LIVE LOAD DISTRIBUTION FACTORS IN TWOGIRDER
BRIDGE SYSTEMS USING PRECAST TRAPEZOIDAL
UGIRDERS
by
Salahudin A. Mensah
B.S., University of Utah, 2006
A thesis submitted to the
University of Colorado Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Civil Engineering
2010
This thesis for the Master of Science
degree by
Salahudin A. Mensah
has been approved
by
Chengyu Li
2 *2
Date
Mensah, Salahudin A. (M.S., Civil Engineering)
Live Load Distribution Factors in TwoGirder Bridge Systems Using Precast
Trapezoidal UGirders
Thesis directed by Assistant Professor Stephan A. Durham
ABSTRACT
Vehicular live load is considered to be a design parameter that is of considerable
importance in terms of economy and safety in the design of highway bridges. Live
load distribution factors (LDFs) critically impact the selection of member sizes,
number of members, and detailing requirements. Consequently, strength and
serviceability requirements are often governed by live load distribution.
The Lever Rule method is often used to determine live load distribution
factors in twogirder bridge systems because of the ranges of applicability implicit in
the simplified equations of the AASHTLO LRFD Bridge Design Specifications.
Since the Lever Rule method for yielding LDFs is typically more conservative than
the simplified equations, less economical bridge designs are produced as a result.
It is therefore necessary to assess whether the Lever Rule method of
determining the lateral distribution of live load is closely reflective of the actual
response of twogirder bridge systems using precast trapezoidal Ugirders.
This study aims to determine the degree to which the AASHTO LRFD Bridge design
code is overly conservative in the application of live load distribution factors for two
girder bridge systems.
Comparisons between live load distribution factors (LDFs) from Finite
Element Analysis (FEA) and the Lever Rule Method of the AASHTO LRFD show
that the Lever Rule Method produces values for LDFs that are closely reflective of
the actual response for shear and to a lesser degree for flexure. The degree of
conservatism for the values produced by the AASHTO LRFD code is 7% for shear
and 25% for moment.
This abstract accurately represents the content of the candidates thesis. I recommend
its publication.
Signed]
StephaifA. Durham
i
ACKNOWLEDGEMENT
I would like to express gratitude to Dr. Stephan A. Durham for the opportunity to
work under his direction and guidance. I would also like to express gratitude to Dr.
Kevin L. Rens and Dr. Chengyu Li for their participation on my graduate advisory
committee. I would like to acknowledge Clint M. Krajnik for his contributions and
expertise.
TABLE OF CONTENTS
Figures.....................................................................ix
Tables....................................................................xiii
Chapter
1. Introduction..............................................................1
1.1 Use of Prestressed Concrete Trapezoidal UGirders in Colorado............1
1.2 TwoGirder Bridge Systems.......................................2
2. Problem Statement........................................................6
2.1 Introduction.............................................................6
2.2 Significance of Research................................................8
2.3 Research Objective......................................................8
3. Literature Review.......................................................10
3.1 Introduction............................................................10
3.2 Methods of Analysis....................................................10
3.3 Approximate Method.....................................................12
3.4 AASHTO LRFD Simplified Equations.......................................13
3.5 Development of AASHTO LRFD Live Load Distribution Factor
Equations...............................................................17
3.6 Refined Method.........................................................23
3.6.1 Finite Element Method................................................24
3.6.2 Grillage Analogy Method..............................................27
vi
3.7 Analytical Studies on Live Load Responses of Box Girder
Bridges.................................................................29
3.8 Experimental Studies on Live Load Responses of Concrete Girder
Bridges.................................................................39
4. Research Plan...........................................................48
5. Finite Element Analysis.................................................50
5.1 Finite Element Modeling Technique.......................................50
5.2 Experimental Study.....................................................51
5.2.1 Introduction..........................................................51
5.2.2 Description of Model Bridge...........................................51
5.2.3 FE Model of Experimental Bridge.......................................54
5.2.4 Live Load Placement of Test Truck.....................................57
5.2.5 Experimental vs. FEA Load Distribution Factors........................58
5.3 Finite Element Model Validation.........................................64
5.4 Finite Element Model...................................................65
5.4.1 Modeling Parameters...................................................67
5.4.2 Moving Load Analysis.................................................70
5.5 Results of Finite Element Analysis.....................................71
5.5.1 Results of Finite Element Analysis for Moment.........................71
5.5.2 Results of Finite Element Analysis for Shear..........................73
5.5.3 Multiple Presence Factors.............................................74
vii
5.5.4 Maximum Response Quantities for Moment and Corresponding Live Load
Position...........................................................74
5.5.5 Maximum Response Quantities for Shear and Corresponding Live Load
Position...........................................................78
5.6 Discussion of Results from Finite Element Analysis....................87
6. AASHTO LRFD Load Distribution Factors..................................91
6.1 Introduction..........................................................91
6.2 Definition of Lever Rule Method.......................................94
6.2.1 Live Load Distribution Factors for Moment and Shear Based on Lever
Rule...............................................................95
6.2.2 Calculated LDFs using Lever Rule Method............................96
7. Comparison of Results..................................................99
7.1 Finite Element Model vs. AASHTO LRFD Lever Rule.......................99
8. Design Comparison.....................................................101
8.1 Design Results using Lever Rule and FE Model.........................102
8.2 Comparison of Results................................................104
9. Conclusions...........................................................105
References
viii
LIST OF FIGURES
Figure
1.1 CDOT Standard Trapezoidal UGirder Sections (Aspire 2008)....................2
1.2 Trinidad Viaduct Typical Deck Section (125 Trinidad Viaduct
Typical Deck Section)........................................................3
1.3 Application of Two Girder Bridge System (125 Trinidad Viaduct
Typical Deck Section)........................................................4
3.1 Illustration of Approximate Method (Tabsh and Sahajwani 1997)..............12
3.2 Finite Element Model (Zokaie 2000).........................................19
3.3 Variation of Girder Spacing in Database Bridges (Zokaie 2000)..............19
3.4 Variations of Distribution Factors with Key Parameters (Zokaie 2000).......20
3.5 Variations of Distribution Factors in Database Bridges (Zokaie 2000).......22
3.6 Schematic of the Finite Element Process (Felippa 2004).....................25
3.7 The Idealization Process in FEM (Felippa 2004).............................26
3.8 Eccentric Beam Model (Chung et al. 2005)...................................27
3.9 Eccentric Beam Model (Barr et al. 2005)....................................27
3.10 Beam and Slab Decks in Grillage Analogy (Hambly 1975).....................28
3.11 Comparison of LDFs in Negative Moment Region (Samaan et al. 2002).........32
3.12 Distribution Factors for Moment LRFD Formula and
Grillage Analysis (Song et al. 2003)........................................34
3.13 Typical FE Mesh for Noncomposite and Composite Bridge
(Samaan et al. 2005)........................................................38
IX
3.14 Comparison of Measured and Calculated Midspan
Moments (Barr et al. 2001).................................................42
3.15 Elevation View of Study Bridge (Hughs and Idriss
2006)......................................................................45
5.1 Eccentric Beam Model.......................................................50
5.2 Crosssection and Plan of Study Bridge (Hughs and Idriss
2006)......................................................................53
5.3 Testtruck Dimensions and Axle Locations (Hughs and Idriss
2006)......................................................................53
5.4 Dimensions of U54 Girder (TxDOT)..........................................55
5.5 Layout of FE Model if Experimental Bridge in SAP2000......................56
5.6 Closed U54 Shape..........................................................56
5.7 Truck Positions to Maximize Moment and Shear (Hughs 2004).................58
5.8 Illustration of Live Load Causing Deflection in Span 5....................59
5.9 Distribution Factors from FEM vs. Position of Live Load
Truck......................................................................60
5.10LDFs from FEM vs. Position of Live Load Truck (Girder 1)..................61
5.11 LDFs from FEM vs. Position of Live Load Truck (Girder 2).................61
5.12 LDFs from FEM vs. Position of Live Load Truck (Girder 3).................62
5.13 LDFs from FEM vs. Position of Live Load Truck (Girder 4).................62
5.14 LDFs from FEM vs. Position of Live Load Truck (Girder 5).................63
5.15 LDFs from FEM vs. Position of Live Load Truck (Girder 6).................63
x
5.16FE Model Typical Deck Section (125 Trinidad
Viaduct Construction Drawings)..............................................69
5.17U78 Girder Section (125 Trinidad Viaduct Construction Drawings)............69
5.18 Maximum Stress Response Quantity in the Bottom Fiber......................75
5.19 Corresponding Stress Quantity in Adjacent Beam............................75
5.20Maximum and Corresponding Stress Responses in Both Beams................76
5.21 Position of HL93 Loading Configuration for Maximum Moment................76
5.22 Position of HL93 Loading Configuration for Moment........................77
5.23 Position of HL93 Loading Configuration for Moment........................77
5.24 Maximum and Corresponding Force Responses
in Both Beams at Interior Supports..........................................79
5.25 Corresponding Force Quantity in Adjacent Beam.............................79
5.26 Maximum and Corresponding Force Responses in
Both Beams at Interior Supports.............................................80
5.27 Maximum Reaction at interior Support Location (Two Trucks)................81
5.28 Loading Configuration Causing Maximum Reaction at Interior
Support.....................................................................82
5.29 Loading Configuration Causing Maximum Reaction at Interior
Support.....................................................................82
5.30 HL93 Loading Configuration Causing Maximum Reaction at Interior
Support.....................................................................83
5.31 2nd Largest Reaction at interior Support Location.........................83
xi
5.32 Loading Configuration Causing 2nd Largest Reaction at Interior
Support.....................................................................84
5.33 Loading Configuration Causing 2nd Largest Reaction at Interior
Support.....................................................................84
5.34 Loading Configuration Causing 2nd Largest Reaction at Interior
Support.....................................................................85
5.35 Maximum Reaction at Abutment Support Location.............................85
5.36 Loading Configuration Causing Maximum Reaction at Abutment Support
Location....................................................................86
5.37 Loading Configuration Causing Maximum Reaction at Abutment Support
Location....................................................................86
5.38 Loading Configuration Causing Maximum Reaction at Abutment Support
Location....................................................................87
5.39 Loading Configuration Causing Maximum Reaction at Abutment Support
Location....................................................................89
5.40 Loading Configuration Causing Maximum Reaction at Abutment Support
Location....................................................................90
6.1 AASHTO LRFD Design Truck (AASHTO LRFD 2007)................................92
6.2 Notational Model for Applying Lever Rule (AASHTO LRFD 2007)................94
6.3 Notational Model for Applying Lever........................................96
7.1 Moment Distribution Factors for Girder 1 and Girder 2......................100
7.2 Shear Distribution Factors for Girder 1 and Girder 2......................100
LIST OF TABLES
Table
2.1 AASHTO LRFD Common Deck Superstructures
(AASHTO 2007)...............................................................15
2.2 AASHTO LRFD LDF Equations for Concrete
Deck on Concrete Spread Box Beams (AASHTO 2007).............................16
5.1 Live Load Distribution Factors from Experimental Study
(Hughs 2004)................................................................59
5.2 Percent Difference between Experimental Bridge and FEA Model...............65
5.2a Relationship between Transverse and Longitudinal Stiffness.................66
5.3 Various Model Parameters...................................................68
5.4 Calculated Live Load Distribution Factors..................................88
5.5 Calculated Live Load Distribution Factors..................................88
6.1 Multiple Presence Factors (AASHTO 2007)....................................93
6.2 LDF Equations for Moment in Exterior Beams................................93
6.3 LDF Equations for Shear in Exterior Beams.................................94
6.4 AASHTO Live Load Distribution Factors...................................96
7.1 Finite Element vs. AASHTO Live Load Distribution Factors....................99
8.1 Model Parameters and Live Load Distribution Factors.........................102
8.2 Design Results Comparing All Models.......................................103
8.3 Design Results from Model A w/Decreased Number of Strands.................103
8.4 Design Results from Model A w/Decreased Girder Depth......................103
8.5 Design Results from Model A w/lncreased Span Length.......................103
xiii
1. INTRODUCTION
1.1 Use of Precast Trapezoidal UGirders
Due to an increase in complexity of the U.S. highway system, the number of
horizontally curved bridges has increased. Curved girders are widely used in bridge
superstructures today. Horizontally curved bridges allow bridges to be constructed in
areas where unique characteristics exist, such as highway onoff ramps, complicated
interchanges, or river crossings where site space is limited. Span lengths of curved
geometry bridges have also increased due to technological advancements in the
design and fabrication of curved girders.
