Structural Behaviour of Spatial Arch Bridges

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2017 Structural Behaviour of Spatial Arch Bridges

Hudecek, Martin

Hudecek, M. (2017). Structural Behaviour of Spatial Arch Bridges (Unpublished doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/27901 http://hdl.handle.net/11023/3839 doctoral thesis

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Structural Behaviour of Spatial Arch Bridges

Martin Hudecek

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

GRADUATE PROGRAM IN CIVIL ENGINEERING

CALGARY, ALBERTA

MAY, 2017

This thesis investigates particular aspects of the structural behaviour of a new form of bridge structure called spatial arch bridge (SAB). SABs are designed to be outstanding pieces of architecture. Due to the enhancement in graphical and structural analysis software, the number of SABs is increasing. However, the complexity of SABs results in certain challenges in structural analysis and design. The unique combination of the arch, hangers, and deck result in the development of out-of-plane loads that significantly influence the structural response. This work focuses on SABs with an inferior deck suspended on flexible hangers (cables) below the arch. Three different spatial arrangements of the arch and deck are developed and analyzed to fill particular gaps in the current state of knowledge. Finite element analysis (FEA) is employed as the main analysis tool. Both linear and nonlinear parametric models are developed to address the objectives of the research. Ratios of primary and secondary geometric variables including arch rise, span, deck reach, and inclination or rotation of the parabolic arch are established to identify the trends in the structural response and to consider the effects of various types of loads. The significance of variability in the bending stiffness of the arch and the deck is studied to determine the optimal combination of these parameters that results in spatial configurations with minimal susceptibility to buckling as a function of particular types of end conditions of the deck. The work describes aspects of advanced composite materials (ACM), such as a low modulus of elasticity and time-dependent material properties in all-composite assemblies, and determines the applicability of commonly used durable structural profiles in SABs taking into account the nonlinear character of the cables and the effect of additional tensioning of the cables. The results of the large number of analyses conducted are summarized in tables accompanied with charts and schematic sketches that are intended to serve as guidelines when analyzing or designing structures similar in nature to the ones described in this work. Design criteria are proposed.

Key words: Spatial arch bridges, parametric models, FEA, geometric ratios, stiffness ratios, end conditions of the deck, flexible hangers, susceptibility to buckling, additional tensioning of cables, advanced composite materials, design guidelines

Preface

The work described in this thesis assesses the structural behaviour of spatial arch bridges (SABs). These structures can be very complex and many aspects can influence their response to applied loads. The thesis is organized into eight chapters to clarify the structural behaviour of SABs in order of increasing complexity. The structural behaviour of SABs is investigated using finite element analysis (FEA), and the significance of certain variables, including geometric and stiffness ratios and material properties, is studied. Each chapter contains an introduction and sections on methodology, results, and discussion. The influence of a change in the geometric variables on structural response in the arch, deck, and cables under several loading conditions is investigated first. The revealed issues are studied in the chapters that follow. The significance of arch and deck stiffness under live loads is investigated, which provides grounds to define the susceptibility to buckling as a function of the end conditions of the deck. The application of advanced composite materials (ACMs) in SABs is examined. Additional tensioning of flexible hangers, transferring the loads from the deck to the arch, is considered in certain configuration. The arrangement of the hangers in the assumed spatial configurations and the response to dynamic loading are discussed as possible directions for future work. The appendix of the thesis provides details, such as large tables or charts, to the analyses presented in the body of text. A complete parametric input file that gathers all necessary parameters and commands is included in the appendix as well. The understanding of the structural behaviour of SABs obtained from the research described in each chapter is summarized at the end of each chapter and can be used by designers of SABs as a guideline. Design criteria are proposed for the spatial configurations studied in this thesis. The proposed guidelines and criteria are not intended to specify exact design procedures. Rather, the intention is to provide credible information that can be used while designing and analyzing these structures.

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Acknowledgements

I would like to express my greatest appreciation to Dr. Nigel Graham Shrive, who guided me and offered invaluable advice throughout my doctoral studies.

I would like to acknowledge and thank the following for their support: Stantec consulting Ltd. and the Natural Sciences and Engineering Research Council of Canada (NSERC), who have supported my research financially via the Industrial Postgraduate Scholarship (IPS) program. I would also like to thank Compute Canada and the University of Calgary for providing the finite element software, Abaqus.

My countless thanks belong to all my colleagues, coworkers, friends, and family members who helped me throughout my studies and with revising this thesis in particular.

Dedication

With love and great memory to my parents Jarmila and Karel†, my brother Tomáš and to my wife Elissa who always supported me throughout my academic pursuit. I would like to dedicate this thesis to my daughter Sophia who is the bright light in our lives.

Table of Contents

Abstract ...... ii Preface ...... iii Acknowledgements...... iv Dedication ...... v Table of Contents ...... vi List of Tables ...... xii List of Figures and Illustrations ...... xxi List of Symbols, Abbreviations and Nomenclature ...... xxxv Epigraph ...... xli Chapter One: Introduction and Motivation...... - 1 - 1.1 Spatial Arch Bridges ...... - 1 - 1.2 Problem Definition ...... - 2 - 1.3 Objectives ...... - 4 - 1.4 Scope of Work ...... - 5 - 1.5 Thesis Layout...... - 6 - Chapter Two: Literature Review ...... - 9 - 2.1 Introduction...... - 9 - 2.2 Definition of Spatial Arch Bridges ...... - 9 - 2.3 History and Development of Spatial Structures ...... - 10 - 2.4 Geometry of Spatial Arch Bridges...... - 17 - 2.4.1 General...... - 17 - 2.4.2 Classification of Spatial Arch Bridges ...... - 18 - 2.4.2.1 Spatial Arch Ribs and Spatial Shells ...... - 18 - 2.4.2.2 Symmetry in Plan-view ...... - 19 - 2.4.2.3 Mutual Position of Arch and Deck ...... - 20 - 2.4.2.4 Arch Bridges with Imposed Curvature...... - 21 - 2.4.2.5 Multiple, Diagonally Rotated, and Elevated Bridges ...... - 21 - 2.4.3 Geometrical Ratios ...... - 22 - 2.4.4 Form Finding Approaches ...... - 24 - 2.4.4.1 Antifunicular Arches ...... - 24 - 2.4.4.2 Interactive Parametric Tools ...... - 25 - 2.4.4.3 Form Finding Methods ...... - 27 - 2.5 Structural Behaviour of Spatial Arch Bridges ...... - 28 - 2.5.1 General...... - 28 -

2.5.2 Structural Response of Different Configuration of SABs ...... - 29 - 2.5.2.1 Linear Analysis ...... - 29 - 2.5.2.2 Nonlinear Analysis ...... - 30 - 2.5.2.3 Arch Bridges with Imposed Curvature...... - 30 - 2.5.2.4 Laterally Loaded Spatial Arches ...... - 31 - 2.5.2.5 SABs with Superior Deck and Dynamic analysis ...... - 33 - 2.5.3 Elastic Instability of Arches ...... - 33 - 2.5.3.1 Development of Elastic Instability Theory ...... - 34 - 2.5.3.2 Elastic Instability in Planar Arches ...... - 35 - 2.5.3.3 Elastic Instability in SABs ...... - 36 - 2.5.4 Effects of Thermal Loads ...... - 38 - 2.6 Materials in Spatial Arch Bridges...... - 41 - 2.6.1 General...... - 41 - 2.6.2 Material Complexity ...... - 41 - 2.6.3 Composite Systems...... - 41 - 2.6.4 Conventional Composite Systems ...... - 42 - 2.6.5 Advanced Composites Systems ...... - 43 - 2.6.5.1 Application of Structural Profiles made of ACM in SABs ...... - 44 - 2.6.5.2 Technology of Suitable Structural Profiles for SABs made of ACMs ... - 46 - 2.6.5.2.1 Pultrusion ...... - 46 - 2.6.5.2.2 Filament Winding ...... - 47 - 2.6.5.2.3 Automated fibre placement ...... - 48 - 2.6.5.3 Examples and Applications of Suitable ACM Structural Profiles ...... - 48 - 2.6.5.4 Connections of Structural Profiles ...... - 50 - 2.6.5.4.1 Material Connections ...... - 50 - 2.6.5.4.2 Structural Connections ...... - 52 - 2.6.5.5 ACM Cables ...... - 53 - 2.6.5.6 Analysis and Theories Describing the Behaviour of ACMs ...... - 55 - 2.6.5.7 Challenges in Structural Profiles made of ACMs ...... - 56 - 2.6.5.8 FE Modeling of ACMs ...... - 59 - 2.7 Conclusion to Chapter Two ...... - 60 - Chapter Three: Development of Models ...... - 62 - 3.1 Introduction...... - 62 - 3.2 Software ...... - 62 - 3.3 Structural Configuration of Developed FE Models ...... - 63 - 3.3.1 Character of Models ...... - 63 - 3.3.1.1 Geometric nonlinearity ...... - 63 - 3.3.1.2 Linear Finite Element Model ...... - 65 - 3.3.1.3 Nonlinear Finite Element Model ...... - 65 - 3.3.1.4 Comparison of Linear and Nonlinear Models ...... - 66 - 3.3.2 Structural Components ...... - 68 - 3.3.3 Structural Connections and Boundary Conditions ...... - 69 - 3.3.4 Discretization ...... - 72 -

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3.3.5 Applied Loads...... - 73 - 3.4 Methodology of FE Modeling and Evaluating of Results ...... - 74 - 3.4.1 Pre-processing and Development of Universal Parametric Model...... - 74 - 3.4.1.1 Superstructure Geometry...... - 75 - 3.4.1.2 Definition of Cross-sectional Parameters and Material Properties ...... - 78 - 3.4.1.3 Definition of Loads ...... - 79 - 3.4.1.3.1 Definition of Static Load ...... - 79 - 3.4.1.3.2 Definition Global Thermal Load ...... - 80 - 3.4.1.4 Additional Tensioning of the Cables ...... - 80 - 3.4.1.5 Definition of Boundary Conditions ...... - 83 - 3.5 Material Properties used in FE Models...... - 86 - Chapter Four: Significance of Geometry ...... - 87 - 4.1 Introduction...... - 87 - 4.2 Objectives ...... - 88 - 4.3 Model Specifications ...... - 88 - 4.3.1 Characteristics of the Models and Applied Materials ...... - 88 - 4.3.2 Reference Configuration...... - 88 - 4.3.3 Boundary Conditions ...... - 89 - 4.3.4 Geometry of Spatial Configurations ...... - 90 - 4.3.5 Cross-sectional Properties ...... - 95 - 4.3.6 Applied Loads...... - 95 - 4.3.7 Structural Response ...... - 96 - 4.4 Results and Discussion ...... - 98 - 4.4.1 Critical Live Load Distribution ...... - 98 - 4.4.2 Comparison of Global Thermal Load with Critical Case of Live Load ...... - 102 - 4.4.2.1 Configuration C01 ...... - 103 - 4.4.2.1.1 Expected Behaviour ...... - 103 - 4.4.2.1.2 Axial Force in the Arch ...... - 103 - 4.4.2.1.3 Combined Bending Moment in the Arch ...... - 105 - 4.4.2.1.4 Torsional Moment in the Arch ...... - 107 - 4.4.2.1.5 Combined Bending Moment in the Deck ...... - 108 - 4.4.2.1.6 Tensile Force in the Cables ...... - 110 - 4.4.2.2 Configuration C02 ...... - 113 - 4.4.2.2.1 Expected Behaviour ...... - 113 - 4.4.2.2.2 Axial Force in the Arch ...... - 114 - 4.4.2.2.3 Combined Bending Moment in the Arch ...... - 116 - 4.4.2.2.4 Torsional Moment in the Arch ...... - 118 - 4.4.2.2.5 Combined Bending Moment in the Deck ...... - 120 - 4.4.2.2.6 Tensile Force in the Cables ...... - 124 - 4.4.2.3 Configuration C03 ...... - 126 - 4.4.2.3.1 Expected Behaviour ...... - 126 - 4.4.2.3.2 Axial Force in the Arch ...... - 127 -

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4.4.2.3.3 Combined Bending Moment in the Arch ...... - 130 - 4.4.2.3.4 Torsional Moment in the Arch ...... - 132 - 4.4.2.3.5 Combined Bending Moment in the Deck ...... - 134 - 4.4.2.3.6 Tensile Force in the Cables ...... - 136 - 4.4.3 Principles of the Structural Behaviour in the Assumed Configurations ...... - 138 - 4.5 Conclusion to Chapter Four ...... - 144 - 4.5.1 Configuration C01 ...... - 144 - 4.5.2 Configuration C02 ...... - 145 - 4.5.3 Configuration C03 ...... - 147 - Chapter Five: The Significance of Boundary Conditions and Bending Stiffness- 150 - 5.1 Introduction...... - 150 - 5.2 Objectives ...... - 152 - 5.3 Model Specification ...... - 152 - 5.3.1 The Geometry of Models ...... - 152 - 5.3.2 Bending Stiffness of the Arch and Deck ...... - 153 - 5.3.3 Boundary Conditions ...... - 154 - 5.3.4 Loads and Materials...... - 157 - 5.3.5 Achieving the Objectives of Chapter Five ...... - 158 - 5.4 Results and Discussion ...... - 159 - 5.4.1 Determining of a Critical Live Load Distribution ...... - 160 - 5.4.2 Structural Behaviour under changing Stiffness ...... - 163 - 5.4.2.1 Configuration C01 ...... - 163 - 5.4.2.1.1 Configuration C01 — Deck Boundary Conditions: BC#01 ...... - 163 - 5.4.2.1.2 Configuration C01 — Deck Boundary Conditions: BC#02 ...... - 166 - 5.4.2.1.3 Configuration C01 — Deck Boundary Conditions: BC#03 ...... - 169 - 5.4.2.2 Configuration C02 ...... - 172 - 5.4.2.2.1 Configuration C02 — Deck Boundary Conditions: BC#01 ...... - 173 - 5.4.2.2.2 Configuration C02 — Deck Boundary Conditions: BC#04 ...... - 175 - 5.4.2.3 Configuration C03 ...... - 178 - 5.4.2.3.1 Configuration C03 — Deck Boundary Conditions: BC#01 ...... - 179 - 5.4.2.3.2 Configuration C03 — Deck Boundary Conditions: BC#04 ...... - 181 - 5.4.3 Susceptibility to Buckling ...... - 185 - 5.4.3.1 Configuration C01 ...... - 186 - 5.4.3.2 Configuration C02 ...... - 189 - 5.4.3.3 Configuration C03 ...... - 192 - 5.5 Conclusion to Chapter Five ...... - 195 - 5.5.1 Configuration C01 ...... - 195 - 5.5.2 Configuration C02 ...... - 195 - 5.5.3 Configuration C03 ...... - 196 - Chapter Six: Application of Advanced Composite Materials in SABs ...... - 198 - 6.1 Introduction...... - 198 -

6.2 Objectives ...... - 199 - 6.3 Model Specifications ...... - 200 - 6.3.1 Nonlinear Model ...... - 201 - 6.3.2 Selection of a Critical Configuration ...... - 201 - 6.3.3 Boundary Conditions ...... - 202 - 6.3.4 Cross-sections ...... - 203 - 6.3.4.1 Principle of a Variation in the Bending and Axial Stiffness ...... - 204 - 6.3.4.2 Section Points ...... - 205 - 6.3.5 Loads ...... - 205 - 6.3.6 Materials ...... - 206 - 6.4 Results and Discussion ...... - 207 - 6.4.1 Creep Analysis ...... - 207 - 6.4.1.1 Susceptibility to Creep-rupture ...... - 207 - 6.4.1.2 Significance of Weight per Unit Length and Modulus of Elasticity ..... - 208 - 6.4.1.3 Significance of Creep on the Stress Distribution ...... - 215 - 6.4.1.4 Effect of Additional Tensioning of Cables ...... - 220 - 6.4.2 The Significance of Variability in the Coefficient of Thermal Expansion.... - 225 - 6.4.3 The Significance of Variability in Deck Stiffness ...... - 231 - 6.4.3.1 Effect of Variability in Deck Stiffness and Deck Reach ...... - 232 - 6.4.3.1.1 Displacement of the Deck...... - 232 - 6.4.3.1.2 Internal Forces in the Deck and Arch ...... - 238 - 6.4.3.2 Effect of Material Properties and Level of Applied GTL ...... - 240 - 6.5 Conclusion to Chapter Six ...... - 242 - Chapter Seven: Design Criteria and Directions of Future Work ...... - 243 - 7.1 Design Criteria ...... - 243 - 7.1.1 Configuration C01 ...... - 243 - 7.1.2 Configuration C02 ...... - 244 - 7.1.3 Configuration C03 ...... - 245 - 7.2 Directions of Future Work ...... - 247 - 7.2.1 Arrangement of the Cables ...... - 247 - 7.2.2 Additional Tensioning of the Cables ...... - 247 - 7.2.3 Response to Dynamic Loading ...... - 248 - Chapter Eight: Summary and Final Remarks ...... - 249 - Reference ...... - 255 - Appendix A: Details to Chapter Three ...... - 269 - A.1. Selection of Finite Elements ...... - 270 - A.2. Verification and Calibration ...... - 278 - A.3. Parametric Input File...... - 302 - Appendix B: Details to Chapter Four ...... - 322 -

