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GRILLAGE ANALYSIS OF HEAVY-DUTY RIVETED STEEL GRATINGS
A Thesis
Presented to
The Graduate Faculty of the University of Akron
In Partial Fulfillment
Of the Requirements for the Degree
Master of Science
Vikas Kumar Cinnam
May 2018
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GRILLAGE ANALYSIS OF HEAVY-DUTY RIVETED STEEL GRATINGS
Vikas Kumar Cinnam
Thesis
Approved: Accepted:
Advisor Dean of the College Dr. Craig C. Menzemer Dr. Don J Visco
Committee Member Dean of the Graduate College Dr. Anil K. Patnaik Dr. Chand Midha
Committee Member Date Dr. Tirumalai S. Srivatsan
Department Chair Dr. Wieslaw Binienda
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ABSTRACT
A heavy-duty riveted steel grating is a lightweight and easy to install open grid
bridge deck system used primarily in movable bridges. These systems may increase the load
carrying capacity of the bridge deck because of the decrease in the dead weight of the
structure. Initially, bridge decks were analyzed according to the principles used in the design
of railways, but with the industrial revolution in the 19th century and the increasing need for
more bridges for transportation, a precise method for analyzing bridge decks was needed.
Empirical methods were available and used by the bridge engineers until ‘Bridge Deck
Behavior’ was issued by E.C. Hambly (1976). Many improvements were made with
research focused on bridge decks and new methods of analyzing deck systems came to light.
There is no method that is simple, yet accurate that can be used at a desk by engineers.
The current research focuses on representing a bridge deck system by a grillage
model and studying its behavior under static H20 truck (AASHTO H20) loading along with
a 30% impact factor. Grillage analysis is one of the methods to represent the bridge structure
with a grillage formed by two-dimensional discrete grillage members representing the main
bearing bar members. Experimental results from the tests conducted on two panels of R-37-
5 (5”x3/8”) Lite Steel Gratings are compared to the results from the model analysis results
from SAP2000 v17. Static behavior of the riveted gratings, as well as stress patterns were
investigated using grillage analogy principles.
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Two different models were prepared in SAP2000 using the principles of grillage
analysis to demonstrate the importance of reticuline bars. Based on the comparison,
guidelines to model grating for grillage analysis are given and these guidelines are used to
estimate the stress patterns and deflection values of a grating with different cross-sectional
dimensions and different span. The idea is to demonstrate that grillage analogy can be used
to predict the behavior of a riveted steel grating.
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DEDICATION
This work is dedicated to the five most important women of my life:
Mrs. Sakkubai Cinnam,
Late Mrs. Ratna Manju, Ms. Harshiva Matcha, Ms. Hemalatha Cherlopalli,
And my love, Sanjana Perumalla Cinnam.
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ACKNOWLEDGEMENTS
I thank everyone from my family for their encouragement and support that kept me
motivated to complete my master’s research. I extend my gratitude to my friends for
understanding me and I apologize for not being able to talk to you for months. I would not
be able to complete this work without the incomparable guidance, advice and support of
my graduate advisor, Dr. Craig Menzemer. He is the one who helped me out in my most
difficult times and kept believing in me, which has made this work possible. I thank Dr.
Ateef Saleeb, for spending hours of his time to make this research a lot simpler for me.
I would like to thank my committee members for their help and input towards the
completion of this research.
Constant motivation from Sanjana Perumalla and Hemalatha Cherlopalli will
always be appreciated. Although there are many individuals who helped me directly or in-
directly in completing this research that are not mentioned here, there is no one
unappreciated.
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TABLE OF CONTENTS LIST OF TABLES .................................................................................................................... xi
LIST OF FIGURES ................................................................................................................. xii
CHAPTER
INTRODUCTION ...................................................................................................... 1
1.1 Background ....................................................................................................... 1
1.2 Problem Statement ............................................................................................ 3
1.3 Justification ....................................................................................................... 4
1.4 History of Heavy Duty Riveted Gratings ............................................................ 4
1.5 Objectives ......................................................................................................... 7
1.6 Outline ................................................................................................................... 8
LITERATURE REVIEW ......................................................................................... 10
2.1 Introduction ..................................................................................................... 10
2.2 Metal Bar Gratings - Flooring System ............................................................ 10
2.3 The Marking System of Metal Gratings ............................................................ 12
2.4 Background ..................................................................................................... 13
2.4.1 Heavy-Duty Riveted Gratings Background........................................... 13
2.4.2 Grillage Analogy Background .............................................................. 16
2.5 Grillage Analysis ................................................................................................ 18
2.5.1 Stiffness Matrix ..................................................................................... 19
viii
2.5.2 Design Principles for Grillage Members ................................................. 20
EXPERIMENTAL TESTS ....................................................................................... 22
3.1 Specifications of R-37-5 (5”x1/4”) Lite Steel gratings ................................... 22
3.2 Loading ........................................................................................................... 29
3.3 Experimental Results and Discussion ............................................................. 29
3.3.1 Load-Micro Strain Trend for Test A ........................................................ 30
3.3.2 Load-Micro strain Trend for Test B ......................................................... 33
3.4 Parametric Studies .......................................................................................... 34
3.4.1 Intermediate Bearing Bars and Reticuline Bars ....................................... 34
GRILLAGE ANALYSIS MODELING ...................................................................... 36
4.1 Load Distribution ............................................................................................ 36
4.2 Load Graph ..................................................................................................... 39
4.3 Grate Models ................................................................................................... 41
4.3.1 Test Panel A modeling – First Approach ................................................. 42
4.3.1.1 Boundary Conditions and loading ..................................................... 44
4.3.1.2 Analysis and Results .......................................................................... 46
4.3.2 Test Panel A modeling – Second Approach ............................................. 48
4.3.2.1 Boundary conditions and loading ...................................................... 50
4.3.2.2 Analysis and Results .......................................................................... 52
4.3.3 Test Panel B – Approach A ...................................................................... 54
4.3.3.1 Boundary conditions and loading ...................................................... 56
4.3.3.2 Analysis and Results .......................................................................... 56
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4.3.4 Test Panel B – Second Approach ............................................................. 58
4.3.4.1 Boundary conditions and loading ...................................................... 60
4.3.4.2 Analysis and Results .......................................................................... 60
COMPARISON AND MODELING OF TEST PANEL C ...................................... 63
5.1 Comparison of Test Panel A - First Approach with the experiments ............. 63
5.2 Comparison of Test Panel A - Second Approach with the experiments ......... 65
5.3 Comparison of Test Panel B - First Approach with the experiments .............. 66
5.4 Comparison of Test Panel B - Second Approach with the experiments ......... 68
5.5 Summary from the comparisons ..................................................................... 70
5.6 Modeling of Test Panel C – Second Approach ............................................... 71
5.6.1 Boundary conditions and loading ............................................................. 74
5.6.2 Analysis and Results ................................................................................ 74
SUMMARY AND CONCLUSION ......................................................................... 80
6.1 Summary and Conclusions .............................................................................. 80
6.2 Recommendations for future work ................................................................. 82
BIBLIOGRAPHY ................................................................................................................... 83
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LIST OF TABLES
Table Page
2.1 Standard Marking of Heavy Duty Riveted Steel Gratings ........................................ 13
3.1 Eight Static Tests on Heavy Duty Riveted Gratings (White, 2009) ..................... 30
3.2 Intermediate Bearing Bars Contributions .................................................................... 32
4.1 Line Loads for Type I............................................................................................ 38
4.2 Line Loads for Type II ................................................................................................. 39
4.3 Load and Load Factor .......................................................................................... 40
4.4 Dimensions of Sub-Components for Test Panel A – First Approach.................... 42
4.5 Dimensions of the Bearing Bars for Second Approach ........................................50
5.1 Comparison of Results with Experiments for Test Panel A – First Approach….. 64
5.2 Comparison of Results with Experiments for Test Panel A – Second Approach... 65
5.3 Comparison of Results with Experiments for Test Panel B – First Approach….. 67
5.4 Comparison of Results with Experiments for Test Panel B – Second Approach..69
5.5 Cross-Sectional Dimensions of the Sub-Components for Test Panel C ............. 71
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LIST OF FIGURES Figure Page
1.1 Veteran’s Memorial Bridge, Bay City, Michigan .................................................... 5
1.2 LaSalle Street Bridge, Chicago, Illinois ............................................................... 5
1.3 Robert Moses Causeway Southbound Bridge, Captree State Park, New York… 6
1.4 Grosse Ile Bridge, Wayne County, Michigan ......................................................... 6
2.1 Standard Marking of Steel Gratings (MBG 532) ................................................. 12
(a) Welded or Pressure-Locked Grating ................................................................ 12
(b) Riveted Grating ................................................................................................. 12
3.1 Sectional View of R-37-5 Lite Grating ................................................................. 23
3.2 Side View of the Support Arrangement ...................................................................... 24
3.3 Reticuline Bars Geometry .......................................................................................... 24
3.4 Laboratory Setup for R-37-5 (5-inch x ¼-inch) Lite Steel Grating for Test A.. 25
3.5 Laboratory Setup for R-37-5 (5-inch x ¼-inch) Lite Steel Grating for Test B.. 26
