BEHAVIOR AND ANALYSIS OF

HIGHLY SKEWED STEEL I-GIRDER BRIDGES

by

KONLEE BAXTER DOBBINS

B.S., University of Virginia, 2001

A thesis submitted to the

Faculty of the Graduate School of the

University of Colorado in partial fulfillment

of the requirements for the degree of

Master of Science

Civil Engineering

2013

ii

This thesis for the Master of Science degree by

Konlee Baxter Dobbins

has been approved for the

Department of Civil Engineering

by

Kevin Rens, Chair

Cheng Yu Li

Yail Jimmy Kim

November 12, 2013

iii

Dobbins, Konlee B. (M.S., Civil Engineering)

Behavior and Analysis of Highly Skewed Steel I-Girder Bridges

Thesis directed by Professor Kevin Rens

ABSTRACT

Skewed bridge supports for steel I-girder bridges, introduce complexities to the

behavior of the girder system that can be difficult to accurately model and analyze. In

addition there have been some reported shortfalls in the 2D-grid analysis method

typically used by engineers to design steel girder bridges with significant skews. Some

improvements have been suggested to bridge the gap in the inaccuracies of 2D-grid and

2D-frame analyses. These improvements include overwriting the girder torsional

stiffness to include warping effects, overwriting the equivalent beam stiffness of cross-

frames using a more accurate method of calculating the stiffness, including locked-in

cross-frame forces due to dead load fit detailing, and more accurately calculating flange

lateral bending stresses with staggered cross-frame layouts. This thesis examines these

improvements, compares different levels of analysis, provides recommendations for these

methods of analysis, and explains the behavior of the girder system during erection.

The form and content of this abstract are approved. I recommend its publication.

Approved: Kevin Rens

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ACKNOWLEDGMENTS

I would like to thank my committee members Dr. Kevin Rens, Dr. Cheng Yu Li,

and Dr. Jimmy Kim for reviewing my work. I would like to thank Parsons for funding

my master’s degree. I would also like to thank my coworkers who provided

encouragement and guidance, especially Steve Haines, with his help providing

information about the Geneva Road Bridge project. And most of all, I would like to

thank my family and in particular my wife, Kirsi Petersen, for her patience and

encouragement during the long nights and weekends spent away from the family.

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TABLE OF CONTENTS

CHAPTER

I. INTRODUCTION .............................................................................................1

Effects of Skew ..................................................................................................1

Continuation of Previous Thesis ........................................................................2

II. LITERATURE REVIEW ..................................................................................4

Introduction ........................................................................................................4

Relevant Documents ..........................................................................................5

III. THEORETICAL BACKGROUND .................................................................10

Suggestions to Simplify Structure Geometry in Skewed Bridges ...................10

Framing Plan – Cross-Frame Layout ...............................................................11

Rotations and Deflections ................................................................................12

Detailing – NLF vs. SDLF vs. TDLF ..............................................................14

Analysis Methods.............................................................................................17

Improvements to 2D Modeling ........................................................................19

Preferred Analysis Method for Straight Skewed Girders ................................28

IV. ANALYTICAL PLAN ....................................................................................33

Example Bridge Description ............................................................................33

Analysis Models...............................................................................................40

V. ANALYTICAL RESULTS – COMPARING MODELS ...............................47

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VI. CONCLUSIONS .............................................................................................75

Recommended Method of Analysis .................................................................75

General Recommendations for Future Work ...................................................77

REFERENCES .................................................................................................................80

APPENDIX A ...................................................................................................................81

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LIST OF FIGURES

Figure

III.1 Typical Fit-up Procedure for Skewed I-Girders. ..................................................... 16

III.2 Lateral Bending Moment, Ml, in a Flange Segment Under Simply Supported and

Fixed-End Conditions. .............................................................................................. 24

III.3 Conceptual Configurations Associated with Dead Load Fit (TDLF or SDLF)

Detailing .................................................................................................................... 26

III.4 Matrix of Grades for Recommended Level of Analysis for I-Girder Bridges ......... 31

IV.1 Typical Section of the Geneva Road Bridge ........................................................... 34

IV.2 Layout of the Geneva Road Bridge ......................................................................... 35

IV.3 Elevation Layout of the Geneva Road Bridge ......................................................... 36

IV.4 Girder Elevation of the Geneva Road Bridge .......................................................... 38

IV.5 Framing Plan of the Geneva Road Bridge ............................................................... 39

IV.6 Underside of the Geneva Road Bridge .................................................................... 40

V.1 Major-Axis Bending Stress of Girder 1 Top Flange ............................................... 49

V.2 Axis Bending Stress of Girder 1 Bottom Flange ...................................................... 49

V.3 Vertical Girder Displacements Along Girder 1 and Girder 3 ................................... 50

V.4 Cross-Frame Axial Forces Along Bay 2 Between Girder 2 and Girder 3 ................ 51

V.5 Major-Axis Bending Stress of Girder 1 .................................................................... 56

V.6 Vertical Girder Displacements Along Girder 1 and Girder 3 ................................... 57

V.7 Cross-Frame Axial Forces Along Bay 2 Between Girder 2 and Girder 3 ................ 58

V.8 Flange Lateral Bending Stress .................................................................................. 62

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V.9 Girder Layover at Bearings ....................................................................................... 65

V.10 Major-Axis Bending Stress of Girder 1 .................................................................. 66

V.11 Vertical Girder Displacements along Girder 1 and Girder 3 .................................. 67

V.12 Cross-Frame Axial Forces Along Bay 2 Between Girder 2 and Girder 3 .............. 68

V.13 Flange Lateral Bending Stress Along Girder 1 ....................................................... 71

V.14 Girder Layover at Start and End Bearings for 3D Models ..................................... 73

V.15 Girder Layover at Start and End Bearings for 3D TDLF Models and from Field

Data ........................................................................................................................... 74

1

CHAPTER I

INTRODUCTION

Effects of Skews

Skewed bridge supports and horizontal curvature in steel I-girder bridges exhibit

torsional forces that can introduce unexpected stress, displacements, and rotations during

construction. As the skew angle or degree of curvature increases, the difficulty of

constructing steel I-girder bridges increases. The sequence of erection and assumptions

made during fabrication can introduce forces and deflections that were not accounted for

during the design. In many cases, these forces and deflections are negligible; however, in

some cases they can be significant and unaccounted for if following today’s standard

design practice and codes. Many of today’s more commonly used structural software

take into account the effects of horizontal curvature on steel superstructures. However,

accurately capturing the effects of skewed supports seems to be lacking in these software

(NCHRP, 2012).

Many reports and research papers lump the effects of horizontal curvature and

skews together and tend to provide all encompassing guidelines that address both aspects.

Many of the effects of horizontal curvature and skews are similar in nature; however,

they can act in opposite directions or in different locations and affect the design

differently. Therefore, it is important to understand the effects of each separately. This

thesis will focus on the effects of skews only.

Skews at bridge supports alter the behavior of girders. Historically, skews were

avoided whenever possible because the effects were not well understood. Over time,

advances in structural analysis and results from case studies have made the effects a bit

2

clearer. One report in particular, National Cooperative Highway Research Program

(NCHRP) Report 725 “Guidelines for Analysis Methods and Construction Engineering of

Curved and Skewed Steel Girder Bridges”, has taken great strides in highlighting the

shortcomings of today’s standard practice, specifications, and guidelines for highly

skewed steel I-girder bridges. More is discussed on these shortcomings and how to

accurately account for them in the Literature Review and Theoretical Background

sections.

Continuation of Previous Thesis

This thesis follows up and expands on a fellow University of Colorado Denver

graduate student’s thesis “Crossframe Analysis of Highly-Skewed and Curved Steel I-

Girder Bridges” that touched on a variety of similar topics and provided a case study

example. That thesis focused on cross-frame design by looking at different framing plan

and cross-frame configurations (x-frame vs. k-frame) to find the most efficient design. It

also included some background research, a literature review, theoretical background, and

analysis of cross-frames in highly skewed and curved steel I-girder example bridges

(Schaefer, 2012).

The theoretical background is predominantly based on the American Association

of State Highway Transportation Officials (AASHTO) and National Steel Bridge

Alliance (NSBA) Steel Bridge Collaboration Document G13.1, Guidelines for Steel

Girder Bridge Analysis. That thesis is a good source for background information on

cross-frame types, framing plan configurations, and specification requirements from

AASHTO and can be used as a precursor. It also provides a list of several curved and/or

3

skewed bridges, with framing plans and member sizes included for each bridge listed, in

the Denver, CO metro area (Schaefer, 2012).

The conclusions that can be taken from “Crossframe Analysis of Highly-Skewed

and Curved Steel I-Girder Bridges” include:

Staggered cross-frame configurations induced the least amount of forces within its

cross-frames.

Contiguous cross-frame configurations induced the most forces within its cross-

frames.

A stiffer transverse system will accumulate more force than a flexible system.

The K-frame type cross-frame performs better than the X-frame. The diagonal

members in a K-frame cross-frame absorb significantly less force than the

diagonal members in an X-frame.

The double angle and WT-members are less slender, more flexible, and thus

attract fewer loads than single angles that have to meet slenderness requirements

(Schaefer, 2012).

This thesis focuses on the effects that displacements and detailing have on the

design of highly skewed steel I-girders and the most accurate design methods that should

be used with common steel I-girder structural software.

4

CHAPTER II

LITERATURE REVIEW

Introduction

The literature review in “Crossframe Analysis of Highly-Skewed and Curved

Steel I-Girder Bridges” includes the history of design specifications that contributed to

today’s codes and standard practice for the design of skewed or horizontally curved steel

girders. The list includes:

AASHTO Guide Specifications for Horizontally Curved Steel Girder Highway

Bridges, 1980

NCHRP Project 12-38, 1993

AASHTO Guide Specifications for Horizontally Curved Steel Girder Highway

Bridges, 1993

NCHRP Project 12-52, 1999

AASHTO Guide Specifications for Horizontally Curved Steel Girder Highway

Bridges, 2003

AASHTO/NSBA – G13.1 Guidelines for Steel Girder Bridge Analysis 1st Edition,

2011

One very important document missing from the list is from the research of

NCHRP Project 12-79, “NCHRP Report 725 – Guidelines for Analysis Methods and

Construction Engineering of Curved and Skewed Steel Girder Bridges.” NCHRP Report

725 points out several deficiencies in the latest AASHTO Load and Resistance Factor

Design (LRFD) Bridge Design Specifications, the latest guidelines (AASHTO/NSBA –

5

G13.1 Guidelines for Steel Girder Bridge Analysis 1st Edition), and standard practices

assumed with the most commonly used 1D and 2D analysis software.

Relevant Documents

The following literature review and theoretical background focuses on the G13.1

Guidelines and NCHRP Report 725, while briefly discussing contributions from other

research papers.

AASHTO/NSBA – G13.1 Guidelines for Steel Girder Bridge Analysis

In the Forward of this document, it states “the document is intended only to be a

guideline, and only offers suggestions, insights, and recommendations but few, if any,

‘rules.’” The purpose of the document is to provide engineers, particularly less

experienced designers, with guidance on various issues related to the analysis of common

steel girder bridges. The document focuses on presenting the various methods available

for analysis of steel girder bridges and highlighting the advantages, disadvantages,

nuances, and variations in the results. The guidelines are, to a certain extent, all-

encompassing for steel girder bridges, while briefly discussing the effects of different

variations such as skews and horizontal curvature. The general behavior and suggested

analysis methods are discussed; however, it does not go into great detail. At the time this

document was released there had been very few guideline resources for the design of

skewed and horizontally curved steel girders and their corresponding cross-frames.

The contents include:

1. Modeling descriptions

2. History of steel bridge analysis

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3. Issues, objectives, and guidelines common to all steel girder bridge analyses

4. Analysis guidelines for specific types of steel girder bridges

Of particular interest are the sections on skewed bridges. These sections include

information on the behavior, constructability analysis issues, predicted deflections,

detailing of cross-frames and girders for the intended erected position, cross-frame

modeling in 2D, geometry considerations, and analysis guidelines for skewed steel I-

girder bridges (AASHTO/NSBA, 2011).

NCHRP Report 725 – Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges

This report contains guidelines on the appropriate level of analysis needed to

determine the constructability and constructed geometry of curved and skewed steel

girder bridges. The report also introduces improvements to 1D and 2D analysis that

require little additional computational costs. The research for this report was performed

under NCHRP Project 12-79. The objectives and scope of NCHRP Project 12-79

include:

1. An extensive evaluation of when simplified 1D or 2D analysis methods are

sufficient and when 3D methods may be more appropriate.

