
Curve Effect
The curve effect is greatly influenced by the increase in aspect ratio. This is shown by
the fact that uplift occurs in the obtuse corner at -42o central angle. The curve effect is nonnegligible for small central angles. As with the bridges with aspect ratio 1.0, the curve effect is
greater for negative central angles than for positive central angles.
Coupled Skew-Curve Effect
For bridges with high negative central angle and high skew, the coupled skew-curve
effect is pronounced. This is seen by noticing that the curve for -48 degree central angle crosses
the curves next to it. This is most likely because the obtuse corner at the first abutment becomes
much closer the center of gravity than all other corners as shown in the figure below.
Figure 4.5: Extreme case with central angle 36 degrees and skew 50 degrees to illustrate the close
proximity of the obtuse corner to the bridge center of gravity.
For bridges with positive central angle, the coupled skew-curve effect is minimal since
the slope of the curves does not vary much with changing central angle.
Design Curve
The skew effect is not well predicted for aspect ratio 2.0. As an important note, the
design curve did not change at all as the aspect ratio changed since skew correction factors are
36
not dependant on aspect ratio. At an aspect ratio of 2.0, the skew response of the non-curved
bridge is higher than the design curve shows. The design curve prediction is non-conservative in
this case.
37
4.3.3 Aspect Ratio = 4.0
A model of a single-span box girder with a length of 160 feet, a width of 40 feet, a girder
spacing of 8 feet, and an overhang of 4 feet was parametrically varied between skew angles 0 to
60 degrees and central angles -48 to 48 degrees. The curve below shows the reaction vs. skew
relationship for multiple central angles. The reaction value plotted is the percent of the abutment
reaction that is borne by the obtuse corner. This value is obtained by dividing the obtuse corner
reaction by the sum of the reactions at the abutment.
Aspect Ratio 4.0, Kbearing = 500,000 k/in
140
120
100
Ro
80
60
40
20
0
-20 0
-40
20
40
60
Design
48
42
36
30
24
18
12
6
0
-6
-12
-18
-24
-30
-36
-42
-48
Abutment Skew [deg]
Figure 4.6: Obtuse Corner Reaction Percentage (Ro) vs. Abutment Skew for central angles
between -36 and 36 degrees. The curve with the circle markers represents the non-curved bridge
(central angle = 0) and the dashed line represents the spine model results with Caltrans
modification factors applied (see Section 3.1.2 to clarify plots).
38
Skew Effect
At an aspect ratio of 4.0, the skew effect is much more severe than at lower aspect ratios.
This can be seen by the fact that percent of abutment reaction in the obtuse corner reaches 40% at
a skew angle of 15o. The skew effect reaches a peak at about 40 degree skew.
Curve Effect
The curve effect is also more severe than at lower aspect ratios. This is seen in the fact
that uplift occurs at a central angle of 30o in the non-skewed case. The curve effect is not
negligible at small central angles. As with the bridges with aspect ratios 1.0 and 2.0, the curve
effect is greater for negative central angles than for positive central angles.
Coupled Skew-Curve Effect
For bridges with high negative central angle and high skew, the coupled skew-curve
effect is more pronounced than in lower central angles. This is the same effect observed and
discussed in the previous pages for aspect ratio 2.0. For bridges with a positive central angle, the
coupled skew-curve effect is minimal. This is evidenced by the fact that the slope of the curves
does not vary much with the changing central angle.
Design Curve
The design curve does not accurately approximate the 3D response for aspect ratio 4.0.
At only 25 degree skew, the non-curved model response is 50% higher than the design
approximation. The design curve is highly non-conservative for most cases in this aspect ratio.
39
4.3.4 Aspect Ratio = 8.0
A model of a single-span box girder with a length of 200 feet, a width of 25 feet, a girder
spacing of 5.5 feet, and an overhang of 1.5 feet was parametrically varied between skew angles 0
to 60 degrees and central angles -48 to 48 degrees. Below is a screenshot of an example model.
The curve below shows the reaction vs. skew relationship for multiple central angles. The
reaction value plotted is the percent of the abutment reaction that is borne by the obtuse corner.
This value is obtained by dividing the obtuse corner reaction by the sum of the reactions at the
abutment.