B.1. Determination of Critical LL Distribution ...... - 323 - B.2. Comparison of Global Thermal Load with Critical Case of Live Load ...... - 325 - B.3. Summary Tables of the Structural Response ...... - 379 - Appendix C: Details to Chapter Five ...... - 397 - C.1. Determining Critical Live Load Distribution...... - 398 - C.2. Structural Behaviour under changing Stiffness...... - 400 - C.3. Susceptibility to Buckling ...... - 417 - Appendix D: Details to Chapter Six ...... - 421 - D.1. Selection of a Critical Configuration ...... - 422 - D.2. Analysis AN#01A: Assumed Cross-sectional Properties ...... - 423 - D.3. Analysis AN#01B-I: Effect of Creep...... - 424 - D.4. Analysis AN#01C: Effect of Additional Tensioning of Cables...... - 427 - D.5. Analysis AN#02: Effect of the Coefficient of Thermal Expansion ...... - 430 - D.6. Analysis AN#03A: Effect of the Variability in Deck Stiffness ...... - 439 - D.7. Analysis AN#03B: Effect of the Material Properties on Stress Distribution ..... - 447 -

List of Tables

Table 3-1: Material properties of assumed materials...... - 86 -

Table 4-1: Primary and secondary variables of the assumed spatial configurations ...... - 90 -

Table 4-2: Range of primary and secondary variables ...... - 90 -

Table 4-3: Cross-sectional properties of the arch, deck, and cables ...... - 95 -

Table 4-4: List of structural responses evaluated in Chapter Four ...... - 97 -

Table 4-5: Critical patterns of LL distribution in the three spatial configurations ...... - 99 -

Table 5-1: Combinations of primary and secondary variables in individual spatial configurations ...... - 153 -

Table 5-2: Geometric properties of the assumed structural sections for the arch and deck ...... - 154 -

Table 5-3: Assumed types of deck boundary conditions ...... - 155 -

Table 5-4: Proposed optimal mechanical ratios of deck and arch stiffness resulting in a combination, which is the least susceptible to buckling ...... - 185 -

Table 6-1: Range of primary and secondary variables assumed in Chapter Six ...... - 202 -

Table 6-2: Definition of deck boundary condition BC#02B ...... - 203 -

Table 6-3: E(EFF) for the GFRP profiles and cables...... - 206 -

Table 6-4: Results of analysis AN#01A ...... - 208 -

Table 6-5: EI of the arch resulting in satisfactory deck deflections ...... - 208 -

Table 6-6: EI(V) and U2 of the deck ...... - 210 -

Table 6-7: EI(H) and U3 of the deck ...... - 210 -

Table 6-8: Parameters of models in analysis AN#01B-II ...... - 215 -

Table 6-9: Deck vertical displacement as a function of changing parameters of cables ...... - 217 -

Table 6-10: The significance of changes of the cable properties on deck vertical displacement ...... - 217 -

Table 6-11: Resulting parameters of Equation 3-10 providing an optimal distribution of additional tensile forces in the cables ...... - 221 -

xii

Table 6-12: The EI of the arch and U2 in the deck achieved from models M(REF), M001, and M001B ...... - 221 -

Table 6-13: Stress change and ratio of stress change due to +ve and –ve GTL in the individual structural components for GFRP, CFRP, and steel ...... - 227 -

Table 6-14: Stress reduction in individual structural components when GFRP and CFRP is employed instead of steel in in C01B ...... - 229 -

Table 6-15: Vertical deck displacement at the midspan for GFRP, CFRP, and steel at three levels of applied GTL ...... - 229 -

Table 6-16: Critical DSRs that result in the maximum and minimum vertical displacement of the deck at the midspan at three levels of deck reach...... - 234 -

Table 6-17: Changes in deflections and stresses at critical DSRs ...... - 234 -

Table 6-18: The magnitude and differences in change of SM(COMB) in the arch and deck for M001 and M006...... - 239 -

Table 6-19: Stress fluctuations in the decks assuming deck profiles of M001 and M006 for all three materials and three levels of applied GTL ...... - 240 -

Table A-1: A comparison of the analytical and FE models for nonlinear cable behaviour ...... - 287 -

Table A-2: A comparison of the AM and FE models for suspended cables under thermal loads ...... - 293 -

Table A-3: Parameters of the assumed configuration used to verify the entire structure ...... - 297 -

Table A-4: Cross-sectional properties of the main structural components ...... - 298 -

Table A-5: Summary of verification of entire structure ...... - 301 -

Table B-1: Configuration C01 — A comparison of SM(COMB) in the arch due to two different patterns of distribution of LL over the deck ...... - 323 -

Table B-2: Configuration C02 — A comparison of SM(COMB) in arch due to two different patterns of distribution of LL over the deck ...... - 323 -

Table B-3: Configuration C03 — A comparison of SM(COMB) in arch due to two different patterns of distribution of LL over the deck ...... - 324 -

Table B-4: Configuration C01 — A comparison of SF1 in the arch ...... - 325 -

Table B-5: Configuration — A comparison of the differences between SF1 in the arch...... - 325 -

xiii

Table B-6: Configuration C01 — A comparison of the of SF(COMB) in the arch ...... - 326 -

Table B-7: Configuration C01 — A comparison of the differences between SF(COMB) in the arch...... - 326 -

Table B-8: Configuration C01 — A comparison of SM(COMB) in the arch ...... - 329 -

Table B-9: Configuration C01 — A comparison of the differences in SM(COMB) in the arch - 329 -

Table B-10: Configuration C01 — A comparison of SM3 in the arch...... - 330 -

Table B-11: Configuration C01 — A comparison of the differences between SM3 in the arch...... - 330 -

Table B-12: Configuration C01 — A comparison of U(COMB) in the arch ...... - 331 -

Table B-13: Configuration C01 — A comparison of the differences between U(COMB) in the arch...... - 331 -

Table B-14: Configuration C01 — A comparison of SF1 in the deck ...... - 334 -

Table B-15: Configuration C01 — A comparison of the differences between SF1 in the deck ...... - 334 -

Table B-16: Configuration C01 — A comparison of SF(COMB) in the deck ...... - 337 -

Table B-17: Configuration C01 — A comparison of the differences between SF(COMB) in the deck ...... - 337 -

Table B-18: Configuration C01 — A comparison of SM(COMB) in the deck ...... - 340 -

Table B-19: Configuration C01 — A comparison of the differences in SM(COMB) in the deck ...... - 340 -

Table B-20: Configuration C01 — A comparison of U(COMB) in the deck ...... - 341 -

Table B-21: Configuration C01 — A comparison of the differences between U(COMB) in the deck ...... - 341 -

Table B-22: Configuration C01 — A comparison of CF in cables ...... - 344 -

Table B-23: Configuration C01 — A comparison of the differences between CF in cables . - 344 -

Table B-24: Configuration C02 — A comparison of SF1 in the arch ...... - 345 -

Table B-25: Configuration C02 — A comparison of the differences between SF1 in the arch...... - 345 -

xiv

Table B-26: Configuration C02 — A comparison of SF(COMB) in the arch ...... - 346 -

Table B-27: Configuration C02 — A comparison of the differences between SF(COMB) in the arch...... - 346 -

Table B-28: Configuration C02 — A comparison of SM(COMB) in the arch ...... - 349 -

Table B-29: Configuration C02 — A comparison of the differences in SM(COMB) in the arch ...... - 349 -

Table B-30: Configuration C02 — A comparison of SM3 in the arch...... - 350 -

Table B-31: Configuration C02 — A comparison of the differences between SM3 in the arch...... - 350 -

Table B-32: Configuration C02 — A comparison of U(COMB) in the arch ...... - 351 -

Table B-33: Configuration C02 — A comparison of the differences between U(COMB) in the arch...... - 351 -

Table B-34: Configuration C02 — A comparison of SF(COMB) in the deck...... - 353 -

Table B-35: Configuration C02 — A comparison of the differences between SF(COMB) in the deck ...... - 353 -

Table B-36: Configuration C02 — A comparison of SM(COMB) in the deck ...... - 355 -

Table B-37: Configuration C02 — A comparison of the differences in SM(COMB) in the deck ...... - 355 -

Table B-38: Configuration C02 — A comparison of U(COMB) in the deck ...... - 356 -

Table B-39: Configuration C02 — A comparison of the differences between U(COMB) in the deck ...... - 356 -

Table B-40: Configuration C02 — A comparison of CF in cables ...... - 359 -

Table B-41: Configuration C02 — A comparison of the differences between CF in cables ...... - 359 -

Table B-42: Configuration C03 — A comparison of SF1 in the arch ...... - 360 -

Table B-43: Configuration C03 — A comparison of the differences between SF1 in the arch...... - 360 -

Table B-44: Configuration C03 — A comparison of SF(COMB) in the arch ...... - 361 -

Table B-45: Configuration C03 — A comparison of the differences between SF(COMB) in the arch...... - 361 -

Table B-46: Configuration C03 — A comparison of SM(COMB) in the arch ...... - 364 -

Table B-47: Configuration C03 — A comparison of the differences between SM(COMB) in the arch...... - 364 -

Table B-48: Configuration C03 — A comparison of SM3 in the arch...... - 365 -

Table B-49: Configuration C03 — A comparison of the differences between SM3 in the arch...... - 365 -

Table B-50: Configuration C03 — A comparison of the U(COMB) in the arch ...... - 366 -

Table B-51: Configuration C03 — A comparison of the differences between U(COMB) in the arch...... - 366 -

Table B-52: Configuration C03 — A comparison of the SF(COMB) in the deck ...... - 371 -

Table B-53: Configuration C03 — A comparison of the differences between SF(COMB) in the deck ...... - 371 -

Table B-54: Configuration C03 — A comparison of SM(COMB) in the deck...... - 373 -

Table B-55: Configuration C03 — A comparison of the differences in SM(COMB) in the deck ...... - 373 -

Table B-56: Configuration C03 — A comparison of the U(COMB) in the deck...... - 374 -

Table B-57: Configuration C03 — A comparison of the differences between U(COMB) in the deck ...... - 374 -

Table B-58: Configuration C03 — A comparison of CF in cables ...... - 378 -

Table B-59: Configuration C03 — Comparison of differences in CF in cables ...... - 378 -

Table B-60: Configuration C01 — The significance of GTL and LL identifying the importance of primary and secondary variable...... - 380 -

Table B-61: Configuration C01 — The trend of sensitivity of the structure to GTL and LL - 381 -

Table B-62: Configuration C01 — The trend of structural response showing the maximum magnitude under GTL and LL ...... - 383 -

Table B-63: Configuration C02 — The significance of GTL and LL identifying the importance of primary and secondary variable...... - 386 -

Table B-64: Configuration C02 — The trend of sensitivity of the structure to GTL and LL ...... - 387 -

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Table B-65: Configuration C02 — The trend of structural response showing the maximum magnitude under GTL and LL ...... - 389 -

Table B-66: Configuration C03 — The significance of GTL and LL identifying the importance of primary and secondary variable...... - 392 -

Table B-67: Configuration C03 — The trend of sensitivity of the structure to GTL and LL ...... - 393 -

Table B-68: Configuration C03 — The trend of structural response showing the maximum magnitude under GTL and LL ...... - 395 -

Table C-1: Configuration C01 — A comparison of the effects of two types of LL distribution (LL50 and LL100) affecting the magnitude of SF1 in the arch ...... - 398 -

Table C-2: Configuration C02 — A comparison of the effects of two types of LL distribution (LL50 and LL100) affecting the magnitude of SF1 in the arch ...... - 398 -

Table C-3: Configuration C03 — A comparison of the effects of two types of LL distribution (LL50 and LL100) affecting the magnitude of SF1 in the arch ...... - 399 -

Table C-4: Configuration C01 — BC#01: Combinations of arch and deck stiffness organized in ascending order for SF1 in the arch ...... - 400 -

Table C-5: Configuration C01 — BC#01: Combinations of arch and deck stiffness organized in ascending order for SM(COMB) in the arch ...... - 400 -

Table C-6: Configuration C01 — BC#02: Combinations of arch and deck stiffness organized in ascending order for SF1 in the arch ...... - 400 -

Table C-7: Configuration C01 — BC#02: Combinations of arch and deck stiffness organized in ascending order for SM(COMB) in the arch ...... - 401 -

Table C-8: Configuration C01 — BC#03: Combinations of arch and deck stiffness organized in ascending order for SF1 in the arch ...... - 401 -

Table C-9: Configuration C01 — BC#03: Combinations of arch and deck stiffness organized in ascending order for SM(COMB) in the arch ...... - 401 -

Table C-10: Configuration C01 — Arrangements of arch and deck cross-sections for three types of deck BCs that result in the maximum and minimum axial force ...... - 402 -

Table C-11: Configuration C01 — Arrangements of arch and deck cross-sections for three types of deck BCs that result in the maximum and minimum bending moment ...... - 403 -

Table C-12: Configuration C01 — SF1 in the arch: BC#01 ...... - 404 -

Table C-13: Configuration C01 — SM(COMB) in the arch: BC#01 ...... - 404 -

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Table C-14: Configuration C01 — SF1 in the arch: BC#02 ...... - 405 -

Table C-15: Configuration C01 — SM(COMB) in the arch: BC#02 ...... - 405 -

Table C-16: Configuration C01 — SF1 in the arch: BC#03 ...... - 406 -

Table C-17: Configuration C01 — SM(COMB) in the arch: BC#03 ...... - 406 -

Table C-18: Configuration C02 — BC#01: Combinations of arch and deck stiffness organized in ascending order for SF1 in the arch ...... - 407 -

Table C-19: Configuration C02 — BC#01: Combinations of arch and deck stiffness organized in ascending order for SM(COMB) in the arch ...... - 407 -

Table C-20: Configuration C02 — BC#04: Combinations of arch and deck stiffness organized in ascending order for SF1 in the arch ...... - 407 -

Table C-21: Configuration C02 — BC#04: Combinations of arch and deck stiffness organized in ascending order for SM(COMB) in the arch ...... - 408 -

Table C-22: Configuration C02 — Arrangements of arch and deck cross-sections for three types of deck BCs that result in the maximum and minimum axial force ...... - 408 -

Table C-23: Configuration C02 — Arrangements of arch and deck cross-sections for three types of deck BCs that result in the maximum and minimum bending moment ...... - 409 -

Table C-24: Configuration C02 — SF1 in the arch: BC#01 ...... - 410 -

Table C-25: Configuration C02 — SM(COMB) in the arch: BC#01 ...... - 410 -

Table C-26: Configuration C02 — SF1 in the arch: BC#04 ...... - 411 -

Table C-27: Configuration C02 — SM(COMB) in the arch: BC#04 ...... - 411 -

Table C-28: Configuration C03 — BC#01: Combinations of arch and deck stiffness organized in ascending order for SF1 in the arch ...... - 412 -

Table C-29: Configuration C03 — BC#01: Combinations of arch and deck stiffness organized in ascending order for SM(COMB) in the arch ...... - 412 -

Table C-30: Configuration C03 — BC#04: Combinations of arch and deck stiffness organized in ascending order for SF1 in the arch ...... - 412 -

Table C-31: Configuration C03 — BC#04: Combinations of arch and deck stiffness organized in ascending order for SM(COMB) in the arch ...... - 413 -

Table C-32: Configuration C03 — Arrangements of arch and deck cross-sections for three types of deck BCs that result in the maximum and minimum axial force ...... - 413 -

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Table C-33: Configuration C03 — Arrangements of arch and deck cross-sections for three types of deck BCs that result in the maximum and minimum bending moment ...... - 414 -

Table C-34: Configuration C03 — SF1 in the arch: BC#01 ...... - 415 -

Table C-35: Configuration C03 — SM(COMB) in the arch: BC#01 ...... - 415 -

Table C-36: Configuration C03 — SF1 in the arch: BC#04 ...... - 416 -

Table C-37: Configuration C03 — SM(COMB) in the arch: BC#04 ...... - 416 -

Table C-38: Ranking of SF1 and SM(COMB) in C01 for BC#01 ...... - 417 -

Table C-39: Ranking of SF1 and SM(COMB) in C01 for BC#02 ...... - 417 -

Table C-40: Ranking of SF1 and SM(COMB) in C01 for BC#03 ...... - 418 -

Table C-41: Ranking of SF1 and SM(COMB) in C02 for BC#01 ...... - 418 -

Table C-42: Ranking of SF1 and SM(COMB) in C02 for BC#04 ...... - 419 -

Table C-43: Ranking of SF1 and SM(COMB) in C03 for BC#01 ...... - 419 -

Table C-44: Ranking of SF1 and SM(COMB) in C03 for BC#04 ...... - 420 -

Table D-1: A comparison of stresses in all structural components of three spatial configurations; C01, C02, and C03 considering the three types of material ...... - 422 -