2. A guidelines document providing recommendations on the level of construction

analysis, plan detail, and submittals suitable for direct incorporation into

specifications or guidelines.

Of particular interest for this thesis are the sections pointing out the deficiencies

of 1D or 2D analysis used in standard practice and the proposed improvements for

analyzing skewed bridges. The report focuses on problems that can occur during, or

7

related to, the construction. The key construction engineering considerations for skewed

steel girder bridges include:

1. The prediction of the deflected geometry at the intermediate and final stages of

the construction,

2. Determination and assessment of cases where the stability of a structure or unit

needs to be addressed,

3. Identification and alleviation of situations where fit-up may be difficult during the

erection of the structural steel, and

4. Estimation of component internal stresses during the construction and in the final

constructed configuration.

AASHTO LRFD Bridge Design Specifications, Customary U.S. Units, 6th Edition (2012)

This specification is used in every state throughout the United States as the

national standard that engineers are required to follow for bridge design. Many states

include their own amendments to this specification and additional guidelines, but it’s still

the standard that the nation’s bridge designs are based upon. The specifications have also

been adopted or referenced by other bridge-owning authorities and agencies in the United

States and abroad. Since its first publication in 1931, the theory and practice have

evolved greatly resulting in 17 editions of the Standard Specifications for Highway

Bridges with the last edition appearing in 2002 and six editions to date of the load-and-

resistance factor design (LRFD) specifications (AASHTO, 2012).

As the national standard, the specifications are a bit lacking in providing

requirements or guidance for designing highly skewed bridges. In Section 4 – Structural

8

Analysis and Evaluation, equations are provided to adjust the live load distribution

factors for moment and shears using approximate methods of analysis. The approximate

method of analysis involves line girder or 1D analysis of “typical” bridges within a set

range of applicability for girder design. Section 6 – Steel Structures, includes

commentary on the effects that skews have on girder and cross-frame deflections,

rotations, and potential additional stresses. However, in many cases, it recommends

performing a more refined analysis to more accurately capture the effects of skews and

leaves a fair amount to engineering judgment to decide when a refined analysis is

necessary and to what amount of detail. The AASHTO/NSBA G13.1 Guidelines,

NCHRP Report 725, and several other reports and research papers help bridge that gap

and provide more guidance.

Other Reports

There are many more research reports, presentations, and short articles on the

effects of skews on steel I-girder bridges and experiences during construction. The

authors include structural engineers, professors, fabricators, and construction managers.

Several of these authors also contributed to NCHRP Report 725. Some articles, such as

“Design and Construction of Curved and Severely Skewed Steel I-Girder East-West

Connector Bridge over I-88”, describe the challenges and lessons learned during the

design and construction of a specific bridge. In the presentation “Erection of Skewed

Bridges: Keys to an Effective Project”, the chief engineer for High Steel Structures Inc.,

presents three case studies of highly skewed steel girder bridges and the experiences from

the point of view of the fabricator. The presence of large skews and the assumptions

made on fit-up detailing during erection affect all stages of design and construction.

9

Engineers, fabricators, and contractors all need to understand the movements, forces

required for fit-up, and corresponding locked-in stresses that occur during different stages

of construction.

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CHAPTER III

THEORETICAL BACKGROUND

Suggestions to Simplify Structure Geometry in Skewed Bridges

Skews present complexities in design, detailing, fabrication, and erection that

translate into increased costs for steel girder bridges. As per NSBA/AASHTO Steel

Bridge Collaboration, skew angles should be eliminated or reduced wherever possible.

The bridge designer should work closely with the roadway designer to improve and

simplify roadway alignments. Once the alignment is set, a few suggestions for

eliminating or reducing skews include:

Lengthening spans to locate the abutments far enough from the roadways below

to allow for the use of radial abutments or bents while maintaining adequate

horizontal clearance. Designers should consider the cost of a longer span versus

the cost associated with the complications of skew in the bridge.

Retaining walls may allow the use of a radial abutment in place of a header slope.

Typically these walls are of variable height and require odd-shaped slope

protection behind the wall. Designers should consider the cost of the walls versus

the cost associated with the complications of skew in the bridge.

Use integral radial interior bent instead of a skewed traditional bent cap to

maintain adequate vertical clearance in cases where a traditional radial bent would

have insufficient vertical clearance and where the vertical profile of the bridge

cannot be raised.

11

Use dapped girder ends with inverted-tee bent caps to maintain adequate vertical

clearance at expansion joint locations instead of an integral bent cap

(AASHTO/NSBA 2011).

In many cases, highly skewed supports cannot be avoided for a number of

reasons. Typically, geometry constraints in highly congested highway interchanges leave

very little wiggle room to eliminate or reduce large skews. Where large skews cannot be

avoided, design engineers, detailers, fabricators, and contractors all need to understand

the stresses and deflections that occur during different stages of construction.

Framing Plan – Cross-Frame Layout

Cross-frames or diaphragms should be placed at bearing lines that resist lateral

force. Wind loads and other lateral forces are transferred from the deck and girders

through the cross-frames at supports to the bearings and down to the substructure. As per

AASHTO LRFD Bridge Design Specifications 6.7.4.2 – Diaphragms and Cross-Frames

for I-Section Members, cross-frames at supports can either be placed along the skew or

perpendicular to the girder:

Where support lines are skewed more than 20 degrees from normal, intermediate diaphragms or cross-frames shall be normal to the girders and may be placed in contiguous or discontinuous lines.

Where a support line at an interior pier is skewed more than 20 degrees from normal, elimination of the diaphragms or cross-frames along the skewed interior support line may be considered at the discretion of the Owner. Where discontinuous intermediate diaphragm or cross-frame lines are employed normal to the girders in the vicinity of that support line, a skewed or normal diaphragm or cross-frame should be matched with each bearing that resists lateral force (AASHTO, 2012).

As research has shown, placing a cross-frame normal to the girders and at the

bearing location of a skewed support, provides an alternate load path and attracts a

12

significant amount of force in that cross-frame. NCHRP Report 725 referred to these

cross-frames at or near the supports as providing “nuisance stiffness” transverse load

paths and should be avoided if possible. Therefore, standard practice is to provide a

cross-frame along the skew at the supports.

Staggered cross-frame layout configurations appear to induce the least amount of

forces within its cross-frames. This is especially true and desirable for interior cross-

frames closest to highly skewed supports. Staggered cross-frames allow for more

flexibility in the system and therefore attract less load. Also, K-frame type cross-frames

tend to be the better choice over X-frame type cross-frames (Schaefer, 2012).

Rotations and Deflections

The root of the complications due to skew is the out of plane rotations and

deflections at the skewed supports that cause twisting in the girders as vertical loads are

applied. In 1D line girder analysis, the effects of a skew are not captured. When

analyzing a single girder in a single span, as vertical loads (such as dead loads from the

steel self-weight, concrete deck weight, and miscellaneous superimposed dead loads and

live loads from vehicular traffic) are applied, the girder deflects downward with the max

deflection occurring at mid-span. There are, theoretically, no lateral deflections or

twisting. However, as cross-frames are attached connecting skewed girders together, the

girders start to twist near the supports. The differential deflection between two adjacent

girders causes a twisting motion, also known as layover. This twisting motion can be

counterbalanced by specifying certain detailing that essentially forces the girders to be

twisted in the opposite direction when connecting the cross-frames to the girders during

erection. This is known as Steel Dead Load Fit (SDLF) and Total Dead Load Fit (TDLF)

13

detailing, where the goal is to have vertically plumb girder webs at the specified

construction stage. This will be discussed in more detail in the next section.

Per the AASHTO/NSBA Steel Bridge Collaboration Document G12.1

“Guidelines for Design Constructability”:

The problem for cross-frames at skewed piers or abutments is the rotation of the girders at those locations. In a square bridge, rotation of the girders at the bearings is in the same plane as the girder web. If supports are skewed, girder rotation due to non-composite loads will be normal to the piers or abutments. This rotation displaces the top flange transversely from the bottom flange and causes the web to be out of plumb (AASHTO/NSBA, 2003).

Where end cross-frames are skewed parallel to the support, which is typically

standard practice, these end cross-frames contribute to the rotations and transverse

movements described above. The end cross-frames are very stiff in the axial direction

along the skewed support and flexible in the weak axis direction, which allow these

rotations normal to the support.

The movements of simple span straight girders on non-skewed supports are

predictably uniform. With downward deflection between supports due to vertical dead

loads, the top flange compresses. At the supports, the top flange deflects toward mid-

span. Conversely, the bottom flange is in tension and deflects away from mid-span at the

expansion supports that are free to move longitudinally. The ends of girders also rotate

due to the length changes in the flanges. For girders on skewed supports the movement

becomes more complex by adding transverse deflections and twisting rotations. The

rotation normal to the pier as described in AASHTO/NSBA G12.1 is a bit of a

generalization and is true for bearings that are at the same elevation at the given support.

If the bearing elevations differ along the given support, the axis of rotation will be in the

14

plane including the actual centerline of bearing, but slope to intersect the centers of

rotation at adjacent bearings. This describes the theoretical movements. Actual

movements will vary slightly since the members are framed together and restrained by

the deck and bearings, therefore some distortions will result (Beckmann and Medlock,

n.d.).

The transverse movements and twisting causes the ends of girders to be out of

plumb as vertical loads are applied if the girders are not detailed to counteract these

movements during construction. For more detailed information on rotations and

deflections in skewed steel girder bridges, the article “Skewed Bridges and Girder

Movements Due to Rotations and Differential Deflections” is recommended.

Detailing – NLF vs SDLF vs TDLF

As the skew angle increases, the transverse flange movement increases. For

strength, serviceability, and aesthetic reasons, it is typically desirable to detail the girders

with sizeable skews to counteract these girder end movements and be plumb at certain

dead load cases. However, each bridge needs to be evaluated for several factors,

including constructability and girder design at different stages of construction, to

determine the most economic design. Fabrication and construction must follow the fit-up

condition assumed during the design of the girders and cross-frames. Otherwise,

unintended locked-in forces or movements that were not considered during design can

arise.

The designer generally has three choices of conditions for which the girders and

cross-frames shall be designed:

15

No-Load Fit (NLF) condition – the girder webs are theoretically plumb/vertical

before any load is applied.

Steel Dead Load Fit (SDLF) condition – the girder webs are theoretically

plumb/vertical under the steel dead after the cross-frames are installed and before

the concrete deck is poured.

Total Dead Load Fit (TDLF) condition – the girder webs are theoretically

plumb/vertical under the total dead load in the final condition (Beckmann and

Medlock, n.d.).

Each detailing method affects deflected geometry, can create fit-up issues,

produce stability effects and second-order amplification, and affect component internal

stresses during construction. Construction plans and submittals for these complex

geometries with high skews need to clearly state the fit-up method assumed during design

and construction (NCHRP, 2012).

In SDLF and TDLF detailing methods, the cross-frames do not fit-up with the

connection work points on the initially fabricated girders. During fit-up of cross-frames

with the girders, the girders are forced into place by twisting the girders. A girder is

much more flexible twisting about its longitudinal axis than a cross-frame deforming

axially. As the dead load is applied, the girders deflect and rotate back to plumb.

AASHTO/NSBA G 12.1 Guidelines for Design Constructability describes the process of

SDLF or TDLF fit-up as seen in Figure III.1.

16

Figure III.1 Typical Fit-up Procedure for Skewed I-Girders (AASHTO/NSBA, 2003)

17

For the designer, the biggest concern is the presence of any unaccounted forces

and correctly modeling the structure at different stages of construction. SDLF and TDLF

detailing introduces locked-in forces during erection when the girders are forced to fit up

with the stiffer cross-frames. In many cases, especially for straight girders on skewed

supports, locked-in forces are relieved as the dead loads are applied. However, it can be

dangerous to assume that this occurs in all cases. For example, as with curved girders

with radial/nonskewed supports, the locked-in forces from fit-up and forces due to

differential deflections between adjacent girders can be additive. Or in highly skewed

straight bridges, if the first intermediate cross-frames are too close to the bearing line, the

locked-in cross-frame forces near the acute corners tend to be additive with the dead load

effects (NCHRP, 2012).