Aspect Ratio 8.0, Kbearing = 500,000 k/in
250
200
Ro
150
100
50
0
-50
-100
0
20
40
60
Design
48
42
36
30
24
18
12
6
0
-6
-12
-18
-24
-30
-36
-42
-48
Abutment Skew [deg]
Figure 4.7: Obtuse Corner Reaction Percentage (Ro) vs. Abutment Skew for central angles
between -36 and 36 degrees. The curve with the circle markers represents the non-curved bridge
40
(central angle = 0) and the dashed line represents the spine model results with LRFD modification
factors applied (see Section 3.1.2 to clarify plots).
Skew Effect
At this aspect ratio, the skew effect observed is more severe than in lower aspect ratios.
The non-curved case reaches 40% obtuse corner reaction at about 8o skew angle. At 40 degree
skew, the obtuse corner is bearing over 100% of the load, which means that the sum of the other
reactions is a negative number. This suggests that a large amount of uplift is occurring in the
acute corner and at the inside obtuse support. The skew effect peaks at about 45o skew.
Curve Effect
The curve effect is much more severe than it was at lower aspect ratios and is not
negligible for all central angles. This can be seen by observing that uplift occurs in the obtuse
corner at only -24o central angle. As with the bridges with aspect ratios 1.0, 2.0, and 4.0, the
curve effect is greater for negative central angles than for positive central angles.
Coupled Skew-Curve Effect
The coupled skew-curve effect is pronounced for all central angles. This is can be seen
by noticing the varying slopes of the curves they fan away from each other and draw near to each
other depending on the skew and central angle. For positive central angles with skew between 0
and 40 degrees, the bridge curvature augments the response. This is evidenced by the slight
spreading out of the plot curves for positive central angle from 0 to 40 degree skew. After 40
degree skew, as the plot curves slope downwards, the bridge curvature causes the curves to come
together. For negative central angles, the coupled skew-curve response is similar to the response
at aspect ratio 4.0. The coupled response has decreased in severity from the response at aspect
ratio 4.0 since the curve for central angle -48 degrees intersects the 0 central angle curve at about
60 degree skew, whereas it crossed around 57 degree skew for the bridge with aspect ratio 4.0.
41
Design Curve
The design curve does not provide a good approximation for the response at this aspect
ratio. Since the skew correction factors are not dependant on aspect ratio, they do not capture the
increase in the skew effect at higher aspect ratios.
4.3.5 General Observations
Skew Effect
The skew effect increases as aspect ratio increases. For large aspect ratios, the reaction
vs. skew curves peak somewhere between 35 and 45 degrees.
Curve Effect
The curve effect increases as aspect ratio increases. The curve effect is very significant at
all aspect ratios. The curve effect is not negligible for central angles less than 12 o
Coupled Skew-Curve Effect
The coupled skew-curve effect is noticeable in both low and high aspect ratios but is not
very significant in models with skew angles less than 35 degrees. It is almost unnoticeable in
models with a positive central angle but is very noticeable in models with negative central angles.
The coupled skew-curve effect does not necessarily increase with aspect ratio.
Design Curve
The design curve, which reflects the spine model results with an applied skew correction
factor, nearly approximates the skew effect for aspect ratios near 1.0. However, for aspect ratios
of 2.0 and higher, the design curve predictions are non-conservative. For high aspect ratios, the
design curve does not reasonably approximate the obtuse corner reaction for any skew angle
above 10 degrees.
42
CHAPTER 5 : FINDINGS AND INTERPRETATIONS
5.1 General
According to the analysis of the bridge configurations above, aspect ratio plays a
prominent role in the behavior of the skew effect, the curve effect, the skew-curve effect, and the
accuracy of the design curve. For bridges with aspect ratios greater than 1.0, the 3D spine
analysis with LRFD modifications may yield non-conservative results since the full 3D response
is difficult to predict. Stiffer bearings such as PTFE bearings will most likely give a similar
bridge response to the curves above whereas the skew and curve responses will change if lessstiff bearings are used. The coupled skew-curve effect depends on aspect ratio and is not
necessarily increased as aspect ratio is increased. For this study, the coupled skew-curve effect
was observed most prominently when aspect ratio was 4.0. However, this effect is very small at
low skew angles (below 35 degrees) and low central angles (below 30 degrees).
5.2 Discussion
The values of the reactions at the obtuse corners of the bridges analyzed above are clearly
affected by skew angle, central angle, and some coupling that occurs when the two are combined
together.