3.6 Dial Gauge Locations .................................................................................................. 28
3.7 Strain Gauge Locations ............................................................................................... 28
3.8 General Static Test Layout of R-37-5 (5-inch x ¼-inch) Lite Steel Grating
For Test B ........................................................................................................... 29
3.9 Load vs Micro-Strain for Tests A ...................................................................... 31
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3.10 Distribution pattern in (a) Compression, (b) Tension ....................................... 32
3.11 Micro-Strain Trend on Test Panel B ................................................................. 33
3.12 Sub-Components View Under the Loading ........................................................... 35
3.13 Strain Distribution Directly Under the Load… ................................................. 35
4.1 Load Distribution Type I .................................................................................... 37
4.2 Load Distribution Type II .......................................................................................... 37
4.3 Bar Numbering for Live Load… ............................................................................ 38
4.4 Time History Graph ............................................................................................ 40
4.5 Cross-Section of Sub-Components
(a) Main Bearing Bar, (b) Intermediate Bearing Bar, (c) Transverse Bar,
(d) Reticuine Bar, (e) Rivet.................................................................................. 43
4.6 Reticuline Bar Geometry with Rivets .................................................................. 44
4.7 Model for Test Panel A – First Approach using Loading Type I ........................ 45
4.8 Model for Test Panel A – First Approach using Loading Type II ........................ 45
4.9 Load vs Deflection for Main Bearing Number 4 – Test Panel A
First Approach using Loading Type I ................................................................. 46
4.10 Deflection Across the Grating at 41.6-kips Load – Test Panel A
First Approach using Loading Type I ................................................................. 47
4.11 Load vs Deflection for Main Bearing Number 4 – Test Panel A
First Approach using Loading Type II ..................................................................... 47
4.12 Deflection Across the Grating at 41.6-kips Load – Test Panel A
First Approach using Loading Type II ..................................................................... 48
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4.13 Geometry of Sub-Components for Second Approach
(a) Main Bearing Bar at Intersection, (b) Intermediate Bearing Bar at
Intersection, (c) First Main Bearing at Intersection (d) Last Main Bearing
Bar at Intersection ................................................................................................. 49
4.14 Model for Test Panel A – Second Approach using Loading Type I ................ 51
4.15 Model for Test Panel A – Second Approach using Loading Type II ................ 51
4.16 Load vs Deflection for Main Bearing Number 4 – Test Panel A
Second Approach Loading Type I ...................................................................... 52
4.17 Deflection Across the Grating at 41.6-kips Load – Test Panel A
Second Approach using Loading Type I ............................................................ 53
4.18 Load vs Deflection for Main Bearing Number 4 – Test Panel A
Second Approach using Loading Type II ................................................................. 53
4.19 Deflection Across the Grating at 41.6-kips Load – Test Panel A
Second Approach using Loading Type II ................................................................ 54
4.20 Model for Test Panel B – First Approach using Load Type I ......................... 55
4.21 Model for Test Panel B – First Approach using Load Type II .......................... 55
4.22 Load vs Deflection for Main Bearing Number 4 – Test Panel B
First Approach using Load Type I ...................................................................... 56
4.23 Deflection Across the Grating at 41.6-kips Load – Test Panel B
First Approach using Load Type I ...................................................................... 57
4.24 Load vs Deflection for Main Bearing Number 4 – Test Panel B
First Approach using Load Type II ........................................................................... 57
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4.25 Deflection Across the Grating at 41.6-kips Load – Test Panel B
First Approach using Load Type II ........................................................................... 58
4.26 Model for Test Panel B – Second Approach using Load Type I ....................... 59
4.27 Model for Test Panel B – Second Approach using Load Type II........................ 59
4.28 Load vs Deflection for Main Bearing Number 4 – Test Panel B
Second Approach using Load Type I .................................................................. 60
4.29 Deflection Across the Grating at 41.6-kips Load – Test Panel B
Second Approach using Load Type I ..................................................................61
4.30 Load vs Deflection for Main Bearing Number 4 – Test Panel B
Second Approach using Load Type II ...................................................................... 61
4.31 Deflection Across the Grating at 41.6-kips Load – Test Panel B
Second Approach using Load Type II ...................................................................... 62
5.1 Cross-Section Dimensions of the Sub-Components for Test Panel C
(a) Main Bearing Bar, (b) Intermediate Bearing Bar, (c) Transverse Bar,
(d) Reticuline Bar, (e) Rivet ................................................................................ 72
5.2 Model for Test Panel C – Second Approach using Load Type I......................... 73
5.3 Model for Test Panel C – Second Approach using Load Type II .......................... 73
5.4 Load vs Deflection on Main Bearing Bar Number 4 Loading Type I ................ 75
5.5 Load vs Deflection on Main Bearing Bar Number 4 Loading Type II ................ 75
5.6 Deflection Pattern on the Riveted Grating at 41.6-kips Load… .......................... 76
5.7 Distribution of Strain Across Test Panel C Loading Type I .............................. 77
5.8 Distribution of Strain Across Test Panel C Loading Type II ................................ 78
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CHAPTER I
INTRODUCTION
1.1 Background
Bridges have existed since the start of human history. They are structures that create a
passage over obstacles, rivers, valleys which enrich transportation and business all
around the world. The very first bridges were constructed using wood logs and ropes.
The need for stronger and more durable bridges has opened doors to many
construction methods and materials. In the early stages, there was not a wide
exposure to the behavior of structural systems. With the advancement in construction
materials, methods, equipment and the changing needs of the transportation system,
steel girder bridge construction and analysis methods have evolved. The invention of
the automobile in the 19th century and its mass production has also increased the
transportation needs.
From the early 1900s through the 1940s, simple medium-span bridges typically
consisting of a single-span or multi-span straight bridges were constructed, with the
alignment normal to the obstacle so as to reduce the span length. Many of these
bridges were constructed using timber and steel beam-deck superstructure or cast-in
place concrete. In the 1950s, span lengths of the bridges increased to meet
transportation industry requirements. Steel members that were used in the bridge
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construction for deck superstructures, consisted of either rolled steel beams or
fabricated steel plate girders. By the 1960s, steel box girders were introduced which
exhibited better torsional propertied in curved bridge structures.
‘Bridge deck behavior’ by E.C. Hambly, first published in 1976, has provided
valuable guidance for bridge engineers. By then, the processing power and storage
capacity of computers had increased by more than 1000 times, which made the
analysis and data storage easier. Before this revolution, a significant amount of time
was spent in theoretical analysis and experimental research to design a bridge.
The development of steels with high strength, improved ductility and increased
toughness provides for greater performance of the steel bridges, and these advances
made through research are reflected in the AASHTO specifications.
Today, there are many available commercial software suites which will enable the
engineer to analyze complex systems and aid in understanding the physical behavior
of the bridge. There are many computer modeling techniques in practice and one of
those is the grillage method, which has been in use since the latter half of the 20th
century. This method involves representing the actual structure by an equivalent
grillage of beams. To visualize this arrangement, it is a two-dimensional grillage
formed by interconnecting discrete one-dimensional elements. For analysis purposes,
the bending and stiffness of the structure is assumed to be concentrated in these
3
beams. The elements are chosen so that the behavior is similar to the actual structure
under the same loading. That means that the grillage arrangement deforms like the
actual structure when acted upon by similar loads. It is quite natural that the actual
loading also should be replaced by point loads or line loads on the grillage.
1.2 Problem Statement
In the latter half of the 19th century, two methods were commonly used to analyze
bridge superstructures, grillage analysis and finite element analysis. Due to its wide
range of applications and the flexibility, finite element analysis has become more
prominent. Though grillage analysis is a simpler concept, it is not widely used today.
When the bridge superstructure is complicated, or if immediate results are required
so as to take decisions in the design phase, grillage analysis can be used to get
solutions without a great loss of accuracy. Ohio Gratings Inc. has sponsored The
University of Akron, since 2009 to conduct research in the field of riveted gratings
and between 2011 and 2014, fatigue testing programs were conducted to understand
the behavior and resistance of the grating using H20 loading. There was additional
research done to determine whether the gratings can be modeled and analyzed using
finite element analysis, and the conclusions were positive. This present thesis focuses
on determining the applicability of grillage analysis on the R-37-5/8 Lite steel grating
under H20 loading with an impact factor of 30%. The range of variation of the
solution from the sample testing data is studied and guidelines to model a riveted
grating for grillage analysis are given.
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1.3 Justification
Finite element analysis is a complicated process which needs attention to every detail
in each and every phase. If there is any minute error in the input data, it can lead to
erroneous results. These software packages need a level of expertise and practice.
When solutions are needed immediately, and pinpoint accuracy is not that important,
the grillage analogy can be used to represent many structures with an equivalent 2D
grid arrangement. Software package that offers 2D and 3D frame analysis may be
used as to perform grillage analysis as a type of frame analysis. Therefore, once
understood, the concept and rules to be followed to represent a structure using a
grillage analogy, can save a lot of time and effort while giving the feel of structural
behavior without a great loss of accuracy.
1.4 History of Heavy Duty Riveted Gratings
Heavy duty riveted grating is commonly used in movable bridges, where the dead
weight of the bridge structure should be minimized and the live load capacity needs
to be assured. According to the website “historicbridges.org”, there are over 1000
riveted grating bridge deck structures in the USA alone. These structures performed
beyond expectations for many years. The Veterans Memorial Bridge in Bay City,
Michigan, the LaSalle Street Bridge in Chicago, Illinois and the Robert Moses
Causeway Southbound Bridge in New York are a few among these bridges with
5
riveted grating deck superstructure. Figure 1.1 shows the Veterans Memorial Bridge
in Bay City, Michigan.
Figure 1.1 Veterans Memorial Bridge, Bay City, Michigan
This four-lane bridge was installed in 1994 and has five inch deep heavy duty riveted
grating bridge deck. Inspection revealed that it is still in good condition after 23 years.