Analysis Methods

The level of detail for the girder and cross-frame analysis is an important decision

to make and is often left to engineering judgment. 3D finite element analysis (FEA)

provides the most accurate results when done correctly. However, it is by far the most

complex and time-consuming and with a large number of variables, it leaves a lot of

room for error. 1D and 2D simplified analysis are much less time-consuming and

therefore preferred by engineers for the design of non-complex structures. What

constitutes a structure to be complex and where to draw the line is often a topic of debate

among engineers. AASHTO LRFD Bridge Design Specifications provide criteria for

determining if using a simplified method of linear analysis is acceptable. When a refined

method of analysis is required or recommended, there are still a good number of methods

to choose from including 2D-grid and 3D-FEA. It is ultimately left up to engineering

18

judgment to choose an appropriate refined method of analysis and understand the basic

assumptions and methodology of the software used (AASHTO, 2012). Even in cases

where 1D or 2D methods of analysis are deemed acceptable, NCHRP Report 725 has

made light of some assumptions that can turn out to be quite erroneous. NCHRP Report

725 has also exposed some assumptions typically made by most 1D or 2D analysis

software that can significantly alter the results. The following sections provide a brief

overview of the different methods of analysis.

1D – Line Girder Analysis Method

Line girder analysis, as the name suggests, isolates and analyzes one single girder

line. Loads are distributed to each girder by way of distribution factors. Effects on girder

moments and shear from skews no greater than 60 degrees are accounted for with

additional factors in AASHTO LRFD Bridge Specifications. The effects of the cross-

frames are not taken into account. This method is adequate for fairly simple structures

with little to no skew angle.

2D – Grid Analysis Method

In plan grid or grillage analysis, the structure is divided into plan grid elements

with three degrees of freedom at each node. This method is most often used in steel

bridge design and analysis (AASHTO/NSBA, 2011). The effects of the cross-frames are

taken into account; however, most common 2D software, such as DESCUS and MDX,

use equivalent beam element properties when modeling the cross-frames. As discussed

in NCHRP Report 725, how these common 2D software compute the equivalent beam

element properties for the cross-frames and the equivalent torsional constant properties of

the girders, isn’t typically accurate especially in cases of high skews or high degrees of

19

horizontal curvature. These inaccuracies and how to account for them will be explained

in greater detail in later sections. As the most commonly used method of analysis, it is

vital to keep the inaccuracies to a minimum.

3D – Finite Element Analysis Method

In the 3D-FEA method, the bridge superstructure is fully modeled in all three

dimensions. The model typically includes modeling the girder flanges as beam elements

or plate/shell elements; modeling the web as plate/shell elements; modeling each member

of the cross-frames as beam or truss elements; and modeling the deck as plate/shell

elements. This method is arguably the most accurate; however, it is typically very time-

consuming and complicated. Therefore, it is mostly only used for very complex

structures or for performing refined local stress analysis of a complex detail. There are

other complicating factors, such as the output reporting the stresses in each element

instead of moments and shears that the engineer typically checks against the required

limits in AASHTO or local state specifications. The engineer would need to convert the

stresses into moments and shears if so desired. When and how to use refined 3D finite

element analysis is a controversial issue, and this method has not been fully incorporated

into the AASHTO specifications to date (AASHTO/NSBA, 2011).

Improvements to 2D Modeling

Cross-Frame Modeling

Most designers use the methods described in the AASHTO/NSBA (2011) G13.1

document for finding the equivalent beam stiffness of cross-frames in 2D analysis

models. There are two approaches here:

20

1. Calculate the equivalent moment of inertia based on the flexural analogy method.

In a model of the cross-frame, a unit force couple is applied to one end to find the

equivalent rotation that is then used to back-calculate the equivalent moment of

inertia.

2. Calculate the equivalent moment of inertia based on the shear analogy method. In

a model of the cross-frame, a unit vertical force is applied to one end to find the

equivalent deflection that is then used to back-calculate the equivalent moment of

inertia (AASHTO/NSBA, 2011).

Both methods use Euler-Bernoulli beam theory equations. The issue with using

one of these methods is the flexural analogy method only accounts for the flexural

stiffness, while the shear analogy only accounts for shear stiffness. In cases where either

the flexure or shear is considered negligible, using the appropriate method above is

acceptable. However, in cases where both flexure and shear are present, the equivalent

moment of inertia should account for both flexural and shear stiffness. Differential

deflection of adjacent girders might primarily engage the shear stiffness of the cross-

frames, while differential rotation of adjacent girders might be more likely to engage the

flexural stiffness of the cross-frames (AASHTO/ NSBA, 2011).

NCHRP Report 725 recommends a more accurate approach for calculating the

cross-frame equivalent beam stiffness. This approach includes an equivalent shear area

for a shear-deformable beam element representation (Timoshenko beam theory) of the

cross-frame. In the report, it compares the equivalent stiffness results from the flexural

analogy method, shear analogy method, pure bending (Timoshenko) method, and 3D-

FEA calibrated to a test bridge and finds that the pure bending (Timoshenko) method

21

provides the most accurate overall results. This is due to the fact that the Timoshenko

beam theory element is able to represent both flexure and shear deformations.

In the pure bending (Timoshenko) method, the equivalent moment of inertia is

determined first based on pure flexural deformation. This is similar to the flexural

analogy method except that the constraints are modeled differently and the corresponding

end rotation is equated from the beam pure flexure solution M/(EIeq/L) versus the Euler-

Bernoulli beam rotation equation M/(4EIeq/L) used in the flexural analogy method. This

results in a substantially larger equivalent moment of inertia and that EIeq represents the

“true” flexural rigidity of the cross-frame. The cross-frame is supported as a cantilever at

one end and is subjected to a force couple at the other end, producing a constant bending

moment and corresponding end rotation. In the second step of this method, the cross-

frame is still supported as a cantilever but is subjected to a unit transverse load at its tip.

The Timoshenko beam equation for the transverse displacement is:

∆3

which is used to find the equivalent shear area (NCHRP, 2012).

As per NCHRP Report 725, the Timoshenko beam element provides a closer

approximation of the physical cross-frame behavior compared to the Euler-Bernoulli

beam for all types of cross-frames (including X and K type cross-frames) that are

typically used in I-girder bridges. Not only are the calculated forces more accurate but

the deflections and rotations are more accurate. Predicting deflections and rotations

22

during construction becomes much more important as skew angles increase (NCHRP,

2012).

The fabricator can more accurately fabricate the girders for the appropriate final

orientation and fit-up method. The contractor more accurately understands the

deflections and rotations to expect during construction and the forces necessary for the

chosen fit-up method. The engineer can more accurately and efficiently design the

girders and cross-frames for the expected movements and locked-in forces from fit-up

and final condition loads.

I-Girder Torsion Modeling

Current practice in 2D-grid models substantially underestimates the girder

torsional stiffness. This is due to software only considering St. Venant torsional stiffness

of the girders while neglecting warping torsional stiffness. This practice tends to

discount the significant transverse load paths in highly skewed bridges, since the girders

are so torsionally soft that they are unable to accept any significant load from the cross-

frames causing torsion in the girders. As a result, the cross-frame forces can be

significantly underestimated (NCHRP, 2012).

NCHRP Report 725 provides some equations to calculate an equivalent torsional

constant, Jeq that includes both the St. Venant and warping torsional stiffness. It should

be noted that these equations were based in part on prior research developments by

Ahmed and Weisgerber (1996), as well as the commercial implementation of this type of

capability within the software RISA-3D. In this approach, an equivalent torsional

constant must be calculated for each unbraced length and girder sectional property. The

equation for the equivalent torsion constant for the open-section thin-walled beam

23

associated with warping fixity as each end of a given unbraced length (cross-frame

spacing) is:

1sinh cosh 1

sinh

Where Lb is the unbraced length between the cross-frames, J is the St. Venant torsional

constant, and p2 is defined as GJ/ECw. Assuming warping fixity at the intermediate

cross-frame locations leads to a reasonably accurate characterization of the girder

torsional stiffness (NCHRP, 2012).

I-Girder Flange Lateral Bending Modeling

AASHTO LRFD Bridge Specifications section C4.6.1.2.4b provides a simplified

equation to calculate the lateral moment for a horizontally curved girder based on the

radius, major-axis bending moment, unbraced length, and web depth. For other

conditions that produce torsion, such as skew, AASHTO suggests other analytical means

which generally involve a refined analysis. However, Section C6.10.1 provides a coarse

estimate by stating:

The intent of the Article 6.10 provisions is to permit the Engineer to consider flange lateral bending effects in the design in a direct and rational manner should they be judged to be significant. In absence of calculated values of fl from a refined analysis, a suggested estimate for the total unfactored fl in a flange at a cross-frame or diaphragm due to the use of discontinuous cross-frame or diaphragm lines is 10.0 ksi for interior girders and 7.5 ksi for exterior girders. These estimates are based on a limited examination of refined analysis results for bridges with skews approaching 60 degrees from normal and an average D/bf ratio of approximately 4.0. In regions of the girders with contiguous cross-frames or diaphragms, these values need not be considered. Lateral flange bending in the exterior girders is substantially reduced when cross-frames or diaphragms are placed in discontinuous lines over the entire bridge due to the reduced cross-frame or diaphragm forces. A value of 2.0 ksi is suggested for fl for the exterior girders in such cases, with the suggested value of 10 ksi retained for the interior girders. In all cases, it is suggested that the recommended values of fl be proportioned to dead and live load in the same proportion as the unfactored major-axis dead and

24

live load stresses at the section under consideration. An examination of cross-frame or diaphragm forces is also considered prudent in all bridges with skew angles exceeding 20 degrees (AASHTO, 2012).

NCHRP Report 725 recommends a more accurate but simplified method of

calculating lateral bending stress than the coarse estimates provided above. Their method

includes a local calculation in the vicinity of each cross-frame, utilizing the forces

delivered to the flanges from the cross-frames placed in discontinuous lines. The

approximate calculation takes the average of pinned and fixed end conditions as shown in

Figure III.2 below.

Figure III.2 Lateral Bending Moment, Ml, in a Flange Segment Under Simply Supported and Fixed-End Conditions (NCHRP, 2012)

Calculation of Locked-In Forces Due to Cross-Frame Detailing

Regardless the type of analysis used (2D-grid, 2D-frame, or 3D-FEA), the

analysis essentially assumes a NLF condition unless the locked-in forces are accounted

for in the model. Any lock-in forces, due to the lack of fit of the cross-frames with the

girders in the undeformed geometry in SDLF or TDLF, add to or subtract from the forces

determined from the analysis. Typically for straight skewed bridges, the locked-in forces

25

tend to be opposite in sign to the internal forces due to dead loads. Therefore the 2D-grid

or 3D-FEA analysis solutions for cross-frame forces and flange lateral bending stresses

are conservative when SDLF or TDLF initial fit-up forces are neglected. However, these

solutions can be prohibitively conservative for highly skewed bridges (NCHRP, 2012).

TDLF or SDLF detailing is first and foremost a geometrical calculation for the

detailer and fabricator. Yet, they can significantly affect the locked-in cross-frame

forces. Figure III.3 shows four configurations that visually explain how the locked-in

forces can be calculated. Configurations 1 and 4 are used by structural detailers.

Configurations 2 and 3 are theoretical geometries that technically never take place in the

physical bridge, but are used to calculate the internal locked-in forces. The differential

camber shown in Configuration 1 is detailed to counterbalance the eventual differential

deflection that occurs under the corresponding dead load. This differential camber

induces the twisting shown in Configuration 3 from the cross-frames being forced into

place and released. The deflections due to the twisting are approximately equal and

opposite to the deflections at these locations under the corresponding total or steel dead

load (NCHRP, 2012).

For cases where the initial lack-of-fit effects are important, the designer can

simply include an initial stress or strain similar to a thermal stress or strain. Calculating

the initial strains and stresses associated with SDLF or TDLF detailing of the cross-

frames involves finding the nodal displacements between Configurations 2 and 4 and

applying the corresponding stresses to the cross-frame ends. In 3D-FEA, the calculated

axial strains from the nodal displacements are converted into stresses simply by

multiplying the strains by the elastic modulus of the material. The stresses are then

26

multiplied by the cross-frame member areas to determine the axial forces. In 2D-grid

models that use equivalent beam elements for the cross-frames, the displacements

calculated above are converted into beam end displacements and end rotations.

Assuming fixed-end conditions, the end displacements are used to calculate the fixed-end

forces, which are then applied to the equivalent cross-frame beam element.

