Finite Strip Analysis of Continuous Thin-Walled Box Girder Bridges

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.

TABLE OF CONTENT

Pages

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

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 - - - -

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

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- - - - -

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

5.1.0 Numerical Example - - - - - - - 95

5.2.0 Bridge Description and Loading - - - - - 96

5.3.0 Results of the Free Vibration Response Analysis for the Longitudinal Interpolation Functions - - - - - 97

5.4.0 Results of the Integration Scheme - - - - - 98

5.5.0 Convergence Test Using Three Finite Strip Models - - - 98

5.6.0 Comparison of Main Analysis Results With Beam Theory Solution- 109

5.7.0 Detailed Analytical Results - - - - - - 115

5.7.1 Deflection - - - - - - - - 115

5.7.2 Twisting Moment - - - - - - - 115

5.7.3 Longitudinal Bending Moment - - - - - 115

5.7.4 Transverse Bending Moment - - - - - - 116

5.7.5 Longitudinal Normal Stress - - - - - - 116

5.7.6 Transverse Normal Stress - - - - - - 128

5.7.7 Shear Stress Distribution in Webs - - - - - 128

CHAPTER 6

6.0 SUMMARY OF RESEARCH, CONCLUSION AND RECOMMENDATION - - - - - - 130

6.1.0 Summary of Research - - - - - - 130

6.2.0 Conclusion - - - - - - - - 134

6.3.0 Recommendation - - - - - - - 136

References - - - - - - - - - 137

Appendixes - - - - - - - - - 166

Appendix 1 MATLAB Computer Program for the Solution of the Beam Vibration Differential Equation - - - 166

Appendix 2 MATLAB Program for Evaluation of Integrals for Finite Strip Analysis of Continuous Box Girder Bridges- - - - -- 167

Appendix 3 Main Program for Finite Strip Analysis of Thin-Walled Box Girder Bridges - - -168