Figure 1.2 LaSalle Street Bridge, Chicago, Illinois
The riveted grating bridge deck superstructure in the Figure 1.2 was installed as a
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part of 1971 renovation project. This bridge is in a location with an average traffic of
27,000 vehicles per day. It was demonstrated that the bridge was in good working
condition even after 37 years in 2008.
Figure 1.3 Robert Moses Causeway Southbound Bridge, Captree State Park, New
York
The Robert Moses Causeway Southbound Bridge was constructed in 1951 and was
still in service in the year 2007.
Figure 1.4 Grosse Ile Bridge, Wayne County, Michigan
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The Grosse Ile Bridge was constructed during 1912 and 1913. It is a privately owned
bridge connecting an Island. The bridge deck was replaced with a welded steel grating in
1986, and developed cracks over the years and hence the deck was again replaced with
heavy duty riveted grating in the year 2007. The picture above is taken in 2007.
1.5 Objectives
Hota V. S. Gangarao (1987) studied open grid decks by testing 26 open grid panels.
These grids were comprised of a few riveted gratings and others were welded. He
concluded in his work that riveted gratings have more strength and outperform
AASHTO specifications. He also suggested that if the fatigue behavior is studied
further and if the cross-sections could be decreased, cost savings would be realized.
Peter C. Fetzer (2013) studied the behavior of open grid steel bridge decks unde r
service and fatigue loads. Eight different types of open grid bridge deck specimens
were used as part of the experiments and he concluded that gratings performed well
in smaller spans when compared to the larger spans. Godwin A. Arthur (2014)
studied heavy duty riveted steel gratings under H20 loading and proposed an S-N
curve. Using Finite Element Analysis, fatigue life of the heavy duty riveted gratings
was estimated.
This research at the University of Akron has the following objectives:
• Develop grillage analysis models using the software package SAP2000.
• Compare two grillage models, one with reticuline bars and other without
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reticuline bars, with the results from experimentally obtained values. This will
help to decide on how simple the specimens can be modeled while not losing
accuracy in the results.
• Based on the findings from the two models, model another grating with a
different span.
• Study the stress distribution under static H20 loading with an impact factor of
30% along the components of the riveted grating.
1.6 Outline
As part of the study of the heavy duty riveted steel gratings using grillage analysis, a
number of approaches were tried and they are all presented in this thesis. Chapter 1
provides a basic introduction to the evolution of bridges and bridge analysis in the
United States. The problem statement for this current research and its objectives are
also given. Chapter 2 provides background on grillage analysis, introduction to
gratings and the nomenclature of gratings according to NAAMM (National
Association for Architectural Metal Manufacturers). The design principles used to
represent any structure using a grillage are also given in chapter 2.
Chapter 3 presents all the information regarding the specifications of the riveted steel
grating used for this thesis. Experiments conducted are explained in chapter 3 along
with the loading. The results obtained from the tests performed on the sample gratings
are also presented in this chapter. Strain patterns with respect to the applied loading
is shown in the form of graphs and discussed.
9
Models of two steel test gratings for grillage analysis are explained in detail in
chapter 4. Analysis of test panels with two different approaches are explained in
detail and the results are shown.
Results from sample testing are compared to the results from the model analysis in
chapter 5. Based on the findings from the comparisons test panel C is modeled and
the analysis run to results are discussed.
Chapter 6 gives the conclusion and summary along with the recommendations for
future work.
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CHAPTER II
LITERATURE REVIEW
2.1 Introduction
Open grids, filled decks or partially filled decks are the bridge deck systems commonly
used. These bridges are performing well under service load conditions all around the United
States. Among the three different types of grid deck systems (namely riveted, welded and
pressure locked), riveted steel gratings are still used due to its strength, life and reliability.
Heavy-duty riveted steel gratings are made of main bearing bars, which are usually
rectangular in cross-section, separated by a definite distance and held together by cross
beams and reticuline bars. These reticuline bars are riveted to the main bearing bars at
regular points of intersection. As in the present case, a lite version of the steel grating has
alternate main bearing bars replaced by intermediate bearing bars which are of smaller
cross-section compared to that of main bearing bars.
2.2 Metal Bar Gratings - Flooring System
Metal gratings are used for supporting a wide variety of loads from human traffic to
vehicular loads. The Heavy-Duty Grating manual of the National Association of
Architectural Metal Manufacturers (NAAMM) sets rules for their design and use. The use
11
of open grid bridge deck systems over other types of bridge deck systems have emerged due
to the following.
• Riveted gratings are light weight and of high gauge in supporting vehicular weights.
• Relative ease of installation and less disruption to normal operations. Also, replacing the
damaged parts of the bridge deck is simplified as compared to other deck types.
• Maximum quality control in the shop.
• Open Grid decks are redundant, strong and cost effective.
• Minimize the need for snow removal.
There are several issues related with the utilization of grid decks associated with the
behavior over time, because of the development of localized cracks, primarily in welded
systems. Given the localized cracking of welded grids, routine maintenance is required to
keep them in good condition. Plates utilized for repairs result in poor aesthetics. While
welded decks have been utilized widely because of the shedding of snow, enhanced welding
techniques and the simplicity of creation. Issues related with welded decks however include
welding defects and stress concentrations which gives rise to local fatigue cracking, as
contrasted with a riveted deck utilized under comparable conditions (Gangaroo, 1987).
12
2.3 The Marking System of Metal Gratings
The standard marking system for metal gratings has five identifiers that describe geometrical
properties. These standards are given by NAAMM and they are used in the industry.
Figure 2.1 Standard marking of Steel Gratings (MBG 532)
(a) – Welded or Pressure Locked Grating, (b) – Riveted Grating
In this research, as R-37-5 (5x3/8 inch) Steel Lite grating was used. The principal parameter,
R, depicts the joining method which for this situation is Riveted. The number (37) speaks to
the spacing between the main bearing bars. It is interpreted as, 37 times 1/16th of an inch, or
the spacing from bar to bar to bar is 37/16-inches. The third parameter (5) provides the
separation of the rivets in inches. The fourth item (5 x 3/8) provides the cross-section of the
main bearing bars. The fifth parameter (Steel) is the material utilized for the grating. The
additional parameter (Lite) means that the alternating main bearings are replaced by
intermediate bearing bars that have a smaller cross-section that in turn will reduce both the
13
weight and the cost. Table 2.1 gives a rundown of the standard nomenclature identifying
gratings.
Table 2.1 Standard Marking of Heavy Duty Riveted Steel Gratings
Parameter Standard Marking
Description
1 R Riveted
2 37 Bearing bars spaced 2 -5/16 between faces
3 5 Rivets spaced 5in on centers
4 5 x 3/8 Bearing Bar size
5 STEEL Material
2.4 Background
2.4.1 Heavy-Duty Riveted Gratings Background
Rivets were the favored mechanical fasteners, and were used for a long time until the moment
that the presence of bolts. They provide for positive attachment and confinement, safely
transmitting the associated loads. Rivets still find typical use in the aerospace and rail
applications. Rivets are used to join longitudinal and reticuline bars in the steel grating that
were tested under static H20 loading. There has been a diminishing use of rivets because the
bolts provide more predictable clamping under open cross sections in certain circumstances.
However, the use of decks and controlled fabrication make the use of rivets acceptable. The
present layout methodology used for heavy-duty riveted steel gratings is given in the
NAAMM Manual, which uses an elastic analysis approach following allowable stress design
concepts. When the stresses exceed and a section of the structure yields, the excess load is
simply transferred to the sections adjacent to it. This redistribution of the loads among the
sections is the principal cause for the loads to be managed effectively.
14
Different testing methods have been utilized in the outline and investigation of open grid
decks keeping in mind the end goal to exploit the additional load carrying capabilities
happening after yield. A plastic mechanism method utilizes virtual work concepts. Such
failure is joined by extensive deflections with the behavior withdrawing from the elastic
hypothesis. Yield line investigation along with the virtual work has shown the redundancy
and quality of open grid deck gratings (Cannon 1969). Cannon (1969) tried two grating decks
and utilized the yield line theory to decide how the gratings collapse and calculate the
required load. Minute outline conditions were produced for both, a square grating and a
rectangular grating which rely upon the dispersing of the network and the plastic limit. It
was resolved that gratings had less strength than slabs as they displayed more noteworthy
quality than that anticipated by yield line hypothesis. The report showed that the hypothetical
yield line collapse load gives a decent, upper bound, measure of the genuine collapse load
of the gratings. Vukov (1986) has proposed an upper bound approach for orthogonal grid
systems using kinematic mechanisms and the results were in good correlation with the
laboratory results of Cannon (1969).
According to the AASHTO LRFD specifications article 4.6.2.1.1, approximate methods can
be used for the analysis of grid deck members. Therefore, as one of the approximate methods,
the bridge deck can be represented by a combination of strips perpendicular to the supports.
15
Huang (2001) attempted to understand the behavior of grid decks by utilizing analytical, trial
and numerical strategies. Four open and three filled grid decks that were welded were used
to measure their structural conduct tentatively. The expository techniques incorporated the
utilization of the traditional orthotropic thin plate analysis and the hypothesis of beams on
elastic foundations. Three dimensional limited component models were produced for both
the open and filled grid decks. Results from the finite element analysis were contrasted, and
established the behavior to be similar to orthotropic plate theory. Parametric investigations
were conducted to determine what influences the deck behavior. Concrete filled decks were
stiffer than their open framework partners by around 33%. Results from the FEM models
likewise demonstrated general concurrence with test results and in this manner FEM can be
utilized as a device to investigate and configuration of open network decks. Further, a
traditional orthotropic hypothesis gives sensible outcomes for filled grid decks but, not open
grid decks.