Figure III.3 Conceptual Configurations Associated with Dead Load Fit (TDLF or SDLF) Detailing (NCHRP, 2012)

27

Figure III.3 (Continued) Conceptual Configurations Associated with Dead Load Fit (TDLF or SDLF) Detailing (NCHRP, 2012)

28

The behavior of the end cross-frames at skewed bearing lines is slightly different,

however the locked-in forces due to cross-frame fit-up is calculated in the same manner

following the configurations in Figure III.3. The girders cannot displace vertically at the

bearings and the skewed cross-frames impose a twist in the girder ends. The top flange

of the girders at the bearing line can only displace significantly in the direction normal to

the plane of the cross-frame. In order for the skewed end cross-frame to fit up with the

girders in Configuration 2, the cross-frame has to rotate about its longitudinal axis and be

strained into position to connect them with the rotated connection plates in the initial

cambered no-load, plumb geometry of the girders (NCHRP, 2012). Again, this is a

theoretical configuration that technically would not occur in the physical bridge. It is

used to calculate the displacements and the corresponding forces.

Preferred Analysis Method for Straight Skewed Girders

NCHRP Report 725 provides recommendations on the analysis and detailing

method that should be used for various levels of skews and horizontal curvature. For

straight skewed steel I-girder bridges, the recommendations are prominently based on the

skew index, IS. The skew index is a measure of the severity of the skew based on the

skew angle, the span length, and the bridge width measured between fascia girders.

tan

Straight skewed I-girder bridges are divided into three groups: Low (IS < 0.30), Moderate

(0.30 ≤ IS < 0.65), and High (IS ≥ 0.65). Bridges with a low skew index of less than 0.30

are not as sensitive to the effects of skews. As the skew index increases above 0.30,

responses associated with lateral bending of the girder flanges becomes significant. At

29

this point, the stress ratio of flange lateral bending stress over major-axis bending stress

(fl/fb) increases above 0.30 where cross-frames are staggered. This is considered a large

flange bending effect. As the skew index increases into the High category above 0.65,

the skew effects can significantly influence the major-axis bending responses. Below this

level the vertical components of the forces from the cross-frames are too small to

noticeably influence the major-axis bending response (NCHRP Appendix C, 2012).

NCHRP Report 725 provides a matrix of grades for traditional 2D-grid and 1D-

line girder analysis for several different levels of skew and horizontal curvature as seen in

the Figure III.4. For straight skewed bridges with a high skew index (IS ≥ 0.65), 2D-grid

and 1D-line girder analysis receive really poor grades. However, it should be noted that

the recommended improvements to 2D-grid analysis, as described in previous sections,

dramatically improve grades and percentage of error, especially for solutions of cross-

frame forces and flange lateral bending stresses. The grades are based on the percentage

of normalized mean error of the results for each structure response. The break-down of

grades include:

A: 6% or less normalized mean error, reflecting excellent accuracy;

B: between 7% and 12% normalized mean error, reflecting reasonable

agreement;

C: between 13% and 20% normalized mean error, reflecting significant

deviation from the accurate benchmark;

D: between 21% and 30% normalized mean error, reflecting poor

accuracy; and

30

F: over 30% normalized mean error, reflecting unreliable accuracy and

inadequate for design (NCHRP, 2012).

NCHRP Report 725 also provides recommendations for cross-frame detailing

methods for straight skewed I-girder bridges based on the skew index. In general TDLF

detailing is preferred in order to keep layover to a minimum and ensure the web is plumb

in the final TDL condition. Layover is defined as the relative lateral deflection of the

flanges from the twisting motion of the girders. For IS < 0.30, TDLF is typically the

preferred option.

The total dead load (TDL) cross-frame forces and girder flange lateral bending

stresses will essentially be canceled out by the TDLF locked-in forces. With a low skew

index level, the forces required for cross-frame fit-up during steel erection are very

manageable. Ensuring that the first intermediate cross-frames are a minimum distance

offset from centerline of bearing, will help alleviate nuisance stiffness effects and reduce

fit-up forces by providing enough flexibility at the end of girder to force the girders into

position with the relatively stiff cross-frames. The recommended minimum offset

distance from the bearing centerline is:

a ≥ max(1.5D, 0.4b)

where D is the girder depth and b is the second unbraced length within the span from the

bearing line (NCHRP, 2012).

31

Figure III.4 Matrix of Grades for Recommended Level of Analysis for I-Girder Bridges (NCHRP, 2012)

32

For straight skewed I-girder bridges with a higher skew index of IS > 0.30, TDLF,

SDLF, or detailing between SDLF and TDLF are typically good options. As the skew

index increases, the force required for cross-frame fit-up increases and becomes much

more difficult to erect. If SDLF detailing is used, excessive layover in the final TDL

condition may become a concern for bridges with large skews and long spans. Besides

TDLF cross-frame detailing, layover can be addressed with the use of beveled sole plates

and/or using bearings with a larger rotational capacity (NCHRP, 2012).

33

CHAPTER IV

ANALYTICAL PLAN

The analytical plan involves applying the theories and recommendations

discussed in the Theoretical Background chapter of this thesis towards an example

bridge. Several different models of the example bridge superstructure were created and

analyzed and then the results are compared. The models include a conventional 2D-grid

base model, an improved 2D-grid model, a 2D-frame base model, an improved 2D-frame

model, a 3D-FEA NLF-detailing model, and a 3D-FEA TDLF-detailing model. The

results for major-axis bending stresses, vertical displacements, cross-frame forces, flange

lateral bending stresses, and girder layover at bearings are all compared in the Analytical

Results chapter.

Example Bridge Description

The Geneva Road Bridge in Utah was analyzed as the example bridge used in this

thesis. The bridge is a part of the SR-114 Geneva Road Design-Build Project which was

undertaken to improve travel between Provo and Pleasant Grove, Utah. The project

involved reconstruction and widening work of about four miles of SR-114 and new

construction of a bridge over the Union Pacific Railroad and Utah Transit Authority

tracks. Parsons served as the designer and teamed with the contractor, Kiewit, to design

and build the bridge. The owner is the Utah Department of Transportation.

The Geneva Road Bridge has a 103’-4” wide deck that includes four lanes of

traffic (two in each direction), two 10’ shoulders, a 14’ median, and a sidewalk on each

side. The single span bearing to bearing length is 254’-5 ¼”. The skew angle is almost

62 degrees and its skew index, IS = 0.65, puts it right on the edge of the most severe

34

category as per NCHRP Report 725 and as seen in Figure III.4. There are nine steel plate

I-girders spaced at 11’-0 ¾” on center and 7’-5” overhangs. All structural steel conforms

to AASHTO M 270-50W, which is a weathering steel with a yield stress of 50 ksi. See

Figure IV.1 for the typical superstructure section and Figures IV.2 and IV.3 for the plan

and elevation layouts.

The deck overhangs appear to be a bit large compared to the girder spacing;

however the plans explicitly state that the sidewalks shall never be converted to travelled

lanes. The analysis in this thesis focuses on the behavior of the structure due to dead

loads during construction; therefore the exterior girders appear to take a larger amount of

load in the analytical results. The relatively small amount of live load due to pedestrian

loads distributed to the exterior girder compared to the much larger vehicular live loads

distributed to the interior girders, balances out the total end design load among all girders.

Typically, the preferred ratio of overhang length to girder spacing is between 0.3 and 0.5

for overhangs that could potentially see large vehicular live loads.

Figure IV.1 Typical Section of the Geneva Road Bridge (Parsons, 2011)

Fig

ure

IV

.2 P

lan

Lay

out

of t

he

Gen

eva

Roa

d B

rid

ge (

Par

son

s, 2

011)

35

36

Figure IV.3 Elevation Layout of the Geneva Road Bridge (Parsons, 2011)

The girder sections and lengths are the same for all nine steel plate I-girders.

Girder 1 only differs by location of the splice; however, splice location is irrelevant for

the purposes of this thesis. The web is constant at 105” x ¾”. The top flange width is a

constant 30” and the thickness varies from 1 ½” at the ends to 1 ¾” at the middle section.

The bottom flange width remains constant at 32” with a thickness that varies from 1 ¾”

at the ends to 2 ¼” at midspan. See Figure IV.4 for Girder Elevations.

The cross-frames are K-type cross-frames with WT members for the bottom, top,

and diagonal chords. The interior cross-frames are continuous where possible as seen in

the framing plan in Figure IV.5. After an initial analysis in the original design, the first

interior cross-frame near each obtuse corner of the framing plan was removed. Those

cross-frames attracted a significant amount of load due to the behavior of wide and highly

skewed bridges tending to find an alternate load path by spanning between the obtuse

corners in addition to spanning along the centerline of the girders. These cross-frames at

37

or near the supports provide “nuisance stiffness” transverse load paths especially at the

obtuse corners (NCHRP, 2012). The idea to remove the first interior cross-frame at the

obtuse corners came from the article, “Design and Construction of the Curved and

Severely Skewed Steel I-Girder East-West Connector Bridges over I-88.” The article

explains how non-skewed cross-frames that frame directly into skewed supports provide

alternate load paths and also refer to these effects as “nuisance stiffness.” These cross-

frames were removed to mitigate these effects (Chavel et al, 2010).

The next few cross-frames that are in-line with the removed cross-frames on the

Geneva Road Bridge, still experienced significant loads in the analysis and required

larger member sizes. See the framing plan in Figure IV.5 for the location of the stiffer

type 2 cross-frames. As per recommendations from NCHRP Report 725,

AASHTO/NSBA G13.1, and Schaefer’s thesis, all of which were published after the

Geneva Road Bridge was designed, the cross-frames could have been staggered

(discontinuous) and pushed back a distance a ≥ max(1.5D, 0.4b) offset from the bearing

line to the first interior cross-frame in order to reduce the cross-frame loads and mitigate

nuisance stiffness effects. However, arranging the cross-frames in continuous lines could

significantly reduce the lateral flange bending stresses.

The Geneva Road Bridge has already been designed and constructed. The

designers used the commonly used 2D-grid steel girder structural analysis software,

MDX, for the majority of the superstructure analysis. The design has been checked and

construction occurred without any issues that would have compromised the integrity of

the structure. The bridge is open to traffic and there have been no reported issues to date.

Fig

ure

IV

.4 G

ird

er E

leva

tion

of

the

Gen

eva

Roa

d B

rid

ge (

Par

son

s, 2

011)

38

39

Figure IV.5 Framing Plan of the Geneva Road Bridge (Parsons, 2011)

40

The intent of using this bridge in this thesis isn’t to recommend a better layout or

a better design method but rather to gain a better understanding of the behavior of the

girders and cross-frames during construction. The method of construction is known and

the behavior was witnessed with some recorded field data, which helped verify the

modeled behavior.

Figure IV.6 Underside of the Geneva Road Bridge

After precast panel and deck rebar installation and before the cast-in-place concrete deck pour (with permission from Kiewit).

Analysis Models

Six different analysis models were created and analyzed. The steel girder analysis

software, MDX, is used for two 2D-grid models and the 3D structural analysis software,

41

LARSA 4D with the Steel Bridge Module, is used for the other four models that include

2D-frame and 3D-FEA. The results are compared in the next chapter – Analytical

Results. As described in the Theoretical Background chapter, improvements to the 2D

models include:

Adjusting the equivalent beam stiffness assumed for cross-frames,

Adjusting the torsional stiffness to include warping stiffness, and

Calculating more accurate lateral flange bending stresses.

All 2D and 3D models assume NLF detailing by default, meaning no initial locked-in

cross-frame forces are included in the analysis. The final improvement includes adding

the locked-in cross-frame forces due to TDLF detailing for the 3D-FEA model.

The theory behind calculating more accurate lateral flange bending stresses is

based on assuming a staggered cross-frame layout is used. Since the cross-frames are

continuous, the lateral flange bending stresses will not be calculated as per the outlined

2D-grid improvements in the analysis of the example bridge. This improvement would

have been a post-processing step and will continue to be one unless 2D-grid software

companies choose to re-write their code and implement it directly into the software.

2D-Grid Base Model – MDX

This model was used for the original design and is left unchanged without any

improvements implemented for comparison purposes. In the MDX software program,

the user runs through a wizard to input various geometric and load parameters. The user

runs through five modes or input phases in the process of creating a girder system design

model:

42

1. Layout Mode – the user provides general layout information to establish the

framing plan.