Skew Effect
The skew effect increases significantly as aspect ratio increases. Based on the results
presented in Chapter 4, the skew effect can be isolated by only considering the non-curved data
for all aspect ratios. The plot below shows this comparison.
43
140
Skew Effect for Varying Aspect Ratios
120
Design
AR = 1.0
80
AR = 4.0
Ro
100
AR = 2.0
AR = 8.0
60
40
20
0
0
20
40
Abutment Skew [deg]
60
Figure 5.1: A plot showing the skew effect for multiple aspect ratios (AR in the legend). The data
used to create these curves is the same data shown in Chapter 4. The design curve is plotted in a
dashed line
As is shown in Figure 5.1, the skew effect is clearly dependent on aspect ratio and the design
curve does not provide a sound prediction at high aspect ratios.
Neither the LRFD Specifications nor the Caltrans Amendments provide equations
specifically for approximating reactions. In general, the skew shear correction factors are thought
to apply to reactions as well as to shear. The LRFD equation that applies to the box-girder bridges
analyzed in this report is the Caltrans correction factor shown below.
𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛⁡𝐹𝑎𝑐𝑡𝑜𝑟 = 1.0 +
𝜃
50
(5.1)
44
The variable  is the skew angle in degrees. This study has found that the skew effect cannot be
accurately predicted based solely on the skew angle itself. The results of this study suggest that
the skew correction equation grows steadily less accurate as aspect ratio increases.
A simple equation that would better approximate this effect by adding dependence on
aspect ratio is:
𝑆𝑘𝑒𝑤⁡𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛⁡𝐹𝑎𝑐𝑡𝑜𝑟 = 1.0 +
𝜃
∗
50
𝐴𝑅 0.9
(5.2)
where  is the skew angle and AR is the aspect ratio (Length/Width). Below is a plot showing
how this equation approximates the true response for three of the four aspect ratios considered in
this study.
120
Proposed Correction Formulas
Compared to 3D Model Response
AR = 1.0
AR = 2.0
100
AR = 4.0
80
Pred: AR = 1.0
Ro
Pred: AR = 2.0
60
Pred: AR = 4.0
40
20
0
0
10
20
30
Abutment Skew [deg]
40
Figure 5.2: A plot of skew angle vs. percent of abutment reaction at obtuse corner with dotted
lines showing approximation by proposed formula.
45
The equation fits the data very closely for aspect ratios between 1.0 and 4.0 from skew angles 0 to
40 degrees. The equation approximation is too conservative for the aspect ratio 8.0 and for skew
angles above 40 degrees. Therefore, this equation is recommended to approximate the skew
effect for aspect ratios between 1.0 and 4.0 from skew angles 0o to 40 o.
Curve Effect
The curve effect increases as aspect ratio increases. Based on the results presented in
Chapter 4, the curve effect can be isolated by only considering the non-skewed data for all aspect
ratios. Below is a plot that shows the effect of aspect ratio for non-skewed box girder bridges.
Curve Effect for Varying Aspect Ratios
80
AR = 1.0
60
AR = 2.0
AR = 4.0
40
AR = 8.0
Ro
20
0
-50
-30
-10
-20
10
30
-40
-60
Central Angle [deg]
Figure 5.3: Plot showing the curve effect for various aspect ratios. When the curves show
negative y-axis values, uplift is occurring at the obtuse corner.
50
46
This plot shows that the curve effect is greatly dependant on aspect ratio.
The LRFD Specification allows curved bridges with central angles below 12 degrees to
be idealized as straight spine models. The results from this study show that, at aspect ratios of 1.0
and above, small central angles can produce an increase in the reaction in the obtuse corner that is
not negligible.
Furthermore, the effect of horizontal alignment curvature is different for negative and
positive central angles. This is easier to think about in terms of “inside corner” and “outside
corner.”
Figure 5.4: Figure showing the obtuse inside and obtuse outside corners of a bridge model. For
any skew angle, a positive central angle will produce exactly the same curve effect on the outside
corner as a negative central angle will produce on the inside corner.
The higher the central angle, the more the reaction forces will shift to the outside corner and the
less reaction force will be resisted at the inside corner. This happens because the center of mass of
the bridge is closer to the outside corner. A bridge with this configuration requires a force couple
at each end to provide torsion resistance. This accounts for the difference in the curve effect
between negative and positive central angles.