Mahama (2003) analyzed the present metal bar grating design provisions utilizing analytical,
numerical and experimental approaches. Real accentuation was put on creating limited
component models of welded gratings utilizing ANSYS, consolidating both material and
geometrical nonlinearities to foresee the collapse load of the gratings. Information from both
diagnostic and finite element analysis were compared with the physical tests, which gave an
understanding into the breaking point of the metal gratings.
Bejgum (2006) did an evaluation of the present techniques for metal bar gratings and tested
four heavy-duty riveted and two welded metal gratings under static loads. The gratings were
16
loaded over and a simulated tire patch of 20in x 20in was utilized to recreate loading
condition. Strain and deflection information was obtained as loads were applied until the
point that the LVDTs went offscale. A nonlinear finite element analysis in view of the
models made for design loads, was produced and adjusted with research facility information.
Collapse loads of the gratings were found to have related well with analytical (within 20%)
and the finite element analysis.
Arthur (2014) found that the there are no design provisions in AASHTO LRFD
specifications for heavy-duty riveted gratings based for fatigue evolution. Therefore, to study
the fatigue behavior of heavy-duty riveted gratings, two full size R-37-5 (5x1/4) Lite Steel
gratings were used to evaluate behavior under standard H20 loading. Finite element analysis
models were also generated to compare the experimental results. Twenty-six smaller panels
were then used for fatigue testing at varying stress ranges. Fracture mechanics approach was
utilized in the research. Arthur (2014), using this approach, proposed an effective width that
can be used to estimate the fatigue strength and thereby fatigue life of the riveted gratings.
The results from his observations were also utilized to propose an S-N curve and design
guide for heavy-duty riveted steel gratings.
2.4.2 Grillage Analogy Background
First ever research on developing the guidelines for grillage idealization was done by
Lightfoot (1963). The use of a grid framework to replace a plate was in use before, but there
were no specific guidelines to follow. Lightfoot (1963) utilized the concepts of Matrix
multiplication in slope deflection equations, which included transverse forces and
deflections in the transverse direction.
17
Davison and Hughes (1974) gave equations for edge moments and interaction forces, the
knowledge of which was utilized to analyze various beams using simple beam theory. The
suggestions given proved to be extremely useful in desk calculations and if computers were
used, these suggestions proved to make the computations faster. Stiffness matrices were
formulated based on the assumptions and the equations proposed were used to calculate
various required parameters like deflections, stress and moments using simple beam
hypothesis. Working examples on ship deck idealization were also presented as part of the
publication to support the guidelines.
Jager (1982) made an attempt to provide guidelines for grillage idealization of various types
of structures. The research focuses on idealization of slab, beam-slab, cellular and voided
slab bridge structures. Varying depth bridge deck idealization was also included. Jager
(1982) proposed formulae for moment of inertia, torsional inertia for all the above-mentioned
cases and showed that grillage analysis can be adopted to any kind of structure even with
complications like heavy skew, edge stiffening and isolated supports.
Yaseer (2014) from the University of Kufa studied the concept of grillage analysis with the
help of two models. The deflection and stress results were the result of the moment caused
due to applied normal forces on the steel I beams. Results from the two models were
compared with the experimental results and found to be in good correlation when warping
and shear stresses were taken into consideration. One of the two models included beams with
18
different cross-sections which implies that they have different stiffness matrices. It was also
concluded that grillage analysis can be utilized at the design stage to any structure with a
single material or a structure with different materials just by taking into consideration of the
changes in the mechanical properties.
2.5 Grillage Analysis
A comparable grillage of interconnected beams can be developed to give a satisfactory
dispersion of forces and deflections inside the steel structure. Although the technique is
essentially surmised, it has the benefit of generality. At the joints of the grillage, any ordinary
type of boundary condition might be considered to solve any condition. This measure of
value, joined with economy in registering, input planning and translation of output, makes
the grillage analogy a well-known and broadly utilized strategy in workplaces.
The grillage anology includes the portrayal of successfully a three-dimensional steel
structure by a two-dimensional collection of discrete one-dimensional interconnected beams
in bending and torsion. This method can be applied to a practical problem and can be
compared with the results obtained from other methods and the available experimental work
data. The proposed method in this research has been to provide acceptable results in the past
for various structures. But, this method was always used to analyze concrete slabs, concrete
slabs with steel reinforcement. This research attempts to apply this methodology on steel
only structure to understand if there are any changes in the modeling required.
19
2.5.1 Stiffness Matrix
To construct a stiffness matrix of a steel structure or any structure, the individual stiffness
matrices of the members must be first formulated, and are typically 6x6. These formulations
should ideally include flexural rigidity (EI) for longitudinal deformations and torsional
rigidity (GJ) to account for torsional deformations. Transverse deformations are also
accounted by the inclusion of shearing rigidity (GA).
The general sign convention used while constructing a stiffness matrix:
· Moments and rotations are considered to be positive in clock-wise directions from the local
coordinates (right hand rule)
· Forces and deformations are considered positive downwards.
A typical stiffness matrix [K’] equation can be given as the following, which depends on the
Force vector {F’} and displacement vector {δ’} in the local coordinates:
{F’} = [K’]{δ’} (1)
After formulating individual stiffness matrices and transferring them all to global
coordinates, the following governing matrix equation in the global coordinates is formed:
(2)
20
Where,
2.5.2 Design Principles for Grillage Members
For any structure to be represented by an equivalent grillage, the elastic rigidities of the
proposed grillage members must be equal to the elastic rigidities of the structure under
consideration. The elastic rigidities for any structure can be broken down to three elastic
section rigidities:
· Bending (or flexural) rigidity (EI)
· Torsional rigidity (GJ)
· Shearing rigidity (GA)
These can be calculated from basic equations that are known for rectangular or I shaped
members. If any other complex sections are involved, careful assessment has to be made.
The whole idea of maintaining the elastic rigidities of the grillage member equal to the
structure under consideration is because of the idea that the grillage member has to deform
or rotate in the same amount as that of the actual structure under similar loading conditions.
The other main design principle that needs to be taken care is that the grillage member
21
position should coincide with the center line of the actual member that it is representing.
Once all the members are assembled into a grillage structure, boundary conditions should be
applied to best suit reality. This is an approximation and even these boundary conditions are
to be applied at the center line. Then, the loads should be idealized to line or point loads that
can be applied on the grillage members and the analysis is then performed to get the
anticipated results.
22
CHAPTER III
EXPERIMENTAL TESTS
This chapter deals with the experimental setup portion of the study. Setup for the experiment
is described in detail along with the various grillage models generated to obtain the simplest
design. While performing various analysis on the grillage model, the structure is assumed to
be continuous as observed in real situations for bridges. AASHTO LRFD specification with
a few modifications were followed during the laboratory testing phase and the test panels
under consideration are R-37-5 (5”x3/8”) Lite Steel gratings.
3.1 Specifications of R-37-5 (5”x1/4”) Lite Steel gratings.
The grating under consideration is regarded as a heavy-duty steel grating with a total span of
130.5-inch span length over three supports. The main bearing bars are of a rectangular cross-
section with 5-inch depth and 1/4-inch thickness dimension. Alternate bearing bars as in case
of a regular R-37-5 (5”x1/4”) Steel grating are replaced by intermediate bearing bars to reduce
the dead weight of the structure to accommodate more live loads. It is estimated that this
replacement of alternate bearing bars can save upto 30% of the total deal load of the structure.
It is also observed even after the replacement; the Lite gratings still serve exceptionally well
under the anticipated vehicular loads. These intermediate bearing bars also have a rectangular
23
cross-section with 1.5-inch depth and 1/4-inch thickness. Transverse bars are the sub-
components of this grating responsible for the transfer of loads and deflections among the
adjacent bearing bars. The geometry of the assembly is shown in the figure 3.1
Figure 3.1 Sectional view of R-37-5 lite grating
All the main bearing bars and intermediate bearing bars are connected with each other with
the help of reticuline bars. The importance and behavior of these reticuline bars or connecting
bars are also studied in this research. These are the sinuously bent bars with a cross section of
1.5-inch depth and 0.25-inch thickness. These have a tooth like structure on the top surface
to offer better grip to the vehicles. Supports are W8x24 I section stringers. The complete
structural arrangement can be seen in the figures 3.2, 3.3, 3.4 and 3.5. The static testing was
conducted on two span lengths: A-42-inch and B-65-inch and the results are discussed in this
chapter.
27
A spreader beam was used to apply loads on the grating, 21-inch apart from the central
support for Test A and 3-feet apart for Test B to replicate AASHTO H20 loading condition
and to generate the maximum negative bending moment over the central support. A set of
strain gages and LVDTs were attached to the grating to record tensile strain values and
deflections. The readings from the strain gages and LVDTs are read on Vishay System 5000
data acquisition system. The strain gage placement on the bars was 0.25-inch from the top
and bottom of every bearing bar in the loading plane.
Strain gage locations, LVDTs locations, general test layout of the gratings are shown in the
Figures 3.6, 3.7 and 3.8. LVDT 5 is placed on the top of bottom flange of the support beam
as shown in Figure 3.6.
Figure 3.6 Dial Gauge Location
29
Figure 3.8 General Static Test Layout of R-37-5 (5”x1/4”) Lite Steel Grating for Test B
3.2 Loading
A 10” x 20” center tire patch load was applied on the grating with the help of a high
durometer rubber pad attached to a steel spreader beam. AASHTO LRFD specifications
suggest using 15% impact factor, but here, 30% impact factor was conservatively used to.
