TABLE OF CONTENTS
Title page – – – – – – – – – – i
Approval Page- – – – – – – – – – ii
Certification – – – – – – – – – – iii
Dedication – – – – – – – – – – iv
Acknowledgement – – – – – – – – – v
Abstract – – – – – – – – – – vi
Table of Content – – – – – – – – – vii
List of Figures- – – – – – – – – – ix
List of Plates – – – – – – – – – – xvi
List of Tables – – – – – – – – – – xviii
List of Symbols – – – – – – – – – xx
CHAPTER 1
1.0 INTRODUCTION – – – – – – – – 1
1.1.0 Background – – – – – – – – 1
1.2.0 Structural Response – – – – – – – 4
1.2.1 Primary Responses – – – – – – – 5
1.2.2 Secondary Responses – – – – – – – 5
1.3.0 Statement Of The Problem – – – – – – 5
1.4.0 Research Objectives – – – – – – – 6
1.5.0 Significance of the Study – – – – – – 7
1.6.0 Scope – – – – – – – – – 7
1.7.0 Limitations- – – – – – – – – 8
CHAPTER 2
2.0 LITERATURE REVIEW – – – – – – – 9
2.1.0 Outline of Review – – – – – – – 9
2.2.0 Previous Reviews – – – – – – – 9
2.3.0 Thin–Walled Curved Beam Theory – – – – – 10
2.4.0 BEF/EBEF Method – – – – – – – 12
2.5.0 Finite Segment Method – – – – – – 13
2.6.0 Folded Plate Method – – – – – – – 14
2.7.0 Finite Difference Method – – – – – – 15
2.8.0 Energy Variational Principle – – – – – – 15
2.9.0 Grillage Analogy and Space Frame Methods – – – – 16
2.10.0 Finite Element Method – – – – – – 17
2.11.0 Finite Strip Method – – – – – – – 21
2.12.0 Simplified /Miscellaneous Methods – – – – – 23
2.13.0 Experimental Studies – – – – – – – 26
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2.13.1 Thin-Walled Cross-Section Theory – – – – – 26
2.13.2 Laboratory Experiments and Fields Studies – – – 27
2.14.0 MATLAB Software – – – – – – – 35
2.15.0 Other Computer Methods – – – – – – 35
2.16.0 Important Deductions from Literature Review – – – – 41
2.17.0 Expected Original Contributions – – – – – 42
CHAPTER THREE
3.0 FINITE STRIP FORMULATIONS AND SOFTWARE
DEVELOPMENT – – – – – – – – 43
3.1.0 Preamble – – – – – – – – – 43
3.2.0 Methodology – – – – – – – – 43
3.3.0 Classical Finite Strip in Bending – – – – – – 44
3.4.0 Classical Finite Strip in Plane Stress – – – – – 47
3.5.0 Flat Shell Strip – – – – – – – – 50
3.6.0 Trend in the solution of the Free Vibration Differential
Equation of the Continuous Beam – – – – – 52
3.6.1 Summary of the Methods of Solution of the Free Vibration
Differential Equation of the Continuous Beam – – – –
53
3.6.2 General Solution of the Free Vibration Differential Equation
of the Continuous Beam – – – – – – 54
3.6.3 Element Stiffness Matrix for Beam Vibration – – – 55
3.6.4 Graphical Solution in MATLAB – – – – – 57
3.6.5 Analytical Solution using MATLAB Program – – – 63
3.6.6 Omission of Roots – – – – – – – 65
3.7.0 Development of MATLAB program for the Integration Scheme – 66
3.8.0 Main Program Development, Algorithm and Flow Chart – – 66
3.9.0 Generating a Finite Strip Model – – – – – 70
CHAPTER 4
4.0 EXPERIMENTAL STUDIES AND PROGRAM VALIDATION – – 71
4.1.0 Principles and Procedure – – – – – – 71
4.2.0 Experimental Studies – – – – – – – 71
4.2.1 Dimension of the Prototype Box Girder Models – – – 72
4.2.2 Reinforced Concrete – – – – – – – 74
4.2.3 Form Work – – – – – – – – 75
4.2.4 Experimental Setup and Instrumentation – – – – 75
4.2.5 Load Frame – – – – – – – – 75
4.2.6 16-Channel Data Acquisition System – – – – – 76
4.2.7 Loading – – – – – – – – 77
4.2.8 Measuring Method – – – – – – – 77
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4.2.9 Equipment and Activities in the Experiment – – – – 77
4.2.10 Finite Strip Models for the Prototype Box Girder Models – – 85
4.2.11 Results of Experiments – – – – – – 86
4.3.0 Theoretical Analysis – – – – – – – 91
4.3.1 Bridge Description and Finite Strip Model for the simply
supported Box Girder Bridge- – – – –
91
4.3.2 Results of Deflections, Stresses and Moments at Midspan – – 92
4.3.3 Comparison with Beam Theory Solution – – – – 94
4.4.0 Validation – – – – – – – – 94
CHAPTER 5
5.0 ANALYSIS OF CONTINUOUS MULTI-CELL BOX