47
Therefore, two equations are proposed to approximate this effect.
𝛼
𝐶𝑢𝑟𝑣𝑒⁡𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛⁡𝐹𝑎𝑐𝑡𝑜𝑟⁡𝑎𝑡⁡𝑂𝑢𝑡𝑠𝑖𝑑𝑒⁡𝐶𝑜𝑟𝑛𝑒𝑟 = 1.0 + 100 ∗ 𝐴𝑅
𝛼
𝐶𝑢𝑟𝑣𝑒⁡𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛⁡𝐹𝑎𝑐𝑡𝑜𝑟⁡𝑎𝑡⁡𝐼𝑛𝑠𝑖𝑑𝑒⁡𝐶𝑜𝑟𝑛𝑒𝑟 = 1.0 + ⁡ 50 ∗ 𝐴𝑅 0.4
(5.3)
(5.4)
The variable  is the central angle and AR is the aspect ratio. Below is a plot showing the
correlation between the proposed formula and the data from this study.
Proposed Correction Formulas
Compared to 3D Model Response
AR = 1.0
AR = 2.0
AR = 4.0
Pred: AR = 1.0
Ro
Pred: AR = 2.0
-35
Pred: AR = 4.0
60
50
40
30
20
10
0
-15
5
-10
-20
-30
Central Angle [deg]
25
Figure 5.5: Plot comparing recommended formula with analysis results
The variation of the curve effect with aspect ratio is more difficult to capture with a simple
empirical formula as compared to the skew effect. This formula was calibrated for the inside and
48
outside girders of bridges with central angles between 0 and 30 degrees and aspect ratios between
1.0 and 4.0.
Coupled Skew-Curve Effect
The influence of the coupled skew-curve effect on reactions in the obtuse corner is
negligible at skew angles less than 35 degrees. Furthermore, it is negligible for all skew angles
below 60 degrees at the outside corner of curved bridges with aspect ratios below 4.0. Therefore,
no equation is developed in this report that relates these effects. For aspect ratios between 1.0
and 4.0, skew angles between 0o and 40o, and central angles between 0o and 30o, the additive
formula below is proposed to predict the response of a skewed curved box girder bridge from 3D
spine model results.
𝜃
𝛼
𝑆𝑘𝑒𝑤𝐶𝑢𝑟𝑣𝑒⁡𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛⁡𝐹𝑎𝑐𝑡𝑜𝑟 = 1 + 50 ∗ 𝐴𝑅 0.9 + 100 ∗ 𝐴𝑅⁡
𝑆𝑘𝑒𝑤𝐶𝑢𝑟𝑣𝑒⁡𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛⁡𝐹𝑎𝑐𝑡𝑜𝑟 = 1 +
𝜃
∗
50
𝐴𝑅 0.9 +
𝛼
50
∗ 𝐴𝑅 0.4
(5.5)
(5.6)
The variable  is the central angle,  is the skew angle, and AR is the aspect ratio.
The primary situation where the coupled skew-curve effect is significant is at the inside
obtuse corner. The plots from Chapter 4 are shown side by side in the figure below. The lower
curves (negative central angle) at high skews show a significant coupled skew-curve effect.
49
Aspect Ratio 2.0, Kbearing = 500,000 k/in
80
Design
30
60
24
18
50
12
6
0
-6
-12
-30
0
-36
20
30
Abutment Skew [deg]
-10 0
40
Aspect Ratio 4.0, Kbearing = 500,000 k/in
120
100
Ro
80
60
40
20
0
-20 0
-40
20
40
Abutment Skew [deg]
60
20
40
Abutment Skew [deg]
60
Aspect Ratio 8.0, Kbearing = 500,000 k/in
Design
48
42
36
30
24
18
12
6
0
-6
-12
-18
-24
-30
-36
-42
-48
250
200
150
Ro
140
30
10
-24
10
40
20
-18
0
Design
48
42
36
30
24
18
12
6
0
-6
-12
-18
-24
-30
-36
-42
-48
70
36
Ro
Ro
Aspect Ratio 1.0, Kbearing = 500,000 k/in
50
45
40
35
30
25
20
15
10
5
0
100
50
0
-50
0
-100
20
40
60
Design
48
42
36
30
24
18
12
6
0
-6
-12
-18
-24
-30
-36
-42
-48
Abutment Skew [deg]
Figure 5.6: Comparison of plots of percent abutment reaction at obtuse corner vs. abutment skew
for all four aspect ratios.