The static loads were applied to the grating by an MTS actuator of 55-kips capacity. The
load gradually increases from 0kips to 41.6-kips in increments of 5-kips. The test process
was repeated after the gradual unloading following the same decrements of 5-kips until the
load on the grating reached 0-kips.
3.3 Experimental Results and Discussion
Experiments conducted on the test panels during Test A and Test B are presented here and
discussed. The load positions are as shown in the figure 5.4 and figure 5.5 and the loading is
applied in the increments of 5kips until the maximum load, 41.6kips is reached. The behavior
30
of heavy-duty riveted grating with the effect of varying load conditions, support conditions
is reported with the concentration on the stress distribution.
3.3.1 Load-Micro Strain Trend for Test A
Both test panels A and B were loaded to a maximum load of 41.6kips and unloaded. 5000
Vishay data acquisition unit was used to record that strain values with each load increment.
It includes strain values in both tension and compression at 0.25-inch from the top and
bottom ends of the bearing bars. The load-micro strain relationship is as shown in the figure
3.9 for the eight tests performed as tabulated in table 3.1
Table 3.1 Eight Static Tests on Heavy-Duty riveted Gratings (White, 2009)
Test
No:
Description
1 I-Beam supports were used to support the deck from the center of the
grating on both sides. Loading footprint used is 10-inch x 20-inch.
2 Same test setup as Test-1
3 Additional strain gauges were added on the top of the recticuline and
intermediate bearing bar next to the middle main bearing bar, while
maintaining rest of the test setup similar to Test-1
4 Test setup similar to Test-1 with additional strain gauges on the
recticuline bars and intermediate bar on the bottom on either side of the
middle main bearing bar.
5 Same setup as Test-1 with Alternate Triangular supports instead of I-
31
Beam supports at 21-inch spacing from the middle on either side.
6 Test setup 5 with re-centered Jack and shimmed Triangular supports.
7 Supports are changed to I-Beam Supports and aligned so that the web
is in line with the Intermediate Bar 1. Large Jack is replaced with
smaller Jack and all the other conditions remain the same.
8 I-Beam supports are replaced by modified Triangular Supports with
bottom surface ground down. Large Jack is replaced by smaller Jack
and all the other conditions remain the same.
There is a steady increase in the strain values with increasing load as anticipated. A shift in
the load-micro strain values is observed for test 7 and test 8 which can be explained because
of changing support conditions.
Figure 3.9 Load Vs Microstrain for Tests A
32
As the strain gages placed at 0.25-inch from both top and bottom ends of the bearing bars,
the stress distribution can be obtained in tension and compression regions. Figure 3.10 shows
the strain distribution pattern of the grating in both tension and compression at a load of 30-
kips. The main bearing bars directly under the applied load and the bearing bars that are next
to load offer the maximum resistance. Remaining bearing bars offer are significantly lower
resistance meaning the response is more localized.
Figure 3.10 Strain Distribution in (a) Compression, (b) Tension for Test A
33
3.3.2 Load-Micro strain Trend for Test B
Static tests were also conducted on test panel B shown in figure 3.5 and the results are shown
in figure 3.11. Strain gauges were placed at two positions over the span, first set between
supports and other set over the central support.
Figure 3.11 Micro-strain trend on test panel B
The results follow the expected trend and it is seen that the maximum str ain is observed
directly under the load. The strain values are observed to be influenced by the support
conditions and stiffness of the adjacent panel.
34
3.4 Parametric Studies:
3.4.1 Intermediate Bearing Bars and Reticuline Bars
The total load carrying capacity is provided by all sub-components of the riveted grating, i.e.
main bearing bars, intermediate bearing bars, reticuline bars and transverse bars. Together
they offer stability and structural capacity. The different configurations of the grating are
based on the number of main bearing bars provided and number of intermediate bearing bars
provided. Based on the number of main bearing bars and intermediate bearing bars, different
spacing provides for unique dimension combination. The contribution of individual
components toward the total load carrying capacity is studied in this section. As mentioned
earlier, strain gauges were attached to the intermediate bearing bars and reticuline bars
directly under the load. Five tests each for Test A and Test B were run with loads varying
from 0 to 41.6-kips. The components under observation are shown in figure 3.12 and the
results are shown in the figure 3.13.
It is observed that the reticuline bars have low stress values for all loading cases, and as such,
can be ignored in calculation of the total load carrying capacity. But, they cannot be ignored
or removed as they are the components that provide lateral stability to the grating.
Intermediate bearing bars have shown significant amount of stress, which means that they
contribute to the total load carrying capacity. Table 3.2 shows the percentage of total strain
absorbed by the intermediate bars as compared to the main bearing bars over the central
support.
35
Figure 3.12 Sub-components view under the loading
B1, B2, B3 are the main bearing bars and I1, I2 are the intermediate bearing bars directly
under the load.
Table 3.2 Intermediate bearing bars contribution
Test Number B1+B2+B3 (%) I1+I2 (%)
3 83 17
4 82 18
5 84 16
Figure 3.13 Strain distribution directly under the load
900
800
700
600
500
400
300
200
100
0 B1 I3 B2 I2 B3
Main Bearing Bars Intermediate Bearing Bars
Mic
rost
rain
36
Chapter IV
GRILLAGE ANALYSIS MODELING
The static testing loading was conducted on Test Panel A and Test Panel B. Two approaches
are analyzed to obtain the most time saving method to model heavy duty riveted gratings
while not losing accuracy. The results from these models are compared those obtained from
the test results. After completion of models for panels A and B, a model is developed on a
37 R 5 Lite, 5-inch x 3/8-inch panel which is termed as Test Panel C. All specimens are
modeled and analyzed in SAP2000 v17.
One of the approaches include consideration of complete reticuline bars and the other
approach only considers the reticuline bar geometry at the point of contact with the bearing
bars. Beam elements are chosen to define all sub-components and the models appear to be a
frame work with discrete one-dimensional elements in a 3-dimensional space.
4.1 Load Distribution
Distribution of load on the grillage model is one of the key steps involved in the analysis. A
basic grillage analogy principle is to represent a 3D model with a combination of one
dimensional lines arranged in the form of a frame. Therefore, all the loads applied on a frame
have to be idealized as line loads or point loads that can be applied to the model. There are
a few methods to do this and the method applied in this research is explained here.
37
Area loads were idealized as line loads applied on the main bearing bars. The number of
bearing bars under the load area with the spacing between them have been considered. The
area between two bearing bars is divided into two halves and each of the bearing bars
supports the load from one half of the area. When the area load doesn’t take the whole area
between the bearing bars, then the load is divided between the two bearings by taking
moments.
Two different types of load distribution are applied to the same model to check if there are
any major differences between them. In type I, the loads are distributed only to the main
bearing bars and in type II, loads are distributed between main bearing bars and intermediate
bars. Figure 4.1 shows a Type I distribution and figure 4.2 shows a Type II.
Figure 4.1 Load Distribution Type I
Figure 4.2 Load Distribution Type II
In Type I load distribution, the live load is distributed on two main bearing bars only. The
38
total load in Type II, the total live load is distributed between two main bearing bars and
three intermediate bearing bars. The length of the live load is a standard 20-inches which is
the length of the tire patch. Figure 4.3 shows the numbering of bearing bars and table 4.1
shows the distributed load values on the bearing bars for Type I and table 4.2 for Type II.
Figure 4.3 Bar Numbering for Live Load
Table 4.1 Line Loads for Type I
Bearing Bar Number Line Load Value (kips/in)
7 0.52
9 0.52
39
Table 4.2 Line Loads for Type II
Bearing Bar Number Line Load Value (kips/in)
6 0.06853343
7 0.3117088
8 0.2795
9 0.3117088
10 0.06853343
4.2 Load Graph
Live load is applied on the model for panel in Test A and Test B which gradually increases
from 0 kips to 41.6 kips and is gradually unloaded. Figure 4.4 shows the loading and
unloading pattern that is applied on the specimen as a time history graph option available in
SAP2000. The total loading and unloading time for one complete cycle is 24 seconds. The
time history graph is constructed to show the loading of one tire patch or half of the axle
load. Table 4.3 shows the load and time variation for the graph.
To construct a time history graph in SAP2000, a load factor is calculated. The graph in figure
4.4 shows the load factor on Y-axis which can be calculated with the equation:
Load Factor = (Load)/Maximum load
= load/20.8
40
1.2
1
0.8
0.6
0.4
0.2
0 0 5 10 15
Time in Secs 20 25 30
Figure 4.4 Time History Graph
Table 4.3 Load and Load Factor
Time in seconds Load in lbs Load Factor
0 0 0
1 1000 0.048076923
2 5000 0.240384615
3 10000 0.480769231
4 13800 0.663461538
5 10000 0.480769231
6 5000 0.240384615
7 1000 0.048076923
8 0 0
9 1000 0.048076923
Load
Fac
tor
41
10 5000 0.240384615
11 10000 0.480769231
12 13800 0.663461538
13 15000 0.721153846
14 17500 0.841346154
15 20000 0.961538462
16 20800 1
17 20000 0.961538462
18 17500 0.841346154
19 15000 0.721153846
20 13800 0.663461538
21 10000 0.480769231
22 5000 0.240384615
23 1000 0.048076923
24 0 0
4.3 Grate Models
Several approaches were examined in developing models for test A and test B panels. These
approaches are explained and discussed in detail. The results from both the approaches are
compared to the experimental results for both test panels A and B. Then, the best method
was selected for the analysis of 37 R 5 Lite, 5-inch x 3/8-inch grating with span of 62-inches.