2. Preliminary Analysis Mode – the user provides the loading.

3. Preliminary Design Mode – the user provides design controls to be enforced

on the generation of a set of girder designs based on the preliminary design

forces.

4. Design Mode – the user defines the bracing and can generate bracing and

girder designs after setting up certain parameters.

5. Rating Mode – this final mode is used for tuning the design (MDX, 2013).

The output includes forces, stresses, and displacement results for each girder and

for the girder system that includes the cross-frames. The results are also checked against

the latest AASHTO bridge specifications.

2D-Grid Improved Model – MDX

This model includes any possible recommended improvements to a 2D-grid

analysis. The issue is, given the constraints of the input wizard, there’s very little that

can be manipulated to improve the analysis and better represent the behavior of the girder

and cross-frame system. The software automatically calculates the torsional stiffness, J,

based on the St. Venant pure torsional stiffness by using the section dimensions input.

Warping stiffness is not included in the torsional stiffness and there’s no way to overwrite

this sectional property. In addition, there is no way to add the locked-in cross-frame

forces for TDLF or SDLF detailing.

43

That leaves adjusting the equivalent beam stiffness assumed for cross-frames as

the only improvement that can be implemented in the 2D-grid model. The user has the

option of inputting the cross-frame type (K-type, X-type, or diaphragm) and the

associated member sizes or manually input the equivalent cross-frame properties. If the

first option is chosen, the software automatically converts the cross-frame into an

equivalent beam and calculates the equivalent stiffness using the flexural analogy

method. This method does not account for the shear stiffness. The improved method as

described in the Theoretical Background chapter is implemented in this model. See

Appendix A for calculations.

2D-Frame Base Model – LARSA 4D

This model does not include any improvements and is used as a base model for

comparison purposes. LARSA 4D allows much more flexibility in modeling a structure

compared to commonly used 2D-grid software. There are two methods of modeling a

steel girder structure: 2D-frame and 3D-FEA. 2D-frame models create the structure in

one horizontal plane with each girder modeled as a beam element offset from the deck

and connected with rigid links. The deck is modeled as plate elements and the cross-

frames are modeled as truss or beam elements as appropriate with the connection points

offset from the deck.

LARSA 4D includes a design tool called the Steel Bridge Module that helps

significantly reduce the time required to model the structure and apply the appropriate

loads. The user goes through the module in similar fashion as the MDX wizard to set up

the model, and has the flexibility to adjust the model and add loads manually as deemed

44

appropriate by the user. LARSA 4D also includes a construction staging analysis

function. Typically this function is used to analyze material time effects (time is

considered the fourth dimension in the name) such as creep and shrinkage of concrete and

relaxation of stressed tendons. However, time is irrelevant in steel girder design, except

for considering fatigue but that is based on total stress cycles. The construction staging

analysis can still be a useful tool for steel girder design to determine stresses and

movements as loads are applied and as cross section properties change (composite vs.

noncomposite) at each construction stage.

2D-Frame Improved Model – LARSA 4D

This model includes improvements for the girder torsional stiffness. See

Appendix A for calculations on the equivalent girder torsional stiffness. Other potential

2D improvements were not included in this model. The cross-frames are modeled in 3D,

therefore computing the equivalent beam stiffness is unnecessary. The flange lateral

bending stresses are automatically computed. Locked-in cross-frame forces due to TDLF

detailing are only analyzed in the 3D-FEA TDLF model for ease of comparison with the

3D-FEA NLF model.

3D-FEA NLF Model – LARSA 4D

3D-FEA truly models the structure in three dimensions. The girders are modeled

as a combination of beam elements for the flanges and plate elements for the web. The

cross-frames are again modeled as truss or beam elements and are connected to the

corresponding top and bottom flange beam elements. The warping component of the

girder torsional stiffness is automatically included. The cross-frames are modeled in 3D

45

and therefore do not need to be converted to equivalent beam elements. As with all 2D

models, this model assumes NLF detailing by default and does not include any initial

locked-in cross-frame forces that would be present for TDLF or SDLF detailing methods.

By assuming the NLF detailing method, the results can be compared directly against the

2D models. Further research would need to be conducted to validate the accuracy of this

model with a full-size test bridge. However, this is out of the scope of this thesis and the

3D-FEA NLF model is used as the benchmark and assumed to be the most accurate.

The major-axis bending stresses and lateral bending stresses in the flanges are

determined from the member stresses results. The axial stress at the centroid of the

flange beam members resembles the stress due to major-axis bending. The lateral flange

bending stresses are determined from taking the difference between the axial stress at the

centroid and the average of the top and bottom stress points at one side of the rectangular

flange section. The vertical deflections are taken from the joint displacements results in

the vertical direction along the bottom flange of the girders. The cross-frame axial forces

are taken from the member end forces results in the local member coordinates. The

girder layovers are taken from the lateral joint displacements at the top of the girder ends.

The bottom of the girder is restrained in the lateral direction at the bearings.

3D-FEA TDLF Model – LARSA 4D

The only improvement needed for this model is including the initial locked-in

cross-frame forces due to TDLF detailing. The locked-in cross-frame forces are

calculated by determining the axial strain of the truss type members of the cross-frames

due to the camber differences for total dead load differential deflections. These initial

46

strains are inputted into the model as an equivalent thermal strain load. See Appendix A

for example calculations.

47

CHAPTER V

ANALYTICAL RESULTS - COMPARING MODELS

The primary goal of this thesis is to find the most efficient method of analysis that

accurately models the behavior of highly skewed steel plate I-girder bridges. Six models

were created, analyzed, and results compared. The results include:

Major-axis bending stresses

Vertical displacements

Cross-frame forces

Flange lateral bending stresses

Girder layover at bearings

These are the same results used in NCHRP Report 725 to grade the accuracy of

traditional 2D-grid and 1D-linear analysis as seen in Figure III.4. The results of the

example bridge models in this thesis are compared to the average and worst case results

reported by NCHRP. With a skew index, IS = 0.65, the example bridge is compared to

bridges in the highest skew index category.

The 3D-FEA NLF LARSA 4D model is assumed to be the most accurate and

therefore used as the benchmark against which all other 2D-grid and 2D-frame models

are compared. The 2D-grid MDX models (base and improved) are first compared to the

3D-FEA NLF LARSA 4D model. Next, the 2D-frame LARSA 4D models (base and

improved) are compared to the 3D-FEA NLF LARSA 4D model. Finally, the 3D-FEA

NLF LARSA 4D model is compared to the 3D-FEA TDLF LARSA 4D model.

48

2D-Grid Models

Major-axis bending stresses. The average grade of traditional 2D-grid analyses

for major-axis girder bending stresses as reported by NCHRP is a C and the worst-case

grade is a D. A grade of C means the normalized mean error is between 13% and 20%,

reflecting a significant deviation from the accurate benchmark. A grade of D means the

normalized mean error is between 21% and 30%, reflecting poor accuracy.

Figures V.1 and V.2 compare the unfactored major-axis bending stresses in the

top and bottom flanges, respectively. The stresses are due to the dead loads, including

the weight of the deck, on the noncomposite steel section. The 2D-grid MDX models

(base and improved) are compared to the 3D-FEA NLF LARSA 4D model. Both figures

show similar patterns for the bending stresses along the length of the girder. The

normalized mean error was not calculated due to the jagged lines in the 3D-FEA NLF

LARSA 4D model results; however the results appear to be within 6% error, which

results in a grade of A. The improved MDX model with the updated cross-frame beam

stiffness appears to resemble the benchmark pattern slightly more accurately.

The reasoning behind the jagged line display, which is more prominent in the top

flange, is unknown for the 3D benchmark model. It is most likely due to the influence of

the cross-frames. The reasoning behind the small but noticeable jump at the girder end in

the 3D benchmark model is also not completely known but not unexpected either. The

results in NCHRP Report 725 show a similar spike at the obtuse end of the exterior

girders, but do not explain the reasoning for this spike. The end cross-frames along the

high skew may be providing some equivalent continuity at the ends of the girder and

therefore cranking in a moment. However, this is purely speculation. The results are

49

deemed acceptable and further analysis into the reasoning of the jagged line pattern and

spike at the obtuse end is considered outside of the scope of this thesis.

Figure V.1 Major-Axis Bending Stress of Girder 1 Top Flange

Figure V.2 Major-Axis Bending Stress of Girder 1 Bottom Flange

‐35

‐30

‐25

‐20

‐15

‐10

‐5

0

5

0 0.2 0.4 0.6 0.8 1

Stress (ksi)

Normalized Length

MDX base

MDX Improved

Larsa 3D

‐5

0

5

10

15

20

25

30

0 0.2 0.4 0.6 0.8 1

Stress (ksi)

Normalized Length

MDX base

MDX Improved

Larsa 3D

50

Vertical Displacements. The average grade of traditional 2D-grid analysis for

girder vertical displacements as reported by NCHRP is a C and the worst-case grade is a

D. Figure V.3 compares the vertical displacements due to noncomposite dead loads

among the 2D-grid MDX models (base and improved) and the 3D-FEA NLF LARSA 4D

model along Girder 1 and Girder 3.

Figure V.3 Vertical Girder Displacements Along Girder 1 and Girder 3

‐20

‐18

‐16

‐14

‐12

‐10

‐8

‐6

‐4

‐2

0

0 0.2 0.4 0.6 0.8 1

Vertical Deflection (in)

Normalized Length

Vertical Girder Deflection ‐ Girder 1

MDX base

MDX Improved

Larsa 3D

‐20

‐18

‐16

‐14

‐12

‐10

‐8

‐6

‐4

‐2

0

0 0.2 0.4 0.6 0.8 1

Vertical Deflection (in)

Normalized Length

Vertical Girder Deflection ‐ Girder 3

MDX base

MDX Improved

Larsa 3D

51

The vertical displacements all follow the same pattern and all have a normalized

mean error of 2% or less, which is a grade A level. This is a much better result than the

average grade reported by NCHRP for bridges with similar skew indexes.

Cross-Frame Forces. The average and worst-case grade of traditional 2D-grid

analysis models for cross-frame forces as reported by NCHRP is an F. A grade of F

means the normalized mean error is over 30%, reflecting unreliable accuracy and making

the results inadequate for design. Figure V.3 compares the unfactored cross-frame forces

due to noncomposite dead loads for the 2D-grid MDX base model and 3D-FEA LARSA

4D model for each member of the cross-frames. The cross-frames along bay 2 between

girders 2 and 3 are shown.

Figure V.4 Cross-Frame Axial Forces Along Bay 2 Between Girder 2 and Girder 3

‐250

‐200

‐150

‐100

‐50

0

50

100

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Axial Load

 (kips)

Cross‐frame Number

Top Chord

MDX Base

Larsa 3D

52

Figure V.4 (Continued) Cross-Frame Axial Forces Along Bay 2 Between Girder 2 and Girder 3

‐200

‐150

‐100

‐50

0

50

100

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Axial Load

 (kips)

Cross‐frame Number

Left Diagonal

MDX Base

Larsa 3D

‐100

‐50

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Axial Load

 (kips)

Cross‐frame Number

Left Bottom Chord

MDX Base

Larsa 3D

53

Figure V.4 (Continued) Cross-Frame Axial Forces Along Bay 2 Between Girder 2 and Girder 3

‐100

‐50

0

50

100

150

200

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Axial Load

 (kips)

Cross‐frame Number

Right Diagonal

MDX Base

Larsa 3D

‐100

‐50

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Axial Load

 (kips)

Cross‐frame Number

Right Bottom Chord

MDX Base

Larsa 3D

54

The pattern of cross-frame member forces along bay 2 is similar for both analysis

models. A spike in axial load in all members at cross-frames 11 and 12 can clearly been

seen. This illustrates the increased loads that occur at the obtuse corners. However, a

significant difference in cross-frame forces can clearly be seen between the two models

from the graphs. The normalized mean error in comparison to the 3D-FEA NLF

benchmark model ranges from 7.3% to 21.2% as seen in Table V.1. This would suggest

a grade of D for the worst case. However, some of the worst errors occur at the

controlling cross-frames with the highest forces. With percentages of error well over

30% for these critical cross-frames, the grade should be an F and the method deemed

unacceptable for calculating cross-frame forces.