The reason for the significant coupled skew-curve effect at negative central angles is that
the inside obtuse corner becomes much closer the center of gravity than all other corners as
shown in the figure below.
50
Figure 5.7: Extreme case with central angle 36 degrees and skew 50 degrees to illustrate the close
proximity of the inside obtuse corner to the bridge center of gravity.
The bridge in Figure 5.7 is an extreme case that was developed purely to exemplify the case
when the inside obtuse corner is near the bridge center. As this shows, the skew angle brings the
obtuse corner near to the center of mass of a curved bridge. Therefore, a coupling of the skew
and curve effects is noticeable in the inside obtuse corner.
Design Curve
The design curve shown in the plots was an application of the LRFD skew correction
factors with Caltrans Amendment modifications. The skew correction factors were developed
from previous research that was specifically for shear response. These skew correction factors
depend solely on the bridge type and the skew angle and are not affected by aspect ratio.
For low aspect ratios, the design curve approximates the reactions fairly well for a noncurved case. However, for aspect ratios greater than 1.0, these results show that the design curve
does not accurately predict the reactions and gives a non-conservative estimate. This report
proposes that aspect ratio be considered in the design of skewed curved bridges.
51
CHAPTER 6 : CONCLUSION
6.1 Summary of Findings
The most noteworthy conclusion of this report is that as bridge aspect ratio (length
divided by width) increases, the skew effect and the curve effect are significantly increased as
well. For high aspect ratio bridges, a 3D spine model analysis with LRFD skew correction
factors applied yielded non-conservative results for obtuse corner abutment reaction forces.
Furthermore, for aspect ratios between 2.0 and 8.0, the coupled skew-curve effect is significant in
inside obtuse corners of curved skewed bridges. Moreover, a reduction in bearing stiffness
caused the skew effect to decrease. Lastly, curved box girder bridges with small central angles
(less than 12o) still exhibited significant increases in reactions at the outside corners and
significant decreases in reactions at the inside corners.
6.2 Design Applications
6.2.1 Aspect Ratio Dependency
The data found in this thesis suggests that skewed or curved bridges being designed with
aspect ratios greater than 1.0 should be investigated for skew and curve effects on shear demand
and bearing demand. A simple 3D spine model analysis with LRFD shear correction factors
applied may yield highly non-conservative results. Therefore, a full 3D model such as a beamplate model should be built to accurately predict shear and reaction forces in bridge members.
The formulas for skew and curve correction proposed in this report highlight the importance of
aspect ratio in determining the curve and skew effects. These may serve as a basic estimate of the
affect of aspect ratio on skew and curve effects. However, since these equations were empirically
developed for a limited range of bridge models, they are not recommended for design.
52
6.2.2 Bearing Stiffness
In bridge design, bearing stiffness is highly dependent on bearing type. Plain elastomeric
bearing pads are much softer than PTFE bearings and the skew and curve responses of bridges
designed with these two bearing types should be expected to vary significantly. The study of the
effect of softer bearings found in Appendix B yields the following conclusion. Bridges with
softer bearing pads exhibit a less severe skew effect. See Appendix B and Section 3.2.1 of this
report for details.
6.3 Further Research
This study can be expanded to further quantify other issues not extensively studied here.
These include:

Other bridge configurations such as multi-span bridges

Moment response in girders of curved bridges

Laboratory and field studies

Bearing stiffness and arrangement
This thesis highlights the major role that aspect ratio plays in determining the skew
effect, curve effect, and coupled skew-curve effect on vertical abutment reaction forces. This
study, however, used a limited range of bridge configurations and types. The effect of skew and
horizontal curvature on abutment reactions of multi-span bridges should be explored. This would
provide relevant data for bridges with bents. Single-column and multi-column bents should be
studied since the transverse response of a single-column bent is different from a multi column
bent. The data presented in this study focuses on 4-cell box girder bridges. The conclusions
would be strengthened by the analysis of 1-cell and 2-cell box girder bridges as well as wide
bridges with cell numbers greater than six. Since this study focused on aspect ratios between 1.0
53
and 8.0, the data could be filled out if aspect ratios of 0.3 to 1.0 were investigated. Other bridge
types including slab bridges and beam-slab bridges should also be investigated.