GIRDER BRIDGES – – – – – – – – 95
CHAPTER ONE
1.0 INTRODUCTION
1.1.0 BACKGROUND
The most popular type of Highway Bridge in service is the concrete deck on steel-girder
Bridge (Fu and Lu, 2003; Cao and Shing, 1999; Mabsout et; al., 1997). However, this type of
concrete bridge were not economical for long spans because of the rapid increase in the ratio of
dead to total design load as the span lengths increased and so the box girder bridge, with
hollow sections, was developed as a solution to the problem. Box girder bridges are common in
the western world especially California [Scordelis, 1967; Song et., al 2003]. For instance 3100
reinforced concrete box girder bridges were designed and built in California between 1937 and
1977 [Degenkolb, 1977].
Fig. 1.1 Typical Cross-Section of Single Cell Box Girder Bridge
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Fig. 1.2 Typical Cross-Section of Multispine Box Girder Bridge
Fig. 1.3 Typical Cross-Section of Multicell Box Girder Bridge
A box girder bridge is a particular case of a folded-plate structure in which the plates are
arranged so as to form a closed section [Rockey et. al., 1983; Dong and Sause, 2010]. Box
girder configurations may take the form of single cell (one box), multispine (separate boxes),
or multicell (contiguous boxes or cellular shape) with common flange [Sennah and Kennedy,
2001; Davidson et. al., 2004]. A typical cross-section of reinforced or prestressed concrete
Plate 1.1 Single-Cell Curved Box Girder Flyover Bridge under construction at
Majnu Ka Tila, New Delhi, India.
multicell box Girder Bridge consists of top and bottom slab (or flange) connected
monolithically by vertical webs (or stem) to form a cellular or box-like structure.
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The use of box girder bridges in modern highway has become increasingly popular because of
its stability, high torsional resistance, economy, aesthetic appearance and structural efficiency
[Jawanjal and Kumar, 2006; Scordelis, 1967]. Additionally the hollow section of a box girder
bridge can be used to accommodate services [Ugale et. al., 2006]. Thin-walled box girder
bridges have proven to be very efficient structural solution for medium to long-span bridges
(Huang et. al., 1995). The advent of prestressing increased the practical length for box girder
bridges and also permitted considerably thinner structures. Span as much as 240m has already
been completed and it is expected that longer spans may be achieved in future (Degenkolb,
1977). Good examples of curved and straight box girders are shown in Figs. 1.4, 1.5 and 1.6.
Analytically, however, thin-walled box girder bridge has proved to be a very complex
indeterminate problem. Fundamental contributions to the general solutions were given by
Vlasov (1961a, 1961b, and 1965). Since then, a lot of analytical and experimental studies on
the static, dynamic, and stability analyses of thin-walled box girders has been presented in
journals by many other researchers (Wasti and Scordelis, 2000; Chen, 2002; Wu et al., 2002;
Sung, 2002; Ricciardelli, 2003; Tandon, 2003; Choi and Yoo, 2004; Niezgodzinski and
Kubiak, 2005; Sheng and Xin, 2005; Hughs and Idriss, 2006; Attanayake and Aktan, 2006; Vo
Plate 1.2 Single-Cell Railway Box Girder Flyover Bridge at Mehrauli-Gurgaon
Road, New Delhi, India
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and Lee, 2007; Jianguo and Liang, 2007; Doerrer and Hindi, 2008; Yamaguchf, 2008; Kim and
Shin, 2009; Lubardo, 2009; Hodson, 2010; Kim and Fam, 2011; Halkude and Akim, 2012;
Kasan and Harries, 2013; etc). Also some texts (Degenkolb, 1977; Barker and Puckett, 1997;
Iyengar and Gupta, 1997; Victor, 2007; Raju, 2009) and research reports (Meyer and Scordelis,
1970b; Davinson et al., 2002 and 2004; Kulicki et al., 2005 and 2006 ) have, more explicitly,
presented the elastic analysis, design and construction issues relating to box girder bridges.