As mentioned earlier test panel A has an effective span of 42-inches and test B panel has an
effective span of 65-inches.
Material used for modeling all the sub-components is A36 steel which has an elastic modulus
42
of 29,000ksi and Poisson’s ratio of 0.3.
4.3.1 Test Panel A modeling – First Approach
The first Approach includes modeling of the complete gratings sub-components. The
geometry with dimensions are shown in the figure 4.5. Dimensions of the various sub-
components are tabulated in table 4.4.
Table 4.4 Dimensions of Sub-Components for Test Panel A - Approach A
Sub-Component Dimensions
Main Bearing Bar 5-inch x 1/4-inch
Intermediate Bearing Bar 1.5-inch x ¼-inch
Transverse Bar 1-inch x 1/4-inch
Reticuline Bar 1.5-inch x ¼ -inch
Rivets 0.375 inch Dia
To start the model development, grid lines are generated and the first principle of modeling
any structure using grillage analysis is to position the grid lines at the center line of the
bearing bars. Therefore, while constructing grid lines the spacing is given such that it
includes the face to face spacing of the bearing bars as well as the thickness of the bars. In
this way, grid lines are aligned along the center lines of the bearing bars.
Later, transverse and reticuline bars are constructed by generating additional gridlines. To
model the riveted connections, the rivets are given an extra length so that there is an
outstanding part of the rivets at every end. Geometry of the rivets is shown in figure 4.6.
43
(a) (b)
(c) (d) (e)
Figure 4.5 Cross-section of Sub-Components
(a) Main Bearing Bar (b) Intermediate Bearing Bar (c) Transverse Bar
(d) Reticuline Bar (e) Rivet
44
Figure 4.6 Reticuline Bar Geometry with Rivets
4.3.1.1 Boundary Conditions and loading
Analysis of a structure examine behavior under prescribed loading and requires definition of
idealized boundary conditions to best represent the real-life scenario.
Simply supported boundary conditions tends to yield maximum results and fixed end
boundary conditions give minimum. They act as upper and lower bounds to any real life
condition. The model used in this study is restrained in all directions on the ends and simply
supported in the center to enable continuous behavior of the grating. This is an approximate
representation of the laboratory tests. Distance between the supports is 42-inches. Figure 4.7
shows the complete model as given in SAP2000 for test panel A, the first approach with load
type I. Figure 4.8 shows the model setup for load type II
45
Figure 4.7 Model for Test Panel A – The First Approach with Load Type I
Figure 4.8 Model for Test Panel A – The First Approach with Load Type II
46
Loading on either side of the central support is separated by a distance 21-inches. These line
loads are applied using time history function which are defined as live loads.
4.3.1.2 Analysis and Results
To analyze the grating for gradual increase and decrease of load, linear direct integration
history is selected as the load case. For time history analysis, modal analysis is required and
the modal analysis is done first to decide the most impacting mode number for the time step
determination. The analysis is then set to run.
Figure 4.9 Load vs Deflection for main bearing bar number 4 – Test Panel A First Approach Load Type I
Load in kips experiment model
25000 20000 15000 10000 5000 0
0.06 0.05 0.04 0.03 0.02 0.01
0
Defle
ctio
n in
inch
es
47
Figure 4.10 Deflection across the grating at 41.6kips load– Test Panel A First Approach Load Type I
Figure 4.11 Load vs Deflection for main bearing bar number 4 – Test Panel A First Approach Load Type II
0.06 0.05 0.04 0.03 0.02 0.01
0 1 2 3 4
Main Bearing Bar Number Experiment Analysis
0.06 0.05 0.04 0.03 0.02 0.01
0 0 5000 10000 15000 20000 25000
Load in kips experiment model
Defle
ctio
n in
inch
es
Defle
ctio
n in
inch
es
48
Figure 4.12 Deflection across the grating at 41.6kips load–
Test Panel A First Approach Load Type II
The results from the analysis run in SAP2000 were within 20% of the experimental results.
The maximum deflection at gauge location 4 is 0.05089-inches when loading type I is
applied and is 0.05033-inches when loading type II is applied. The maximum strain value
for the 30-kips load directly under the tire patch is 1028 microstrains.
4.3.2 Test Panel A modeling – Second Approach
Approach B includes modeling of main bearing bars, intermediate bars and transverse bars.
The basic assumption for this approach that makes it different is that at junctions where the
reticuline bars are connected to the bearing bars with rivets, the reticuline and bearing bars
are modeled as one unit. Therefore, cross-section at these joints look like T beam. The
geometry of the bearing bars are shown in figure 4.9.
Analysis Experiment
Main Bearing Bar Number 4 3 2 1
0.06 0.05 0.04 0.03 0.02 0.01
0
Defle
ctio
n in
inch
es
49
(a) (b)
(c) (d)
Figure 4.13 Geometry of Sub-Components for Second Approach
(a) Main Bearing at Intersection, (b) Intermediate Bearing Bar at Intersection,
(c) First Main Bearing bar at Intersection, (d) Last Main Bearing Bar at Intersection
Similar to the first approach, the grid lines are generated based on the face to face spacing
50
of the bearing bars and the thickness of bearing bar and reticuline bars. Bearing bars are
placed taking into account the combination of bearing and reticuline bar cross-sections. At
these locations, T shaped cross-sections are provided to satisfy the assumptions and the
remaining parts of the bearing bars are assumed to have normal rectangular cross-sections.
The Dimensions of the bearing bars at the junction are tabulated in table 4.4.
Table 4.5 Dimension of the Bearing Bars for Second Approach
Property Main Bearing Bar T
Section, in
Intermediate Bearing
Bar T Section, in
Height 5 1.5
Flange Width 0.75 0.75
Depth of Flange 1.5 1.5
Web Thickness 0.25 0.25
4.3.2.1 Boundary conditions and loading
There are three supports provided by wide flange beams. Boundary conditions within the
model are similar to the boundary conditions provided in the laboratory. The distance
between supports is 42-inches. The complete setup for the analysis in SAP2000 is shown in
figure 4.10 and figure 4.11.
51
Figure 4.14 Model for Test Panel A – Second Approach B using Load Type I
Figure 4.15 Model for Test Panel A – Second Approach using Load type II
52
Loading is applied in the form of a time history function at 21-inches center to center distance
from the central support on either side.
4.3.2.2 Analysis and Results
The analysis procedure is same for all the tests performed. Model analysis is first performed
to decide the most critical mode number and the time step for the time history analysis is set
such that it is less than 1/10th of the time interval obtained from modal analysis. Once all the
inputs are given, the analysis is then set to run and the results are read.
Figure 4.16 Load vs Deflection for Main Bearing Bar Number 4 – Test Panel A Second Approach using Load Type I
Load in kips Experiment Model
25000 20000 15000 10000 5000 0
0.06 0.05 0.04 0.03 0.02 0.01
0
Defle
ctio
n in
inch
es
53
Figure 4.17 Deflection across the grating at 41.6kips load– Test Panel A Second Approach using Load Type I
Figure 4.18 Load vs Deflection for Main Bearing Bar Number 4 – Test Panel A Second Approach using Load Type II
Analysis Experiment
4 3 2 1
0.06 0.05
0.04
0.03
0.02
0.01
0
0.06 0.05 0.04 0.03 0.02 0.01
0 0 5000 10000 15000 20000 25000
Load in kips experiment model
Defle
ctio
n in
inch
es
54
Figure 4.19 Deflection across the Grating at 41.6kips Load– Test Panel A Second Approach using Load Type II
The results obtained from this method are comparatively higher than the results obtained
from approach A which make the results less divergent from the experiments. The maximum
deflection on the main bearing bar number 4 is 0.05402-inches from loading type I and
0.0552-inches with loading type II, directly under the tire patch. The maximum strain for 30-
kip load directly under the tire is 1009.2-microstrains.
4.3.3 Test Panel B – Approach A
All the modeling for test panel B is similar to that of test panel A. The only difference
between test panel A and test panel B is the span of the grating. While test panel A has a 42-
in span, test panel B has a span of 65-inches. The complete model setup for test panel B is
shown in figure 4.12 and 4.13.
Analysis Experiment
Main bearing bar number 4 3 2 1
0.06 0.05 0.04 0.03 0.02 0.01
0
Defle
ctio
n in
inch
es
55
Figure 4.20 Model for Test Panel B – First Approach A using Load Type I
Figure 4.21 Model for test panel B – First Approach using load type II
56
4.3.3.1 Boundary conditions and loading
Boundary conditions are the same as test panel A. Simple support conditions are provided at
the center of the grating and fixed supports are provided at the ends, at 65-in from the center
of the simply supported condition on either side.
The 10 x 20-in tire patch loads are represented by line loads applied at 36-in from the center
support on either side.
4.3.3.2 Analysis and Results
Analysis is similar to that of test panel A. Deformation values can be read from the user
interface by placing the cursor on the main bearing bars.
Figure 4.22 Load vs Deflection for Main Bearing Bar Number 4 – Test Panel B First Approach using Load Type I
0.12
0.1 0.08 0.06 0.04 0.02
0 0 5000 10000 15000 20000 25000
Load in kips experiment model
Defle
ctio
n in
inch
es
57
Figure 4.23 Deflection across the Grating at 41.6kips Load– Test Panel B First Approach using Load Type I
Figure 4.24 Load vs Deflection for Main Bearing Bar Number 4 – Test Panel B First Approach using Load Type II
Analysis Experiment
Main bearing bar number 4 3 2 1
0.12
0.1 0.08 0.06 0.04 0.02
0
0.12
0.1 0.08 0.06 0.04 0.02
0 0 5000 10000 15000 20000 25000
Load in kips experiment model
Defle
ctio
n in
inch
es
Defle
ctio
n in
inch
es
58
Figure 4.25 Deflection across the grating at 41.6kips load– Test Panel B First Approach using Load Type II
The maximum deformation directly under the load on main bearing bar number 4 is 0.0851-
in under load type I and is 0.0817-inch using load type II. The maximum strain value for
41.6-kips directly under the tire patch is 1452-microstrains.