Moment response in some girders of skewed bridges is generally thought to be less
severe than in non-skewed bridges. All girders are usually designed for maximum moment. This
practice is considered to be conservative. This, however, is not the case for curved bridges. The
outside exterior girder will carry higher moment than the inside exterior girder since it is longer.
The moment in the outside exterior girder will also be increased due to the torsion produced by
the curve effect as presented in this thesis. Therefore, investigation of the effects of skew and
horizontal curvature on girder moment would be useful. Furthermore, the effect of differential
prestressing to counteract differing girder moments is a topic which deserves additional
consideration because of the many complications that arise due to constructability, camber, and
elastic shortening.
Since all of the data from this study was gathered from computer models, field testing
and laboratory verification would be helpful. A significant body of data currently exists, so a
study of existing curved and skewed bridges which highlights the effects of aspect ratio on
abutment reactions would refine the conclusions presented in this study.
This study did not consider a large range of bearing stiffnesses. The data presented in
Chapter 4 reports results for bridges modeled with stiff bearing elements (500,000 kips/in) and
the data presented in Appendix B compares these results to results for bridges modeled with soft
bearing elements (5,000 kips/in). This range of bearing stiffnesses is not sufficient to formulate
strong conclusions on the effects of bearing stiffness. Furthermore, the effects of bearing
arrangement should also be investigated. Caltrans Memo to Designers 7-1 proposes that
abutment skew be considered in bearing design and that bearing spacing may be varied to
accommodate the distribution of vertical abutment reactions along the bridge width. This,
54
however, introduces differential stiffness across the bridge width and may cause an increase in
the skew effect since the some of the shear force will shift to the stiffer side of the bridge. In
design, abutment bearings may also be shifted toward the obtuse corner so that there is a wider
distance between the edge bearing and the acute corner than the distance between the opposite
edge bearing and the obtuse corner. This is not a common design practice, but it may be
beneficial in preventing a skewed bridge from lifting off the bearing as a result of acute corner
uplift.
55
APPENDIX A: MODELING DETAILS
This appendix contains the details for the modeling of the four major bridge
configurations used in this report. The analysis results obtained above can be verified
independently with the information below. The bridge configurations fall into four major
categories based on aspect ratio. The skew angles, central angles, and bearing stiffnesses were
varied to produce the analysis results presented in this report.
56
Model 1: Aspect Ratio 1.0
57
Model 2: Aspect Ratio 2.0
58
Model 3: Aspect Ratio 4.0
59
Model 4: Aspect Ratio 8.0
60
APPENDIX B: EFFECT OF SOFTER BEARINGS
This appendix presents the same bridge data that was presented in Chapter 4 except that
the bridge bearings were modeled to be softer. In Chapter 4, the bearings were modeled with a
stiffness of 500,000 kips/in. This was taken to correspond to the stiffness of stiff bearings used in
bridge design such as PTFE bearings. All the bridges in Chapter 4 were also analyzed with soft
bearings with a stiffness of 5,000 kips/in. This stiffness was taken to correspond to the stiffness
of soft bearings such as plain elastomeric bearings.