The Vlasov’s theory is extremely rigorous in its application and is not easily amenable to
computer manipulation. So, researchers and practicing engineers often shy away from the
direct application of the Vlasov’s theory. The current trend is to use the simplified methods (for
practicing engineers), the modified form of the Vlasov’s theory, or the refined methods (for
researchers) like the finite element (FEM) and finite strip (FSM) methods, etc. Most of the
methods used in the analysis of thin-walled box girder bridges, are complicated. However, the
refined methods lend themselves well to computer programming and, therefore, are usually
preferred. Different types of commercially available software are also used for the analysis of
thin-walled box girder bridges. Commercially available software is usually limited by the
scope of output and cost of the product.
Plate 1.3 Single-Cell Railway Box Girder Flyover Bridge at Railway Station,
Mehrauli-Gurgaon Road, New Delhi, India
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Several studies on the analysis of thin-walled box girder bridges have been presented in the
literature but not much has been covered in the analysis of continuous thin-walled multi-cell
box girder bridges. There is the need to produce a reliable and less cumbersome tool for the
accurate prediction of the static response of continuous thin-walled multi-cell box girder
bridges. Therefore, the present research study is concerned with the finite strip analysis of
continuous thin-walled box girder bridges including the effects of shear deformation.
MATLAB Computer program will be developed for the analysis. Experimental studies will be
conducted to validate the developed computer program and to study the effect of flange width
on the static response of thin-walled box girder bridges under service load. A numerical study
of displacement and stress distributions will be carried out to demonstrate the application of the
theoretical formulations and the developed MATLAB computer program to the analysis of a
typical continuous thin-walled multi-cell box girder bridge subjected to self weight and
vehicular loads .
1.2.0 STRUCTURAL RESPONSES
A complicated state of responses develops when a box girder bridge is loaded particularly
when the bridge is curved. Both primary and secondary responses are set up.
1.2.1 Primary Responses: primary responses of the box girder bridges to the actions of
external loads include [Jawanjal and Kumar, 2006):
· Longitudinal bending
· Transverse bending
· Torsion
· Distortion
· Warping
1.2.2 Secondary Responses: Secondary responses also develop due to cross section
distortion (Okeil and El-Tawil, 2004). They include,
· Warping torsion moment
· Bimoment
When subjected to self weight and other symmetrical loadings the actual responses of the box
girder bridges reduces to longitudinal and transverse bending only, so long as the supports are
not skewed and the bridge is not curved in plan (Jawanjal and Kumar, 2006). Under
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asymmetrical loading all the above primary responses are induced. The true distribution of
these responses in a box girder bridge is a complex indeterminate problem.
1.3.0 STATEMENT OF THE PROBLEM
Most analytical methods for box girder bridges are complex and so computer-aided analysis is
the modern trend. Obviously there are a lot of commercially available computer packages.
These commercial packages provide user-friendly data-input platforms and elegant and easy to
follow display formats. A few of them are research oriented packages while majority are
geared towards structural designs in general or bridge design in particular. Packages geared
towards design are basically finite element implementations with strict code interpretation.
They offer a loading module according to the latest code provisions but may lack the ability to
incorporate special effects. Commercial packages, in general, do not provide an insight into the
formulations and solution methods. Besides, most of them are usually very expensive. It is
difficult for private researchers and small companies to procure the license.
It is, therefore, deemed necessary to ignore the commercially available computer packages and
develop a software package that will be tailored to the needs of the present research study.
Specially developed software packages with available source code enhance the learning
process because they usually show how the steps in the theoretical development are
implemented in the programs.