4.3.4 Test Panel B – Second Approach
The procedure for modeling test panel B using second approach is similar to the modeling
procedure for test panel A using second approach. The complete test setup is shown in
figures 4.26 and 4.27.
Analysis Experiment
Main bearing bar number 4 3 2 1
0.12
0.1 0.08 0.06 0.04 0.02
0
Defle
ctio
n in
inch
es
59
Figure 4.26 Model for Test Panel B- Second Approach using Load Type I
Figure 4.27 Model for Test Panel B- Second Approach Load Type II
60
4.3.4.1 Boundary conditions and loading
Boundary conditions are similar to test panel B first approach. Simple support boundary
conditions are provided along the center support and fixed boundary conditions are provided
at the ends 65-inches from the central support on either side.
Two types of live loads represented by line loads are applied on test panel B at 36-in from
the central support on either side.
4.3.4.2 Analysis and Results
Settings for the analysis are similar to that of test panel A.
Figure 4.28 Load vs Deflection for Main Bearing Bar Number 4 – Test Panel B Second Approach Load Type I
Load in kips experiment model
25000 20000 15000 10000 5000 0
0.12
0.1 0.08 0.06 0.04 0.02
0
Defle
ctio
n in
inch
es
61
Figure 4.29 Deflection across the Grating at 41.6kips load– Test Panel B Second Approach Load Type I
Figure 4.30 Load vs Deflection for Main Bearing Bar Number 4 – Test Panel B Second Approach Load Type II
Analysis Experiment
Main bearing bar number 4 3 2 1
0.12
0.1 0.08 0.06 0.04 0.02
0
0.12
0.1 0.08 0.06 0.04 0.02
0 0 5000 10000 15000 20000 25000
Load in kips experiment model
Defle
ctio
n in
inch
es
Defle
ctio
n in
inch
es
62
Figure 4.31 Deflection across the Grating at 41.6kips load– Test Panel B Second Approach Load Type II
The maximum deflection directly under the tire patch on main bearing number 4 is 0.0971-
in when loading type I is used and is 0.1129-inch when load type II is used. The maximum
strain value directly under the load for 41.6kips load is 1475-microstrains.
Analysis Experiment
Main bearing bar number 4 3 2 1
0.12
0.1 0.08 0.06 0.04 0.02
0
Defle
ctio
n in
inch
es
63
CHAPTER V
COMPARISON AND MODELING OF TEST PANEL C
In this chapter, the results from the analysis conducted in SAP2000 are compared with the
experimental results, and modeling of test panel C is discussed. By comparing the percentage
of variation between results from the grillage analysis models and values obtained from the
experiments, the best method is selected. This best method is then used model test panel C.
5.1 Comparison of Test Panel A - First Approach with the experiments.
For first approach, all the sub components of the riveted grating are modeled in SAP2000
using beam elements. The analysis is run with loads represented in two ways and the results
are found to be in reasonable agreement. Table 5.1 shows the percentage of variation of the
deflection results from the model with experiments for test panel A first approach on the
main bearing number 4.
64
Table 5.1 Comparison of results with Experiments for Test Panel A – First Approach
Load Loading I Loading II
Experiment Model % of
variation
Experiment Model % of
Variation
0 0 0 0 0 10.56391
1000 0.00266 0.00229 13.90977 0.00266 0.002379 8.338129
5000 0.0139 0.012148 12.60432 0.0139 0.012741 10.01102
10000 0.02723 0.025024 8.101359 0.02723 0.024504 0.484346
13800 0.03737 0.034133 8.662028 0.03737 0.037189 10.84367
15000 0.0403 0.035268 12.48635 0.0403 0.03593 7.303133
17500 0.04724 0.041451 12.25445 0.04724 0.04379 8.262477
20000 0.0541 0.047965 11.34011 0.0541 0.04963 11.07774
20800 0.0566 0.050891 10.08657 0.0566 0.05033 10.56391
First approach has produced results with a variation less than 16% for test panel A. The microstrain on the riveted grating directly
under the tire patch at 30kips is observed to be 1028 μe from the model and is observed to be 935 μe from the experiment. The
percentage of variation in the strain values is 9.94% for main bearing bar number 4 at 30kips. All other values are also observed
to be within 20% variation.
65
5.2 Comparison of Test Panel A - Second Approach with the experiments.
In the second approach, only the sub-components believed to be active participants in carrying load are modeled. Therefore,
recticuline bars, primarily responsible for lateral stability are not modeled. Only main bearing bars, intermediate bearing bars and
transverse bars are modeled as beam members in SAP2000. The boundary conditions and loading conditions are same as in first
approach. Two loading representations are studied and they are observed to be in reasonable agreement. Table 5.2 compares of
deflections from the experiment and model using second approach. The results are taken directly under the tire patch on main
bearing bar 4.
Table 5.2 Comparison of results with Experiments for Test Panel A – Second Approach
Load Loading I Loading II
Experiment Model % of variation Experiment Model % of Variation
0 0 0 0 0
1000 0.00266 0.00253 4.887218 0.00266 0.002529 4.924812
5000 0.0139 0.013 6.47482 0.0139 0.013299 4.323741
10000 0.02723 0.0246 9.658465 0.02723 0.02496 8.336394
13800 0.03737 0.0334 10.62349 0.03737 0.0329 11.96147
15000 0.0403 0.03701 8.163772 0.0403 0.03687 8.511166
66
17500 0.04724 0.04324 8.467401 0.04724 0.04296 9.060119
20000 0.0541 0.0499 7.763401 0.0541 0.05008 7.430684
20800 0.0566 0.05402 4.558304 0.0566 0.0552 2.473498
All the results from second approach are within 12% and they are within 20% for the other main bearing bars. The microstrain on
main bearing bar 4 directly under the tire patch at 30kips is observed to be 1009 microstrains. It is within 8% of the value observed
with the sample in the laboratory. All other microstrain values with different load values on all the main bearing bars are within
20% of the experimentally obtained values.
For test panel A, second approach yielded better results as compared to first approach.
5.3 Comparison of Test Panel B - First Approach with the experiments.
Test panel B has a span of 65-in. All modeling procedures and assumptions are identical as in the case of test panel A – first
approach. The results from the SAP2000 grillage analysis are tabulated along with the results obtained in the laboratory. Table 5.3
shows the comparison of the results on main bearing bar number 4.
67
Table 5.3 Comparison of results with Experiments for Test Panel B – First Approach
Load Loading I Loading II
Experiment Model % of
variation
Experiment Model % of
Variation
0 0 0 0 0
1000 0.004775 0.0039 18.32461 0.004775 0.0037 22.51309
5000 0.024875 0.0201 19.19598 0.024875 0.0199 20
10000 0.04875 0.0446 8.512821 0.04875 0.0431 11.58974
13800 0.066895 0.0571 14.64235 0.066895 0.0568 15.09081
15000 0.072125 0.0628 12.92894 0.072125 0.0619 14.17678
17500 0.084563 0.0736 12.9643 0.084563 0.0729 13.79208
20000 0.097 0.0803 17.21649 0.097 0.0792 18.35052
20800 0.10132 0.0851 16.00869 0.10132 0.0817 19.36439
68
All the results from first approach for test panel B for both loadings are within 20%. The maximum microstrain value on the main
bearing bar number 4 directly under 41.6-kips load is 1452-μe from the model and is 1435-μe form experiments. The value
obtained from the grillage analysis varies from the value observed from the experiments by less than 2%.
5.4 Comparison of Test Panel B - Second Approach with the experiments
Similar to test panel A, only the sub-components that contribute to the load carrying capacity in a are modeled for test panel B
second approach. Two load representations are applied to this model and the analysis was conducted. Table 5.4 shows the
comparison of results obtained from the model with the results obtained from the experiments.
69
Table 5.4 Comparison of results with Experiments for Test Panel B – Second Approach
Load Loading I Loading II
Experiment Model % of variation Experiment Model % of Variation
0 0 0 0 0
1000 0.004775 0.0047 1.570681 0.004775 0.0053 -10.9948
5000 0.024875 0.0233 6.331658 0.024875 0.0273 -9.74874
10000 0.04875 0.0467 4.205128 0.04875 0.0544 -11.5897
13800 0.066895 0.0644 3.729726 0.066895 0.0751 -12.2655
15000 0.072125 0.07 2.946274 0.072125 0.0816 -13.1369
17500 0.084563 0.0817 3.385641 0.084563 0.0953 -12.697
20000 0.097 0.0933 3.814433 0.097 0.1086 -11.9588
20800 0.10132 0.0971 4.165022 0.10132 0.1129 -11.4291
All results obtained from the model are within 20%. The deflections sobtained from the model for test panel B using second
approach are much closer to the values obtained from the experiment as compared to the results obtained from first approach. The
maximum microstrain value directly under 41.6-kips load on main bearing bar number 4 is observed to be 1475-microstrain from
70
the model vs 1435-microstrain from the experiments.