With increases in the use of curved girders throughout the U.S. highway system,
the use of precast, trapezoidal box girders are also increasing. In the state of
Colorado, designers benefit from using a broad range of structure types for its
bridges. Although consultants are permitted to use the structural type including
materials fabricated using structural steel, castinplace concrete, or precast concrete 
that will result in the most optimal design for the conditions at a particular bridge site,
precast concrete trapezoidal Ugirders are quickly becoming the preferred alternative
for superstructure type when complicated geometry is required.
The Colorado Department of Transportation (CDOT) has adopted the use of
precast, trapezoidal girders for longsegment construction. The success of two
projects namely the Park Avenue Ramp over 125 and the 1225 Ramp over Parker
Road helped stimulate the development of standards for precast trapezoidal Ugirder
sections by CDOT. The standards shown in Figure 1.1 were developed between 1995
and 2000 to be used for either straight or horizontally curved segments (Aspire 2008).
The standard sections can be pretensioned, posttensioned, or a combination of both.
irv irir
Figure 1.1 CDOT Standard Trapezoidal UGirder Sections (Aspire 2008)
1.2 TwoGirder Bridge Systems
TwoGirder bridge systems are used frequently in Colorado to construct cost effective
structures of increasing complexity. The structures are often longspanning and are
usually located in areas where the consideration for aesthetics is significant.
Examples where twogirder bridge systems, using precast trapezoidal Ugirders have
been successfully used include the recently completed 1270 Ramp over 125 in
Denver, the State Highway SH(58) Ramp A Bridge in Golden, and the 125 Trinidad
Viaduct, which is currently under construction. A common typical section used in
2
twogirder bridge systems is illustrated in Figure 1.2. The typical section was used in
the construction of ramp sections on the 125 Trinidad Viaduct. The sections
consisted of both straight and curved members.
Figure 1.2 Trinidad Viaduct Typical Deck Section (125 Trinidad Viaduct
Construction Drawings)
Twogirder bridge systems using precast trapezoidal Ugirders are often used to
simplify confusing and stressful visualizations w hich are characteristic of large,
multilevel ramps and interchanges. Simplification is achieved by using fewer girders
which, subsequently, require fewer piers to support the girders. Since fewer girders
are used, the number of pier columns within each pier can also be reduced. Due to
the decrease in a number of structural elements, the design of the system is simplified
and the aesthetics of the system are vastly improved. Usual application of a two
3
girder bridge system using trapezoidal Ugirders is shown in the typical section of
Figure 1.3.
(LOCKNG AAD STATON)
(SIMILAR TO "UNIT 3". EXCEPT AS SHOWN)
Figure 1.3 Application of Two Girder Bridge System (125 Trinidad Viaduct
Construction Drawings)
In twogirder bridge systems such as those shown in Figures 1.2 and 1.3, both
girders are classified as exterior beams for design purposes. The bending moments
and shear forces in the members due to the distribution of live load are often
calculated by using the Lever Rule method as specified in the current AASHTO
LRFD. Using the Lever Rule to determine live load distribution generally results in
more conservative values than what may be expected of the structure in actuality and
less economical bridge designs may be produced as a result.
4
This thesis investigates the behavior and response of a twogirder bridge system
using precastprestressed trapezoidal Ugirders under the influence of vehicular live
load. Consequently, live load distribution factors for moment and shear are
developed. The distribution factors are compared with live load distribution factors
produced by the AASHTO LRFD Lever Rule method. The comparison encompasses
the effects of live load on parameters that include flexural response, slenderness, and
span length.
5
2. PROBLEM STATEMENT
2.1 Introduction
Vehicular live load significantly impacts the design of highway bridges. Lateral
distribution factors for live load moment and shear are commonly determined using
AASHTO LRFD Bridge Design Specifications (AASHTO LRFD 2007). In two
girder bridge systems, both girders are classified as exterior beams for design
purposes. The bending moments and shear forces in precastprestressed trapezoidal
Ugirders due to the lateral distribution of vehicular wheel loads (referred to as live
load in the AASHTO LRFD) in twogirder bridge systems are calculated by using the
Lever Rule method or a refined method of analysis as specified in the current
AASHTO LRFD. The Lever Rule method is considered an approximate distribution
factor method. The method consists of the statical summation of moments about one
support to determine the reaction at another support, assuming that the supported
transverse deck is hinged at interior supports. Because the transverse cross section is
statically determinate, the method uses direct equilibrium in determining the lateral
load distribution.
For typical beamslab bridge systems where the supporting components consists
of Precast Concrete Boxes and the type of deck consists of castinplace concrete slab
or a precast concrete deck slab, the distribution of live loads per lane for moment and
shear in exterior longitudinal beams are shown in Eq. 2.1 (in LRFD Table 4.6.2.2.2d
1) and Eq. 2.2 (in LRFD Table 4.6.2.2.3b1) respectively.
6
9 e 9interior
(2.1)
e = 0.97 +
de
2&5
for 0 < de < 4.5 and 6.0 < 5 < 18.0
9 & 9interior
e
0.8 + ^
10
for 0 < de < 4.5
(2.2)
where g is distribution factor for both moment and shear, e is correction factor for
distribution, gmtenoris interior distribution factor, S is spacing of supporting
components, and de is horizontal distance from the centerline of the exterior web of
exterior beam at the deck level to the interior edge of curb or traffic barrier.
These simplified live load distribution factor (LDF) equations were developed
based on a comprehensive study that takes into consideration many key parameters
such as beam spacing, span length, longitudinal beam stiffness, and slab thickness.
The limitations of the simplified equations suggest that the more a bridge cross
section deviates from the sections used in the development of the simplified
equations, the less accurate the equations become.
The use of the Lever Rule to determine the fraction of wheel load to a beam
element is required as a result of not meeting the ranges of applicability specified for
use of the simplified equations Eq. 2.1 and Eq. 2.2. This generally results in more
7
conservative values for LDFs when compared to the simplified equations. Since the
typical section described in this study does not conform to the limitations of which
the simplified equations were derived, application of the lever rule is required.
2.2 Significance of Research
The safety and economy in the design of highway bridges are impacted significantly
by its response to vehicular live load. As such, vehicular live load on highway bridges
is considered to be a design parameter that is of considerable importance when
selecting member sizes, number of members, and detailing requirements.
Consequently, strength and serviceability requirements are often governed by live
load distribution.
Since the Lever Rule method for yielding LDFs is typically more conservative
than the simplified equations, less economical bridge designs are produced as a result.
Therefore it is necessary to assess whether the Lever Rule method of determining the
lateral distribution of live load is closely reflective of the actual response of two
girder bridge systems using precast trapezoidal Ugirders.
2.3 Research Objective
The objective of this study is to (1) investigate the behavior and response of a two
girder bridge system using precastprestressed trapezoidal Ugirders under the
influence of vehicular live load. As a result of this investigation, (2) live load
distribution factors for moment and shear will be developed. (3) The new distribution
8
factors will then be compared with LDFs produced by the AASHTO LRFD Lever
Rule method.
Ultimately, the goal of this study is to determine the degree to which the
AASHTO LRFD Bridge design code is overly conservative in the application of live
load distribution factors to two girder bridge systems. This information will enable
bridge designers to produce more economical bridge designs while continuing to
maintain a reasonable margin of safety.
9
3. LITERATURE REVIEW
3.1 Introduction
This section consists of a review and composition of the available literature to
document the research relevant to the effects of vehicular live load in highway bridge
design. The studies reviewed in this section place particular emphasis on the
evaluation and development of the AASHTO LRFD live load distribution equations
as they relate to prestressed concrete spread box girder bridges.
3.2 Methods of Analysis
When determining the effects of live load on highway bridges, the methods in which
the structural analysis is performed are critical. Guidance concerning methods of
analysis in determining force effects and modeling requirements suitable for the
design and evaluation of highway bridges is provided in the current AASHTO LRFD
(2007). Although most of the guidelines pertain to elastic analysis methods, inelastic
analysis provisions are also provided. The analysis methods are divided into two
basic categories considered Approximate Method and Refined Method respectively.
Many researchers have used refined analysis methods such as Finite Element
Analysis (FEA) in the development of more simplified or approximate methods of
analysis.
Zokaie et al. (1991) conducted a parametric and sensitivity study on the
distribution of vehicular wheel loads in highway bridges using FEA methods which
resulted in the development of simplified live load distribution equations that have
10
been adopted in the current AASHTO LRFD bridge design specifications. Aswad
(1994) examined the impact of the LRFD simplified equations on the design of beam
and slab Igirder superstructure type using both finite element and grillage analogy
refined methods. The research concluded that using refined methods of analysis
results in significantly reduced values for live load distribution factors when
compared to the simplified equations. In specific, the percentage reduction in the
amount of strands required and the minimum required release strength for interior
beams decreased by a factor of two. The reductions are attributed to better analytical
tools and the study concludes that refined analytical tools should be adopted as a
viable means of analyzing structures due to its economic advantages.
Chen and Aswad (1996) conducted an extensive review of the AASHTO LRFD
distribution factor formulas for modem prestressed concrete bridges including spread
box girder bridges containing large spanto depth ratios. The study describes a
general refined analysis procedure for predicting vehicular live load response to
simply supported bridges. The study compared the AASHTO LRFD simplified
equations to the results of the refined analysis. The analyses concluded that use of the
FEA may result in a LDF reduction of 18% or less for Ibeams and to a lesser extent
for spread box beams when compared to the use of the simplified equations of the
AASHTO LRFD.
