ABSTRACT
Thin-walled box girder bridges are very efficient structural solution for long-span bridges because of their high torsional resistance, stability, economy, and aesthetic appearance. Though, this type of bridge is not common in Nigeria, it is envisaged that they may offer economic advantages in future high way projects in Nigeria. Analytically, however, the box girder bridge is a very complex indeterminate problem. Therefore, the primary objective of this research was to produce a reliable and less cumbersome tool for accurate prediction of the static response of continuous thin-walled box girder bridges including the effects of shear deformation. A MATLAB computer program was developed for the finite strip analysis of continuous thin-walled box girder bridges. Using six prototype thin-walled box girder bridge models made in the scale 1:10, experimental study was 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. Experimental results showed that the effects of shear deformation increases as the flange width increases. Validation of the theoretical formulations, which is synthesized in the proposed finite strip computer program, was carried out by comparison with both the experimental results and the theoretical analysis results in published literature. A numerical study of displacement and stress distributions was carried out to demonstrate the application of the theoretical formulations and developed computer program to the analysis of a typical continuous thin-walled multi-cell box girder bridge subjected to self weight and vehicular loads. Based on the results of analysis, displacement and stress distributions were plotted. Several useful inferences were made from the plots. The results of all analyses were compared to the beam theory solution which does not include the effects of shear deformation. Results of analyses showed, among other things, that the effects of shear deformation were more pronounced in deflection than in stresses. Also, a MATLAB computer program was developed for the solution of the beam vibration differential equation used in the finite strip analysis of continuous structures. The results obtained from this program are stored as input data to the main program. The solution of this equation is highly susceptible to omission of roots. The causes of omission of roots were studied with the developed program and a graphical approach in MATLAB. The principles and programs developed in this research could be used in practice for the analysis of continuous thin-walled multi-cell box girder bridges, folded plates, and box beams.
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
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.
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
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
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
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
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 thin- walled 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
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 thin- walled 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.
This material content is developed to serve as a GUIDE for students to conduct academic research
FINITE STRIP ANALYSIS OF CONTINUOUS THIN-WALLED BOX GIRDER BRIDGES
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