FINITE STRIP ANALYSIS OF CONTINUOUS THIN-WALLED BOX GIRDER BRIDGES

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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  â€“  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.



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