Tabsh and Sahajwani (1997) developed a simple approximate method to
determine live distribution factors for irregular slab on Ibeam bridges. The method
considers the transverse and longitudinal effects of truck loads and is based on
isolating strips of the decks slab in the transverse direction directly under the wheel
11
loads. The strips are then treated as beams on elastic supports as shown in Figure 3.1.
The accuracy of the approximate method was verified using a detailed refined three
dimensional finite element model.
Wh eel Loads
Far Abutment
~Slab Strip
rSteel Beam
Near Abutment
Figure 3.1 Illustration of Approximate Method (Tabsh and Sahajwani 1997)
3.3 Approximate Method
The mechanism in which loads are transferred from the deck slab to the longitudinal
members to the supports is complex and involves multiple parameters in three
dimensions. To facilitate the ease of evaluation, load distribution factors were
introduced. Load distribution factors (LDF) are considered an approximate method in
that it enables bridge designers to determine the fraction of load that is supported by a
single, onedimensional longitudinal member, using simplified equations. In other
words, load distribution factors reduce the spatial dimensionality of a three
dimensional system by uncoupling the transverse and longitudinal load effects
thereby reducing the overall complexity in the analysis of the system. The fraction of
12
load response (moment, shear, reaction, displacement) that a girder receives is
determined by multiplying the maximum single girder response to the load
distribution factor. For example, an individual girders flexural response to live load
is determined by first calculating the moment caused by a lane of traffic (truck plus
lane or tandem plus lane) acting on the girder and applying to it a live load
distribution factor.
3.4 AASHTO LRFD Simplified Equations
The current AASHTO LRFD Bridge Design Specifications provide approximate
simplified equations for determining lateral live load distribution factors. These
equations take into account several parameters which studies have shown
significantly influences the distribution of live load in highway bridges (Zokaie et al.
2000). The parameters considered in the development of the simplified equations in
the current code include superstructure type (i.e. wood deck on steel beams), cross
section type (i.e. deck and girder type), load response type (i.e. moment, shear,
reaction), number of traffic lanes loaded at a time, and girder spacing. Span length,
relative stiffness between the deck slab and the girders, thickness of the deck slab,
girder location (i.e. interior vs. exterior beams), number of beams, continuity
boundaries (i.e. simple span vs. continuous), diaphragms or crossframe conditions,
skewed supports, and curvature were also considered to be important in the
development of the simplified equations.
13
The simplified equations provided in the code contain a set of LDF formulas for
moment and shear for both exterior and interior beams. The equations use the
variables discussed previously to determine the live load distribution factors. The
factors are further modified to account for skew effects (i.e. shear is increased and
moment is decreased). Table 3.2 provided from the current AASHTO LRFD contains
simplified equations for moment and shear for interior and exterior girders
respectively. These equations are applicable for crosssection type c, which
corresponds to a concrete deck on concrete spread box beams. Partial crosssection
types from AASHTO LRFD Table 4.6.2.2.11 are provided in Table 3.1. From Table
3.2, it is observed that the simplified equations were developed to accommodate a
large amount of variability in bridge types and geometry. This is illustrated in the
ranges of applicability provided in the last column of Table 3.2. When a structure
meets the ranges of applicability developed for use with the simplified equations, the
implications suggest that the distribution of live load determined by the simplified
equations are accurate for that particular structure type and geometry. However, the
converse also implies that the simplified equations become less accurate and therefore
inapplicable as the ranges of applicability are exceeded.
14
Table 3.1 AASHTO LRFD Common Deck Superstructures (AASHTO 2007)
Supporting Components___________Typo Of Deck
Typical CrossSection
Steel Beam
Castinplace concrete slab,
precast concrete slab, steel
grid glued/spiked panels
stressed wood
[l
Jl
Closed Steel or Precast Concrete
Boxes
C'astinplace concrete slab
P,
dj
j]
(b)
Open Steel or Precast Concrete
Boxes
Castinplace concrete slab,
precast concrete deck slab
k
Jl
u
(O)
CastinPlace Concrete Muhicell
Box
Monolithic concrete
k
Jl
I
m
CastinPlace Concrete Tee Beam
Monobthic concrete
k
Jl
lOJ
(e)
IT
Precast Solid. Voided or Cellular
Concrete Boxes with Shear Keys
Castinplace concrete
overlay
J]
ii niii
Precast Solid, Voided or Cellular
Concrete Box with Shear Keys and
with or without Transverse Post
Tensionmg
Integral concrete
k
TTnl
P/T
(g)
15
Table 3.2 AASHTO LRFD Live Load Distribution Factor Equations for
Concrete Deck on Concrete Spread Box Beams (AASHTO 2007)________________
C ategory Distiibution Factor Formulas Range of Applicability
Live Load Distribution per Lane for Moment in Interior Beams One Design Lane Loaded ( 5 \*3Sl Sd Uo) 1.12 or) Two oi More Design Lanes Loaded ( S Sd (6.3J (i2.or J 6 0 < S < 18.0 20< I < 140 18id <65 ,Yk>3
Use Lew Rule S > 18 0
Lir e Load Distiibution per Lane for Moment in Exterior Longitudinal Beams One Design Lane Loaded Lever Rule Two oi More Design Lanes Loaded 0 < d, < 4 5 6 0< S <18 0
Use Lever Rule S >18.0
Live Load Distribution per Lane for Shear in Interior Beams One Desien Lane Loaded (W Two oi More Desien Lanes Loaded (Sf ]' (7 4.) (12.01 J 6 0< S <18.0 20 < I < 140 18 < d < 65 Nh>i
Use Lew Rule S > 18.0
Lir e Load Distiibution per Lane for Shear in Exterior Beams One Design Lane Loaded Lever Rule Two oi More Design Lanes Loaded 0
Use Lew Rule S > 18 0
Where,
d = depth of beam (in.)
de = horizontal distance from the centerline of the exterior web of the
exterior beam to the interior edge of curb or traffic barrier (in.)
L = span Length (ft)
Nb = number of beams
S = spacing of beams (ft)
16
In addition to the limitations imposed by the ranges of applicability in Table 3.2,
the simplified equations are further restricted for use by the following limitations as
per the AASHTO LRFD Specifications (2007):
Width of deck must be constant
Number of beam must be greater than four (unless otherwise stated)
Stiffness in all beams are the same
All beams are parallel to each other
The roadway part of the overhang, de, does not exceed 3 ft (0.91
meters)
Curvature in plan is such that the arc span divided by the girder radius
is less than 0.3 radians
Cross section is consistent with one of the cross sections shown in
AASHTO LRFD Table 4.6.2.2.11
3.5 Development of AASHTO LRFD Live Load Distribution Factor
Equations
The AASHTO LRFD live load distribution factor equations were developed based on
the need to produce more accurate distribution factors than those contained in the
previous AASHTO Standard specifications (AASHTO Standard 1996). The current
simplified equations were developed largely as a result from the National Cooperative
Highway Research Program (NCHRP) 1226 project (Zokaie et al 1991). The models
17
used in the development of the LRFD simplified equations contain constant spacing,
girder inertia, skew, and did not include consideration for the effects of end nor
intermediate diaphragms. For models that were continuous, all spans were modeled
equally in length.
In order to achieve a higher level of accuracy in the new equations, the effects of
various parameters including spacing, span length, girder stiffness, St. Venant
torsional constant, number of girders, and edge distance were studied. These
parameters are important in that they allow for the incorporation of a broader range of
bridges than those used in the previous standard specification equations. A database
containing several hundred existing bridges from various states was randomly
assembled for the purposes of conducting a parametric study. The bridges were
comprised of the following superstructure types: reinforced concrete Tbeam, steel
and prestressed concrete Igirder, multicell box girder, spread box beams, adjacent
box beams, and slab bridges. Finite element models were assembled for comparison
and validation of the existing S/D formulas from the AASHTO Standard
specifications as well as the new LRFD simplified equations. A typical idealized
model used in the study is shown in Figure 3.2.
The database was used to assess the range and variation of each parameter and to
identity key parameters. As a sample, Figure 3.3 shows the frequency of variation of
the database with respect to girder spacing. Using the database of existing bridges,
average values of the parameters were determined thus a hypothetical bridge was
created. The hypothetical bridge was used in a sensitivity study to determine the
importance of key parameters on live load distribution.
18
Figure 3.2 Finite Element Model (Zokaie 2000)
091 1.85 27i 366 457
Girder Spacing (m)
Figure 3.3 Variation of Girder Spacing in Database Bridges (Zokaie 2000)
19
Wheel Line! per Girder
3
2 5
0 46 1 37 1 85 2.37 2.8 3.99
GWer Spacing (m)
2
f
0
I
\
0  i ........* 
0 0236 0.0661 0.1406 0.3542 0.5602
Girder Stillness .Kg (m4)
Moment Mullple lane
Momonl One line
Sheer Multiple Lane
MShear One Lane
4Moment Mian pie Lane
Moment One Lane
AShear Mutbpie Lane
KShear One Lane
2.5
2.5 f
2 *
0 5 
0 . 1
152 176.5 229
Slab Thickness (mm)
Moment Multiple Lane
Moment One Lane
A Shear Multiple Lane
XShear One Lane
6.9 11.69 14.63 20.11 25.6 45.72
Span langti (m)
Momert Multiple Lane
Mower* one Lane
ShearMultiple Lane
xShearOne Lana
Figure 3.4 Variations of Distribution Factors with Key Parameters (Zokaie
2000)
20
This was achieved by varying one parameter at a time from maximum to
minimum while holding all other parameters to a fixed average value. The process
was then repeated for all parameters. The conducted study revealed that the key
parameters for each bridge type are girder spacing, span length, girder stiffness, and
slab thickness. Figure 3.4 illustrates the variation in distribution factors with key
parameters.
The simplified formulas were developed by modeling each key parameter by an
exponential function of the form axb where a and b are constants and x is the value
of each parameter. In order for each parameter to be considered separately, it is
assumed the effects of the various parameters are mutually exclusive. The final
distribution factor equation was modeled by an exponential formula which assumed
the following form:
g = (a)(Sb')(Lb2){tbi )()
(3.1)
where,
g = wheel load distribution factor
S = girder spacing
L = span length
t = slab thickness
21
Fine tuning of the formulas for certain parameters allowed the formula to take the
following form:
g = c + (a)(Sbi)(Lb2 )()
(3.2)
The constant c is used to account for an adjustment in the mean value while
preserving the overall accuracy of the formula. Figure 3.5 shows more accurate
predictions of live load distribution factors obtained from the developed simplified
equations. In order to assure conservative results, the constants in the equations were
adjusted such that the average value computed using the formulas to the distribution
factor obtained using the refined analysis was always greater than 1.0.
Moment OlalrlbuUon Ratio* lor Beam and Slab Bildgn
Figure 3.5 Variations of Distribution Factors in Database Bridges (Zokaie 2000)
3.6 Refined Method
22
When bridges do not meet the limitations and conditions specified for use with the
approximate method, a refined method of analysis becomes necessary. Some of the
more common refined analysis methods suggested by the AASHTO LRFD Bridge
Design Specifications include the following: Classical force and displacement
methods, Finite difference method, Finite element method, Foldedplate method, and
the Grillage analogy method.