Normalized Mean Error 

Top Chord  14.6%

Left Diagonal  13.9%

Left Bottom Chord  21.2%

Right Diagonal  14.0%

Right Bottom Chord  7.3%

Table V.1 Normalized Mean Error for Cross-Frame Forces in the 2D-Grid MDX Model

Flange Lateral Bending Stresses. The average and worst-case grade of

traditional 2D-grid analysis models for girder flange lateral bending stresses as reported

by NCHRP is an F. Responses to flange lateral bending are not provided in the MDX

results. In order to determine the flange lateral bending stresses, post-processing using

the cross-frame forces would need to be completed. Since the cross-frame forces results

received a grade of F for the 2D-grid model, calculation results for flange lateral bending

stresses would be inaccurate as well. Therefore, these calculations were not performed.

55

Girder Layover at Bearings. The average grade of traditional 2D-grid analysis

models for girder layover at bearings as reported by NCHRP is a C and the worst-case

grade is a D. MDX does not produce this output, therefore there is nothing to compare.

Girder layover at bearings would be calculated by hand using the differential deflection

output. Because girder layover and vertical displacements are directly related, NCHRP

gave them the same grades.

2D-Frame Models

Major-axis bending stresses. Figure V.5 compares the unfactored major-axis

bending stresses in the top and bottom flanges. The stresses are due to the dead loads,

including the weight of the deck, on the noncomposite steel section. The 2D-frame

LARSA 4D models (base and improved) are compared to the 3D-FEA NLF LARSA 4D

model. Both graphs show similar patterns for the bending stresses along the length of the

girder. The normalized mean error was not calculated due to the jagged lines in the 3D-

FEA NLF LARSA 4D model results; however the results appear to be within 12% error,

which results in a grade of B. It is surprising that the 2D-frame models have a higher

percentage of error; however, with a grade of B, they are considered acceptable for

analysis results.

Vertical Displacements. Figure V.6 compares the vertical displacements due to

noncomposite dead loads among the 2D-frame LARSA 4D models (base and improved)

and the 3D-FEA NLF LARSA 4D model along Girder 1 and Girder 3.

56

Figure V.5 Major-Axis Bending Stress of Girder 1

‐35

‐30

‐25

‐20

‐15

‐10

‐5

0

5

0.0 0.2 0.4 0.6 0.8 1.0

Stress (ksi)

Normalized Length

Top Flange

Larsa 2D base

Larsa 2D Improved

Larsa 3D

‐5

0

5

10

15

20

25

30

0.0 0.2 0.4 0.6 0.8 1.0

Stress (ksi)

Normalized Length

Bottom Flange

Larsa 2D base

Larsa 2D Improved

Larsa 3D

57

The vertical displacements all follow the same pattern and all have a normalized

mean error of less than 8%, which is a grade level of A to B. The critical maximum

deflection near mid-span is off by as much as 14% for both girders. However, with a

maximum difference of 2.37”, the error can be made up by specifying a large enough

haunch in the plans. These results are considered acceptable for the example bridge.

Figure V.6 Vertical Girder Displacements Along Girder 1 and Girder 3

‐20

‐18

‐16

‐14

‐12

‐10

‐8

‐6

‐4

‐2

0

0.0 0.2 0.4 0.6 0.8 1.0

Vertical Deflection (in)

Normalized Length

Vertical Girder Deflection ‐ Girder 1

Larsa 2D base

Larsa 2D improved

Larsa 3D

‐20

‐18

‐16

‐14

‐12

‐10

‐8

‐6

‐4

‐2

0

0.0 0.2 0.4 0.6 0.8 1.0

Vertical Deflection (in)

Normalized Length

Vertical Girder Deflection ‐ Girder 3

Larsa 2D base

Larsa 2D improved

Larsa 3D

58

Cross-Frame Forces. Figure V.7 compares the unfactored cross-frame forces

due to noncomposite dead loads for the 2D-frame LARSA 4D models (base and

improved) and 3D-FEA LARSA 4D model for each member of the cross-frames. The

cross-frames along bay 2 (between girders 2 and 3) are shown.

The pattern of cross-frame member forces in the graph along bay 2 is similar for

all models, except for the end cross-frames in the 2D-frame models. The spike in axial

load in all members at cross-frames 11 and 12 illustrates the increased loads that occur at

the obtuse corners and is much more accurately represented in both 2D-frame LARSA

4D models versus the 2D-grid MDX models.

Figure V.7 Cross-Frame Axial Forces Along Bay 2 Between Girder 2 and Girder 3

‐250

‐200

‐150

‐100

‐50

0

50

100

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Axial Load

 (kips)

Cross‐frame Number

Top Chord

Larsa 2D Base

Larsa 2D Improved

Larsa 3D

59

Figure V.7 (Continued) Cross-Frame Axial Forces Along Bay 2 Between Girder 2 and Girder 3

‐200

‐150

‐100

‐50

0

50

100

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Axial Load

 (kips)

Cross‐frame Number

Left Diagonal

Larsa 2D Base

Larsa 2D Improved

Larsa 3D

‐150

‐100

‐50

0

50

100

150

200

250

300

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Axial Load

 (kips)

Cross‐frame Number

Left Bottom Chord

Larsa 2D Base

Larsa 2D Improved

Larsa 3D

60

Figure V.7 (Continued) Cross-Frame Axial Forces Along Bay 2 Between Girder 2 and Girder 3

‐100

‐50

0

50

100

150

200

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Axial Load

 (kips)

Cross‐frame Number

Right Diagonal

Larsa 2D Base

Larsa 2D Improved

Larsa 3D

‐200

‐150

‐100

‐50

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Axial Load

 (kips)

Cross‐frame Number

Right Bottom Chord

Larsa 2D Base

Larsa 2D Improved

Larsa 3D

61

A significant difference in cross-frame forces can clearly be seen for the end and

first interior cross-frames between the improved 2D-frame LARSA 4D model and the 3D

benchmark. The “improvements” to the girder torsional stiffness appears to provide

inaccurate results for cross-frame forces near the girder ends and provide no noticeable

improvement over the 2D-frame base model. The normalized mean error ranges from

3.1% to 6.8% for the 2D-frame base model and from 6.0% to 12.0% for the 2D-frame

improved model as seen in Table V.2. This results in a grade of A and B for the two

models respectively.

Normalized Mean Error 

Member  Base  Improved

Top Chord  3.1% 7.8%

Left Diagonal  3.1% 12.0%

Left Bottom Chord  6.8% 6.0%

Right Diagonal  3.2% 12.0%

Right Bottom Chord  4.1% 11.7%

Table V.2 Normalized Mean Error for Cross-Frame Forces in the 2D-Frame LARSA 4D Models

Flange Lateral Bending Stresses. Figure V.8 compares the unfactored flange

lateral bending stresses due to noncomposite dead loads for the 2D-frame LARSA 4D

base model and 3D-FEA LARSA 4D model for Girders 1 and 3. The pattern of lateral

bending stresses is similar for both models in the top flange; however, they appear to

differ significantly in the second half of the bottom flange near the obtuse corner.

62

Figure V.8 Flange Lateral Bending Stress

‐8

‐6

‐4

‐2

0

2

4

0.0 0.2 0.4 0.6 0.8 1.0

Stress (ksi)

Normalized Length

Girder 1 Top Flange

Larsa 2D base

Larsa 3D

‐8

‐6

‐4

‐2

0

2

4

0.0 0.2 0.4 0.6 0.8 1.0

Stress (ksi)

Normalized Length

Girder 1 Bottom Flange

Larsa 2D base

Larsa 3D

63

Figure V.8 (Continued) Flange Lateral Bending Stress

‐4

‐3

‐2

‐1

0

1

2

3

4

5

6

0.0 0.2 0.4 0.6 0.8 1.0

Stress (ksi)

Normalized Length

Girder 3 Top Flange

Larsa 2D base

Larsa 3D

‐4

‐3

‐2

‐1

0

1

2

3

4

5

6

0.0 0.2 0.4 0.6 0.8 1.0

Stress (ksi)

Normalized Length

Girder 3 Bottom Flange

Larsa 2D base

Larsa 3D

64

The normalized mean error for the top flange is 13.2% for Girder 1 and 13.8% for

Girder 3, which results in a grade of C. In contrast, the normalized mean error for the

bottom flange is much worse at 90.3% for Girder 1 and 54.5% for Girder 3 and results in

a grade of F. It is important to note that the lateral bending stresses are relatively small,

making the percentage of error a bit inconsequential. The lateral bending stresses are less

than 2 ksi everywhere except near the end of girders near the obtuse end. Because the

first interior cross-frame in bay 1 near the obtuse corner was removed, this area could be

considered as having discontinuous cross-frames. As per AASHTO LRFD Bridge

Design Specifications C6.10.1, “in absence of calculated values of fl from a refined

analysis, a suggested estimate for the total unfactored fl in a flange at a cross-frame or

diaphragm due to the use of discontinuous cross-frame or diaphragm lines is 10.0 ksi for

interior girders and 7.5 ksi for exterior girders.” It continues, “in regions of the girders

with contiguous cross-frames or diaphragms, these values need not be considered”

(AASHTO, 2012). Therefore, 7.5 ksi for exterior and 10 ksi for the interior could

conservatively be assumed for the flange lateral bending stress near the obtuse corners.

The rest of the cross-frames are considered contiguous and therefore the flange lateral

bending stress can be considered negligible.

Girder Layover at Bearings. Figure V.9 compares the girder layover at bearings

under dead loads on the noncomposite girders for both 2D-frame models (base and

improved) and the 3D-FEA NLF model. The girder layover is the horizontal transverse

displacement measured at the top of the girder web with respect to the bottom of the

girder web. The bearings are fixed in the transverse direction.

65

The normalized mean error for the 2D-frame base model is 7.3% at the start

bearing and 9.1% at the end bearing, which results in a grade of B. The normalized mean

error for the 2D-frame improved model is 10.9% at the start bearing and 14.5% at the end

bearing, which results in a grade of B and C respectively. The significance of the

percentage of error depends on the cross-frame fit-up detailing method used and the

rotational capacity in the bearings. If NLF detailing is used, the bearings would need to

be able to handle the large transverse rotations.

Figure V.9 Girder Layover at Bearings

‐6 ‐4 ‐2 0 2 4 6

1

2

3

4

5

6

7

8

9

Girder Layover (in)

Girder Number 3D End

3D Start

2D‐Frame Imp End

2D‐Frame Imp Start

2D‐Frame Base End

2D‐Frame Base Start

66

3D-FEA Model with TDLF Detailing

The 3D-FEA TDLF model includes the initial locked-in cross-frame forces due to

TDLF detailing. Results are compared to the 3D-FEA NLF model that by default

assumes no initial cross-frame forces and the girder webs are plumb in the no-load case.

Results at different construction stages are also compared.

Major-axis bending stresses. As seen in Figure V.10, the major-axis bending

stress due to dead loads on the noncomposite girder sections are very similar. The

patterns are almost identical and the results at each data point differ very slightly.

Figure V.10 Major-Axis Bending Stress of Girder 1

‐35

‐25

‐15

‐5

5

0.0 0.2 0.4 0.6 0.8 1.0

Stress (ksi)

Normalized Length

Girder 1 Top Flange

Larsa 3D TDLF

Larsa 3D NLF

0

5

10

15

20

25

30

0.0 0.2 0.4 0.6 0.8 1.0

Stress (ksi)

Normalized Length

Girder 1 Bottom Flange

Larsa 3D TDLF

Larsa 3D NLF

67

Vertical Displacements. The vertical displacements due to dead loads on

noncomposite girder sections are also very similar between the 3D-FEA NLF and 3D-

FEA TDLF models. There is no noticeable difference between the exterior girders.

There is a slight difference between the interior girders that is most noticeable closest to

the centerline of the bridge (Girder 5). This is to be expected as per the results from

NCHRP Report 725. Girder 1 and Girder 3 vertical displacements are compared in

Figure V.11.

Figure V.11 Vertical Girder Displacements along Girder 1 and Girder 3

‐20

‐15

‐10

‐5

0

0.0 0.2 0.4 0.6 0.8 1.0

Vertical Deflection (in)

Normalized Length

Girder 1

Larsa 3D TDLF

Larsa 3D NLF

‐20

‐15

‐10

‐5

0

0.0 0.2 0.4 0.6 0.8 1.0

Vertical Deflection (in)

Normalized Length

Girder 3

Larsa 3D TDLF

Larsa 3D NLF

68

Cross-Frame Forces. Figure V.12 compares the unfactored cross-frame forces

due to the girder, cross-frame, and deck noncomposite dead loads for the 3D-FEA NLF

and TDLF models, and the forces due to only the steel girder and cross-frame

noncomposite dead loads for the 3D-FEA TDLF model. For most of the cross-frame

members with TDLF detailing, the maximum force in the cross-frames occurs during the

fit-up of the cross-frames with the girders. The girders are twisted and forced into an out

of plumb orientation and as dead loads are applied, the cross-frame forces are offset or

relieved as the girders twist back into the vertically plumb position under total dead

loads.