Table D-2: Occurrence of stress level status in three different spatial configurations .. - 423 -

Table D-3: Cross-sectional properties assumed in AN#01A ...... - 423 -

Table D-4: Fluctuation of stresses in the arch ...... - 424 -

Table D-5: Fluctuation of stresses in the deck...... - 424 -

Table D-6: Fluctuation of stresses in the cables ...... - 424 -

Table D-7: Maximum stresses in the deck obtained from four section points in models M(REF), M001, and M001B...... - 427 -

Table D-8: Stress change in the deck at four designated section points in model M001 and M001B compared to reference model M(REF) ...... - 427 -

Table D-9: Maximum stresses in the arch obtained from four section points in models M(REF), M001, and M001B...... - 427 -

Table D-10: Stress change in the arch at four designated section points in model M001 and M001B compared to reference model M(REF) ...... - 427 -

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Table D-11: Cross-sectional properties of the arch assumed in analysis AN#02 ...... - 430 -

Table D-12: Cross-sectional properties of the deck assumed in analysis AN#02 ...... - 430 -

Table D-13: Cross-sectional properties of the cables assumed in analysis AN#02...... - 431 -

Table D-14: Stress fluctuation at designated section points in the deck at 0°C ...... - 431 -

Table D-15: Stress fluctuation at designated section points in the deck at +50°C ...... - 431 -

Table D-16: Stress fluctuation at designated section points in the deck at -50°C ...... - 432 -

Table D-17: Stress fluctuation at designated section points in the arch at 0°C ...... - 435 -

Table D-18: Stress fluctuation at designated section points in the arch at +50°C ...... - 435 -

Table D-19: Stress fluctuation at designated section points in the arch at -50°C...... - 435 -

Table D-20: Deck stiffness ratios, shape, and dimensions of developed deck cross-sections for GFRP profiles...... - 439 -

Table D-21: A comparison of the magnitude and differences in change of SM(COMB) in the arch and the deck taking into account models M001 and M006 ...... - 443 -

Table D-22: A comparison of the magnitude and differences in change of SF1 in the deck and the arch taking into account models M001 and M006 ...... - 445 -

Table D-23: Cross-sectional properties of the deck profile with DSR 15:1 assumed in analysis AN#03B taking into account GFRP, CFRP, and Steel profiles...... - 447 -

Table D-24: Cross-sectional properties of the deck profile with DSR 1:3 assumed in analysis AN#03 taking into account GFRP, CFRP, and Steel profiles ...... - 447 -

Table D-25: Comparison of maximum SM(COMB) in the deck of configuration C01B .. - 448 -

Table D-26: A comparison of the maximum stresses in cables taking into account the effect of two deck profiles, three materials, and three levels of applied GTL ...... - 450 -

Table D-27: A comparison of the maximum stresses in deck profiles of models M001 and M006 and taking into account three different materials at 0°C ...... - 452 -

Table D-28: A comparison of the maximum stresses in deck profiles of models M001 and M006 and taking into account three different materials at +50°C...... - 452 -

Table D-29: A comparison of the maximum stresses in deck profiles of models M001 and M006 and taking into account three different materials at –50°C ...... - 452 -

List of Figures and Illustrations

Figure 1-1: A schematic flow-chart showing the mutual relation of individual chapters ...... - 8 -

Figure 2-1: The Arch in the Necropolis of Abydos (Perrot, 1883)...... - 11 -

Figure 2-2: Examples of Roman arches: a) the Arch of Titus in Rome, a triumph arch (Anthony, 2005) and b) the Pont du Gard Roman aqueduct near Nimes in France, a utilitarian structure (Beard, 2015)...... - 12 -

Figure 2-3: Schwandbach Bridge, Switzerland 1933 (Chriusha, H., 2011) ...... - 15 -

Figure 2-4: Examples of work of Santiago Calatrava: a) Margaret Hunt Hill Bridge, Dallas USA (Karchmer, 2012) and b) La Devesa Bridge, Ripoll, Spain (Plasencia, 2012) ...... - 16 -

Figure 2-5: The Gateshead Millennium Bridge, Newcastle, UK, opened in 2001 (Perez, 2014) ...... - 17 -

Figure 2-6: Examples of shell spatial bridges: a) Matadero footbridge in Madrid, Spain (Tamorlan, 2011); and b) Bridge of piece in Tbilisi, Georgia (Kavtaradze, 2009) ...... - 19 -

Figure 2-7: Example of a true SAB: Campo de Volantin footbridge, Bilbao, Spain (Garcia, 2014) ...... - 20 -

Figure 2-8: Example of a non-true SAB: Bac de Roda bridge, Barcelona, Spain (Roletschek, 2015) ...... - 20 -

Figure 2-9: a) a ridge with a superior deck: Endarlatsa bridge, Guipúzcoa, Spain (Urruzmendi, 2014) and b) a bridge with an inferior deck: Mayfly Bridge, Szolnok, Hungary (Akela, 2012) ...... - 21 -

Figure 2-10: Example of a diagonal arch bridge: Hulme Arch, Manchester, UK (Peel, 2004) ...... - 22 -

Figure 2-11: Example of a true antifunicular SAB with a superior deck: Ripshorst Pedestrian Bridge, Oberhausen, Germany (SBP, 1997) ...... - 25 -

Figure 3-1: Principle of geometric nonlinearity taking into account large displacements and small strains ...... - 64 -

Figure 3-2: A comparison of forces in cables in the linear and nonlinear models, assuming material with high E...... - 67 -

Figure 3-3: A comparison of forces in cables in the linear and nonlinear models, assuming material with low E ...... - 67 -

xxi

Figure 3-4: Example of a structural configuration representing a concept of SAB with an inferior deck assumed in the thesis ...... - 68 -

Figure 3-5: Example of cable end to connection to arch or deck ...... - 70 -

Figure 3-6: Detail of cable-to-arch and cable-to-deck connection at centroid axis of structural elements ...... - 71 -

Figure 3-7: Example of fixed connection of arch to the substructure ...... - 72 -

Figure 3-8: Example of a parametric definition of cross-sectional parameters and material properties of a steel arch ...... - 79 -

Figure 3-9: Example of parametric definition of gravity and live loads ...... - 80 -

Figure 3-10: Example of parametric definition of a global thermal load ...... - 80 -

Figure 3-11: Example of parametric definition of LTL utilized for additional cable tensioning ...... - 82 -

Figure 3-12: Example of parametric definition of local direction for deck boundary condition ...... - 84 -

Figure 3-13: A schematic representation of general directions of DOFs shown on a curved deck ...... - 85 -

Figure 4-1: A schematic sketch depicting the boundary conditions of the arch and deck ...... - 89 -

Figure 4-2: Configuration C01: a graphical representation of the relevant geometric variables and a 3D perspective view...... - 92 -

Figure 4-3: Configuration C02: a graphical representation of the relevant geometric variables and a 3D perspective view...... - 93 -

Figure 4-4: Configuration C03: a graphical representation of the relevant geometric variables and a 3D perspective view...... - 94 -

Figure 4-5: A comparison of the bending moment distributions in the reference vertical planar arch f(A)/s = 0.25...... - 99 -

Figure 4-6: A comparison of the bending moment distributions in the arch of C01 with f(A)/s = 0.15 and f(D)/s = 0.25 ...... - 100 -

Figure 4-7: A comparison of the bending moment distributions in the arch of C02 with f(A)/s = 0.15 and ω = 15° ...... - 101 -

Figure 4-8: A comparison of the bending moment distributions in the arch of C03 with f(A)/s = 0.15 and θ = 30° ...... - 101 -

xxii

Figure 4-9: Configuration C01 — Distribution of axial forces in the arch comparing the critical cases of LL100, +ve and –ve GTL, and the unloaded state for three levels of f(A)/s and the most sensitive ratio, f(D)/s = 0.25 ...... - 104 -

Figure 4-10: Configuration C01 — Distribution of axial forces in the arch considering the effect of LL100 for three levels of f(A)/s (0.15, 0.20, and 0.25) and four levels of f(D)/s (0.00, 0.15, 0.20, and 0.25) ...... - 105 -

Figure 4-11: Configuration C01 — Distribution of combined moments in the arch comparing the critical cases of LL100, +ve and –ve GTL, and the unloaded state for three levels of f(A)/s and the most sensitive ratio, f(D)/s = 0.15 ...... - 106 -

Figure 4-12: Configuration C01 — Distribution of combined bending moments in the arch considering the effect of LL100 for three levels of f(A)/s (0.15, .020, 0.25) and four levels of f(D)/s (0.00, 0.15, 0.20, and 0.25) ...... - 106 -

Figure 4-13: Configuration C01 — Distribution of torsional moments in the arch comparing the critical cases of LL100, +ve and –ve GTL, and the unloaded state for three levels of f(A)/s and the most sensitive ratio, f(D)/s = 0.25 ...... - 107 -

Figure 4-14: Configuration C01 — Distribution of torsional moments in the arch under LL100 three levels of f(A)/s (0.15, 0.20, 0.25) and for four levels of f(D)/s (0.00, 0.15, 0.20, and 0.25) ...... - 108 -

Figure 4-15: Configuration C01 — Distribution of combined moments in the deck comparing the critical cases of LL100, +ve and –ve GTL, and the unloaded state for three levels of f(A)/s and the most sensitive ratio, f(D)/s = 0.15 ...... - 109 -

Figure 4-16: Configuration C01 — Distribution of combined moments in the deck considering the effect of LL100 for three levels of f(A)/s (0.15, .020, 0.25) and four levels of f(D)/s (0.00, 0.15, 0.20, and 0.25) ...... - 110 -

Figure 4-17: Configuration C01 — Distribution of axial forces in the cables comparing the critical cases of LL100, +ve and –ve GTL, and the unloaded state for three levels of f(A)/s and the most sensitive ratio, f(D)/s = 0.15 ...... - 111 -

Figure 4-18: Configuration C01 — Distribution of axial forces in the cables under LL100 for three levels of f(A)/s (0.15, 0.20, and 0.25) and four levels of f(D)/s (0.00, 0.15, 0.20, and 0.25) ...... - 112 -

Figure 4-19: Configuration C02 — Distribution of axial forces in the arch comparing the critical cases of LL50, +ve and –ve GTL, and the unloaded state in the arch under three levels of f(A)/s and the most sensitive angle of arch inclination, ω = 45° ...... - 115 -

Figure 4-20: Configuration C02 — Distribution of axial forces in the arch under LL50 for three levels of (A)/s (0.15, 0.20, and 0.25) and four levels of arch inclination ω (0°, 15°, 30°, and 45°) ...... - 116 -

xxiii

Figure 4-21: Configuration C02 — Distribution of combined moments in the arch comparing the critical cases of LL50, +ve and –ve GTL, and the unloaded state under three levels of f(A)/s and the most sensitive angle or arch inclination, ω = 30° ...... - 117 -

Figure 4-22: Configuration C02 — Distribution of combined bending moments in the arch under LL50 for three levels of f(A)/s (0.15, 0.20, and 0.25) and four levels of arch inclination ω (0°, 15°, 30°, and 45°) ...... - 117 -

Figure 4-23: Configuration C02 — Distribution of torsional moments in the arch comparing the critical cases of LL50, +ve and –ve GTL, and the unloaded state for three levels of f(A)/s and the most sensitive inclination of the arch, ω = 45° ...... - 119 -

Figure 4-24: Configuration C02 — Distribution of torsional moments in the arch under LL50 for three levels of f(A)/s (0.15, .020, 0.25) and four levels of arch inclination (0°, 15°, 30°, 45°) ...... - 119 -

Figure 4-25: Configuration C02 — Distribution of combined moments in the deck comparing the critical cases of LL50, +ve and –ve GTL, and the unloaded state for f(A)/s = 0.15, ω = 45° ...... - 120 -

Figure 4-26: Configuration C02 — Distribution of the moments vertical plane in the deck comparing the critical cases of LL50, +ve and –ve GTL, and the unloaded state for f(A)/s = 0.15, ω = 45° ...... - 121 -

Figure 4-27: Configuration C02 (f(A)/s = 0.15, ω = 45°) — Distribution of moments vertical plane in the deck comparing critical case of LL100, +ve and –ve GTL and unloaded state ...... - 122 -

Figure 4-28: Configuration C02 — Distribution of vertical moments in the deck comparing LL100, +ve and –ve GTL, and the unloaded state in the arch for four levels of ω (0°, 15°, 30°, and 45°) and the most sensitive ratio, f(A)/s = 0.15 ...... - 122 -

Figure 4-29: Configuration C02 — Distribution of vertical moments in the deck comparing LL50, +ve and –ve GTL, and the unloaded state in the arch for four levels of ω (0°, 15°, 30°, and 45°) and the most sensitive ratio, f(A)/s = 0.15 ...... - 123 -

Figure 4-30: Configuration C02 — Distribution of moments in the horizontal plane of the deck comparing the critical cases of LL50, +ve and –ve GTL, and the unloaded state for f(A)/s = 0.15, ω = 45° ...... - 124 -

Figure 4-31: Configuration C02 — Distribution of axial forces in the cables comparing the critical cases of LL50, +ve and –ve GTL, and the unloaded state for three levels of f(A)/s (0.15, 0.20, 0.25) and the most sensitive arch inclination, ω = 15° ...... - 125 -

Figure 4-32: Configuration C02 — Distribution of axial forces in the cables under LL50 for three levels of f(A)/s (0.15, 0.20, and 0.25) and four levels of ω (0°, 15°, 30°, and 45°) ...... - 126 -

xxiv

Figure 4-33: Configuration C03 — Distribution of axial forces in the arch comparing the critical cases of LL100, +ve and –ve GTL, and the unloaded state for three levels of f(A)/s and the most sensitive angle of arch rotation, θ = 15° ...... - 127 -

Figure 4-34: Configuration C03 — Distribution of axial forces in the arch under LL100 for three levels of f(A)/s (0.15, 0.20, and 0.25) and four levels of arch rotation θ (0°, 15°, 30°, and 45°) ...... - 128 -

Figure 4-35: Configuration C03 — Distribution of axial forces in the arch comparing the critical cases of LL100, +ve and –ve GTL, and the unloaded state in the arch for three levels of f(A)/s and the most sensitive angle of arch rotation, θ = 45°...... - 130 -

Figure 4-36: Configuration C02 — Distribution of the combined moments in the arch comparing the critical cases of LL100, +ve and –ve GTL, and the unloaded state for three levels of f(A)/s and the most sensitive angle or arch inclination, θ = 45° ...... - 131 -

Figure 4-37: Configuration C03 — Distribution of the combined bending moments in the arch under LL100 for three levels of f(A)/s (0.15, 0.20, and 0.25) and four levels of arch rotation θ (0°, 15°, 30°, and 45°) ...... - 132 -

Figure 4-38: Configuration C03 — Distribution of torsional moments in the arch comparing the critical cases of LL100, +ve and –ve GTL, and the unloaded state for three levels of f(A)/s and the most sensitive angle of arch rotation, θ = 45°...... - 133 -

Figure 4-39: Configuration C03 — Distribution of torsional moments in the arch under LL100 for three levels of f(A)/s (0.15, 0.20, 0.25) and four levels of arch rotation (θ = 0°, 15°, 30°, 45°) ...... - 134 -

Figure 4-40: Configuration C03 — Distribution of the combined moments in the deck comparing the critical cases of LL100, +ve and –ve GTL, and the unloaded state for three levels of f(A)/s and the most sensitive angle of arch rotation, θ = 15°...... - 135 -

Figure 4-41: Configuration C03 — Distribution of the combined bending moments in the deck under LL100 for three levels of f(A)/s (0.15, 0.20, and 0.25) and four levels of arch rotation θ (0°, 15°, 30°, and 45°) ...... - 136 -

Figure 4-42: Configuration C03 — Distribution of tensile forces in the cables considering the effect of LL100 and taking into account three levels of (A)/s ratio (0.15, 0.20, and 0.25) and four levels of arch rotation θ (0°, 15°, 30°, and 45°) ...... - 137 -

Figure 4-43: Schematic sketch showing the change in cable forces as α changes ...... - 139 -

Figure 4-44: Distribution of the uneven changes in magnitude of α along the span, shown on section cuts made at the midspan of the bridge structure in a deformed state for a) C01, b) C02, and c) C03 ...... - 140 -