The AASHTO LRFD also suggests the following guidelines when using refined
methods of analysis:
A minimum of five, and preferably nine, nodes per beam span should be used
For FEA involving plate and beam elements, relative vertical distances should
be maintained between various elements
For both positive and negative flexure, assume the slab is fully effective for
stiffness
For FEA, elements should be properly discretized to account for shear lag
Aspect ratio of finite elements and grid panels should not exceed 5.0
For grillage analysis, composite properties of longitudinal members assuming
an effective slab width should be used
St. Venant torsional constant, J, must be determined
The two most common methods used to study the behavior of bridges and in the
determination of live load distribution include the Finite Element Method and the
Grillage Analogy Method. In this section, the application of grillage analogy and
23
finite element method are reviewed in the context of lateral distribution of vehicular
live loads.
In past practices, grillage analysis methods have primarily been used over FE
methods in determining overall bridge behavior. The method gained its popularity
due its simplicity and ease of use. The method is also relatively inexpensive and easy
to comprehend and implement. However, inherent limitations of grillage analysis are
prevalent due the structural members of the system being in one plane. Limitations
such as its inability to model important physical effects, such as shear lag, distortion
of beam members, torsional and warping effects, and local bending effects,
significantly hinders its use when a model of increasing complexity is warranted. Due
to increases in computer technology and advances in threedimensional FE software,
the FEM is replacing grillage analysis methods as the more practical refined method
of analysis in the analysis and design of highway bridges.
3.6.1 Finite Element Method
Felippa (2004) schematizes the role of Finite Element Method (FEM) in numerical
solution in Figure 3.6. The figure illustrates the following key simulations steps:
idealization, discretization, and solution. Idealization consists of the translation of the
physical system to a mathematical model as shown in Figure 3.7. In the illustration,
the physical system shown as roof truss is directly idealized by the mathematical
model represented by a pinjointed bar assembly.
24
IDEALIZATION DISCRETIZATION SOLUTION
Figure 3.6 Schematic of the Finite Element Method Process (Felippa 2004)
Felippa (2004) describes the finite element method as the dominant
discretization technique in structural mechanics that can be interpreted from either a
physical or mathematical perspective. He explains the concept of the physical FEM as
the breakdown of the mathematical model into nonoverlapping components of
simpler geometry referred to as finite elements. Each elements response is expressed
in terms of a finite number of degrees of freedom which is characterized as the value
of an unknown function corresponding to a set of nodal points. The response of the
mathematical model is therefore an approximation of the discrete model which is
obtained by the collection and assemblage of all elements of the system.
25
Physical System
Figure 3.7 The Idealization Process in FEM (Felippa 2004)
The Eccentric Beam Model is a FEA threedimensional modeling technique
that is often employed in the analysis of highway bridges. This modeling technique
which is illustrated in Figure 3.8 and Figure 3.9 considers the primary structural
members which include the deck slab and longitudinal girders for the distribution of
vehicular live load. The eccentric beam model idealizes the concrete slab as
quadrilateral (four node) shell elements and the beams as twonode isoparametric
beam elements. Rigid links are used to account for composite behavior between the
centers of the slab and beam elements. Chen and Aswad (1996), Mabsout et al.
(1997), Chan and Chan (1999), Chen (1999), Barr et al. (2001), Chung et al. (2005)
and Eom and Nowak (2006) studied the effects of live load distribution on highway
bridges using FEA eccentric beam modeling techniques.
26
Quadrilateral Shell
Figure 3.8 Eccentric Beam Model (Chung et al. 2005)
T Frame I Dement
911 Element
I RipU Lsk
Figure 3.9 Eccentric Beam Model (Barr et al. 2001)
3.6.2 Grillage Analogy Method
Hambly (1975) describes the Grillage Analogy Method as a way to idealize complex
threedimensional bridge behavior as twodimensional structural systems. This is
achieved by modeling the superstructure as skeletal members that span longitudinally
as beams and transversely as slab segments which are connected and restrained at
their joints. The transverse and longitudinal members lie within the same plane as
shown in Figure 3.10.
27
Figure 3.10 Beam and Slab Decks in Grillage Analogy (Hambly 1975)
The grillage analogy method employs classical matrix structural analysis
methods in which the unknowns are expressed in terms of displacements at the joints.
The solution consists of determining the compatibility of deformations at the nodes in
order to restore equilibrium in a linearelastic system. When using a grillage analogy
to determine force effects, the following considerations must be taken into account:
the geometry of the members must be arranged such that the correct magnitude for
moments and deflections are achieved, the correct member properties are used such
that the bending and torsional inertias accurately represent the actual structure, the
effects of cantilevers are appropriately addressed, and the appropriate bearing
supports are selected. Where the grillage is formulated with proper regard to the
nature of the bridge superstructure, the true behavior of the superstructure is
28
accurately predicted. However, if due care is not taken, inherent inaccuracies will
exist (OBrien and Keogh 1999). A prevalent source of inaccuracy that may occur in
the grillage is the discontinuity between moments being balanced by a discontinuity
of torques in the beams in the opposite direction. Excessively large discontinuities
may result in inaccurate grillage results. Although inaccuracies may exist in the
analysis of bridges using grillage methods, an advantage to using grillage analogy is
that it allows shear and moment values to be obtained directly without the need for
stress integration as in Finite Element Methods. Aswad (1994), Song et al. (2003) and
Gupta and Misra (2005) have used grillage analogy methods with success in
determining live load contributions in prestressed concrete girder bridges.
3.7 Analytical Studies on Live Load Responses of Box Girder Bridges
Lounis and Cohn (1995) studied the effects of optimization procedures for the design
and analysis of prestressed concrete box girder bridges. The analyses were carried out
using nonlinear finitestrip methods and finitedifference techniques. An approximate
analysis method used to determine live load moment was proposed. The study took
into account a concrete box girder bridges sensitivity to changes in deck thickness as
well as changes in top and bottom flanges.
Sennah and Kennedy (1999) conducted a parametric study using FEA to analyze
120 bridges of varying geometries. The bridges considered in the study consisted of
composite steelconcrete multicell box girder superstructures. The parameters
considered ranged from number of cells, span length, number of lanes, and the effects
of cross bracings. An experimental analysis was carried out using a simply supported
29
threecell box girder bridge. The tests were used to validate results from the FEA
model. The parametric study was used to develop moment and shear live load
distribution factors subjected to AASHTO truck loading. St. Venant torsional
stiffness for composite multicell box girders was also evaluated. The results from the
investigation concluded the following:
1. An empirical formula was deduced that took into account the effects of
torsional resistance using the ratio between web thickness to bottom flange
thickness.
2. The lateral load transfer distribution of moments, shear forces, and deflections
is significantly enhanced by the effects of crossbracing systems.
3. Using the proposed LDF simplified equations resulting from this study would
result in more economical and reliable bridge designs.
Samaan et al. (2002) analyzed the effects of the distribution of wheel loads on
continuity in the analysis of steel spreadbox girder bridges. The study consisted of a
parametric evaluation of 60 continuous bridge prototypes of various geometries
subjected to a number of loading conditions. The parameters considered in the study
include span length, number of spread boxes, and number of lanes. An FEA model
was created using ABAQUS software. The model was then substantiated using
previously verified experimental results obtained by testing five composite concrete
deck steel threecell bridges indicated in Sennah and Kennedy (1999). The loading
conditions considered in the study include the AASHTO design truck and the
30
AASHTO lane load. These conditions were used to determine maximum moment at
midspan and maximum shear at the supports. To determine distribution factors for
maximum positive and negative moment regions, maximum stresses were obtained
from simple beam bending formulas.
The composite section was used to determine moments of inertia. Once
distribution factors were obtained, results from the parametric study were used to
determine the effects of key parameters on the distribution factors used in the design.
The distribution factors were then modified and calibrated and used to develop new
simplified distribution factor equations. A comparison of stress distribution in the
negative moment regions for the three bridge prototypes is illustrated in Figure 3.11.
The image shows good agreement between the FEA model used in the study with the
proposed simplified equations. The illustration also reveals that other methods such as
the AASHTO simplified equations yield overly conservative results and in some
instances, unconservative results. The study did not consider the effects of varying
span lengths or bridges with three or more spans. Additionally, the study did not take
into consideration the effects of skew and curvature. The study concludes that the key
parameters affecting load distribution in continuous steel spreadbox girder bridges
are the span length, number of lanes, and the number of girders.
31
1.0
2L1033b
Jlj60fh
Bricge prototype
4l~206b
Figure 3.11 Comparison of LDFs in Negative Moment Region (Samaan et al.
2002)
Song et al. (2003) found that the AASHTO LRFD imposes strict limitations
which include requirements for a prismatic cross section, large span lengthtowidth
ratio, and small plan curvature on its use of its live load distribution factors for the
design of highway bridges. The study aimed to investigate the live load distribution
characteristics of concrete box girder bridges and the limitations imposed by the
AASHTO LRFD specifications. Song et al. assert that the strict limitations place
severe restrictions on the routine design of box girder bridges, the types of which are
frequently constructed in the state of California and of which the geometry frequently
exceeds the limitations. In specific, the study analyzed concrete box girder bridges
taking into consideration the limitations discussed above by comparing distribution
factors from refined and simplified levels of analysis. The refined method of choice
32
for this study is the grillage analogy method. The grillage model was substantiated by
a three dimensional FEA model. The model consisted of twospan continuous box
girder bridge with span lengths of 99 ft (30.2 meters) each. The cross section of the
bridge contained four cells with equal spacing between girders, an overall depth of
4.1 ft (1.2 meters), and an edgetoedge width of 40 ft (12.2 meters). The intent of the
study was to assess whether a three dimensional refined analysis model is warranted
whenever the specification limitations are exceeded. The model considered loading
conditions applied offcenter to the girders so that torsion, bending moments, and
shear forces are maximized. In comparison to the FEA model, the results of the
grillage model were within 5% for bending moment. The closeness of the two models
suggests that the grillage model is able to simulate with reasonable accuracy the
discontinuity in the shear force at the supports. The grillage model was then used to
determine distribution factors under AASHTO HL93 vehicular loading. The
accuracy of the LRFD specification equations were assessed using an acceptance
ratio. The acceptance ratio compares the distribution factor obtained from grillage
analysis to the distribution factors calculated using the simplified specification
equations. An acceptance ratio less than 1 indicates that the LRFD specification
equation is conservative. The study considered one restriction at a time in the
following order: (1) nonprismatic crosssections; (2) small plan aspect ratio; and (3)
curved bridges. Figure 3.12 illustrates histogram plots of the distribution factors for
moment for LRFD specification equations and grillage respectively. The comparison
illustrated here indicates that with respect to larger plan aspect ratios, the distribution
33
factors from the grillage analysis are lower than the distribution factors calculated
using the LRFD formulas.