Figure V.12 Cross-Frame Axial Forces Along Bay 2 Between Girder 2 and Girder 3

‐250

‐200

‐150

‐100

‐50

0

50

100

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Axial Load

 (kips)

Cross‐frame Number

Top Chord

3D TDLF Steel Only

3D TDLF Steel+Deck

3D NLF Steel+Deck

69

Figure V.12 (Continued) Cross-Frame Axial Forces Along Bay 2 Between Girder 2 and Girder 3

‐200

‐150

‐100

‐50

0

50

100

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Axial Load

 (kips)

Cross‐frame Number

Left Diagonal

3D TDLF Steel Only

3D TDLF Steel+Deck

3D NLF Steel+Deck

‐100

‐50

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Axial Load

 (kips)

Cross‐frame Number

Left Bottom Chord

3D TDLF Steel Only

3D TDLF Steel+Deck

3D NLF Steel+Deck

70

Figure V.12 (Continued) Cross-Frame Axial Forces Along Bay 2 Between Girder 2 and Girder 3

‐100

‐50

0

50

100

150

200

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Axial Load

 (kips)

Cross‐frame Number

Right Diagonal

3D TDLF Steel Only

3D TDLF Steel+Deck

3D NLF Steel+Deck

‐150

‐100

‐50

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Axial Load

 (kips)

Cross‐frame Number

Right Bottom Chord

3D TDLF Steel Only

3D TDLF Steel+Deck

3D NLF Steel+Deck

71

Flange Lateral Bending Stresses. Figure V.13 compares the unfactored flange

lateral bending stresses due to noncomposite dead loads along Girder 1 for the 3D-FEA

NLF and 3D-FEA TDLF models. The TDLF model results include two construction

stages – steel girder and cross-frame dead loads only and steel plus deck dead loads.

Figure V.13 Flange Lateral Bending Stress Along Girder 1

‐8

‐6

‐4

‐2

0

2

4

6

0.0 0.2 0.4 0.6 0.8 1.0

Stress (ksi)

Normalized Length

Top Flange

3D TDLF Steel Only

3D TDLF Steel+Deck

3D NLF Steel+Deck

‐8

‐6

‐4

‐2

0

2

4

6

0.0 0.2 0.4 0.6 0.8 1.0

Stress (ksi)

Normalized Length

Bottom Flange

3D TDLF Steel Only

3D TDLF Steel+Deck

3D NLF Steel+Deck

72

As expected the flange lateral bending stress is minimal at the total dead load case

when using TDLF detailing. The results for the cross-frame forces and flange lateral

bending stress, clearly shows the advantage of using TDLF detailing.

Girder Layover at Bearings. Figure V.14 compares the girder layover at

bearings under dead loads on the noncomposite girders for the 3D-FEA NLF and TDLF

models with girders, cross-frames, and deck dead loads and the 3D-FEA TDLF model

with the steel girders and cross-frame dead loads only.

The girder layover under total noncomposite dead load using the TDLF detailing

is close to zero indicating the girder webs are nearly plumb in the total dead load

condition. Figure V.14 also indicates that the initial layover with the initial locked-in

cross-frame forces due to TDLF detailing is in the opposite direction than the layover due

to dead loads.

Field data was gathered during construction that indicates the girder layover. A 4’

level was used to measure how far out of plumb the girder webs were over the height of

the level. The data was used to determine the girder layover in relation to the full height

of the girder. The field data is believed to have been taken just after girder and cross-

frame erection; however, additional dead loads may have been present, such as the

precast deck panels. Figure V.15 compares the girder layover taken from field data

against the 3D-FEA TDLF models at the steel only load case and steel plus deck load

case. The field data mostly follows the same pattern along the bearing lines and appears

to be slightly less than the 3D-FEA TDLF steel only load case. The method of measuring

the field data leaves room for human error that would affect the accuracy; however, the

73

intent of this figure is to validate the behavior found in the 3D-FEA TDLF models with

the actual behavior of the bridge.

Figure V.14 Girder Layover at Start and End Bearings for 3D Models

‐6 ‐4 ‐2 0 2 4

1

2

3

4

5

6

7

8

9

Girder Layover (in)

Girder Number

Start Bearing Line

3D TDLF Steel+Deck

3D TDLF Steel Only

3D NLF Steel+Deck

‐4 ‐3 ‐2 ‐1 0 1 2 3 4

1

2

3

4

5

6

7

8

9

Girder Layover (in)

Girder Number

End Bearing Line

3D TDLF Steel+Deck

3D TDLF Steel Only

3D NLF Steel+Deck

74

Figure V.15 Girder Layover at Start and End Bearings for 3D TDLF Models and from Field Data

‐1 0 1 2 3 4

1

2

3

4

5

6

7

8

9

Girder Layover (in)

Girder Number

Start Bearing Line

3D TDLF Steel+Deck

3D TDLF Steel Only

Field Data

‐3.0 ‐2.5 ‐2.0 ‐1.5 ‐1.0 ‐0.5 0.0 0.5

1

2

3

4

5

6

7

8

9

Girder Layover (in)

Girder Number

End Bearing Line

3D TDLF Steel+Deck

3D TDLF Steel Only

Field Data

75

CHAPTER VI

CONCLUSIONS

Recommended Method of Analysis

The primary goal of this thesis is to determine the most efficient method of

analysis that accurately models the behavior of highly skewed steel plate I-girder bridges.

By implementing some improvements to 2D methods of analysis as described in NCHRP

Report 725, the hope is that 2D type methods could provide very accurate results. The

improvements appear to be fairly straightforward and simple in theory, but the

application ended up being far from simple.

NLF detailing is assumed by default in all software. However, for straight steel I-

girder bridges, TDLF detailing is the preferred option. If TDLF detailing is chosen, it is

very important to include any locked-in forces, which will counterbalance to a certain

extent the cross-frame loads, lateral deflections, and rotations caused by the dead loads.

If these locked-in forces are not included, the designer is assuming NLF detailing, which

can lead to overly conservative cross-frames forces and lateral flange bending stresses in

cases of very large skews and with the presence of nuisance stiffness cross-frames close

to the bearing line. Correctly modeling the behavior for the chosen detailing method is

important to develop an accurate and efficient design.

The 3D-FEA method is considered the most accurate method; however it still

comes with its own limitations. The biggest limitation is its complexity and the amount

of detail that is required to create a 3D-FEA model. Creating the model, running through

the analysis, and sorting through the massive amount of output data can be very time-

76

consuming. Modeling an already complex structure with a complex method and

interpreting the output can also increase the chance for human error.

The 3D-FEA method was assumed to be the most accurate for analyzing the

example and was used as the benchmark to compare all other models. The field data

measurements of girder layovers during construction provided a very loose validation of

the software. The results for deflections, rotations, stresses, and general overall behavior

of the highly skewed steel I-girder example bridge in the 3D-FEA model were as

expected. The girder ends twisted and rotated about the centerline of the bearing support

and the framing system generally behaved as previously described in the Theoretical

Background section. However, a full-sized test bridge with stress gauges and with

similar geometry is needed to truly validate the assumptions made in the creating and

analyzing the 3D-FEA model. This was considered outside the scope of this thesis.

The 2D-grid method of analysis for highly skewed I-girder bridges appears to be

the least accurate for calculating cross-frame forces for nuisance stiffness cross-frames

and lateral flange bending stresses. 2D-grid software, such as MDX and DESCUS, are

very powerful and useful tools when used in the right context. But until the software

companies update their software to include better equivalent estimations of the girder

torsional stiffness and equivalent beam stiffness of the cross-frames, designers need to do

a significant amount of post-processing calculations to check for additional cross-frame

forces and later flange bending stresses that may not have been captured in the 2D-grid

software analysis. NCHRP Report 725 repeatedly encouraged the bridge software

industry to implement these improvements into the software.

77

The 2D-frame method using the LARSA 4D software or something similar,

currently appears to be the most efficient method to use for highly skewed steel I-girder

bridge design. LARSA 4D contains a steel bridge module that makes creating the model

much easier, similar to the 2D-grid software models. Software like LARSA 4D that has

3D capabilities has the flexibility to manually override certain section properties and add

user specified loads to better model the behavior of highly-skewed steel I-girder bridges.

With this flexibility, the necessity of post-processing can be eliminated or greatly

reduced. Compared to 3D-FEA models, 2D-frame model output is much more

manageable and therefore less time-consuming and efficient for the designer.

General Recommendations and Future Work

Current Recommendations

Recommendations for designers currently analyzing highly skewed steel I-girder

bridges include:

Use the 2D-frame method with software that includes a bridge module for easily

creating the geometry.

Include improvements to the 2D-frame model in certain situations as outlined by

NCHRP Report 725. Adjusting the equivalent girder torsional constant that

includes warping capacity should be used for staggered cross-frame layouts.

When contiguous cross-frames are used throughout each span, a more detailed

analysis should be used to analyze the cross-frame forces and lateral flange

bending stresses near the girder ends.

78

Include initial locked-in cross-frame forces when TDLF or SDLF detailing is used

to get a more efficient design.

The 3D-FEA method should be used on a limited basis to verify behavior and

check localized stresses. It is a good tool to use to check cross-frame forces and

lateral flange bending stresses; however it is too cumbersome to use as an all-

encompassing analysis.

The 2D-grid method is not recommended for steel I-girder bridges with a high

skew index until improvements are made to the software or the designer decides

to accompany this analysis with extensive post-processing.

Future Work Considerations

Future work considerations and recommendations include:

Encourage the bridge software industry to implement the recommended

improvements to 2D-grid software.

Encourage the 3D-capable bridge software industry to implement improvements

to steel bridge design modules to accurately and easily include appropriate

locked-in cross-frame forces for SDLF or TDLF detailing and to automatically

update the internally calculated equivalent girder torsional stiffness.

Research and analyze more highly skewed bridges with different cross-frame

layouts. Specifically, analyze the girders and cross-frames with contiguous cross-

frame layouts and nuisance stiffness cross-frame near bearing supports. Adjust

the 2D analysis method improvements accordingly.

79

Research fit-up practices typically used in highly-skewed steel I-girder erection.

Determine at what point the fit-up forces for TDLF detailing become too large.

Research and analyze more innovative cross-frame configurations, including

partially skewed cross-frames, lean-on bracing, temporary bracing, and different

connection and bearing plate detailing.

80

REFERENCES

AASHTO. American Association of State Highway and Transportation Officials (2012). AASHTO LRFD Bridge Design Specifications, Customary U.S. Units, 6th Edition with 2012 and 2013 Interim Revisions and 2012 Errata.

AASHTO/NSBA G12.1. American Association of State Highway and Transportation

Officials / National Steel Bridge Alliance Steel Bridge Collaboration (2003). G 12.1 Guidelines for Design for Constructability.

AASHTO/NSBA G13.1. American Association of State Highway and Transportation

Officials / National Steel Bridge Alliance Steel Bridge Collaboration (2011). G 13.1 Guidelines for Steel Girder Bridge Analysis, 1st Edition.

Ahmed, M.Z. and Weisberger, F.E. (1996). “Torsion Constant for Matrix Analysis of

Structures Including Warping Effect,” International Journal of Solids and Structures, Elsevier, 33(3), 361-374.

Beckmann, F., and Medlock, R.D. Skewed Bridges and Girder Movements Due to

Rotations and Differential Deflections. Chavel, B., Peterman, L., and McAtee, C. (2010). Design and Construction of the

Curved and Severely Skewed Steel I-Girder East-West Connector Bridges over I-88. 27th Annual International Bridge Conference 2010. IBC-10-24.

MDX (2013). MDX NetHelp. http://www.mdxsoftware.com/. April 2013. NCHRP Report 725. Transportation Research Board of the National Academy of

Sciences (2012). National Cooperative Highway Research Program Report 725: Guidelines for Analysis Methods and Construction Engineering of Curved and Skewed Steel Girder Bridges. Project 12-79.

Parsons Corporation (2011). Design Plans for SR-114 Geneva Road, Roadway

Widening: Geneva Road over UPRR & UTA. Signed by registered Professional Engineer: Haines, Steve. Owner: Utah Department of Transportation.