Data From Chapter 4
Soft Bearings
Design
36
30
24
18
12
6
0
-6
-12
-18
-24
-30
-36
0
10
20
30
Abutment Skew [deg]
% Abut. Rxn. @ Obtuse Corner
% Abut. Rxn. @ Obtuse Corner
Aspect Ratio 1.0, Kbearing = 500,000 k/in
50
45
40
35
30
25
20
15
10
5
0
50
40
30
20
10
0
-10 0
20
40
Abutment Skew [deg]
30
24
18
12
6
0
-6
-12
-18
-24
-30
-36
10
20
30
Abutment Skew [deg]
40
Aspect Ratio 2.0, Kbearing = 5,000 k/in
60
% Abut. Rxn. @ Obtuse Corner
% Abut. Rxn. @ Obtuse Corner
60
36
0
Design
48
42
36
30
24
18
12
6
0
-6
-12
-18
-24
-30
-36
-42
-48
70
Design
40
Aspect Ratio 2.0, Kbearing = 500,000 k/in
80
Aspect Ratio 1.0, Kbearing = 5,000 k/in
45
40
35
30
25
20
15
10
5
0
70
60
50
40
30
20
10
0
-10 0
20
40
Abutment Skew [deg]
Design
48
42
36
30
24
18
12
6
0
-6
-12
-18
-24
-30
-36
-42
-48
60
61
Aspect Ratio 4.0, Kbearing = 5,000 k/in
Aspect Ratio 4.0, Kbearing = 500,000 k/in
120
100
80
60
40
20
0
-20 0
20
40
60
Design
48
42
36
30
24
18
12
6
0
-6
-12
-18
-24
-30
-36
-42
-48
100
% Abut. Rxn. @ Obtuse Corner
% Abut. Rxn. @ Obtuse Corner
140
20
0
0
200
150
100
50
0
0
20
40
Abutment Skew [deg]
60
20
40
60
Abutment Skew [deg]
Aspect Ratio 8.0, Kbearing = 5,000 k/in
Design
48
42
36
30
24
18
12
6
0
-6
-12
-18
-24
-30
-36
-42
-48
% of Abut. Rxn. @ Obtuse Corner
Aspect Ratio 8.0, Kbearing = 500,000 k/in
% of Abut. Rxn. @ Obtuse Corner
40
Abutment Skew [deg]
250
-100
60
-20
-40
-50
80
Design
48
42
36
30
24
18
12
6
0
-6
-12
-18
-24
-30
-36
-42
-48
140
120
100
80
60
40
20
0
-20 0
-40
-60
20
40
60
Design
48
42
36
30
24
18
12
6
0
-6
-12
-18
-24
-30
-36
-42
-48
Abutment Skew [deg]
As can be easily discerned from the plots above, the skew effect is decreased with softer bearings.
62
APPENDIX C: EFFECT OF SPAN LENGTH
Some studies have suggested that span length alone (not aspect ratio) determines the
skew effect. A cursory investigation was performed to explore the effect of span length.
Below is a comparison of results from this variation.
L = 80’, W = 40’, D/S = 0.05
L = 80’, W = 80’, D/S = 0.05
70
60
50
40
30
20
10
0
-10 0
D/S=0.05 , AR = 1.0
Design
48
42
36
30
24
18
12
6
0
-6
-12
-18
-24
-30
-36
-42
-48
20
40
Abutment Skew [deg]
60
% Abut. Rxn. @ Obtuse Corner
% Abut. Rxn. @ Obtuse Corner
D/S = 0.05, AR = 2.0
80
45
40
35
30
25
20
15
10
5
0
Design
36
30
24
18
12
6
0
-6
-12
-18
-24
-30
-36
0
10
20
30
Abutment Skew [deg]
40
L=160’, W = 80’ D/S = 0.025
% Abut. Rxn. @ Obtuse Corner
D/S = 0.025, AR = 2.0
70
60
50
40
30
20
10
0
-10 0
20
40
Abutment Skew [deg]
Design
48
42
36
30
24
18
12
6
0
-6
-12
-18
-24
-30
-36
-42
-48
60
In these curves, D/S represents cross section depth divided by span length. From the data above,
we see that aspect ratio has more effect than span length on determining the severity of the skew
effect, curve effect, and coupled skew-curve effect. It is also noteworthy that the effect of width
63
(with span length held constant) and depth to span ratio do not affect the skew effect as much as
aspect ratio.
64
REFERENCES
AASHTO (2012), AASHTO LRFD Bridge Design Specifications, 6th Edition with Interims,
American Association of State Highway and Transportation Officials, Washington, D.C.
National Research Council. NCHRP Report 620: Development of Design Specifications and
Commentary for Horizontally Curved Concrete Box-Girder Bridges. Washington, DC:
The National Academies Press, 2009.
Song, Shin-Tai, Y. H. Chai and Susan E. Hida (2003), Live Load Distribution Factors for
Concrete Box-Girder Bridges, Journal of Bridge Engineering, ASCE, Vol. 8, No. 5, pp.
273-280.
Wallace, Mark (1975), Skewed Concrete Box-Girder Parameter Studies, California Department
of Transportation.
Zokaie, Toorak, Mish, K. D., and Imbsen, R. A. (1993) Distribution of Wheel Loads on Highway
Bridges, Phase 3, Final Report to NCHRP Project 12-26 (2).
Zokaie, Toorak, M.ASCE. (2000) AASHTO-LRFD Live Load Distribution Specifications, Journal
of Bridge Engineering, Vol. 5, No. 2, May 2000, pp. 131-138