Figure 3.12 Distribution Factors for Moment for LRFD Formula and Grillage
Analysis (Song et al. 2003)
The results of the study indicate that for bridges outside of the specification limits,
the current LRFD specification equations for concrete boxgirder bridges generally
provide conservative values for bending moment and shear forces. The study
concludes that the current LRFD simplified equations are applicable to concrete box
girder bridges that are non prismatic in typical section (nonparallel girders), to
bridges with a lengthtowidth ratio approaching unity, and to curved bridges with
angular changes that exceed 34 degrees. The study disclaims the overall conservatism
of the LRFD simplified equations due to the limited set of bridges used in the study.
34
Huo et al. (2004) introduced and reviewed a simplified method for the calculation
of live load distribution factors for moment. The method, known as Henrys Method,
which has been in use in the State of Tennessee for nearly 40 years, was originally
developed in the 1960s. The method assumes that all beams, interior and exterior
alike, have an equal distribution of live loads. The method achieves simplicity by
limiting the input requirements to roadway width, number of traffic lanes, number of
girders, and the multiple presence factor of the bridge. The advantages to the use of
the Henrys Method lie within its ease of use as well as its flexibility in range of
application. The study was conducted by selecting twentyfour actual bridges to be
built or that were currently under construction. The 24 bridges encompassed six
superstructure types including concrete box beam bridges, Ibeam/ bulbtee bridges,
castinplace concrete Tbeam bridges, castinplace concrete box beam bridges, steel
Ibeam bridges, and steel open boxgirder bridges. The study focused on the effects of
the superstructure type as well as the effects of key parameters including span length,
beam spacing, slab thickness, and beam stiffhess.A finite element model was created
using ANSYS software to investigate the live load distribution factors for moment in
seven bridges representing each of the superstructure types mentioned above.
Modeling techniques were varied depending on the superstructure type. Each model
was then subjected to several loading conditions with AASHTO HL93 loading in
order to maximize moments in the bridge. A calculation of the distribution factor was
obtained by taking the ratio of the controlling moment to the critical moment
response of a single beam loaded by one truck axle loads. A comparison of the
distribution factors from the refined analysis, AASHTO Standard specification
35
formulas, AASHTO LRFD formulas, and Henrys Method was performed for each
bridge superstructure type. The results of the study on the effects of superstructure
type indicate that for precast concrete box beam bridges, distribution factors from
Henrys method were smaller than those calculated using AASHTO Standard
equations but reasonably close with the values obtained from the LRFD equations.
For precast concrete Ibeam and bulbtee bridges, Henrys method yielded smaller
values than LRFD equations and significantly smaller than AASHTO Standard
specification formulas. In general, the distribution factors obtained using Henrys
method stayed within close proximity to the values obtained from the AASHTO
methods. Modification factors were introduced to adjust the distribution factors to the
appropriate superstructure type. The study concludes that Henrys method produces
values for live load distribution that are reasonable and reliable for a wide range of
bridges.
Samaan et al. (2005) studied the effects of live load distribution on curved
continuous steel spreadbox girder bridges. The study consisted of a parametric
evaluation of 240 continuous curved box girder bridges of various geometries
subjected to a number of loading conditions. It is assumed that the construction of the
bridges is fully shored therefore the model assumes full composite action between the
concrete deck and the steel box beams. The parameters considered in the study
include effects of crossbracing, spantoradius of curvature ratio, span length,
number of boxes, and number of lanes, web slope, number of bracings, and truck
loading type. An FEA model was created using ABAQUS software. A typical finite
element mesh is shown in Figure 3.13. The model shown in the figure was verified
36
and substantiated using previously verified experimental results obtained by testing
five 1/12 linear scale composite concrete deck steel threecell bridges indicated in
Sennah and Kennedy (1999). The loading conditions considered in the study include
the AASHTO design truck and the AASHTO lane load. These conditions were used
to determine maximum moment at midspan and maximum shear at the supports. This
study defines the load distribution factor as the ratio of the maximum value of a
structural response quantity obtained from the FEA models used in the parametric
study to the value obtained for an idealized straight girder bridge with two equal
spans. Upon obtaining distribution factors from the FEA models, results from the
parametric study were used to determine the effects of key parameters on the
distribution factors used in the design.
37
Figure 3.13 Typical FE Mesh for Noncomposite and Composite Bridge (Samaan
et al. 2005)
As a result of the parametric study, expressions were developed for the
distribution factors for both dead and live load in terms of maximum longitudinal
tensile and compressive stresses. Additionally, expressions were developed for
deflection as well. The derived empirical formulas were calibrated using best fit least
squares statistical methods. The study did not consider the effects of varying span
lengths or bridges with three or more spans. Additionally, the study did not take into
consideration the effects of skew and curvature. The results from the investigation
concluded the following:
38
1. The empirical formulas were found to be a simple and reliable method to
determine live load responses in curved continuous composite boxgirder
bridges.
2. Using the developed expressions will result in the conservative design of
bridges subjected to AASHTO HL93 loading.
3. Key parameters identified by the study include spantoradius of curvature
ratio, span length, number of boxes, and number of lanes.
4. Web slope effects on live load distribution were determined to be
negligible.
3.8 Experimental Studies on Live Load Responses of Concrete Girder
Bridges
Laman and Schwarz (2000) conducted an experimental study of live load on three
prestressed concrete Igirder bridges. The study aimed to develop service level
stresses and subsequently live load distribution factors from fieldbased response data
for comparison to code based distribution factors for the purposes of facilitating more
accurate design and evaluation of prestressed concrete Igirder bridges. The bridges
included in the study consisted of three simple span structures with span lengths of
33.8 ft (10.3 meters), 76.4 ft (23.3 meters), and 102.7 ft (31.3 meters), with outtoout
cross sections widths of 45.5 ft (13.87 meters), 44.5 ft (13.57 meters), and 43.6 ft
(13.3 meters), respectively. The superstructure of the bridges consisted of six PADOT
Igirders. There were slight differences in girder type, spacing, overhang, and skew.
However, the parametric differences in each bridge in terms of number of girders,
39
slab thickness, support conditions, and surface quality was kept at a minimum so that
a comparison of test results between the three bridges may be obtained with
reasonable results. The live load test was carried out using demountable strain
transducers placed longitudinally at midspan underneath each girder. This enabled the
bridges response to both a normal truck as well as weighed test truck to be obtained.
Both static and dynamic tests were conducted on the bridges. Service level stresses
were obtained using cumulative distribution functions (CDF) of measured stresses in
each girder. CDFs for each girder and each bridge were extrapolated to determine
stress levels corresponding to several occurrence intervals. Normal vehicular traffic
and tandemaxle test trucks were measured in terms of strain at the bottom flange of
each girder near midspan and used to determine girder distribution factors and live
load stresses in each bridge. The distribution factors obtained from AASHTO LRFD
and AASHTO Standard specification formulas exceed the largest onelane
distribution factor value measured from normal truck traffic. The AASHTO code
equations are also conservative when compared to the twolane distribution factor
values obtained from experimental data. The study found that a twoweek recurrence
interval of live load stresses in the girders bottom flange exceeds the stress induced
by the HS20 design truck in some interior girders.
Barr et al. (2001) presents an evaluation of live load distribution factors for
moment for a series of threespan prestressed concrete girder bridges. The response of
one bridge under a static live load test was used to evaluate the reliability of a finite
element model. The model was varied 24 times and used to evaluate live load
distribution factor equations for flexure from the AASHTO LRFD Specifications,
40
AASHTO Standard Specifications, and the Ontario Highway Bridge Design Code.
Another use of the FEA models were to investigate the effects of lifts, intermediate
diaphragms, end diaphragms, continuity, skew angle, and load type on distribution
factors. The study was part of an investigation on highperformance concrete in
prestressed concrete girders. As part of the study, Washington State Department of
Transportation (WSDOT) designed and performed a live load test on the
SRI 8/SR156 Bridge. The bridge consisted of three spans with lengths of 80.1 ft
(24.4 meters), 136.8 ft (41.7 meters), and 80.1 ft (24.4 meters). The angle of skew
was 40 degrees. The design compressive strength at release and at 56 days was 7.4 ksi
(51 MPa) and 9.9 ksi (68.9 MPa) respectively. The Washington State girders
W74MG were used for all girders and made continuous over the piers. Differences in
camber were accounted for by adding lifts to the deck at the girders. A twoaxle dump
truck weighing 35.5 kips (158 kN) was used in the load test. The truck was placed at
various locations both transversely as well as longitudinally so as to determine the
bridges maximum response to live load. Measured moment values were calculated
from recorded strain values during loading. A finite element model was then
developed using SAP2000 software which consisted of frame elements, shell
elements, and rigid constraints so that strains measured from the live load test could
be accurately reproduced. Figure 3.14 compares the FEA and measured midspan
moments for girders when a single truck is placed at midspan. The image shows that
the FE moment arid the measured moment are close in value. The largest discrepancy
is less than 6% in maximum moment in each girder.
41
Figure 3.14 Comparison of Measured and Calculated Midspan Moments (Barr
et al. 2001)
The FEA was validated by the close agreement between the calculated and
measured moments for both transverse and longitudinal profiles. Twentyfour model
variations were then produced to determine the effects of lifts, intermediate
diaphragms, end diaphragms, continuity, skew angle, and load type and in all cases,
the live load distribution factors calculated from the LRFD specification equations
were conservative. The values calculated were up to 28% larger than the values
obtained from the FE model. The presence of lifts, continuity, and end diaphragms
significantly decreased the calculated distribution factors, while the adding
intermediate diaphragms had almost no effect on the distribution factors. The addition
of continuity had the effect of increasing or decreasing the distribution factors. In all
cases, the presence of skew decreased the distribution factors. Lane loading produced
distribution factors that were consistently an average of 10% lower than those
42
calculated from truck loading. The study suggests that using the distribution factors
from the FE model would result in reduction in release strength from 7.4 ksi (51
MPa) to 6.4 ksi (44.1 MPa) or alternatively, the bridge could have been designed for
39% higher live load.