Schaefer, A.L. (2012). Crossframe Analysis of Highly-Skewed and Curved Steel I-Girder

Bridges. Thesis submitted to the University of Colorado Denver. ProQuest LLC.

81

APPENDIX A

Appendix A includes calculations for the equivalent beam stiffness of the cross-

frames and the steel plate girder design calculations that include the equivalent girder

torsional stiffness constant used in the 2D models.

UCD Master's Thesis ‐ Skewed Steel I‐Girders

Equivalent Cross‐Frame Stiffness Calculations

Calculations By:  K. Dobbins

10/18/2013

Equivalent Beam Stiffness for Cross‐Frames

Constants:

Es = 29000 ksi

Crossframe Type 1

height btwn working pts 85 in

width btwn working pts 127 in

weight ‐ top chord 26.5 plf

weight ‐ bott chord 26.5 plf

weight ‐ diagonals 26.5 plf

weight ‐ connection plates 0.726 kips

Total weight 1.548

STAAD Output

Nodal Displacements

  Horizontal Vertical Resultant

Node L/C X (in) Y (in) Z (in) (in) rX (rad) rY (rad) rZ (rad)

1 1 UNIT LOAD COUPLE 0 0 0 0 0 0 0

2 1 UNIT LOAD COUPLE 0 0 0 0 0 0 0

3 1 UNIT LOAD COUPLE ‐0.00056 0.00084 0 0.00101 0 0 0

4 1 UNIT LOAD COUPLE 0.00056 0.00084 0 0.00101 0 0 0

5 1 UNIT LOAD COUPLE 0.00028 0.00021 0 0.00035 0 0 0

1 2 UNIT SHEAR 0 0 0 0 0 0 0

2 2 UNIT SHEAR 0 0 0 0 0 0 0

3 2 UNIT SHEAR ‐0.00042 0.0024 0 0.00244 0 0 0

4 2 UNIT SHEAR 0.00042 0.0024 0 0.00244 0 0 0

5 2 UNIT SHEAR 0.00042 0.00104 0 0.00113 0 0 0

Ɵ 0.00001318 rad

Ieq 28250 in4

Aseq 7.268 in2

Rotational

82

UCD Master's Thesis ‐ Skewed Steel I‐Girders

Equivalent Cross‐Frame Stiffness Calculations

Calculations By:  K. Dobbins

10/18/2013

Crossframe Type 2

height btwn working pts 83.5 in

width btwn working pts 104 in

weight ‐ top chord 59.5 plf

weight ‐ bott chord 59.5 plf

weight ‐ diagonals 59.5 plf

weight ‐ connection plates 1.633 kips

Total weight 3.359

Nodal Displacements

  Horizontal Vertical Resultant

Node L/C X (in) Y (in) Z (in) (in) rX (rad) rY (rad) rZ (rad)

1 1 UNIT LOAD COUPLE 0 0 0 0 0 0 0

2 1 UNIT LOAD COUPLE 0 0 0 0 0 0 0

3 1 UNIT LOAD COUPLE ‐0.0002 0.00025 0 0.00033 0 0 0

4 1 UNIT LOAD COUPLE 0.0002 0.00025 0 0.00033 0 0 0

5 1 UNIT LOAD COUPLE 0.0001 0.00006 0 0.00012 0 0 0

1 2 UNIT SHEAR 0 0 0 0 0 0 0

2 2 UNIT SHEAR 0 0 0 0 0 0 0

3 2 UNIT SHEAR ‐0.00013 0.00077 0 0.00078 0 0 0

4 2 UNIT SHEAR 0.00013 0.00077 0 0.00078 0 0 0

5 2 UNIT SHEAR 0.00013 0.00035 0 0.00037 0 0 0

Ɵ 0.00000479 rad

Ieq 62510 in4

Aseq 16.557 in2

Rotational

83

UCD Master's Thesis ‐ Skewed Steel I‐Girders

Equivalent Cross‐Frame Stiffness Calculations

Calculations By:  K. Dobbins

10/18/2013

Crossframe Type 3

height btwn working pts 85 in

width btwn working pts 127 in

weight ‐ top chord 26.5 plf

weight ‐ bott chord 34 plf

weight ‐ diagonals 34 plf

weight ‐ connection plates 0.726 kips

Total weight 1.713

Nodal Displacements

  Horizontal Vertical Resultant

Node L/C X (in) Y (in) Z (in) (in) rX (rad) rY (rad) rZ (rad)

1 1 UNIT LOAD COUPLE 0 0 0 0 0 0 0

2 1 UNIT LOAD COUPLE 0 0 0 0 0 0 0

3 1 UNIT LOAD COUPLE ‐0.00056 0.00075 0 0.00093 0 0 0

4 1 UNIT LOAD COUPLE 0.00044 0.00075 0 0.00087 0 0 0

5 1 UNIT LOAD COUPLE 0.00022 0.00016 0 0.00027 0 0 0

1 2 UNIT SHEAR 0 0 0 0 0 0 0

2 2 UNIT SHEAR 0 0 0 0 0 0 0

3 2 UNIT SHEAR ‐0.00042 0.00194 0 0.00199 0 0 0

4 2 UNIT SHEAR 0.00033 0.00194 0 0.00197 0 0 0

5 2 UNIT SHEAR 0.00033 0.00081 0 0.00088 0 0 0

Ɵ 0.00001176 rad

Ieq 31641 in4

Aseq 9.521 in2

Rotational

84

UCD Master's Thesis ‐ Skewed Steel I‐Girders

Equivalent Cross‐Frame Stiffness Calculations

Calculations By:  K. Dobbins

10/18/2013

Crossframe Type 4 (End)

height btwn working pts 85 in

width btwn working pts 230.125 in

weight ‐ top chord 26.5 plf

weight ‐ bott chord 26.5 plf

weight ‐ diagonals 26.5 plf

weight ‐ connection plates 0.818 kips

Total weight 2.372

Nodal Displacements

  Horizontal Vertical Resultant

Node L/C X (in) Y (in) Z (in) (in) rX (rad) rY (rad) rZ (rad)

1 1 UNIT LOAD COUPLE 0 0 0 0 0 0 0

2 1 UNIT LOAD COUPLE 0 0 0 0 0 0 0

3 1 UNIT LOAD COUPLE ‐0.00102 0.00275 0 0.00294 0 0 0

4 1 UNIT LOAD COUPLE 0.00102 0.00275 0 0.00294 0 0 0

5 1 UNIT LOAD COUPLE 0.00051 0.00069 0 0.00086 0 0 0

1 2 UNIT SHEAR 0 0 0 0 0 0 0

2 2 UNIT SHEAR 0 0 0 0 0 0 0

3 2 UNIT SHEAR ‐0.00138 0.00918 0 0.00928 0 0 0

4 2 UNIT SHEAR 0.00138 0.00918 0 0.00928 0 0 0

5 2 UNIT SHEAR 0.00138 0.00366 0 0.00391 0 0 0

Ɵ 0.00002400 rad

Ieq 28104 in4

Aseq 4.917 in2

Rotational

85

UCD Masters Thesis ‐ Geneva Road

Steel Plate Girder Design

By: Konlee Dobbins

10/7/2013

Steel Girder Layout and Section Properties

Overview

Skewed straight steel plate girder bridge

Simple span bridge

Design Code: AASHTO LRFD Bridge Design Specifications, 5th edition, 2010

Live Loads: HL‐93 and Tandem as per AASHTO (no permit trucks considered)

Design Parameters Value Unit Comments

Roadway width 82 ft

Barrier width 2.000 ft

Left Sidewalk Width 6.667 ft

Right Sidewalk Width 10.667 ft

Deck width 103.333 ft

Number of Design Lanes 6 integer part of roadway width/12'

Span 1 Length 254.4375 ft Brg to Brg

Haunch and Top Flange 6 in Bott of deck to bott of top flange, constant

Assumed Avg Haunch 4.5 in

Deck thickness 8.5 in

Overhang deck thickness at edge 8.5 in

FWS Asphalt overlay 3.43 in 40 psf

sacrificial deck thickness 0.5 in

Design deck thickness 8 in

Barrier area 4.667 ft2

Barrier weight 0.7 klf includes 0.05 klf for chain‐link fence

Concrete strength, f'c 4 ksi

Reinf steel fy 60 ksi

Structural steel fy 50 ksi M270 Grade 50W

Reinf Conc unit weight 0.15 kcf includes extra 0.005 for rebar

Conc Unit weight for Ec 0.145 kcf

Asphalt unit weight 0.14 kcf

Steel unit weight 0.49 kcf

Es 29000 ksi

Ec 3644 ksi 33000*wc^1.5*(f'c)^0.5

n = Es/Ec 8.0

Future ADTT 2500 trucks/day

Layout

Number of girders 9

Girder spacing 11.0625 ft

Overhang 7.417 ft

overhang/spacing 0.67

86

UCD Masters Thesis ‐ Geneva Road

Steel Plate Girder Design

By: Konlee Dobbins

10/7/2013

Section Properties

Section 1 at ends

Section 2 between S1 & S3

Section 3 at midspan

Steel girder only S1 S2 S3

top flange width 30 30 30 in

top flange thickness 1.5 1.75 1.75 in

web thickness 0.75 0.75 0.75 in D/tw<150 140.0

web height 105 105 105 in

Bottom flange width 32 32 32 in

Bottom flange thickness 1.75 2 2.25 in

Girder depth 108.25 108.75 109 in

Girder Area 179.75 195.25 203.25 in2

top flange cog, y 107.5 107.875 108.125 in

web cog, y 54.25 54.5 54.75 in

bottom flange cog, y 0.875 1 1.125 in

Girder COG from bott, y 50.95 51.32 49.54 in A*y/AGirder COG from top of deck, y ‐65.548 ‐65.685 ‐67.709 in

Major Moment of inertia, Ix 357558 403157 423492 in4

Stop 6240 7019 7122 in3

Sbot 7017 7856 8548 in3

Top Flange Moment of Inertia, Iytf 3375 3938 3938 in4

Bott Flange Moment of Inertia, Iybf 4779 5461 6144 in4

Minor Moment of Inertia, Iy 8157 9403 10085 in4

Torsional Constant, J 105.7 153.7 189.9 in4

Warping Constant, Cw 22487732 26133499 27473512 in6

Shear Modulus, G 11154 11154 11154 ksi

p 0.001344 0.001504 0.001630 1/in

87

UCD Masters Thesis ‐ Geneva Road

Steel Plate Girder Design

By: Konlee Dobbins

10/7/2013

Equivalent Torsional Constant S1 S2 S3

Brace length 1  Lb 25 N/A N/A in

Equivalent Torsional Constant, Jeq 1122714 N/A N/A in4

Brace length 2  Lb 272 N/A N/A in

Equivalent Torsional Constant, Jeq 9610 N/A N/A in4

Brace length 3  Lb 247 247 247 in

Equivalent Torsional Constant, Jeq 11627 13549 14278 in4

Brace length 4  Lb 240 240 240 in

Equivalent Torsional Constant, Jeq 12308 14340 15109 in4

Brace length 5  Lb 227 227 227 in

Equivalent Torsional Constant, Jeq 13743 16008 16863 in4

Brace length 6  Lb 293 293 293 in

Equivalent Torsional Constant, Jeq 8299 9682 10212 in4

Brace length 7  Lb 248 248 248 in

Equivalent Torsional Constant, Jeq 11534 13442 14165 in4

Brace length 8  Lb 245 245 245 in

Equivalent Torsional Constant, Jeq 11816 13768 14508 in4

Brace length 9  Lb 251 251 251 in

Equivalent Torsional Constant, Jeq 11263 13126 13833 in4

Brace length 10  Lb 276 276 276 in

Equivalent Torsional Constant, Jeq 9337 10888 11480 in4

Brace length 11  Lb 45 N/A N/A in

Equivalent Torsional Constant, Jeq 346604 N/A N/A in4

Brace length 12  Lb 41 N/A N/A in

Equivalent Torsional Constant, Jeq 417508 N/A N/A in4

Brace length 13  Lb 37 N/A N/A in

Equivalent Torsional Constant, Jeq 512630 N/A N/A in4

Brace length 14  Lb 33 N/A N/A in

Equivalent Torsional Constant, Jeq 644403 N/A N/A in4

Brace length 15  Lb 29 N/A N/A in

Equivalent Torsional Constant, Jeq 834392 N/A N/A in4

88