Figure 4-45: Transformation of angle α and γ resulting in larger cable forces in the configuration with a larger arch rise f(A) as a function of arch and deck stiffness...... - 141 -

xxv

Figure 4-46: Comparison of spatial out-of-plane displacement of the arch two cases with different f(A)/s ratios in configuration C02 resulting transformation of angle α and γ. Scale factor = 5.0 ...... - 143 -

Figure 5-1: The types of BCs of the deck that apply to C01: a) BC#01: Roller and one pin, b) BC#02: Two pins, c) BC#03: Fixed both ends...... - 156 -

Figure 5-2: The types of BCs of the deck that apply to C02 and C03: a) BC#01: Roller and one pin, b) BC#04: Roller and two pins...... - 157 -

Figure 5-3: A comparison of a distribution of SF1 in the arch in C01, C02, and C03 for LL100 and LL50 ...... - 161 -

Figure 5-4: A comparison of a distribution of SM(COMB) in the arch in C01, C02, and C03 for LL100 and LL50 ...... - 161 -

Figure 5-5: A comparison of a distribution of U2 in the arch in C01, C02, and C03 for LL100 and LL50 ...... - 162 -

Figure 5-6: A comparison of a distribution of U3 in the arch in C01, C02, and C03 for LL100 and LL50 ...... - 162 -

Figure 5-7: Configuration C01: A comparison of the distributions of SF1 in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#01 .... - 165 -

Figure 5-8: Configuration C01: A comparison of the distributions of SM(COMB) in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#01 ...... - 166 -

Figure 5-9: Configuration C01: A comparison of the distributions of SF1 in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#02 .... - 168 -

Figure 5-10: Configuration C01: A comparison of the distributions of SM(COMB) in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#02 ...... - 169 -

Figure 5-11: Configuration C01: A comparison of the distributions of SF1 in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#03 .... - 171 -

Figure 5-12: Configuration C01: A comparison of the distributions of SM(COMB) in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#03 ...... - 172 -

Figure 5-13: Configuration C02: A comparison of the distributions of SF1 in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#01 .... - 174 -

xxvi

Figure 5-14: Configuration C02: A comparison of the distributions of SM(COMB) in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#01 ...... - 175 -

Figure 5-15: Configuration C02: A comparison of the distributions of SF1 in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#04 .... - 177 -

Figure 5-16: Configuration C02: A comparison of the distributions of SM(COMB) in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#04 ...... - 178 -

Figure 5-17: Configuration C03: A comparison of the distributions of SF1 in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#01 .... - 180 -

Figure 5-18: Configuration C03: A comparison of the distributions of SM(COMB) in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#01 ...... - 181 -

Figure 5-19: Configuration C03: A comparison of the distributions of SF1 in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#04 .... - 183 -

Figure 5-20: Configuration C03: A comparison of the distributions of SM(COMB) in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#04 ...... - 184 -

Figure 5-21: Configuration C01 — A comparison of the effects of different deck BCs on the distribution of SF1 in the arch for the combination of arch and deck stiffness that results in the smallest SF1 and SM(COMB) ...... - 187 -

Figure 5-22: Configuration C01 — A comparison of the effects of different deck BCs on the distribution of SM(COMB) in the arch for the combination of arch and deck stiffness that results in the smallest SF1 and SM(COMB) ...... - 187 -

Figure 5-23: Configuration C01 — A comparison of the effects of different deck BCs on the distribution of U2 in the arch for the combination of arch and deck stiffness that results in the smallest SF1 and SM(COMB) ...... - 188 -

Figure 5-24: Configuration C01 — A comparison of the effects of different deck BCs on the distribution of U3 in the arch for the combination of arch and deck stiffness that results in the smallest SF1 and SM(COMB) ...... - 188 -

Figure 5-25: Configuration C02 — A comparison of the effects of different deck BCs on the distribution of SF1 in the arch for the combination of arch and deck stiffness that results in the smallest SF1 and SM(COMB) ...... - 190 -

Figure 5-26: Configuration C02 — A comparison of the effects of different deck BCs on the distribution of SM(COMB) in the arch for the combination of arch and deck stiffness that results in the smallest SF1 and SM(COMB) ...... - 190 -

xxvii

Figure 5-27: Configuration C02 — A comparison of the effects of different deck BCs on the distribution of U2 in the arch for the combination of arch and deck stiffness that results in the smallest SF1 and SM(COMB) ...... - 191 -

Figure 5-28: Configuration C02 — A comparison of the effects of different deck BCs on the distribution of U3 in the arch for the combination of arch and deck stiffness that results in the smallest SF1 and SM(COMB) ...... - 191 -

Figure 5-29: Configuration C03 — A comparison of the effects of different deck BCs on the distribution of SF1 in the arch for the combination of arch and deck stiffness that results in the smallest SF1 and SM(COMB) ...... - 193 -

Figure 5-30: Configuration C03 — A comparison of the effects of different deck BCs on the distribution of SM(COMB) in the arch for the combination of arch and deck stiffness that results in the smallest SF1 and SM(COMB) ...... - 193 -

Figure 5-31: Configuration C03 — A comparison of the effects of different deck BCs on the distribution of U2 in the arch for the combination of arch and deck stiffness that results in the smallest SF1 and SM(COMB) ...... - 194 -

Figure 5-32: Configuration C03 — A comparison of the effects of different deck BCs on the distribution of U3 in the arch for the combination of arch and deck stiffness that results in the smallest SF1 and SM(COMB) ...... - 194 -

Figure 6-1: Spatial Configuration C01B...... - 202 -

Figure 6-2: Schematic representation of deck boundary conditions BC#02B (Restricted Pin-connection) in C01B ...... - 203 -

Figure 6-3: Principle of constant-bending-stiffness (CBS) ...... - 204 -

Figure 6-4: Locations of section points in the cross-section of arch and deck ...... - 205 -

Figure 6-5: U2 of the deck, considering deck: W = 2.0m, H = 0.4m in all materials ... - 209 -

Figure 6-6: U3 of the deck, considering deck: W = 2.0m, H = 0.4m in all materials ... - 209 -

Figure 6-7: U2 of the arch, considering deck: W = 2.0m, H = 0.4m in all materials .... - 211 -

Figure 6-8: U3 of the arch, considering deck: W = 2.0m, H = 0.4m in all materials .... - 211 -

Figure 6-9: Stresses in cables (Cable Ø = 50mm in all materials) ...... - 212 -

Figure 6-10: Distribution of stresses in the arch comparing the effect of GFRP, CFRP, and steel at four section points ...... - 213 -

Figure 6-11 Distribution of stresses in the deck comparing the effect of GFRP, CFRP, and steel at four section points ...... - 214 -

xxviii

Figure 6-12: U2 of the deck taking into account creep in the arch and deck and five levels of axial stiffness of the cables ...... - 216 -

Figure 6-13: U3 of the deck taking into account creep in the arch and deck and five levels of axial stiffness of the cables ...... - 216 -

Figure 6-14: U2 of the arch taking into account creep in the arch and deck and five levels of axial stiffness of the cables ...... - 218 -

Figure 6-15: U3 of the arch taking into account creep in the arch and deck and five levels of axial stiffness of the cables ...... - 219 -

Figure 6-16: Stress in the cables taking into account creep in the arch and the deck and five arrangements of cables axial stiffness and susceptibility to creep...... - 220 -

Figure 6-17: U2 of the deck taking into account creep in the structural components and additional tensioning of cables ...... - 222 -

Figure 6-18: U3 of the deck taking into account creep in the structural components and additional tensioning of cables ...... - 222 -

Figure 6-19: U2 of the arch taking into account creep in the structural components and additional tensioning of cables ...... - 223 -

Figure 6-20: U3 of the arch taking into account creep in the structural components and additional tensioning of cables ...... - 223 -

Figure 6-21: Stress distribution in the cables, taking into account additional tensioning of the cables and creep in the arch, deck, and cables ...... - 224 -

Figure 6-22: Stress change in all three materials due to the application of a +ve thermal load in the arch, deck, and cables ...... - 226 -

Figure 6-23: Stress change in all three materials due to the application of a –ve thermal load in the arch, deck, and cables ...... - 227 -

Figure 6-24: Distribution U2 of the arch for GFRP, CFRP, and Steel at three levels of GTL ...... - 228 -

Figure 6-25: Distribution U3 of the arch for GFRP, CFRP, and Steel at three levels of GTL ...... - 228 -

Figure 6-26: Distribution U2 of the deck for GFRP, CFRP, and Steel at three levels of GTL ...... - 229 -

Figure 6-27: Distribution U3 of the deck for GFRP, CFRP, and Steel at three levels of GTL ...... - 230 -

xxix

Figure 6-28: C01B assuming only GFRP; a comparison of the deck displacement in the vertical direction at three levels of deck reach to span ratio ...... - 233 -

Figure 6-29: Schematic sketch capturing the mechanism of deck displacement in C01B assuming the deck profile with DSR = 15:1 (vertically flexible and horizontally stiff deck) ...... - 235 -

Figure 6-30: Schematic sketch capturing the mechanism of deck displacement in C01B assuming the deck profile with DSR = 1:3 (vertically stiff and horizontally flexible deck) ...... - 236 -

Figure 6-31: Distribution of SM(COMB) in the deck for M001 and M006 at three levels of ratio: f(D)/s = 0.25, 0.20, and 0.15 ...... - 239 -

Figure A-1: Selection of the “tension-only” material behaviour assigned to the truss elements utilized for modeling of hangers in the linear models...... - 274 -

Figure A-2: Definition of geometric quantities in a generalized profile of a beam element cross-section...... - 276 -

Figure A-3: A graphical interpretation of a connector section “U-joint” used to model the connection of the cables to the arch and deck ...... - 278 -

Figure A-4: A spatial configuration assumed to verify beam elements that resist the combined shear forces and bending and torsional moments ...... - 280 -

Figure A-5: A comparison of the displacements of a free end cantilever beam ...... - 281 -

Figure A-6: A comparison of number of elements required to achieve a consistent level of displacement ...... - 282 -

Figure A-7: A configuration assumed to verify beam elements that resist thermal loads ...... - 283 -

Figure A-8: Mesh sensitivity analysis of B32 indicating the suitable number of elements for flexible cables ...... - 284 -

Figure A-9: An example of a setting of the incremental steps for the nonlinear analysis ...... - 285 -

Figure A-10: Configuration of a weightless cable loaded by a concentrated force at its midspan assumed to verify the cable modeled with beam elements with reduced bending stiffness ...... - 286 -

Figure A-11: A comparison of the deformed shapes of the cables using the parabola and catenary equations...... - 289 -

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Figure A-12: A comparison of the deformed shape of a cable modeled using B32 elements with reduced bending stiffness and AM using a parabolic curve ...... - 290 -

Figure A-13: Cable deformed under self-weight: related symbols...... - 290 -

Figure A-14: A comparison of the displacements of a cable, modeled using B32 elements, exposed to three levels of thermal load ...... - 293 -

Figure A-15: A comparison of the distributions of the tensile force in a cable, modeled using B32, exposed to three levels of thermal load ...... - 294 -

Figure A-16: A configuration used to verify the entire structure ...... - 296 -

Figure B-1: Configuration C01 — A comparison of the distributions of SF(COMB) in the arch comparing critical case of LL, +ve and –ve GTL and unloaded state for three levels of f(A)/s and most sensitive ratio, f(D)/s = 0.15...... - 327 -

Figure B-2: Configuration C01 — A comparison of the distributions of SF(COMB) in the arch considering the effect of LL100...... - 328 -

Figure B-3: Configuration C01 — A comparison of the distributions of U(COMB) in the arch comparing critical case of LL, +ve and –ve GTL and unloaded state in the arch for three levels of f(A)/s and the most sensitive ratio, f(D)/s = 0.15 ...... - 332 -

Figure B-4: Configuration C01 — A comparison of the distributions of U(COMB) in the arch considering the effect of LL...... - 333 -

Figure B-5: Configuration C01 — A comparison of the distributions of SF1 in the deck comparing critical case of LL, +ve and –ve GTL and unloaded state for three levels of f(A)/s and most sensitive ratio, f(D)/s = 0.15...... - 335 -

Figure B-6: Configuration C01 — A comparison of the distributions of SF1 in the deck considering the effect of LL100...... - 336 -

Figure B-7: Configuration C01 — A comparison of the distributions of SF(COMB) in the deck comparing critical case of LL, +ve and –ve GTL and unloaded state for three levels of f(A)/s and most sensitive ratio, f(D)/s = 0.15 ...... - 338 -

Figure B-8: Configuration C01 — A comparison of the distributions of SF(COMB) in the deck considering the effect of LL100 ...... - 339 -

Figure B-9: Configuration C01 — A comparison of the distributions of U(COMB) in the deck comparing critical case of LL, +ve and –ve GTL and unloaded state for three levels of f(A)/s and most sensitive ratio, f(D)/s = 0.15...... - 342 -

Figure B-10: Configuration C01 — A comparison of the distributions of U(COMB) in the deck considering the effect of LL100 ...... - 343 -

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Figure B-11: Configuration C02 — A comparison of the distributions of SF(COMB) in the arch comparing critical case of LL, +ve and –ve GTL and unloaded state for three levels of f(A)/s and most sensitive angle of arch inclination, ω = 15° ...... - 347 -

Figure B-12: Configuration C02 — Distribution of SF(COMB) in the arch considering the effect of LL50 ...... - 348 -

Figure B-13: Configuration C02 — A comparison of the distributions of U(COMB) in the arch considering the effect of LL50...... - 352 -

Figure B-14: Configuration C02 — A comparison of the distributions of SF(COMB) in the deck considering the effect of LL50 ...... - 354 -

Figure B-15: Configuration C02 — A comparison of the distributions of U(COMB) in the deck comparing critical case of LL, +ve and –ve GTL and unloaded state for three levels of f(A)/s and most sensitive arch inclination, ω = 15° ...... - 357 -

Figure B-16: Configuration C02 — A comparison of the distribution of U(COMB) in the deck considering the effect of LL50...... - 358 -

Figure B-17: Configuration C03 — A comparison of the distributions of SF(COMB) in the arch comparing critical case of LL, +ve and –ve GTL and unloaded state for three levels of f(A)/s and most sensitive angle of arch inclination, θ = 15° ...... - 362 -

Figure B-18: Configuration C03 — A comparison of the distributions of SF(COMB) in the arch considering the effect of LL100...... - 363 -

Figure B-19: Configuration C03 — A comparison of the distributions of U2 in the arch comparing critical case of LL, +ve and –ve GTL and unloaded state for three levels of f(A)/s and most sensitive angle of arch rotation, θ = 45° ...... - 367 -

Figure B-20: Configuration C03 — A comparison of the distributions of U3 in the arch comparing critical case of LL, +ve and –ve GTL and unloaded state for three levels of f(A)/s and most sensitive angle of arch rotation, θ = 45° ...... - 368 -

Figure B-21: Configuration C03 — A comparison of the distributions of U2 in the arch considering the effect of LL100...... - 369 -

Figure B-22: Configuration C03 — A comparison of the distributions of U3 in the arch considering the effect of LL100...... - 370 -

Figure B-23: Configuration C03 — A comparison of the distributions of SF(COMB) in the deck considering the effect of LL100 ...... - 372 -

Figure B-24: Configuration C03 — A comparison of the distributions of U(COMB) in the deck comparing critical case of LL, +ve and –ve GTL and unloaded state for three levels of f(A)/s and most sensitive angle of arch rotation, θ = 15° ...... - 375 -

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Figure B-25: Configuration C03 — A comparison of the distributions of U2 of the deck comparing critical case of LL, +ve and –ve GTL and unloaded state taking into account the most sensitive configuration: f(A)/s = 0.15 and θ = 15° ...... - 376 -

Figure B-26: Configuration C03 — A comparison of the distributions of U2 in the deck considering the effect of LL100...... - 377 -

Figure B-27: Configuration C01— A graphical representation of the spatial out-of-plane displacement of the arch and deck considering the most sensitive configuration (f(A)/s = 0.15, f(D)/s = 0.25). Scale factor 1.0 ...... - 379 -

Figure B-28: Configuration C02— A graphical representation of the spatial out-of-plane displacement of the arch and deck considering the most sensitive configuration (f(A)/s = 0.25, ω = 45°). Scale factor = 5 ...... - 385 -

Figure B-29: Configuration C03 — A graphical representation of the spatial out-of-plane displacement of the arch and deck considering deformed shape of the most sensitive configuration (f(A)/s = 0.15, θ = 45°). Scale factor = 10.0 ...... - 391 -

Figure D-1: Distribution of stresses in the deck comparing the effect of GFRP, CFRP, and steel at four section points a) SP01: Top, b) SP02: Bottom, c) SP03: RHS, and d) SP04: LHS ...... - 425 -