Barnes et al. (2003) reported results from liveload tests performed on a high
performance concrete bridge to provide comparisons of the field measurements
obtained with the values calculated using the AASHTO simplified equations. The
bridge consisted of seven spans which were simply supported. The deck thickness of
the bridge was 7 in. (178 mm) and the overall length of the bridge was 798 ft (243.2
meters). The bridge contained five AASHTO BT54 girders per span which were
spaced at 8.75 ft (2.7 meters). The total roadway width was 40 ft (12.2 meters). Span
5, which was 114ft (34.7 meters) in length, was instrumented and subjected to live
load tests. In carrying out the comparison study, static and dynamic tests were
performed. Static tests were done by either loading a single truck or two trucks
simultaneously. The load test trucks were identical in configuration and weight. The
bridge was instrumented during construction. Upon completion of construction, the
static and dynamic tests were performed. The bridges structural responses including
strains, stresses, and deflections, were measured during the tests. These values were
then compared to the AASHTO simplified equations. The results of the study show
that for this particular structure, the ratio of the dynamic load effects to static load
effects were lower than the dynamic load allowance value specified in the code. The
distribution factors obtained from the tests indicate the loads are distributed to all
girders at least as well as predicted by the AASHTO formulas. In other words, for all
43
cases, the LDF of the AASHTO specifications were found to be conservative when
compared to the experimental values. The same is true for strains, stresses, and
deflection. These values also exhibited conservative predictions when compared to
measured values. The researchers indicate that this is due to the additional stiffness
provided by traffic barriers on the bridges, which are normally neglected during
design. The study also reveals that for exterior girders in bridges with diaphragms,
which are required by AASHTO to assume the deck and beams to be treated as a rigid
body for live load distribution considerations, the bridge did not behave as a rigid
body under multiple lane loadings. In fact, the study suggests that the AASHTO code
equations produce the most conservative estimate of live load distribution factor for
exterior girders because of this assumption. The amount of stresses in the bottom
flange of the exterior girders calculated with the AASHTO code equations were 85%
larger than the measured stresses. The closest match between the AASHTO code
equations and the measured values are with respect to deflections. The study shows
that the AASHTO assumption that deflections be calculated assuming all girders
deflect equally when loaded corresponded to what was measured during the tests. The
deflections calculated with the assumption were 20% or lower than the measured
values for deflections. The overall results indicate that the simplified equations
yielded values that were conservative when compared to the measured values for
prestressed concrete girder bridges. Furthermore, bridges with diaphragms may
require use of more complex analysis methods in order to reduce the conservatism of
the code.
44
Hughs and Idriss (2006) conducted an evaluation of shear and moment liveload
distribution factors for a new, prestressed concrete, continuous, spread boxgirder
bridge. The purpose of the research was to evaluate the accuracy of the AASHTO
Standard Specification and LRFD simplified equations for live load distribution
factors by comparing them with results from a finite element model validated though
field tests. The study bridge consists of five spans with a total length of 642 ft (195.7
meters). The bridge has a skew angle of 12 degrees with an overall width of 53 ft
(16.2 meters). Each span consists of six 54 in. (1.4 meters) prestressed concrete
trapezoidal Ugirders. The bridge design considered simply supported behavior under
noncomposite dead loads and continuous behavior under live load. An elevation
view the study bridge is shown in Figure 3.15.
The girders for the new bridge used in the study were constructed of highstrength
prestressed concrete with a minimum 28day compressive strength of 10 ksi (68.9
MPa). The prestressing consisted of 0.6 in. (15 mm) diameter, lowrelaxation, 7wire
strands with a yield strength of 170 ksi (1.2 GPa). The deck and diaphragms used
normal, 4,000 psi (27.6 MPa) concrete. The deck was constructed of 8 in. (203 mm)
thick castinplace concrete.
495.7 m <64?'0 3/4') (back of backwa.il to back of backwall>
Span 1 Span 2 SpQn 3 Span 4 Span 5
42.14 n r30.94 ni *40.70 m r40.70 nj 40.31 n
<138'3') (133'6*> <133'60
I' II 1 r^~
<Westbound 110
Traffic Direction
Figure 3.15 Elevation View of Study Bridge (Hughs and Idriss 2006)
45
Fiber optic sensors were instrumented in span 5 during construction to measure
moment and shear distribution resulting from the live load test truck. The test truck
used for the liveload test was a three axle dump truck containing a total weight of
56,760 lbs (252.5 kN). The location for maximum moment distribution and maximum
shear distribution was determined using influence lines. The determined locations in
each girder were chosen at the centerline of span 5 and at the abutment for maximum
shear distribution. Optical sensors embedded in the girders were used to measure the
moment and shear forces.
A threedimensional finite element model was created using SAP2000 software
for the purposes of comparing the AASHTO simplified equations. The model
consisted of frame elements representing the girders, shell elements for the deck, and
displacement restraints were used to model abutments and pier supports. Rigid links
between the deck and girders were used to simulate composite behavior. The
measured results from the liveload test were used to confirm the accuracy of the
finite element model. The maximum distribution factors obtained from the FE model
for both moment and shear were close in value to the distribution factors obtained
from the liveload test. The percent differences of the load test maximum girder
distribution factors for moment to the values calculated using the FE model are 12.2%
for interior girders and 5.55% for exterior girders, respectively.
Upon validating the model with the liveload test, the model was then loaded with
various transverse AASHTO HS20 truck loading combinations to maximize moment
and shear. Comparison of live load distribution factors for moment from the FE
model to the predicted values from the AASHTO formulas indicate that, for interior
46
girders, the AASHTO standard specification equations were 26.8% more
conservative and the LRFD equations were unconservative by 1.7% For exterior
girders, the AASHTO standard specification equations yielded values that were
47.0% more conservative than the values obtained from the FE model. The LRFD
equation values were 75.7% more conservative than the FE model for exterior
girders. In general, the study concludes that the LRFD specifications predictions of
live load distribution factors are accurate or conservative when compared to the FE
model for all girders of the study bridge. The distribution factors were most accurate
for interior girders but overly conservative for exterior girders. The results of the
study indicate that for the study bridge, although use of the AASHTO LRFD
simplified equations for live load would result in a safe design, the exterior girders
would be overdesigned. The overdesign of the exterior girders would translate into a
direct loss of economy which is undesirable. Further research is suggested to correct
this deficiency.
I
47
4. RESEARCH PLAN
A literature review was first conducted to review and assemble the available literature
relevant to the development and implementation of the AASHTO LRFD live load
distribution factor equations. The literature selected for review in this study focuses
on approximate and refined methods of analysis as well as on analytical modeling and
experimental testing. Particular emphasis is placed on LRFD simplified equations
with respect to box girder bridges.
A general plan for the proposed research presented herein may be described as
follows:
1. A refined analysis method using a Finite Element Method (FEM)
modeling technique is described in detail.
2. The results from an experimental study conducted by Hughs (2004)
are introduced into this study to test the validity of the FE modeling
technique developed in (1).
3. A FEM model using the parameters described in (2) is developed
using SAP2000 software, and the modeling techniques of (1), and
compared to the results obtained from the experimental liveload tests
of (2). Closeness of the values obtained from the experimental and
refined analysis indicates the reliability of the modeling technique for
use in determining live load distribution factors for this study.
4. With the modeling technique substantiated from (3), a new model is
created using LARSA 4D software conforming to the parameters of
the bridge described in this study.
48
5. The FE model from (4) is then used to develop new live load
distribution factors for moment and shear.
6. Live load distribution factors are then hand calculated using the
AASHTO LRFD simplified equations (Lever Rule).
7. The distribution factors presented in (5) are compared to the AASHTO
LRFD simplified equations in (6) for the bridge parameters identified
in this study.
8. A design comparison is conducted using PGSuper software and the
values for live load distribution factors obtained from (5) and
compared to a design using the values from (6).
9. The designs from (8) will compare spantodepth ratios by first
varying span lengths while keeping the depth of the girder constant
and second, maintaining a constant span length while varying the
girder depth.
49
5. FINITE ELEMENT ANALYSIS
5.1 Finite Element Modeling Technique
The modeling technique employed for use in this thesis is the Eccentric Beam Model.
The Eccentric Beam Model is a Finite Element Analysis (FEA) threedimensional
modeling technique that is often used in the analysis of highway bridges. This
modeling technique which is illustrated in Figure 5.1 considers the primary structural
members which include the deck slab and longitudinal girders for the distribution of
vehicular live load. The eccentric beam model idealizes the concrete slab as
quadrilateral (four node) shell elements and the beams as twonode isoparametric
beam elements. Rigid links are used to account for composite behavior between the
centers of the slab and beam elements.
Node
J Frame Element
Shell Element
1 Rigid Link
o Moment End Release
Figure 5.1 Eccentric Beam Model
50
5.2 Experimental Study
5.2.1 Introduction
Hughs (2004) conducted an evaluation of shear and moment liveload distribution
factors for a new, prestressed concrete, continuous, spread boxgirder bridge. The
purpose of the research was to evaluate the accuracy of the AASHTO Standard
Specification and LRFD simplified equations for live load distribution factors by
comparing them with results from a finite element model validated though field tests.
The evaluation conducted by Hughs does not consider the effects of various
parameters such as number of girders, span length, and slenderness and thus, solely
compares distribution factors obtained from an FE model to the AASHTO simplified
Specification equations (Standard Spec, and LRFD) for the newly constructed,
prestressed concrete, continuous, spread boxgirder bridge shown in Figure 5.2.
This thesis uses the results from field testing conducted by Hughs (2004) to
verify the results of the FE model developed in this study and to validate the
modeling technique described in section 5.1. The results from Hughs (2004) are used
because experimental live load testing was not possible as part of this study. The FEA
results are compared directly to the field testing results of the experimental bridge.
Validation of the modeling technique and the results of the FE analysis are necessary
in order to ensure the model accurately represents the behavior of the actual structure.
5.2.2 Description of Experimental Bridge
The I10 Bridge over University Avenue and Main Street in Las Cruces, New Mexico
was selected to test the validity of the FE modeling technique described in section
51
5.1. The study bridge consists of five spans with a total length of 642 ft (195.7
meters). The bridge has a skew angle of 12 degrees with an overall width of 53 ft
(16.2 meters). Each span consists of six 54 in. (1.37 meters) prestressed concrete
trapezoidal Ugirders. The bridge design considered simply supported behavior under
noncomposite dead loads and continuous behavior under live load. A crosssection
and plan of the study bridge is shown in Figure 5.2. The girders for the bridge used in
the study were constructed of highstrength prestressed concrete with a minimum 28
day compressive strength of 10 ksi (68.9 MPa). The prestressing consisted of 0.6 in.
(15 mm) diameter, lowrelaxation, 7wire strands with a yield strength of 170 ksi
(1.12 GPa). The deck and diaphragms used normal, 4,000 psi (27.6 MPa) concrete.
The deck consists of 8 in. (203 mm) thick castinplace concrete.
Fiber optic sensors were instrumented in span 5 during construction to
measure moment and shear distribution resulting from the live load test truck. The
test truck used for the liveload test was a three axle dump truck containing a total
weight of 56,760 lbs (252.5 kN). Dimensions and distribution of the axles are shown
in Figure 5.3. Weight distribution for the test truck is as follows: front axle is 14,970
lbs (66.6 kN), middle axle is 20,895 lbs (92.9 kN), and rear axle is 20,985 lbs (92.9
kN). The determined locations in each girder were chosen at the centerline of span 5
and at the abutment for maximum shear distribution. Optical sensors embedded in the
girders were used to measure the moment and shear forces.