There is the need to produce a reliable and less cumbersome tool for the accurate prediction of
the static response of continuous thin-walled multi-cell box girder bridges. This will be
achieved by programming the more straightforward and less complicated method in the
computer to include the effects of shear deformation. The Finite Strip Method is considered
very suitable for the present study and MATLAB is considered the software of choice.
MATLAB is a high-level language and interactive environment that enables you to perform
computationally intensive tasks faster than with the traditionally programming languages such
as C, C++, FORTRAN and BASIC. It has an inventory of routines, called as functions, which
minimize the task of programming even more.
For a complex indeterminate problem, as is obtainable in the analysis of thin-walled box girder
bridges, it a valid scientific approach – to verify theoretical formulations and computer
programs with experimental studies or literature results or both. Thereafter, valid parametric
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studies could be carried out with the developed computer programs. This being the case,
experimental study is also deemed very necessary for the present research.
1.4.0 RESEARCH AIM AND OBJECTIVES
The aim of the present research is to carry out the finite strip analysis of continuous thin-walled
box girder bridges including the effects of shear deformation.
The objectives of the study are to:
1. Develop a MATLAB computer program for the finite strip analysis of continuous thinwalled
multi-cell box girder bridges including the effects of shear deformation and
compare analytical results obtained by the program with theoretical results in published
literature.
2. Conduct experimental studies to study the effect of flange width on the static response
of thin-walled box girder bridges under service load and to validate the developed
computer program.
3. Use the developed computer program to study displacement and stress distributions in a
typical continuous thin-walled multi-cell box girder bridges subjected to self weight
and vehicular loads.
4. Compare the results of analyses obtained with the developed program and experiment,
to that of the beam theory solution which does not include the effects of shear
deformation.
1.5.0 SIGNIFICANCE OF THE STUDY
Box sections can provide stability for long span bridges and allow large deck overhangs. The
closed box section in the completed bridge has a torsion stiffness that may be 100 times to
more than 1000 times the stiffness of a comparable I-girder section [Fan and Helwig 1999].
Also, the inherent torsional rigidity of curved steel boxes permit shipping and erecting without
external supports.
Box girder bridges may offer economic advantages in future high way projects in Nigeria
where the slab-on-girder bridges, of short to medium spans, dominate the highway systems.
The federal government policy on dredging of major rivers in the country will be more
profitable with the introduction of long span bridges like the box girder bridge because passage
of bigger and wider vessels (ships), under such bridges, will be permitted. Additionally the
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hollow sections of such box girder bridges could be used as sub-ways for pedestrian traffic and
to accommodate services such as public utilities and drainage systems.
More box girder bridge research projects will likely arise in the future in Nigeria. Therefore,
the present study may serve as preliminary work and a leading research program in this area
which will open the door for further research works in the future.
There is the need to improve the existing methods of analysis. Such improvements could be by
seeking simplifications to the existing methods or by seeking to improve the accuracy of
results. The present study seeks to provide a better understanding of the behavior of thinwalled
box girder bridges and to provide a simple tool for the analysis of box girder bridges
without the loss of accuracy of results.
1.6.0 SCOPE
The present study is based on the elastic analysis of continuous thin-walled box girder bridges.
Static response, including the effects of shear deformation, is determined using the finite strip
method. Dynamic and stability analysis are outside the scope of this research study.
1.7.0 LIMITATIONS
The finite strip method (FSM) can be regarded as a degenerate form of the finite element
method (FEM) which is used to primarily model the response of prism-like structures such as
plates and solids. As a degenerate form of the finite element method, there are restrictions on
its application to problems with arbitrary geometry, boundary conditions and material
variations. Therefore, the finite strip method is limited to the analysis of prismatic isotropic
structures, like the Box Girder Bridges or Folded Plates, with constant cross–section and end
boundary conditions that do not change transversely.
Box girder bridges are not common in Nigeria. In addition, the equipment/facilities available in
our Structural Engineering Laboratory are inadequate to carry out experiments which will
capture all the peculiarities of box girder behavior under load. Therefore, experimental/field
studies will be difficult to conduct locally in the present research work. The alternative of
conducting such an experiment abroad is limited by a number of factors which include:
availability of adequate laboratory space, fund, and travel permits.
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