Figure D-2: Distribution of stresses in the arch comparing the effect of GFRP, CFRP, and steel at four section points a) SP01: Top, b) SP02: Bottom, c) SP03: RHS, and d) SP04: LHS ...... - 426 -

Figure D-3: A comparison of distribution of stresses in the deck at four designated section points, taking into account creep in the arch, deck, and cables and additional tensioning of cables ...... - 428 -

Figure D-4: Comparison of distribution of stresses in the arch at four designated section points, taking into account creep in the arch, deck, and cables and additional tensioning of cables ...... - 429 -

Figure D-5: Fluctuation of stresses in the deck at three levels of thermal load (0°C, +50°C , and -50°C), taking into account effect of GFRP, CFRP, and steel at section points: a) SP#01 — Top, b) SP#02 — Bottom ...... - 433 -

Figure D-6: Fluctuation of stresses in the deck at three levels of thermal load (0°C, +50°C , and -50°C), taking into account effect of GFRP, CFRP, and steel at section points: a) SP#03 — RHS, b) SP#03 — LHS ...... - 434 -

Figure D-7: Fluctuation of stresses in the arch at three levels of thermal load (0°C, +50°C , and -50°C) taking into account effect of GFRP, CFRP, and steel at section points: a) SP#01 — Top, b) SP#02 — Bottom ...... - 436 -

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Figure D-8: Fluctuation of stresses in the arch at three levels of thermal load (0°C, +50°C , and -50°C) taking into account effect of GFRP, CFRP, and steel at section points: a) SP#03 — RHS, b) SP#03 — LHS ...... - 437 -

Figure D-9: Distribution of stress in cables at three levels of GTL (0°C, +50°C, and -50°C) taking into account GFRP, CFRP, and steel ...... - 438 -

Figure D-10: A comparison of the distributions U2 of the deck taking into account there levels of f(D)/s ratio ...... - 440 -

Figure D-11: A comparison of the distributions U3 of the deck taking into account there levels of f(D)/s ratio ...... - 441 -

Figure D-12: A comparison of the distributions of stress in the cables along the span, taking into account there levels of f(D)/s ratio ...... - 442 -

Figure D-13: A comparison of the distributions of SM(COMB) in the deck, taking into account models M001 and M006 at three level of ratio f(D)/s: 0.25, 0.20, and 0.15...... - 443 -

Figure D-14: A comparison of the distributions of SM(COMB) in the arch, taking into account models M001 and M006 at three level of ratio f(D)/s: 0.25, 0.20, and 0.15 ...... - 444 -

Figure D-15: A comparison of the distributions of SF1 in the deck, taking into account models M001 and M006 at three level of ratio f(D)/s: 0.25, 0.20, and 0.15 ...... - 445 -

Figure D-16: A comparison of the distributions of SF1 in the arch, taking into account models M001 and M006 at three level of ratio f(D)/s: 0.25, 0.20, and 0.15 ...... - 446 -

Figure D-17: Distribution of SM(COMB) in the deck taking into account ratio f(D)/s = 0.25, three materials and three levels of GTL: a) 0°C, +50°C, and c) -50°C ...... - 449 -

Figure D-18: Distribution of stresses cables in configuration C01B taking into account ratio f(D)/s = 0.25, three materials and three levels of GTL: a) 0°C, +50°C, and c) -50°C .... - 451 -

Figure D-19: A comparison of stress distribution in the deck at section point located at left- hand-side (LHS) of the deck cross-section taking into account the effect of GFRP, CFRP, and steel at 0°C ...... - 453 -

Figure D-20: A comparison of stress distribution in the deck at section point located at left- hand-side (LHS) of the deck cross-section taking into account the effect of GFRP, CFRP, and steel at +50 ...... - 453 -

Figure D-21: A comparison of stress distribution in the deck at section point located at left- hand-side (LHS) of the deck cross-section taking into account the effect of GFRP, CFRP, and steel at –50°C ...... - 454 -

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List of Symbols, Abbreviations and Nomenclature

Symbol Definition 1 Global longitudinal direction 2 Global vertical direction 3 Global transversal direction A Cross-sectional area a Ratio of /

AA Cross-sectional𝑇𝑇0 𝑊𝑊0 area of the arch ACM Advanced Composite Materials AM Analytical model BC Boundary Conditions BFRP Basalt-Fibre-Reinforced-Polymer

C(i) Coordinate of individual cables along the span CAS Constant-axil-stiffness CBS Constant-bending-stiffness CF Axial force in the cable

CF(H) Horizontal component of the cable force

CF(V) Vertical component of the cable force CFRP Carbon-Fibre-Reinforced-Polymer C-S Cross-sectional CTE Coefficient of Thermal Expansion DL Dead Load (permanent load) DOF Degree of Freedom ds Differential increment in the tangential direction DSR Deck Stiffness Ratio dx Differential increment in the horizontal direction dy Differential increment in the vertical direction E Modulus of elasticity

E(EFF) Effective modulus of elasticity EA Axial stiffness

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Symbol Definition

EA(EFF) Effective axial stiffness

EAA Axial stiffness of the arch EI Bending stiffness

EI(EFF) Effective bending stiffness

EI(H) Bending stiffness in the horizontal direction

EI(V) Bending stiffness in the vertical direction F Point load f Cable sag f(A) Arch rise f(A)/s Arch rise to span ratio; primary geometric variable f(AZ) Arch reach in the transversal direction f(D) Deck reach f(D)/s Deck reach to span ratio; secondary geometric variable f(VC) Rise of the camber of the deck in the vertical direction f/s(VC) Deck camber in the vertical direction to span ratio FEA Finite Element Analysis G Shear modulus g(H) Horizontal offset between the arch and deck abutments g(V) Vertical offset between the arch and deck abutments GFRP Glass-Fibre-Reinforced-Polymer GJ Torsional stiffness GTL Global Thermal Load HSS Hollow Structural Section I Second moment of area I11 Second moment of area about the vertical axis of the cross-section I22 Second moment of area about the horizontal axis of the cross-section J Torsional constant k(w) Number of sinusoidal half-waves L Length

L(i) Projected length of the arch segments between individual nodes

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Symbol Definition

LA Length of the parabolic arch

LC Length of the parabolic curve LHS Left-hand-side LL Live Load LL100 Live Load distributed equally over the entire length of the span LL50 Live Load distributed over half of the deck starting deck abutments and ending at midspan LTL Local Thermal Load M Moment m The peak value added to the at midspan (representing the “rise” of the

parabolic curve) 𝑇𝑇𝑇𝑇 MA Moment at the abutment of the arch

MP Moment effect at a general point P n Parameter determining an increase in the curvature of the parabolic distribution

NA Axial thrust in the arch

Ni(ID) Total number of nodes in one cable

Ni(No) Node order in the cable

No(Nodes) Number of nodes in the arch qc Weight of cable per unit length

RH Reaction force in horizontal direction

RHA Horizontal reaction force at the arch abutment RHS Right-hand-side

RV Reaction force in the vertical direction

RVA Vertical reaction force at the arch abutment s Span of the bridge between abutments of the arch s(VC) Span of the deck assuming deck camber in vertical direction S01A First configuration of the arch bending stiffness: the arch is flexible in the vertical direction and flexible in the horizontal direction S01D First configuration of the deck bending stiffness: the deck is flexible in the vertical direction and flexible in the horizontal direction

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Symbol Definition S02A Second configuration of the arch bending stiffness: the arch is stiff in the vertical direction and flexible in the horizontal direction S02D Second configuration of the deck bending stiffness: the deck is stiff in the vertical direction and flexible in the horizontal direction S03A Third configuration of the arch bending stiffness: the arch is flexible in the vertical direction and stiff in the horizontal direction S03D Third configuration of the deck bending stiffness: the deck is flexible in the vertical direction and stiff in the horizontal direction S04A Fourth configuration of the arch bending stiffness: the arch is stiff in the vertical direction and stiff in the horizontal direction S04D Fourth configuration of the deck bending stiffness: the deck is stiff in the vertical direction and stiff in the horizontal direction SAB Spatial Arch Bridge

SF(COMB) Combined shear force taking into account the effect of both the SF2 and SF3 SF1 Axial force in the direction of the longitudinal axis of the structural component SF2 Shear force in the vertical direction SF3 Shear force in the horizontal transversal direction

SM(COMB) Combined bending moment taking into account the effect of both the SM1 and SM2 SM1 Section bending moment about local axis in the horizontal transversal direction (vertical bending moment) SM2 Section bending moment about local axis in the vertical direction (transversal bending moment) SM3 Section torsional moment about the longitudinal axis of a member SP Section point T Torsional moment

T0 Horizontal component of a tensile force in the cable

Tb Base temperature; regarding the additional tensioning of cables

TC Tensile force at arbitrary point in the cable

TR Magnitude of the reaction force in the tangential direction at the cable anchor

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Symbol Definition

U(COMB) Combined displacement taking into account the effect of both the U2 and U3 U1 Displacement in the longitudinal global direction U2 Displacement in the vertical global direction

U2A.CR Vertical displacement at the arch crown

U2D.MS. Vertical displacement of the deck at midspan U3 Displacement in the transversal global direction UR1 Rotation about axis in the longitudinal global direction UR2 Rotation about axis in the vertical global direction UR3 Rotation about axis in the transversal global direction +ve Positive –ve Negative

W0 Weight of the cable per unit length WPUL Weight per unit length x Distance along the span X Global longitudinal direction

Xi(0) Reference coordinate linearly increasing from zero to total span in

longitudinal direction in between abutments of the arch 𝑠𝑠 Xi(0) Horizontal coordinate along the span

Xi(A) A node coordinate of the arch in the longitudinal direction

Xi(C) A node coordinate of the cables in the longitudinal direction

Xi(C-0) Location of the cable along the span

Xi(D) A node coordinate of the deck in the longitudinal direction Y Global vertical direction

Yi(A) A node coordinate of the arch in the vertical direction

Yi(C) A node coordinate of the cables in the vertical direction

Yi(D) A node coordinate of the deck in the vertical direction Z Global transversal direction

Zi(A) A node coordinate of the arch in the transversal direction

Zi(C) A node coordinate of the cables in the transversal direction

Zi(D) A node coordinate of the deck in the transversal direction

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Symbol Definition α Angle between horizontal plane of the deck and the plane of the cables

α(TRAN) Transformed angle α γ The difference in the cable inclination between the deformed and undeformed state

δA Displacement due to axial load ΔT Difference in the applied temperature

δTL Displacement due to thermal load θ Angle of arch rotation about the vertical axis at located at midspan; secondary geometric variable ω Angle of arch inclination from the vertical plane; secondary geometric variable

Epigraph

Learning never exhausts the mind. Leonardo da Vinci

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Chapter One: Introduction and Motivation

1.1 Spatial Arch Bridges The creation of spatial structures has been a desire of human society since ancient times. Great works of architecture were built to celebrate prosperity, development, and achievements in technology. Often times, symbolism inherited in the structure played a key role in the designs. Modern, architecturally appealing structures continue the theme of an outstanding design combined with mastered functionality and efficiency. Spatial structures cover many possible areas and concepts ranging from sculptures to purely functional buildings; they have been the passion of many architects and structural designers. On the boundary of art and efficient design lie spatial arch bridges which, in their configuration, combine arch ribs with a deck connected via a sophisticated system of hangers. Due to their configuration, spatial arch bridges have a light appearance employing architectural elegance with structural and material efficiency. In modern urbanistic developments, spatial bridges fulfill not only the function of a simple, physical connection between two points, but they also, and mainly, carry an attribute of a landmark or symbol of a particular event. In recent years, spatial bridges have been built as millennium bridges, entry gates connecting communities, or monuments celebrating innovation. The complexity of spatial structures results in marvellous forms linking material appearance with a unique distribution of geometrical members. Nevertheless, the combination of spatial elements that employ various structural details leads to certain challenges in structural analysis and design. Understanding the significance of individual factors that influence the mechanics and structural behaviour of spatial configurations is a complex discipline.

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Chapter One: Introduction and Motivation

1.2 Problem Definition

Spatial arch bridges (SABs) can be very complex. Even though, in their configurations, a simple alignment of an inclined planar arch and straight deck fulfills the condition of spatiality, more sophisticated systems employing curved decks with multiple rotated arches exist. SABs can be characterized as structures where the deck is connected to a spatially- shaped arch via a system of hangers. In particular configurations, the arch can be replaced by a spatial shell and the hangers can be either flexible or stiff. The attribute that defines SABs is the presence of out-of-plane loads in the structural system. Loads applied vertically on the deck generate out-of-plane loads in the form of shear forces and bending and/or torsional moments that may occur in all the main structural components of the system, such as the arch/shell, deck, and hangers. Further, the structural system also experiences out-of-plane loads as a result of supporting its own weight. Geometric complexity is further elevated by the configuration of the hangers, which may influence structural behaviour. Certain patterns of hanger configuration, which result in more efficient distributions of internal forces, were investigated in network arch bridges by Tveit (2002). The investigation conducted proved that the configuration of the hangers selected can play a significant role. Nevertheless, this investigation was not carried out for configurations of SABs. Even though efficient three-dimensional (3D) antifunicular (compression only) shapes of arches that resists multidirectional loading or defining the optimal geometric and stiffness ratios have been studied by several authors (Lachauer & Kotnik, 2011; Jorquera & Manterola, 2012; Todisco, 2014; Sarmiento-Comesias, 2015), understanding certain geometric relations is still a challenge and needs to be clarified. Exposing structures to fluctuating temperatures results in the generation of stresses that, in some cases, may reach levels as high as the effects from live loads (Sheriff, 1991); therefore, an increased frequency of temperature change can affect the fatigue life of a structure. The magnitude of stresses induced from thermal effects is related to several factors such as the geometry of the structural configuration, material composition, geographic location, or orientation in respect to the cardinal directions (Dilger, 1983). Most SABs have complex geometries and are comprised of structural steel alone or steel combined with reinforced concrete. Structural steel compared to other materials has the

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Chapter One: Introduction and Motivation highest coefficient of thermal expansion, which results in relatively high thermal strains that may be magnified by particular geometric or material configurations. Geographic location and structure orientation greatly affect the stresses generated within a structure. It is not only the temperature variation between seasons that has an effect on structures; the diurnal fluctuation may also be important, especially in Canada where significant differences can occur. In addition, the orientation of a structure, in combination with the material and geometry, may lead to temperature differences as high as 70°C across one structure at the same moment (Dilger, 1983). The majority of SABs have been built in countries where thermal loads do not represent a significant concern, for example Spain, Italy, the United Kingdom, and Southern China (Sarmiento-Comesias et al., 2013). The design process generally considers thermal analysis; however, the analyses on existing SABs or for specific case studies covered only a small range of diurnal temperature fluctuations relevant to the region where the structures were built, for example the Hulme Arch Bridge (diagonal arch over straight deck, Manchester, UK) by Hussain & Wilson (1998) and the Nanning Bridge (two inclined uneven arches over a curved deck, Nanning, Guangxi, China) by Cheng et al. (2010). Based on the number and depth of studies that consider the effect of thermal loads on SABs, the significance of temperature fluctuation has not been described in detail and more rigorous investigation is required to provide a clear understanding of the effects of thermal loads on SABs. The material complexity of SABs is not as broad as the group of possible geometric combinations; nevertheless, due to the configuration of SABs in space, the material behaviour under multidirectional loading may influence the overall structural response. While the specific stiffness (axial, bending, or torsional) can be achieved in any material via a combination of cross-sections and material properties, the overall response of the structure may differ for different materials. The reason for this difference is the specific weight of different materials and their responses to applied loads (creep in concrete and flexural-torsional buckling in steel section). In modern bridge applications, there is an increasing demand for durable, high performance materials that resist not only the applied loads but also the weather conditions. Long winters are typical in the Canadian climate with frequent cycles of freezing and thawing. For bridges, these temperature fluctuations are often accompanied with de-icing

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Chapter One: Introduction and Motivation salts that accelerate the process of material deterioration. One alternative to meet the demand is advanced composite materials (ACMs). ACMs are increasingly used in bridge applications in the form of reinforcement or strengthening systems; all-composite systems also exist. The outstanding properties of ACMs, such as high stiffness-to-weight ratio and chemical and/or thermal stability, make them a promising choice for bridge structures. However, due to ACMs’ anisotropic characteristics and certain challenges in their structural profiles (creep related deformations, high stress concentrations at connections, or their light weight affecting the dynamic response) combined with the effect of out-of-plane loads, no SABs have been constructed from ACMs as an all-composite structure. Therefore, ACMs and SABs represent a gap in our current knowledge. To conclude, SABs represent relatively new architecturally and structurally appealing forms whose popularity has increased in recent years. Most current research has covered certain aspects related to the structural behaviour of SABs. Nevertheless, the geometric complexity combined with multiple possible material selections creates research gaps that need to be addressed by rigorous investigation.