5.5 Summary from the comparisons
From the comparisons, second approach provides improved results as compared to first
approach. It could be related to the fact that rivets were designed as beam members and not
rivets. Second approach has an advantage on processing time as well. Since there are a
smaller number of sub-components modeled, and much simpler sections, the analysis run
time is significantly lower than first approach. The main purpose of grillage analysis is to
simplify modeling while not losing accuracy in the results. After taking all the factors into
consideration, second approach was applied to study test panel C.
71
5.6 Modeling of Test Panel C – Second Approach
All modeling procedures are similar to the approach discussed in chapter IV, except for cross
sections of the components and the span of the grating. Table 5.5 gives the cross-section
dimensions and figure 5.1 shows the images of the cross sections. The only difference is the
thickness of the main bearing bar, and intermediate bearing bar. The thickness of the main
bearing bar and intermediate bearing bar in test panel C is 0.375-inches as compared to that
of 0.25-inches in test panels A and B. The center to center span of this grating is 62-inches
as compared to 42-inches in test panel A and 65-inches in test panel B. The loading is applied
36-inch distance from the central support on either side of the support. The complete test
setup can be seen in figure 5.2.
Table 5.5 Cross-section Dimensions of the Sub-Components for Test Panel C
Sub-Component Dimensions
Main Bearing Bar 5-inch x 3/8-inch
Intermediate Bearing Bar 1.5-inch x 3/8-inch
Transverse Bar 1-inch x 1/4-inch
Reticuline Bar 1.5-inch x ¼ -inch
Rivets 0.375 inch Dia
72
(a) (b)
(c) (d) (e)
Figure 5.1 Cross-Section Dimensions of the Sub-Components for Test Panel C
(a)- Main bearing bar, (b)-Intermediate bearing bar, (c)- Transverse Bar, (d)- Reticuline
bar,
(e)-Rivet
73
Figure 5.2 Model for Test Panel C- Second Approach using loading type I
Figure 5.3 Model for Test Panel C- Second Approach using loading type II
74
5.6.1 Boundary conditions and loading
Boundary conditions and loading are maintained identical to that of test panel A and test
panel B, for approach 2. Simple support is provided along the middle of the grating and two
fixed end supports are provided on either side ends.
Two 10 x 20-inch tire patch loads are represented by line loads applied at 36 inches from the
center support on the either side.
5.6.2 Analysis and Results
To analyze the grating direct linear integration history was selected. For time history
analysis, modal analysis is required. Complete modal analysis is done first to decide the most
influential mode for time step determination. The time step is set such that it is less than 1/10th
of the period for the most influential mode. The complete analysis is then run.
The deflection results and stress distribution results are shown in the figures 5.4, 5.5, 5.6, 5.7
and 5.8 below and discussed.
75
Load Vs Deflection 0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0 0 5000 10000 15000 20000 25000
Load in Kips
Load Vs Deflection 0.1
0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01
0 0 5000 10000 15000 20000 25000
Load in Kips
Figure 5.4 Load Vs Deflection on Main Bearing Bar Number 4 Load Type I
Figure 5.5 Load Vs Deflection on Main Bearing Bar Number 4 Load Type II
Defle
ctio
n in
inch
es
Defle
ctio
n in
inch
es
76
Figure 5.6 Deflection Pattern on the Riveted Grating at 41.6 Kips load
Loading Type II Loading Type I
Bearing bar number 4 3 2 1
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
Deflection pattern on the grating at maximum load
Defle
ctio
n in
inch
es
77
Figure 5.7 Distribution of Strain across Test Panel C – Load Type I
4 3 2 1
Micro-Strain 1600
1400
1200
1000
2Kips
800
10 Kips
20 Kips
27.6 Kips
30 Kips
35 Kips
600 40 Kips
41.6 Kips
400
200
0
Bearing bar number
Mic
ro-S
trai
n
78
Figure 5.8 Distribution of Strain across Test Panel C – Loading Type II
4 3 2 1
Micro-Strain 1600
1400
1200
1000
2 Kips
10 Kips
800
600
400
200
0
Bearing bar number
20 Kips
27.6 Kips
30 Kips
35 Kips
40 Kips
41.6 Kips
Mic
ro-S
trai
n
79
The maximum deflection on main bearing bar 4 for loading type I is 0.0907 inches and for
loading type II is 0.0909 inches. Both the maximum deflections obtained from two load types
gave similar deflection values. The comparison of the results between the two loading types
on main bearing bar number 4 is shown in figure 5.6.
Stress values are taken from the SAP2000 output and strain is calculated using Young’s
Modulus. That is converted to micro-strain and is shown graphically in figures 5.7 and 5.8.
The maximum strain value on the main bearing bar number 4 directly under the tire patch
for load type I is 1432 micro-strains and is 1417 for load type II. All the results from loading
type I are observed to be in reasonable agreement with loading type II. Either of the loading
methods may be used based on personal preference.
80
CHAPTER VI
SUMMARY AND CONCLUSION
6.1 Summary and Conclusions
Two 37R5 lite panels were modeled using a grillage based on two approaches, and were
tested in the lab under simulated H20 loading. Deflection and stress values from the models
and testing were compared. The comparison of results from the model and experiments
establish the following design guidelines;
(a) Loading should be idealized as line loads applied on the center line of the main bearing
bars.
(b) All the sub-components should be defined as beam members.
(c) Rivets should be defined as beam members and there should be a clamping effect
included.
(d) If reticuline bars are not included in the model, the cross section of the bearing bars
should be increased to reflect the combination.
81
Using the above guidelines, a model of test panel C (37R5 Lite, 5-inch x 3/8-inch) with a
span of 62-inches was constructed and an analysis run in order to estimate the deflections
and stresses. Results obtained for test panel C are summarized as;
(a) The maximum deflection observed is 0.0909-inches. All the deflection values are with-
in the NAAMM limit (deflection limit for the heavy duty riveted grating is taken as
L/400, which gives a maximum deflection limit to be 0.155-inches).
(b) The stress distribution can be accurately modeled if more bearing bars are present in the
design and maximum stress value can be estimated with greater accuracy, if there are an
odd number of main bearing bars.
(c) Intermediate bearing bars also provide resistance to applied loads and must not be
ignored in modeling.
(d) Reticuline bars do not provide significant amount of resistance to the applied loads,
therefore they can be ignored when considering the load carrying capacity.
(e) Reticuline bars are responsible for lateral stability of the riveted grating, therefore, they
cannot be ignored while manufacturing gratings.
(f) Grillage analysis can be used to study and predict the behavior of heavy duty riveted
gratings for different spans and for different geometry.
(g) Grillage analysis can simplify the modeling of samples and can also save run time during
analysis.
82
6.2 Recommendations for future work
The following are recommendations for future work on grillage analysis for heavy duty
riveted gratings
(a) Rivet connections should be simplified for modeling, so that accurate predictions can be
made.
(b) Non-linear analysis should be performed on the riveted grating to predict service life.
(c) Support conditions should be studied in detail to represent the real-life situations
accurately.
(d) Fatigue behavior of the heavy duty riveted gratings should be studied using grillage
analysis.
(e) Corrosion factors should be included for predicting deflection, stress patterns and the life
of the structure.
83
BIBLIOGRAPHY
1. American Association of State Highway and Transportation Officials (AASHTO),
“Standard Specifications for Highway Bridges”, (2002) Washington, D.C
2. American Association of State Highway and Transportation Officials (AASHTO),
“LRFD Bridge Design Specifications”, (2004) Washington, D.C
3. ANSI / National Association of Architectural Metal Manufacturers (NAAMM), “Heavy
Duty Metal Bar Grating Manual”, MBG 532-09 (2009)
4. ANSI / National Association of Architectural Metal Manufacturers (NAAMM),
“Engineering Design Manual” MBG 534-12 (2012)
5. Bejgum, M. (2006), “Testing and Analysis of Heavy Duty Riveted Gratings”, Thesis
Submitted to The Graduate Faculty of The University of Akron, Akron
6. Arthur, G. A. (2014), “Fatigue Behavior and Design of Heavy Duty Riveted Steel
Gratings in Bridge Decks”, Thesis submitted to The Graduate Faculty of The University
of Akron, Akron
7. Yasser, J. N. A. (2014). “GRILLAGE ANALYSIS OF STRUCTURES CONSIST OF
STEEL I-BEAMS IN TWO DIRECTIONS WITH TRANSVERSE SHEAR
EFFECT”, University of Kufa, College of Engineering
8. Jaeger, L. G., & Bakht, B. (1982), “The Grillage Analogy in Bridge Analysis”, Canadian
Journal of Civil Engineering, 224-235
9. Shreedhar, R., & Kharde, R. (2013), “Comparative Study of Grillage Method and Finite
Element Method of RCC Bridge Deck”, International Journal of Scientific &
84
Engineering Research
10. Cannon, J. (1969), “Yield Line Analysis and Design of Grid Systems”, AISC
Engineering Journal, October: 124 -129
11. Vukov, A. (1986), “Limit Analysis and Plastic Design of Grid Systems”, Engineering
Journal – American Institute of Steel Construction INC, 77-83
12. Huang, H. (2001), “Behavior of Steel Grid Decks for Bridges”, Dissertation submitted
to The Graduate Faculty of The University of Delaware
13. Mahama, F. (2003), “Finite Element Analysis of Welded Steel Gratings”, Thesis
submitted to The Graduate Faculty of The University of Akron, Akron
14. Davison, L., & Hughes, O.F. (1974), “A Simplified Method of Grillage Method”, Paper
presented to The Royal Institution of Naval Architects, Australia
15. Hambly, E. C. (1991), “Bridge Deck Behavior”, E & FN SPON (Chapman & Hall), U.K.