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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

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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


4.3.3 Test Panel B – Approach A ...................................................................... 54



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4.3.4 Test Panel B – Second Approach ............................................................. 58



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


First Approach using Loading Type II ..................................................................... 47


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


Second Approach Loading Type I ...................................................................... 52


Second Approach using Loading Type I ............................................................ 53


Second Approach using Loading Type II ................................................................. 53


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


First Approach using Load Type II ........................................................................... 57

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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


Second Approach using Load Type I .................................................................. 60


Second Approach using Load Type I ..................................................................61


Second Approach using Load Type II ...................................................................... 61


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

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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

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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.

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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

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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).

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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

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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.

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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.

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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

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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.

24

Figure 3.2 Side View of the Support Arrangement.

Figure 3.3 Reticuline Bars Geometry

25

Figure 3.4 Laboratory Setup for R-37-5 (5”x1/4”) Lite Steel Grating for Test A

26

Figure 3.5 Laboratory Setup for R-37-5 (5”x1/4”) Lite Steel Grating for Test

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

28

Figure 3.7 Strain Gauge Locations

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


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.


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


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


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.


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


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


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


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


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


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.


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


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


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


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


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


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


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



variation


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


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

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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

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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

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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

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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

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6. Arthur, G. A. (2014), “Fatigue Behavior and Design of Heavy Duty Riveted Steel

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EFFECT”, University of Kufa, College of Engineering

8. Jaeger, L. G., & Bakht, B. (1982), “The Grillage Analogy in Bridge Analysis”, Canadian

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9. Shreedhar, R., & Kharde, R. (2013), “Comparative Study of Grillage Method and Finite

Element Method of RCC Bridge Deck”, International Journal of Scientific &

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10. Cannon, J. (1969), “Yield Line Analysis and Design of Grid Systems”, AISC

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11. Vukov, A. (1986), “Limit Analysis and Plastic Design of Grid Systems”, Engineering

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12. Huang, H. (2001), “Behavior of Steel Grid Decks for Bridges”, Dissertation submitted

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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.