1.3 Objectives

The primary goals of this work are to broaden our fundamental knowledge of the structural behaviour of SABs with an inferior deck and to compile the resulting data in tabular and graphical form so that the data can be used as a guideline. More specific objectives are provided in chapters four, five, and six, which focus on aspects of a particular form of a structural response of SABs.

The main objectives are:

• To investigate the significance of geometric variables, including arch rise, deck reach, angle of arch inclination, and angle of arch rotation, in three different spatial configurations of SABs, while considering thermal and live loads;

• To examine the effect of variability in bending stiffness of the arch and deck and the end boundary conditions of the deck on the magnitude of the internal forces in the arch

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Chapter One: Introduction and Motivation

to determine the level of susceptibility of the arch to buckling, in the three spatial configurations;

• To investigate the effect of material properties, in particular the properties of ACMs, on the structural behaviour of one spatial configuration, while taking into consideration effects such as creep, coefficient of thermal expansion, low modulus of elasticity, additional tensioning of the cables or the ratio of vertical to horizontal bending stiffness of the deck;

• To establish design criteria for the three spatial configurations studied in the thesis, while considering the effect of loads, overall geometry, cross-sectional dimensions, bending stiffness, and end conditions of the deck.

1.4 Scope of Work

Investigations are carried out using finite elements analyses (FEA). Three main configurations that satisfy the definition of SABs are considered first to depict the structural behaviour of SABs at a general level; further, the results are applied to more specific geometries at later stages of the work. The analyses take into account only spatial bridges with inferior decks, i.e., bridges with their deck suspended below the arch. The three main configurations of SABs with inferior decks are as follows: (1) a vertical parabolic arch with a horizontal parabolic deck, (2) a parabolic arch inclined from the vertical plane with a straight deck, and (3) a vertical arch rotated about the vertical axis above a straight deck. From these three basic concepts, the most critical configurations are identified and used for further analyses. All models are created with structural beam elements whose properties were altered to investigate particular parameters of interest. Linear and geometrically nonlinear models are developed to evaluate the structural responses during different stages of the research. The investigated configurations represent pedestrian bridges. The Canadian Highway Bridge Design Code provision was followed as a reference to determine the magnitude of applied loads. Dead loads, live loads, and thermal loads are considered as key factors. Thermal loads are considered because of the Canadian climate. Wind and earthquake loads are not included in the scope of the thesis even though these loads may

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Chapter One: Introduction and Motivation affect the response of the structure significantly: for example, earthquake loads would need to be considered for a SAB being designed in Vancouver, British Columbia. As material selection can influence the response in structural configurations, three different material options are investigated: steel and two types of ACMs in the form of structural hollow profiles. Material nonlinearity is not considered and material properties are taken directly from the literature.

1.5 Thesis Layout

This work investigates the structural behaviour of spatial arch bridges. The complexity of the analyses that describe the structural behaviour increases throughout the thesis. Chapter One introduces the topic, defines the research problem, and outlines the objectives. A literature review is provided in Chapter Two, which is divided into several sections related to the types of analysis considered in this work. Chapter Three describes the finite element models and parametric input files that are used in the subsequent analyses. In this chapter, all alternatives of the models (related to the individual analyses) are described. Chapter Four investigates the structural behaviour of the models and clarifies the significance of the geometric variables (ratios of geometric configurations) that potentially influence the structural behaviour of SABs under live and thermal loading. Linear FE models are examined. The influence of variability in bending stiffness of the arch and deck on distribution and magnitude of the axial forces and bending moments in the arch, affecting the susceptibility of the arch to buckling as a function of the end conditions of the deck, is investigated in Chapter Five. The dependence of the structural behaviour on material composition is evaluated in Chapter Six. The use of ACMs in particular SAB configurations is examined. The significance of relevant material properties, such as the coefficient of thermal expansion and modulus of elasticity, are investigated in detail. The effect of variability of deck stiffness, creep, and additional cable tensioning are also considered in this chapter.

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Chapter One: Introduction and Motivation

Chapter Seven proposes design criteria of the three spatial configurations in question. Based on the newly achieved understanding, several straightforward guidelines are established. Directions of future work are discussed in this chapter. A summary and some final remarks are presented in Chapter Eight. This chapter gathers the main contributions of the thesis and comments on the current state of research and on the application of the results of this research in industrial projects. The appendices of this thesis provide the details for individual chapters such as large tables or charts. The appendices also include a complete parametric input file of the geometrically nonlinear model that is employed for the analyses. The input file contains all relevant parameters and commands, and therefore, can be used as a reference for studying purposes and further analysis. The work deals with a relatively large number of variables, whose significance is studied, with increasing complexity, at several levels. The mutual relation of the individual chapters, with direct focus on particular type of variables, is presented in a simplified form in the flow-chart shown in Figure 1-1.

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Chapter One: Introduction and Motivation

CHAPTERS: ONE, TWO, and THREE

CHAPTER FOUR Input for Chapter Four Geometric variables:  Arch rise  Inclination of the arch  Rotation of the arch  Deck reach

Critical configurations

Input for Chapter Five

CHAPTER FIVE

Mechanical variables:

 Bending stiffness of the arch

 Bending stiffness of the deck

 Boundary conditions of the deck

Critical configurations

Input for Chapter Six

CHAPTER SIX

Materials variables:

 Advanced Composite Materials

 Creep

 Coefficient of thermal expansion

CHAPTER SEVEN: DESIGN CRITERIA

Figure 1-1: A schematic flow-chart showing the mutual relation of individual chapters

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Chapter Two: Literature Review

2.1 Introduction Spatial Arch Bridges (SABs) are structures that can be very complex in terms of their geometric configuration, material composition, and dependence on site-related factors. Demand for functional and aesthetically pleasing structures in urban areas drove the development of such structures. SABs are well accepted from the architectural point of view. The arch is a highly suitable structural component to support horizontally curved decks, which are often times a characteristic of modern spatial bridges (Jorquera, 2009). Architectural and engineering points of view are always present when analyzing and designing spatial structures. Therefore, both perspectives should always be considered (Rippmann, 2016). This chapter provides an overview of the history and general concepts of SABs followed by a rigorous literature review. Chapters of the thesis may contain additional information and literature sources directly related to the topic of a particular chapter.

2.2 Definition of Spatial Arch Bridges

SABs constitute a very broad range of possible configurations. In general, SABs can be defined as structures where the deck is connected to an arch via a system of hangers. Typically, the hangers are not contained in the plane of the arch. The arch may not be defined in a plane; it can be curved in space. Further, in certain configurations, the arch can be replaced by a spatial shell and the hangers can be either flexible or stiff. The classification of SABs is provided in Section 2.4.2. The attribute that defines SABs is the presence of out-of-plane loads in the structural system. Loads applied vertically on the deck and loads resulting from the self- weight of the structure generate out-of-plane loads in the form of shear forces and bending and/or torsional moments, which may occur in all three main components of the structural system, the arch/shell, deck, and hangers. In contrast, bridges with vertical planar arches that experience out-of-plane loads from certain loading actions, such as lateral wind loads or eccentrically applied vertical loads that introduce lateral shear forces in the deck or arch, are not considered SABs (Sarmiento et al., 2013). In SABs, the distribution of the

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Chapter Two: Literature Review generated internal forces is dependent on the particular configuration of the bridge. Certain configurations may comprise rigid hangers and a flexible arch, resulting in better utilization of the axial resistance of the arch. Other configurations may consist of a deck supported by flexible hangers (cables) connected to a stiff arch resulting in large bending stresses in the arch. Descriptions of the individual structural systems and their responses to applied loads are provided in Section 2.5.

2.3 History and Development of Spatial Structures

SABs developed as a structural form from vertical in-plane arches due to the demand for spatially appealing structures. The desire for such structures has been present in a variety of forms and purposes (mainly religious) since ancient times (Bunson, 2012). The complexity of these structures evolved in parallel with the ability of humans to handle tools, understand the properties of obtainable materials, and apply structural mechanics (Nicholson, 2000). Self-standing arches in the form of vaults and projected or rotated arches in the form of domes preceded the development of SABs. The arch, defined as an “anti-funicular” structure (a structure whose self-weight is resisted only by pure compression), is opposite to a “funicular” structure whose load is resisted only by tensile forces. In both anti-funicular and funicular structures, the line of thrust must be contained within the thickness of the element. In funicular structures (cable supported), the line of thrust (tension) lies directly on the centre line of the cable as the cable, due its low bending stiffness, automatically develops a funicular shape. In antifunicular structures (arches, domes, etc.), the ideal shape is much more difficult to achieve without a good understanding of the mechanics of arch action. Therefore, suspension bridges, in their simplest form of suspended ropes connected with cross members, were constructed and were more wide-spread much earlier than bridges that used arches as part of their design (Allen, 2010). Even though the invention of the “true” arch is attributed to the Etruscans, the first known evidence of arches comes from Ancient Egypt, from the era of at least the third dynasty, which is approximately 2600 BC (Perrot, 1883). An example of such an arch, the “Arch in the Necropolis of Abydos,” is shown in Figure 2-1; this arch was built from a combination of limestone and mud bricks.

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Chapter Two: Literature Review

According to the findings from the Abydos excavations (Löbbecke, 2012), the arch has a key stone in the form of a true voussoir. From this evidence, Egyptians understood the mechanics of arches. However, arches were built only in certain cases when constructing more common corbeled vaults was not practical. Arches were built from mud bricks wedged together and bonded with mortar (gypsum plaster) and gravel (Nicholson, 2000). Arches were not more common in Ancient Egypt possibly due to the Egyptians’ lack of understanding of materials and that arches constructed from weak stones or mud bricks may collapse easily.

Figure 2-1: The Arch in the Necropolis of Abydos (Perrot, 1883)

A typical Egyptian arch is known as a corbeled arch, a structure in which the shape is achieved with stones placed on top of each other subsequently shifting inwards. The stability of such a system is achieved from the weight of the stones placed at the ends of the cantilevered stones, creating the curved opening. Oftentimes, integral supports were necessary during the construction of corbeled arches (Löbbecke, 2012). The development and use of true arches on a larger scale was achieved during the era of the Roman Empire. Ancient Romans adopted the arch concept from the Etruscans (Gascoigne, 2001) and mastered building structures, which included arches not only in the form of utilitarian buildings, such as the Pont du Gard aqueduct near Nimes in France, but also in the form of architectural landmarks, such as arches of triumph; examples of such structures are shown in Figure 2-2.

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Chapter Two: Literature Review

a) b)

Figure 2-2: Examples of Roman arches: a) the Arch of Titus in Rome, a triumph arch (Anthony, 2005) and b) the Pont du Gard Roman aqueduct near Nimes in France, a utilitarian structure (Beard, 2015)

The Romans, taking many examples from the art and architecture of the Greeks, advanced the “binder” of construction materials, cement. Ancient Greeks living on the east coast (close to present-day Turkey) developed a form of cement in approximately 200 BC. This cement consisted of lime, which binds sand, water, and clay. This invention was advanced by the Romans who replaced the clay with finely ground volcanic lava from the region of Pozzuoli resulting in “pozzolanic cement,” which represented the strongest type of mortar until recently when Portland cements were developed (Gascoigne, 2001). Mixing pozzolanic cement with small particles of volcanic stones produced concrete, which was used to bind stones and bricks; many great arches and aqueducts were constructed using this concrete. The Romans built many great arch structures and, due to their skill of precisely fitting individual bricks, only a small amount mortar (the component that deteriorates most easily) was used. Thus, most of their structures remained intact for centuries. Nevertheless, the Romans did not have an advanced understanding of the mathematical aspects of the arch (Boyd, 1978). A typical Roman arch was semicircular. This shape was selected for symbolic reasons because the semicircular shape represented unity, perfection, and harmony (Todisco, 2014). In a semicircular configuration, the line of thrust does not follow exactly

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Chapter Two: Literature Review the centre line of the semicircular arch rib. The line of thrust is a conceptual line representing the position of the resultant compression force resulting from an applied dead load (Shrive et al., 2000). From the mechanics of masonry arch action, as long as the line of thrust stays within the arch cross-section, the arch is stable (collapse will not occur). However, even though the arch remains stable, it will crack if the line of thrust occurs outside of the kern region. In Roman arches, the heavy masonry fill placed above the arch stabilized the structure. Without the fill, the line of thrust, which was not contained within the cross- section of the semicircular arch, would result in a bending moment that would cause the arch to collapse. Therefore, the shape of the Roman arches was not a significant problem. The drawback of the Roman semicircular arches was their relatively short span. Typically, Roman arch bridges consisted of several small arches, and in some cases, small arches were constructed in several levels above each other (for example, the Pont du Gard aqueduct shown in Figure 2-2). Each arch was supported by an intermediate pier. After the fall of the Roman Empire (476 A.D.), most of the engineering knowledge was forgotten, and during the medieval times, arch bridges were rarely built; when they were built, the structures were not as stable and durable compared to Roman structures (Mock, 1949). In 1676, Robert Hooke provided the first experimental definition of a “true” (antifunicular) arch: an arch that has a line of thrust contained on the centre line of the actual arch, thereby allowing the arch to resist applied loads only in the axial direction (Heyman, 1998). In his experiment, Hooke explained that ideal shape of a true arch takes the form of a chain hanging freely from its ends: “as hangs the flexible line, so but inverted will stand the rigid arch.” Although Hooke’s solution was very close to the modern definition of a true antifunicular arch, the suggested shape could not be described mathematically at the time. Nevertheless, Hooke proposed a cubic function (y = |ax3|) to describe the shape. The equation for the curve of a hanging flexible line or catenary was derived several years later by David Gregory, who expanded Hooke's assertion (Ruddock, 1979). Today’s mathematical description of a true antifunicular arch, which supports its own weight, is a catenary expressed by a hyperbolic cosine function (y = a*cosh (x/a)). The catenary can be sufficiently approximated using the much simpler quadratic parabola (Sheriff, 1991; Todisco, 2014).

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Chapter Two: Literature Review

Using physical models to achieve optimal geometry was popular among architects who designed funicular structures. Similar to Hooke’s approach, Antonio Gaudi (Spanish architect, 1852-1926) created physical models using weights hanging on ropes to determine the optimal structural shape for the Sagrada Famila, a cathedral in the late Spanish gothic style. The ropes with attached weights represented an upside-down model of the structure. In addition, Heinz Isler (Swiss structural engineer, 1926-2009) created physical models in the design of spatial shells (Larsen, 2003). One of the disadvantages of inverted physical models is their inability to account for varying thicknesses of the structural components or any load that is not vertical (Todisco, 2014). Due to material limitations, masonry and stone arches were able to resist applied loads only in the plane in which they were constructed. Significant changes came with the development of reinforced concrete and structural steel in the late 1800s. The works of Robert Maillart, who is considered to be the pioneer in the field of arch bridges carrying out-of-plane loads (the first SABs), are known for their elegance and efficiency of material use (Billington, 1997). Robert Maillart was a Swiss engineer who used reinforced concrete in his structures. His arch bridges were called “deck-stiffened arches” and typically comprised slab arches (an arch rib has the form of a curved two-way slab) supporting the deck above. In his bridges, the depth of the arch rib would vary as a function of the internal bending moment in the arch (larger arch sections would be located in zones of high bending moments). Maillart’s designs emphasized aesthetics. The primary characteristic of his bridges was that, not only the arch, but all components (i.e. arch, deck, and spandrel walls) contributed to the structural response of the bridge. An example of a bridge by Maillart that carries out-of-plane loads is Schwandbach Bridge, shown in Figure 2-3.