52
Br dge CL
16.1?Tn
<5;
nl
11
1S5.7 n 
Spcn 1
 42.14 n
<138'3'>
Span 2
30.94 m
<10t'6')
1
Span 3
4C.70 m
C133'6')
Span 4
40.70 n
(133'6'7
Span 5
40.31 n
<132'3)
\
1
J2*
skew
"Westbound 11C
Traffic Direction
Figure 5.2 Crosssection and Plan of Study Bridge (Hughs and Idriss 2006)
Front View Side View Rear View
Figure 5.3 Testtruck Dimensions and Axle Locations (Hughs and Idriss
2006)
53
5.2.3 FE Model of Experimental Bridge
The experimental study bridge was modeled using SAP2000 software and the
techniques described in section 5.1. The threedimensional model uses frame
elements to represent the girders, shell elements for the slab, and displacement
restraints representing support conditions (abutments and piers). Nodes are placed at
2 ft (0.61 meters) increments in the longitudinal direction and the frame elements are
connected at those locations. The transverse spacing of the frame elements
correspond with the girder spacing shown in Figure 5.2. The shell elements are also
spaced at 2 ft (0.61 meters) intervals in the longitudinal direction. In the transverse
direction, the shell elements are spaced at various intervals in order to maintain an
aspect ratio as close to 1.0 as feasible. Rigid links are placed at the locations shown in
Figure 5.1. The dimensions for the frame elements are shown in Figure 5.4. The
thickness of the shell elements are 8 in. (203 mm) representing the actual deck
thickness. Figure 5.5 show layout of the FE model from SAP2000. A 12 degree skew
is shown in plan view in Figure 5.2. The FE model developed does not contain a skew
and the longitudinal and transverse members are positioned orthogonal to each other.
Each support shown in Figure 5.5 is restrained from translation in the x,y, and
z directions and restrained from rotation about the xaxis. The rotational restraint is
designed to capture the rigidity effects of the end diaphragms on the actual structure.
The U54 girders were restrained in torsion by increasing the torsional constant in
each member. The torsional constant J was calculated using the closed shape shown
in Figure 5.6. Rigid links were assigned from the centroid of the member to the
54
centroid of the slab. Moment releases were used in the rigid links at the slab
locations. This allows the slab to rotate about the girder webs as shown in Figure 5.1.
Figure 5.4 Dimensions of U54 Girder (TxDOT)
55
Figure 5.5 Layout of FE Model of Experimental Bridge in SAP2000
Figure 5.6 Closed U54 Shape
56
5.2.4 Live Load Placement of Test Truck
Figure 5.7 shows truck positions used to optimize maximum moment and shear forces
in span 5 at midspan and at the abutment respectively. In order to properly represent
the placement of the test truck, static loads representing the weight and load
distribution of the test truck were positioned at the locations shown in Figure 5.7. The
figure shows that the field test truck was placed approximately 1.4 ft (0.43 meters)
from the northern barrier and moved transversely across the bridge at various
distances in a direction parallel to the abutment.
57
5.2.5 Experimental vs. FEA Load Distribution Factors
The results from the live load experimental study (Hughs 2004) described in section
5.2.1 are provided in Table 5.1. The distribution factors are developed based on the
extreme fiber stress per girder at midspan with respect to the summation of midspan
stresses for all girders. The values are calculated based on a truck wheelline position
58
transversely moving south from the north edge of deck. The original and deflected
shapes are shown in Figure 5.8. The figure shows live load being placed in span 5.
Figure 5.8 Illustration of Live Load Causing Deflection in Span 5
Table 5.1 Live Load Distribution Factors from Experimental Study (Hughs 2004)
Transverse Position of Live Load Test Truck (ft) Live Load Distribution Factors
Girder 1 Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
1M 1.18 0.360 0.240 0.140 0.100 0.060 0.060
2M 8.31 0.260 0.300 0.170 0.120 0.072 0.070
3M 16.98 0.150 0.230 0.270 0.150 0.110 0.090
4M 26.51 0.110 0.130 0.210 0.270 0.175 0.125
5M 36.04 0.090 0.095 0.125 0.200 0.300 0.225
6M 42.11 0.080 0.080 0.100 0.140 0.250 0.360
Figure 5.9 shows the live load distribution factors vs. transverse position of
the live load test truck from the finite element model. The values from Table 5.1 were
plotted and combined with the values used in Figure 5.9 for each girder. This was
done for comparison of the results from the experimental study with the values
59
Load Distribution Factor
obtained from the finite element model. The comparisons, which were only done for
flexural responses, are illustrated in Figures 5.10 through 5.15.
Transverse Position of Live Load Test Truck (ft)
Girder 1 * Girder 2 ir* Girder 3
*Girder4 * 'Girder5 Girder6
Figure 5.9 Distribution Factors from FEM vs. Position of Live Load Truck
60
Load Distribution Factor Load Distribution Factor
0.450
0.400
0.350
0.300
0.250
0.200
0.150
0.100
0.050
0.000
T
0 5 10 15 20 25 30 35 40 45
Transverse Position of Live Load Test Truck (ft)
Girder 1: FEM ** Girder 1: Actual
igure 5.10 LDFs from FEM vs. Position of Live Load Truck (Girder 1)
Girder 2: FEM Girder 2: Actual
Figure 5.11 LDFs from FEM vs. Position of Live Load Truck (Girder 2)
61
Load Distribution Factor J Load Distribution Factor
0.300
0.250
0.200
0.150
0.100
0.050
0.000
0 5 10 15 20 25 30 35 40 45
Transverse Position of Live Load Test Truck (ft)
Girder 3: FEM Girder 3: Actual
igure 5.12
LDFs from FEM vs. Position of Live Load Truck (Girder 3)
Girder 4: FEM Girder 4: Actual
Figure 5.13 LDFs from FEM vs. Position of Live Load Truck (Girder 4)
62
Load Distribution Factor  j Load Distribution Factor
0.350
0.300
0.250
0.200
0.150
0.100
0.050
V \
./
./
M
0.000 4
0
5 10 15 20 25 30 35 40
Transverse Position of Live Load Test Truck (ft)
45
Girder 5: FEM
Girder 5: Actual
igure 5.14 LDFs from FEM vs. Position of Live Load Truck (Girder 5)
Transverse Position of Live Load Test Truck (ft)
Girder 6: FEM Girder 6: Actual
Figure 5.15 LDFs from FEM vs. Position of Live Load Truck (Girder 6)
63
5.3 Finite Element Model Validation
The comparisons illustrated in Figures 5.11 through 5.15 show good agreement
between the live load responses measured experimentally and the results from the
finite element model developed in this thesis. Some notable differences are shown in
the exterior girders of Girders 1 and 6. The plots for each girder show the distribution
factors increasing as the truck is transversely positioned from one edge of the deck to
the other. This is true for each girder at each truck position. For the exterior girders,
the deviations between the FEM distribution factors compared to the experimental
results are more pronounced. This is attributed to the skew not being present in the FE
model. In actuality, the skew in the actual bridge will help distribute the load more to
the other girders, which subsequently, will cause the distribution factors to be more
uniform. These differences are visualized in Figures 5.11 and 5.15. Table 5.2 shows
percent differences for live load distribution factors between the experimental results
and the results of the FE model. The largest differences occur in Girder 1 as a result
of the test truck being positioned over girder 6 and vice versa when the truck is
positioned over girder 1.
64
Table 5.2 Percent Difference between Experimental Bridge and FEA Model
Transverse Position of Live Load Test Truck (ft) Percent Difference between Experimental Bridge and FEA Model
Girder 1 Girder 2 Girder 3 Girder 4 Girder 5 Girder 6
1M 1.18 16 2 6 4 1 54
2M 8.31 15 2 1 2 2 39
3M 16.98 18 5 4 9 2 15
4M 26.51 8 9 13 5 3 7
5M 36.04 39 18 10 2 9 3
6M 42.11 55 21 1 14 17 3
In conclusion, the finite element model predictions for the load distribution
factors correspond well with the results from experimental field testing. The accuracy
of the results demonstrate that the modeling scheme developed in section 5.1 is
acceptable for use in the analysis for determining live load distribution factors
developed as part of this study.
5.4 Finite Element Model
Three finite element models are developed in this section in order to determine live
load distribution factors for comparison with the AASHTO LRFD simplified live
load distribution factor equations, namely the Lever Rule Method. The models use the
Eccentric Beam modeling techniques previously discussed in section 5.1 to determine
live load distribution factors. The models are developed using LARSA 4D software.
The results from the FE models are used in a design comparison in chapter 7 to
determine the effects of live load on flexural requirements of a member, span length,
and girder depth in two girder bridge systems using trapezoidal Ugirders.
65
Transverse stiffness k is related to applied loads by angle of twist (f)'\n girders and
deflection 8 in the deck slab. Longitudinal Stiffness k is related to applied loads by
deflection 8in the composite section. Table 5.2a shows that transverse and
longitudinal stiffness is inversely related. Thus, as shear modulus G or rotational
moment of inertia J are increased, the stiffer the transverse section becomes which
results in better load distribution and subsequently a decrease in LDFs. As shown in
the table, increasing the modulus of elasticity E or the moment of inertia I of the deck
slab also results in a more rigid transverse section which also creates better load
distribution which results in lower LDFs. Conversely, if the deck slab length (i.e.
girder spacing) is increased, the transverse section becomes less stiff (more flexible)
and load distribution worsens resulting in higher LDFs. The inverse occurs for
longitudinal stiffness: increasing the modulus of elasticity E or the moment of inertia
I of the composite section results in a more rigid longitudinal section, however,
increasing the span length results in better lad distribution and lower LDFs.
Table 5.2a Relationship between Transverse and Longitudinal Stiffness
Transverse Stiffness Increase G, J of Girder More Stiff Better Distribution LDFs l
Increase E, I of Deck Slab More Stiff Better Distribution LDFs l
Increase Deck Slab Length (i.e. Girder Spacing) Less Stiff Worse Distribution LDFs 'l
Longitudinal Stiffness Increase E, I of Girder More Stiff Worse Distribution LDFs j"
Increase Span Length Less Stiff Better Distribution LDFs l
66
Cracked vs. Gross Deck Section Properties
A cracked section condition in the deck slab occurs when the slab is loaded beyond
its cracking moment capacity. If the slab is loaded sufficiently, horizontal tension
and compression stresses will occur in the bottom and top of the slab respectively and
cracking will occur. These opposing stresses will cause vertical cracks near midspan
due to flexure and diagonal cracks due to shear forces near the support locations. A
cracked deck section is used to account for the fact that concrete is essentially
ineffective in tension and is therefore represented by using reduced section properties.
Cracked section properties are usually achieved by reducing the moment of inertia I
by 50%. Thus, as shown in Table 5.2a, a reduction in moment of inertia (i.e. using
cracked section properties) of the deck slab will result in a less rigid (more flexible)
transverse section which would cause the distribution of load to worsen resulting in
higher LDFs. Twogirder bridge systems such as the one used in this study consists of
deck slabs that are sized such that the loads that are applied rarely cause cracking in
the deck slab to occur. Because of this, gross section properties are often used to
analyze twogirder bridge systems using precast trapezoidal Ugirders. The analysis
carried out in this section also uses gross section properties.
5.4.1 Modeling Parameters
A deck section of the bridge used for the FE models is shown in Figure 5.16. The
section consists of two precast trapezoidal Ugirders. The beams are spaced 236
(7.2 meters) apart. The overhang distances from the center of the girders are 99
67
(3.0 meters). The overall width of the bridge is 43 ft (13.1 meters). The study models
contain two spans of equal lengths so as to capture the effects of continuity in the
system as most applications of two girder bridge systems consists of multispan
structures. Bridge parameters that vary per model are given in Table 5.3.