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Chapter Two: Literature Review

Figure 2-3: Schwandbach Bridge, Switzerland 1933 (Chriusha, 2011)

Schwandbach Bridge was constructed in 1933; it is 6m wide and its span is 37m. The curved deck is supported by the arch, which resists both vertical and lateral loads transferred from the deck via inclined spandrel walls. The edge of the arch adjacent to the outer deck edge is straight. The edge of the arch adjacent to the inner deck edge is not straight. In fact, the shape of the arch in plan-view is designed to be curved to resist the lateral loads transferred from the deck. This bridge represents the first arch carrying out-of- plane loads. The concept of spatial bridges continued to develop in skewed masonry arches, which were arches whose abutments on either side differ by a certain angle, and therefore, out-of-plane loads occurred. In such bridges, the line of thrust was still contained within the arch cross-section (Sarmiento-Comesias et al., 2013). Spatial bridges that are formed with a structural shell (a “flat” space arch that contained two curvatures) have a different structural behaviour and differ from spatial bridges supported by an arch rib. However, the analyses and concepts of shell spatial bridges contributed to understanding bridges that contain arch ribs. The first shell arch bridge, which was designed by Sergio Musmeci, was constructed in Potenza, Italy in 1969 (Ingold & Rinke, 2015). True spatial arch bridges, which comprised a deck and an arch rib, rapidly developed along with use of structural steel and post-tensioned concrete systems. The application of such systems strongly relied on the development and use of computer aided

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Chapter Two: Literature Review systems such as CAD/CAM (Computer Aided Design/Computer Aided Manufacturing), which allowed for more accurate analysis and production of SABs. The popularity of these bridges increased in the late 1980s due to the demand for city landmarks and the celebration of innovation. Typically, bridges of this decade comprised a straight deck and an inclined planar arch. Santiago Calatrava was recognized as the architect who designed most of the modern SABs (Sarmiento-Comesias et al., 2013). He built his first SAB, the Felipe II or Bac de Roda bridge, in 1987 in Barcelona, Spain. This bridge had an arch symmetrical in plan. Calatrava’s first bridge with an arch that was not symmetrical in plan-view was the Gentil footbridge also constructed in 1987. The typical character of Calatrava’s designs is an inclined steel arch supporting either a straight or curved deck in plan-view. Examples of Calatrava’s work are shown in Figure 2-4.

a) b)

Figure 2-4: Examples of work of Santiago Calatrava: a) Margaret Hunt Hill Bridge, Dallas USA (Karchmer, 2012) and b) La Devesa Bridge, Ripoll, Spain (Plasencia, 2012)

The new concept of a diagonal arch was introduced in the early 1990s. A bridge with a diagonal arch is an arch rotated in plan (an arch starting at one corner and ending at the opposite corner) above a straight deck. Examples of this concept are the Hulme Arch Bridge (single span) in Manchester, UK and the Juscelino Kubitschek Bridge (three-span) in Brasilia, Brazil (Sarmiento-Comesias et al., 2013).

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Chapter Two: Literature Review

Between the late 20th century and the beginning of the 21st century, many cities wanted to construct outstanding structures that represented this time period. Several spatial bridges adopted the name “millennium-bridge.” These types of bridges truly represented innovation in design and technology. One unique example is the Gateshead Millennium Bridge, which is the only movable bascule spatial arch bridge in the world. This bridge, shown in Figure 2-5, is a pedestrian/bicycle bridge that comprises an inclined arch and curved deck suspended on flexible hangers (White & Fortune, 2012).

Figure 2-5: The Gateshead Millennium Bridge, Newcastle, UK, opened in 2001 (Perez, 2014)

Recent progress and developments in technology (particularly in design and structural analysis software) allows for learned concepts to be implemented in real structures. Examples of more recent spatial bridges include the Te Rewa Rewa bridge in New Plymouth, New Zealand (2010); Ponte della Musica in Rome, Italy (2011); and Weinbergbrücke in Rathenow, Germany (2014).

2.4 Geometry of Spatial Arch Bridges

2.4.1 General

The geometry of SABs can be very complex depending on the materials selected and the purpose of the structure. In certain cases, the geometry of the arch and deck results from a functional requirement, and the desired structural behaviour must be achieved by

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Chapter Two: Literature Review selecting suitable materials and efficient structural details. An example of such a configuration is the Gateshead Millennium Bridge shown in Figure 2-5. It is a movable bascule SAB over the River Tyne with a parabolic arch and deck. In this structure, the arch and deck geometry was designed to allow for ships to pass underneath the bridge when the bridge is “opened” (Johnson & Curran, 2003). This example demonstrates the level of complexity of the geometry of SABs, and how the shape and proportion of the selected geometry correlates with the practicality of the structure. Despite the large number of possible geometric configurations, it is possible to categorize SABs. The following sections describe the classification, geometric ratios, and approaches used to achieve optimal structural shapes of SABs.

2.4.2 Classification of Spatial Arch Bridges

There are many categories of SABs. The geometric complexity of these structures is due to a variety of factors such as the configuration and the properties of the hangers (flexible vs. stiff), the number of arch ribs and their position relative to the deck, and the symmetry of the individual components.

2.4.2.1 Spatial Arch Ribs and Spatial Shells

Generally, SABs can be divided into two categories: (1) a deck supported with arch rib(s) and (2) a deck supported with shell(s). Shell systems are less common; only a few structures of this type have been built, for example the Matadero footbridge in Madrid, Spain and the Bridge of Peace in Tbilisi, Georgia, shown in Figure 2-6. Even though spatial shell systems have completely different configurations and structural responses than systems that contain ach ribs, they are still considered to be spatial configurations (Sarmiento-Comesias et al., 2013). The behaviour of spatial shells was studied by Strasky & Kalab (2007).

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Chapter Two: Literature Review

Figure 2-6: Examples of shell spatial bridges: a) Matadero footbridge in Madrid, Spain (Tamorlan, 2011); and b) Bridge of piece in Tbilisi, Georgia (Kavtaradze, 2009)

2.4.2.2 Symmetry in Plan-view

In the first category of SABs, bridges with arch ribs, two sub-categories can be defined based on their shape in plan-view. Bridges that are longitudinally not symmetrical in plan-view are called true SABs, and bridges whose longitudinal shape is symmetrical in plan-view are called non-true SABs. Both categories experience out-of-plane loads (Sarmiento-Comesias et al., 2013). Examples of a true and a non-true SAB are shown in Figure 2-7 and Figure 2-8, respectively. Spatial arches curved both in plan-view and in elevation are classified as “warped” arches. In such configurations, the curvature of the

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Chapter Two: Literature Review arch in plan-view typically follows the curvature of the deck, which can be either below or above the arch (Manterola et al., 2011).

Figure 2-7: Example of a true SAB: Campo de Volantin footbridge, Bilbao, Spain (Garcia, 2014)

Figure 2-8: Example of a non-true SAB: Bac de Roda bridge, Barcelona, Spain (Roletschek, 2015)

2.4.2.3 Mutual Position of Arch and Deck

Further classification considers the mutual position of the arch and the deck. Bridges that have the deck supported via stiff hangers (columns or walls) to the arch rib

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Chapter Two: Literature Review below are called bridges with a superior deck. Bridges with a deck suspended below the arch either on rigid or flexible hangers are called bridges with an “inferior” deck. Examples of spatial bridges with superior and inferior decks are shown in Figure 2-9.

a) b)

Figure 2-9: a) a ridge with a superior deck: Endarlatsa bridge, Guipúzcoa, Spain (Urruzmendi, 2014) and b) a bridge with an inferior deck: Mayfly Bridge, Szolnok, Hungary (Akela, 2012)

2.4.2.4 Arch Bridges with Imposed Curvature

Depending on the definition of the space where the arch rib is constructed, planar and non-planar arches are recognized. Both categories experience out-of-plane loads. A specific case of a non-planar arch is an arch bridge with an imposed curvature (ABWIC). In such bridges, the centrelines of both of the arch and the deck are forced to be contained on the surface of the same vertical cylinder. The behaviour of ABWIC was studied by Sarmiento-Comesias et al. (2012).

2.4.2.5 Multiple, Diagonally Rotated, and Elevated Bridges

The general definition of the shape and the supporting system of a bridge can be further varied by inclining or twisting the arch rib(s), changing the number of arch ribs or decks, or changing the relative position of the arch rib(s) and deck(s), which can be either in the same elevation or at different levels.

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Chapter Two: Literature Review

An example of a diagonal arch structure is the Hulme Arch Bridge in Manchester, UK shown in Figure 2-10 (Hussain & Wilson, 1999). In this type of configuration, the arch rib starts at one side of the deck and ends at the diagonally opposite corner. In addition, the plan eccentricity of the deck(s) and arch rib(s), the slope of the members that support the deck and the rigidity of the members, and/or the rotation of the arch rib(s) in relation to the alignment of the deck can also vary. A detailed geometrical classification of SABs was done by Sarmiento-Comesias et al. (2013).

Figure 2-10: Example of a diagonal arch bridge: Hulme Arch, Manchester, UK (Peel, 2004)

2.4.3 Geometrical Ratios

In vertical planar arches, geometrical ratios are derived to achieve optimal force distributions in the arch and deck as a function of the span and the geological conditions at springings of the arch, which determine the allowable magnitude of lateral thrust resulting from the arch. Typically, in vertical planar arches, the main geometrical ratio is the arch rise-to-span ratio. The value of this ratio can vary from 0.10 to 0.50 depending on the factors mentioned above. However, typically the rise-to-span ratio ranges from 0.17 to 0.22

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Chapter Two: Literature Review

(Chen et al., 2000), and from 0.12 to 0.30 (Nettleton, 1977), for steel and concrete bridges, respectively. Even though the magnitude of the axial thrust in the arch is inversely proportional to the rise of the arch (an arch with a high rise results in a smaller axial load), higher arch rises require more material, experience more stresses from wind load, and have a higher buckling length (Nettleton, 1977). Therefore, selecting a proper geometric ratio can influence the amount of required material (cost) and the structural response of the system. In spatial arches, the range of rise-to-span ratio, identified for vertical arches, holds as well; however, other ratios must be introduced to account for lateral and out-of-plane loads. Sarmiento-Comesias et al. (2013) suggests considering the following geometrical ratios for SABs: a) the ratio between the distance from the deck shear centre to the axis joining the deck abutments and the span of the deck; b) the ratio between the arch depth and width; this coefficient is required because of the relevance of the out-of-plane behaviour of the arch; c) the deck and arch rise to span ratios: HA/LA and HD/LD where

“HA” and “HD“ are the depths of the arch and deck, respectively, and “LA” and “LD” are the spans of the arch and deck, respectively; and d) an appropriate relationship between fA (the vertical arch rise), and e (the plan eccentricity between the axis of the arch at the springing points and the axis of the deck at the abutments). In addition, “gA“ and “gD“ (the horizontal reaches of the arch and the deck, respectively) might be relevant for defining the spatial shape of the arch thrust line (i.e. the antifunicular shape of the spatial arch). Sarmiento- Comesias et al. (2013) also indicates mechanical ratios of individual structural components, placing emphasis on all capacities that can be utilized while being loaded in the structure, such as the bending stiffness of the arch and deck (EIA/EID) and the bending and torsional stiffness of the arch and deck (EI/GJ). The importance and complexity of these ratios increases because the cross-section of both the arch and the deck can vary along the centre axis of the component to resist the bending and torsional moments at specific locations within a structural component. The magnitudes of these ratios are highly dependent on the properties of the individual structural components and their significance is yet to be studied. General geometrical ratios for SABs are not available because each SAB is designed to satisfy particular requirements. Thus, it is difficult to obtain optimal ratios that

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Chapter Two: Literature Review apply to all SAB structures. Therefore, applicable geometrical ratios that clarify the significance of certain geometries represent a gap in the state of knowledge in this field.

2.4.4 Form Finding Approaches

The form of the individual components of SABs can vary based on aesthetics and practicality. In some cases, aesthetics govern the design, and hence, the structural and material composition must compensate for the required form. The solution to such an approach is called a “form finding” approach.

2.4.4.1 Antifunicular Arches

Spatial arches were developed from vertical planar arches, which are antifunicular systems. In antifunicular systems, the self-weight is resisted only by compression in the arch (see Section 2.3). The shape of the antifunicular arch is a function of the applied loads that determine the location of the line of thrust (a conceptual line representing the path of the compression force only). To resist the applied loads through uniform compression, the geometric centreline of the arch must match the location of the line of thrust (Shrive et al., 2000). The same principle holds for spatial arches in three-dimensional (3D) space. However, in spatial arches, the approach for seeking antifunicular shapes is more complex because the response of the arch cannot be simply divided into two components as it can be in planar arches (Sarmiento-Comesias et al., 2012). Due to the requirements of aesthetics or location restrictions, SABs oftentimes have curved decks, which do not have a symmetrical shape in plan-view. Therefore, the shape of the true antifunicular arch rib can be very complex. Further, a curved deck may be implemented not only due to the desire for an architecturally pleasing shape or an obstacle but also due to its practicality. In bridges with curved decks, the approach spans continuously connect the bridge superstructure crossing the obstacle (river, road, rail tracks) with pathways (bicycle lanes) having an original direction along the obstacle. Implementing continuous horizontally curved decks avoids using zig-zag shaped staircases or ramps.

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Chapter Two: Literature Review

An example of a true antifunicular SAB, non-symmetrical in plan view with a superior deck is Ripshorst Bridge shown in Figure 2-11. This structure was designed by Jorg Schlaich and constructed in Oberhausen, Germany in 1997.

Figure 2-11: Example of a true antifunicular SAB with a superior deck: Ripshorst Pedestrian Bridge, Oberhausen, Germany (SBP, 1997)

The form finding process can be divided into conceptual and detailed design. The conceptual design of spatial bridges was studied by several authors in recent years, for example Schlaich & Gabriel (1996), Allen (2009), Muttoni (2011), and Corres (2013).

2.4.4.2 Interactive Parametric Tools

A detailed design is closely related to the development of 3D parametric tools, which convert a graphical interface into a mathematical form (vector algebra) and provide a base for linear and nonlinear FEA. Examples of such tools are as follows: Rhinoceros, Grasshopper, Karamba (Todisco, 2014), and SOFiSTiK (Lachauer & Philippe, 2014). Each of these software options provides a different level of input which, when combined, allow for an interactive real-time search of the 3D antifunicular shape of SABs. Rhinoceros is a software for converting the designed geometry into a mathematical form. It is a 3D modeling software based on non-uniform rational B-splines (NURBS). NURBS models are very accurate and, therefore, suitable for any geometry ranging from animation to industrial forms. In the last decade, this software has become popular in architectural applications

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Chapter Two: Literature Review due its compatibility with Grasshopper, software that creates parametric geometries (Lachauer & Kotnik, 2011). Grasshopper is a parametric tool that is well accepted for its ease of use combined with intuitive exploring (interaction) while searching for optimized geometries. Created geometries are stored in the form of a script, which is further processed with FEA software, for example, Karamba or SOFiSTiK. FEA software is compatible as a plug-in tool with Grasshopper, and it converts the parametric script into an FE model where structural elements (mesh) and material properties are assigned. A subsequent linear or nonlinear FEA is conducted to verify whether or not the proposed geometry leads to a convergent solution. This procedure is iterative where a parametric algorithm is typically developed to obtain an efficient solution. The compositions of such algorithms were studied by several authors, for example Todisco (2014), who developed an iterative procedure coded into a program called “SOFIA” (shaping optimal forms with an iterative approach). Although this program could not account for material nonlinearities, geometrical nonlinearities in the form of large displacements were included and could provide accurate solutions. Lachauer & Philippe (2014) proposed algorithms combining graphic statics with force density methods. The force density method was developed in the late 1970s by Linkwitz & Schek (1971). This form finding technique was an equilibrium-based method that replaced member stiffness and lengths of branches with force-over-length ratios to find a solution (via a system of linear equations) to unknown coordinates of “free” nodes in the geometry that was being sought. The tool proposed by Lachauer & Philippe (2014) appeared comprehensive for intuitive exploration of initial shapes; however, it could be limited by the computational capacity of the computers used to perform the required iterations because the number of steps required in more complex structures was large. Such limitations could influence real- time responses while the software evaluated the assumed geometries using the iterative procedure. Algorithms were programed in a sophisticated scripting language such as IronPython. The approach proposed by Lachauer & Philippe (2014), was suitable mainly for structures where self-weight was much larger compared to the magnitude of the live load (LL). In cases where LL governs the design, the proposed approach could still be used for the initial design; however, shape reiteration under all combinations of LL (according to appropriate code provisions) needed to be performed to ensure the safety of the design.