Table 5.3 Various Model Parameters
Designation Number of Spans Span Lengths (ft) Girder Type Girder Depth (in)
Model A 2 110 U78 78
Model B 2 110 U74 74
Model C 2 118 U78 78
The three models (A through C) are intended to assess the degree to which
flexural requirements, girder depth, and span length are affected by live load
distribution. Model A is used as the base model. Therefore the parametric evaluation
considers one parameter at a time while keeping the remaining parameters the same.
Figure 5.17 shows dimensions for a U78 girder section. The section illustrates critical
dimensions as well as typical reinforcement requirements. Because the slope of the
webs remain constant for different girder sections, with the exception of the overall
height and width of the member, all other dimensions remain the same for the U74
girder used in Model B of this study. The modeling techniques described in 5.2.3 are
the same and are not repeated in this section. Model A is used in section 5.4.5 and
5.4.6 for discussion of results for moment and shear live load distribution factors.
68
jht
Figure 5.16 FE Model Typical Deck Section (125 Trinidad Viaduct
Construction Drawings)
(STRANDS NOT SHOWN! ^.,6" HEADED ANCHOR S
(AUTOMATICALLY ENDWELL
TO PLATE*, TYP
Figure 5.17 U78 Girder Section (125 Trinidad Viaduct Construction
Drawings)
69
5.4.2 Moving Load Analysis
The response of a member due to live load is determined in LARSA 4D using the
influence surface method of moving load analysis. This method is usually appropriate
for models that include deck elements that are modeled as plates such as the model
described in section 5.4.1. The moving load analysis uses influence coefficients to
determine where to place vehicles and lane loading to create worse case results for a
particular force effect. The moving load analysis is implemented by first placing unit
loads at specified increments transversely and in the longitudinal direction within a
predefined lane boundary. The response to the unit loads result in influence
coefficients for each unit load location. All of the resulting coefficients together
create the influence surfaces for any given location and force effect. The influence
surfaces are used to determine the critical location of loads that cause maximum force
effects. Maximum response quantities are determined by placing AASHTO LRFD
vehicular loading configurations (HL93 loading) at the location on the influence
surface that causes the maximum force effect.
Using influenced based moving load analysis, if a point of interest and force
effect are selected (i.e. tension in bottom fiber at midspan), LARSA 4D can display
graphically the actual live loading configuration superimposed on the influence
surface. This information is necessary in order to compute accurate live load
distribution factors using the appropriate number of vehicles.
70
5.5 Results of Finite Element Analysis
The results for moment and shear live load distribution factors were determined from
the finite element models described in section 5.4.1 using moving load analysis
procedures described in section 5.4.2.
5.5.1 Results of Finite Element Analysis for Moment
The results for flexural distribution factors due to live load were obtained by (1)
determining the maximum tension response quantity in the bottom fiber of the beam,
(2) determining the corresponding location of the maximum response quantity, and
(3) determining the configuration and placement of the vehicle loading that causes the
maximum tension response. (4) Once the maximum response quantity is determined
from the controlling member, an analysis results case is created that allows the
determination of the concurrent response quantity for the adjacent member(s).
The values for live load distribution factors for moment are obtained using the
relationship described in Eq. 5.1.
Where:
fj = Maximum normal stress response quantity resulting from bending forces
in the bottom fiber of the beam j being considered
Moment LDF =
Wj
(5.1)
n
i=l
71
n
^ /j = The summation of the normal stress responses from bending forces in the
1=1
bottom fiber of all i beams being considered
N = Number of trucks placed transversely on the model
y = Multiple presence factor from AASHTO LRFD
The specific values for live load distribution factors for moment are obtained
using the relationship described in Eq. 5.1 and are provided in section 5.6.
5.5.2 Results of Finite Element Analysis for Shear
The results for shear distribution factors due to live load were obtained by (1)
determining the maximum force response quantity in the joint at the support
locations, (2) determining the corresponding location of the maximum response
quantity, and (3) determining the configuration and placement of the vehicle loading
that causes the maximum force response. (4) Once the maximum response quantity is
determined from the controlling member, an analysis results case is created that
allows the determination of the concurrent response quantity for the adjacent
member(s).
The specific values for live load distribution factors for shear are obtained
using the relationship described in Eq. 5.2.
72
Shear LDF =
(5.2)
Nyrj
n
Zi
1=1
Where:
rj = Maximum force response quantity resulting from forces at the support
location of the joint j being considered
n
= The summation of the force responses at the joint i of all the joints being
i=i
considered
N = Number of trucks placed transversely on the model
y = Multiple presence factor from AASHTO LRFD
The specific values for live load distribution factors for shear are obtained using
the relationship described in Eq. 5.2 and are provided in section 5.6.
5.5.3 Multiple Presence Factors
Note the multiple presence factors used in equations 5.1 and 5.2 are described later in
section 6.1. In moving load analysis, the multiple presence factors are accounted for
in the FE model and are taken as 1.0 when used to calculate LDFs using equations 5.1
and 5.2.
73
5.5.4 Maximum Response Quantities for Moment and Corresponding
Live Load Position
The procedures described in sections 5.5.1 are used to determine the maximum
response quantities for moment load distribution factors. Graphical representation of
these procedures are displayed in Figures 5.18 through 5.23
Figure 5.18 illustrates the maximum stress response quantity in the bottom
fiber of the beam under consideration. Figure 5.19 shows concurrent stress response
quantity for the adjacent beam. The values for stress in the bottom fiber of each beam
are used to calculate the live load distribution factors as described in 5.5.1. Figure
5.20 shows both beams emphasizing the maximum stress quantity in one member and
the concurrent stress quantity in the other.
Figures 5.21 through 15.23 shows various viewpoints of the AASHTO HL93
loading configuration positioned at the location that results in the maximum stress
quantity shown in Figure 5.18. Figures 5.21 through 5.23 show that two lanes of
AASHTO Trucks at midspan of span 1 concurrent with the AASHTO Lane Load in
both spans cause the maximum response quantity for flexure.
74
Figure 5.18 Maximum Stress Response Quantity in the Bottom Fiber
Figure 5.19 Corresponding Stress Quantity in Adjacent Beam
75
Figure 5.20 Maximum and Corresponding Stress Responses in Both Beams
Figure 5.21 Position of HL93 Loading Configuration for Maximum Moment
76
Figure 5.22 Position of HL93 Loading Configuration for Moment
Figure 5.23 Position of HL93 Loading Configuration for Moment
77
5.5.5 Maximum Response Quantities for Shear and Corresponding Live
Load Position
The procedures described in sections 5.5.2 are used to determine the maximum
response quantities for shear load distribution factors. Graphical representation of
these procedures are displayed in Figures 5.27 through 5.38
Figure 5.24, however, illustrates the maximum force response quantity in the
beam at the interior support locations. Figure 5.25 shows corresponding shear
response quantity for the adjacent beam and Figure 5.26 shows both beams
emphasizing the maximum force quantity in one member and the concurrent force
quantity in the other. The values for shear in each beam are not used to calculate the
live load distribution factors as they do not account for the slabs shear force
contribution. In order to obtain maximum response quantities for shear force,
reactions at the support locations are used instead. The images in Figures 5.24
through 5.26 are shown only to illustrate a common source of errors when seeking to
obtain meaningful results.
i
i
i
i
78
Figure 5.24 Maximum and Corresponding Force Responses in Both Beams
at Interior Supports
Figure 5.25 Corresponding Force Quantity in Adjacent Beam
79
iLsi
Figure 5.26 Maximum and Corresponding Force Responses in Both Beams
at Interior Supports
Figure 5.27 illustrates the reactions at the interior support corresponding to
HL93 loading configuration which is placed in a manner to produce the largest
reaction at that location shown in Figures 5.28 through 5.30. In Figures 5.25 through
5.30 it is shown that two traffic lanes consisting of two trucks separated by a
minimum distance of 50 ft (15.2 meters) in one lane and one truck positioned directly
over the support in the adjacent lane are causing the maximum reaction at the joint at
the interior support (318.4 kips, 1416.3 kN). The values from this controlling case
were used to calculate the live load distribution factors for shear.
80
Figures 5.31 through 5.38 are illustrated for the purposes of discussion. In
Figure 5.31, the next largest reaction occurs when HL93 loading is positioned over
the supports as shown in Figures 5.32 through 5.34. However, instead of the two
negative moment vehicles, two lanes are positioned adjacent to each other with the
second truck axle of each truck placed directly over the interior support. The value
produced by this configuration is slightly smaller than the controlling case (311.6
kips, 1386.1 kN). Finally, Figures 5.35 through 5.38 shows the maximum reaction
when the HL93 loading is placed directly over the support locations at the abutment.
The values obtained for maximum reaction for this case is significantly lower than the
other two cases (130.8 kips, 581.8 kN).
81
Figure 5.28 HL93 Loading Configuration Causing Maximum Reaction at
Interior Support
Figure 5.29 HL93 Loading Configuration Causing Maximum Reaction at
Interior Support
82
ten 3J16X
termed Mote Mum* Burton CooAdarta Muonee Burbeo Law GZ JeM 491
Jeafo Factor 3B
Figure 5.30 HL93 Loading Configuration Causing Maximum Reaction at
Interior Support
Figure 5.31 2nd Largest Reaction at interior Support Location
83
Figure 5.32 HL93 Loading Configuration Causing 2nd Largest Reaction at
Interior Support
Figure 5.33 HL93 Loading Configuration Causing 2nd Largest Reaction at
Interior Support
84
Soto Factor 32.
AA
Figure 5.34 HL93 Loading Configuration Causing 2nd Largest Reaction at
Interior Support
I
i Figure 5.35 Maximum Reaction at Abutment Support Location
i
I
i
85
Figure 5.36 HL93 Loading Configuration Causing Maximum Reaction at
Abutment Support Location
Figure 5.37 HL93 Loading Configuration Causing Maximum Reaction at
Abutment Support Location
86
born &SBK
Mbrmd MdM Mura 8M7 Cw^Mli k^ancs flutes lf OZ> ? JgM 1J
inb Factor 32.
Figure 5.38 HL93 Loading Configuration Causing Maximum Reaction at
Abutment Support Location
5.6 Discussion of Results from Finite Element Analysis
Live load distribution factors for moment based on FEA were calculated using Eq.
5.1 and are provided in Table 5.4 for Models A, B, and C. The stress values used in
the calculations are also provided. The distribution factors for shear were calculated
using Eq. 5.2 and are shown in Table 5.5. Shear values were only determined for the
base model. Values for shear in Models B and C were not determined as live load
effects on the parameters discussed in section 5.4 are not considered as part of this
study. Note that the live load distribution factor for moment in Girder 1 is 141%
larger than the distribution factor calculated for moment in Girder 2 for Model A.
This large percent difference indicates that live load is not distributed favorably
87