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Chapter Two: Literature Review

2.4.4.3 Form Finding Methods

The parametric interactive tools described above are usually based on graphical (graphic statics) or numerical (force density or dynamic relaxation) methods. These methods are often combined to achieve the optimal shape of the spatial structure. For a long time, graphic statics was used as the main tool in the design and analysis of spatial projects. This method was developed in the mid-19th century by Culmann (1866). The combination of the reciprocal force and form polygon allows analysis of funicular and antifunicular structures: structures experiencing pure tension or compression, respectively. This method allows development of the form of the structure based on known loadings and imposed maximum loadings, or it determines loads from known geometry. Graphic statics can be described as a combination of vector algebra and descriptive geometry in plane (Lachauer & Kotnik, 2011). Other notable engineers who used this method were Ritter (1847-1906), Maillart (1871-1940), and Gustave Eifel (1832-1923). Graphic statics is a very powerful tool; however, it has limitations, and certain assumptions must be accepted while using it. Graphic statics ignores the material stiffness of members, and its deflection solution is derived from the equilibrium of forces. In addition, all structures are assumed to be statically determinate or kinematic pin-jointed systems in plane only. Different approaches extending the basic concept of graphic statics were introduced to overcome these limitations. Lachauer & Kotnik (2011) used graphic statics with the method of dynamic relaxation for spatial bridge design. Introducing planes as constraints for the free nodes used to determine spatial funicular polygons for the design of curved bridges was done by Laffranchi (1999). Laffranchi suggested that the spatial polygon in space could be balanced by two funiculars. Based on the concept of the funicular polygon, Block & Ochsendorf (2007) developed the concept of funicular surface networks with fixed projection. Rippmann (2016) proposed using Thrust Network Analysis (TNA) for form finding and fabricating the geometry of funicular shell structures (based on non-reinforced masonry shells). In this method, geometrical, rather than numerical or analytical interpretations were used. The TNA method also combined the principles of graphic statics with the force density method. The form finding iterative method for SABs using the principle of zero moments (as used for vertical planar arches) was proposed by Jorquera (2009). In his method for

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SABs with fixed ends, at arch springings fictitious jacks were used during each iteration to enhance the zero bending and transversal moments at these locations. The rise of the arch (the vertical coordinate of the crown) was kept constant to maintain a zero bending moment at the crown. In this configuration, the spatial arch appeared to be a three-hinged vertical planar arch in elevation-view and a two-hinged horizontal planar arch in plan-view. The transversal coordinate of the node at the crown and the vertical and transversal coordinates of all other nodes (except the nodes at the springing points) could be corrected to find equilibrium in the form of zero transversal and vertical bending moments, respectively, at any point. As there was no external load acting on the arch, torsional moments dropped to zero once the antifunicular shape of the arch (having no bending moments) was found. Even though the form finding approaches can be complex, improving software capabilities, and most recently, understanding spatial antifunicular shapes make the procedure more feasible and reliable.

2.5 Structural Behaviour of Spatial Arch Bridges

2.5.1 General

The structural behaviour, i.e., the response of the structure (deflections or change in the distribution of internal forces) to applied loads in SABs is not only a function of the geometry and structural stiffness of the individual members (arch, deck, and hangers), but also the stiffness of the connections between the hangers and the deck and the arch (Sarmiento-Comesias et al., 2012). The complex structural response of SABs, a result of the distinctive geometry, is a challenge during the analysis and design of these structures. Factors, such as planar and/or out-of-plane buckling, must address out-of-plane loads. The distribution of thermal stresses may be influenced by the varying hanger length as a result of the curved arch and curved deck. Both linear and non-linear analyses are typically required to analyse SABs. Only a few comprehensive studies have been carried out to clarify the structural behaviour of SABs. The following sections present the most current state of understanding of the structural behaviour of SABs. Typical principles for the structural response of SABs are described and gaps in the research are identified.

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Chapter Two: Literature Review

2.5.2 Structural Response of Different Configuration of SABs

The structural behaviour of SABs can be quite complex and different from traditional vertical planar arches. SABs experience out-of-plane loads that introduce internal axial forces and bending and torsional moments in the arch. These internal forces cause the arch to behave like a transversally loaded beam rather than an axial loaded arch rib in some geometric configurations, such as an arch rotated about a vertical axis (Husain & Wilson, 1999; Johnson & Curran, 2003), or like a cantilever beam in other configurations, such as an arch inclined from a vertical plane (Sarmiento-Comesias et al., 2013). The deck typically experiences shear forces and bending about its longitudinal axis in the vertical direction which, in the case of SABs, is extended to a deck that also resists bending about the longitudinal axis in a transversal direction accompanied with a torsional moment (Sarmiento-Comesias, 2015). In certain configurations, torsional and flexural stiffness can be un-coupled to analyse the contribution of each component separately (Arena & Lacarbonara, 2009). As a rule-of-thumb, a structural response of a SAB employs all the mechanical contributions of a selected geometry. In a selected geometry, different parameters, such as bending and torsional stiffness of the main structural components and the ratios of the stiffness in vertical and horizontal direction, play a key role (Sarmiento- Comesias et al., 2013). The significance of these parameters can be analysed considering either the linear or nonlinear character of the models.

2.5.2.1 Linear Analysis

Jorquera (2007) studied linear behaviour, and he examined the effect of different inclinations of the planar arch ribs, deck curvatures, and eccentricities on the structural response of SABs with decks suspended from planar arches (inferior decks). He also investigated the problem of straight and curved decks and their relations to hanger eccentricity. Hangers were assumed to have pinned connections, and therefore, they were able to carry only axial loads. The significance of the location (eccentricity) of the hangers in relation to the shear centre of the deck was also examined. As a result of his investigations, Jorquera (2009) developed an iterative algorithm for finding an antifunicular shape of the arch in SABs with inferior decks.

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Chapter Two: Literature Review

2.5.2.2 Nonlinear Analysis

Liaghat et al. (2011) presented work done on a pedestrian bridge called “Ponte della Musica” constructed in Rome in 2011. The bridge was architecturally designed by Buro Happold Consulting Engineers in association with Powell–Williams Architects. The structural analyses and designs were carried out by M. P. Petrangeli and Associates. The structural system comprised two inclined planar arches (by 15° outwards from the deck) connected to the deck by rigid hangers. As a consequence of the inclined arches, the torsional stiffness of the bridge was lower than the torsional stiffness in more usual arch bridge designs featuring vertical arches (Lacarbonara, 2013). Liaghat, the chief engineer of the project, indicated aspects of nonlinear analysis that considered in-plane and out-of- plane buckling that occurred on the bridge. The project took 11 years to finish and incorporated many new advances in engineering and science. Arena & Lacarbonara (2009) conducted the aeroelastic instability analysis of the Ponte della Musica. In this work, Arena & Lacarbonara (2009) proposed a 3D parametric model to examine nonlinear responses to static and dynamic loads. Arch geometry represented a parameter of the model. Modal shapes of both the deck and the arch were achieved including out-of-plane bending, shear, and torsion. Strains within the model were functions of displacement gradients. The proposed model considered nonlinear extensional- flexural-torsional coupling to evaluate the aeroelastic response to wind loads. The calculations of the flutter speed and the critical flutter modal shape were conducted together. Arena & Lacarbonara (2009) used a direct total Lagrangian formulation to derive the equation of motion by adopting linearly hyperelastic constitutive equations for all structural members.

2.5.2.3 Arch Bridges with Imposed Curvature

Sarmiento-Comesias et al. (2012) conducted a study to develop a simple analytical model and linear numerical frame-element models that propose design recommendations for arch bridges with imposed curvature (ABWIC). In these configurations, the arch rib and the deck are contained in the same vertical cylinder, and the springings of the arch and the deck abutments are located on the same line. The arch is inclined from a vertical plane, but it is not contained in the plane. The main difference between an arch contained on the

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Chapter Two: Literature Review surface of a vertical cylinder and a conic section cutting through a vertical cylinder is that the arch centre line follows the projected shape of the deck in plan-view. The deck is curved in plan-view, and its shape corresponds to a segment of the cylinder surface (semicircular shape). Sarmiento-Comesias et al. (2012) proved that simplified models that assume a planar arch provide results within acceptable tolerances, and therefore, a planar inclined arch configuration can be used with confidence. A parametric study clarified the significance of the stiffness of the hangers. Parameters, such as bending in two directions or torsional stiffness of the arch, deck, and hangers, were examined. A comparison of the results of the linear and nonlinear analyses showed that a geometrical nonlinear analysis was not necessary if a sufficiently high bending and torsional stiffness of the hangers was selected. The conclusions of this study showed that if the connection of the hangers to the deck was sufficiently rigid and the arch was relatively flexible, the arch rib in an ABWIC can be considered antifunicular. In a similar style, the description of the structural behaviour of bridges with a superior deck was done by Sarmiento-Comesias et al. (2011).

2.5.2.4 Laterally Loaded Spatial Arches

Verlain et al. (2001) presented a paper on the geometric nonlinearity of the Observatory Bridge (“Pont de l'Observatoire”) constructed close to Liege, Belgium in 2002; Santiago Calatrava designed the bridge. The Observatory Bridge is a bowstring bridge with a curved inferior deck. Flexible hangers connect the vertical planar arch to both sides of the deck. The Hulme Arch Bridge in Manchester, UK (a diagonal arch bridge) also experiences significant bending and torsional moments in the arch and deck. Hussain & Wilson (1999) conducted a critical analysis of this bridge, and he found that due to the asymmetric cable arrangement, large out-of-plane bending and torsional moments were generated. In fact, the moments represented the governing load for which the arch must be designed. The abutments of the arch had to be enlarged compared to a conventional planar arch spanning the same distance to resist the moments and internal turning forces. The arch rib, made of hollow structural steel, was filled with concrete at the crown to resist the present torsional moment. Moreover, due to the magnitude of the bending and torsional moments, the arch rib behaved more like a laterally loaded flexural beam than an actual

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Chapter Two: Literature Review arch. A description of this behaviour was provided by other authors and the effect of certain factors was addressed. An example of a multiple diagonal arch bridge is the Juscelino Kubitschek Bridge. This bridge is comprised of three diagonal arches with internal abutments that meet to counteract the horizontal thrust of the adjacent arch ribs. The structural behaviour for a diagonal arch bridge also applies to a multiple diagonal arch bridge because the hangers (flexible cables) are attached on both sides of the deck, which results in large bending and torsional moments in both the deck and the arch rib (Bailey, 2007). Studies carried out by Laffranchi (1997), Jorquera (2007), and Manterola et al. (2011) proved that if horizontal stiffness of the system in a SAB is utilized, the structural behaviour of the arch improves; in other words, the bending and torsional moments in the arch are reduced. Lateral bending and torsional moment analysis was conducted for Gateshead Millennium Bridge by Johnson & Curran (2003). Staged construction and dynamic nonlinear analyses were performed to evaluate the structural response of the bridge. Zhou et al. (2012) studied an optimization method that focused on the force in the flexible hangers of diagonal arches. The optimization method considered both the constrained and unconstrained conditions, and therefore, it was easy to use in the design process. Zhou et al. employed the optimization method in a case study of the Tong-Tai Bridge (190m span) in Zhangjiakou City. Another example of a diagonal spatial arch is the Te-Rewa-Rewa Bridge constructed close to New Plymouth, New Zealand in 2010. This diagonal arch is unique as the hangers are stiff ribs attached to one side of the deck rather than the more typical flexible cables attached to both sides of the deck. Applying stiff hangers utilizes their torsional and bending stiffness to transfer vertical loads from the deck to the arch rib. The resulting torsion is resisted by both the stiff hangers and the deck (Mulqueen, 2011). Some structural details may influence the magnitude of bending moments in the inclined arch. Applied on the Weinbergbrücke (Weinberg-Bridge, a footbridge designed by Schlaich Bergerman Partner for Rathenow Park in Germany), the connections of the flexible hangers to the curved deck are not located on the outer edge girder as in typical configurations, but rather they connect to short cantilevered radial cross-beams extending

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Chapter Two: Literature Review from the deck and pointing towards the centre of the deck’s horizontal curvature. This configuration results in reduced bending moments in the arch rib. Even though the deck cross-beams are exposed to bending moments, whose magnitude depends on the length of the cross-beams, the selected structural detail represents an efficient solution because it is easier to control moment distribution with relatively short radial cross-beams on the deck rather than with the long rib of an inclined arch (SBP, 2014).

2.5.2.5 SABs with Superior Deck and Dynamic analysis

In Sarmiento-Comesias’ doctoral thesis, she describes the structural behaviour of certain groups of SABs and proposes design criteria for bridges made of steel to control out-of-plane responses. This work focuses on arch bridges with imposed curvature (ABWIC) and spatial bridges with superior decks. She examined bridges that were symmetrical and non-symmetrical in plan-view and bridges with flexible and rigid hangers. Sarmiento-Comesias (2015) studied the significance of different deck boundary conditions and the effect of thermal loads for selected configurations. She clarified certain geometric ratios were for ABWIC and dynamic analysis for one configuration of a SAB with a superior deck to establish criteria for design. Commercial finite element modeling (FEM) software was used to carry out proposed analyses. Sarmiento-Comesias (2015) showed that structural behaviour of SABs exposed to out-of-plane loads was not always clear as the relationships between bending and torsional stiffness and connection types and the mutual position of the structural components play significant roles; planar and non-planar inclined arches may experience large bending or torsional moments depending on these relationships. Even though the work of Sarmiento-Comesias (2015) covers a broad field of structural behaviour of SABs accompanied with practical design examples, not all geometric configurations nor the significance of material contributions to the structural responses were included.

2.5.3 Elastic Instability of Arches

Elastic instability of the compression members, commonly called “buckling,” is an issue that must be carefully approached in any structural system exposed to compression

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Chapter Two: Literature Review because this phenomenon typically occurs at the elastic stress level (a level below yielding in steel) and may lead to the collapse of the entire structure (Ghali et al., 2009). Buckling can be defined as an abrupt change of the initial form of equilibrium. Equations for the critical buckling load for simple straight members (columns, compression chords in truss structures, etc.) can be found in structural engineering text books. Elastic instability, or the loss of stability, is caused by the critical load. This critical load can be defined as either the maximum load that the structure can withstand without collapsing or the minimum load that leads to the collapse of the structure. The critical load is related to the state of “stable equilibrium” in the case of deformable systems. In rigid body systems, this equilibrium state is called the “stable position of the structure” (Karnowsky, 2012). Buckling modes (deformed shapes of the components) are typically a function of boundary conditions such as free, pinned, and fixed ends (Nettleton, 1977). Buckling analysis can be done using two methods: linear and nonlinear. Eigenvalue buckling analysis uses a linear method, and this analysis predicts the theoretical buckling strength of an ideal elastic structure. It computes the structural eigenvalues for the given system loading and constraints. This method is also known as classical Euler buckling analysis. More accurate nonlinear buckling analysis uses large deflection, static analysis to predict buckling loads. In principle, at first an arbitrary load is assumed, and via iterations, the threshold value of the critical load is determined. The threshold value can be defined as a magnitude which, if increased by a very small load, results in a very large displacement (Ghali et al., 2009). The use of nonlinear analysis allows for geometric imperfections, load perturbations, material nonlinearities, and gaps to be modeled.

2.5.3.1 Development of Elastic Instability Theory

Kirchhoff (1876) was the first person to introduce the mathematical basis for the stability theory of planar arches. Later, Bryan (1888) conducted the first systematic study of equilibrium stability. In the beginning of the 20th century, the stability of circular arches was investigated by Nikolaee (1918). Later, Federhofer (1934) and Timoshenko & Gere (1961) significantly contributed to the theory of instability of planar arches. The modern approach to elastic instability of planar arches typically considers the Eulerian sense of buckling; Euler buckling is derived from the elastic equation of the arch where the key

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Chapter Two: Literature Review constituents are the second moment of inertia and the boundary conditions of the loaded member.

2.5.3.2 Elastic Instability in Planar Arches

A precise analytical solution to critical loads in planar arches can be found only for the simplest cases, such as a circular arch under uniform pressure normal to the axis of the arch. An analytical approach to the stability of arbitrarily shaped arches includes integrating differential equations of the arch and results in a master equation of stability. Critical loads are then found as roots of these equations. Arbitrarily shaped arches with non-uniform cross-sections are difficult to analytically analyse because the order of corresponding differential equation becomes too high. A comprehensive analytical approach to planar circular and parabolic arches that considers out-of-plane buckling, different arch boundary conditions, and different arch cross-sections is provided by Karnowsky (2012). Today, numerical methods are used to analyse the susceptibility of arches to elastic instability. In planar arches, two kinds of buckling can occur: in-plane and lateral out-of-plane buckling (Nettleton, 1977). In-plane buckling is more common because generally arch bridges have two ribs braced against each other. In the case of the single arch rib, stability is dependent on the boundary conditions (the stiffness of the connection at the arch springings) and cross-sectional properties of the arch rib. Buckling length is a function of the arch boundary conditions. In a two-hinged arch, this length equals half of the arch rib length. In tied-arch bridges, local buckling between hangers is more likely to occur than global buckling of the entire arch rib (Hedefine, 1970). Global buckling in arch bridges was described, for example, by Johnston (1976) and Halpern & Adriaenssens (2014). Tied arches can be designed with either the rib taking all the moments from dead and live loads or, if the arch rib is shallow, the moments from live loads are compensated for by a deep tie (Nettleton, 1977). A magnification factor is typically added to a moment from a governing dead load to account for the additional moment in the arch from a live load. An additional live load deflection is produced by an increase in the live load moment, and this increase in deflection generates an additional increase in the moment. This effect continues in a decreasing trend (Nettleton, 1977). Typically, the moment magnification factor depends on

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Chapter Two: Literature Review the arch rib rise-to-span ratio. A moment magnification factor should not be used in tied- arch bridges because the arch rib and the tie deflect proportionally. The boundary conditions of the arch and the level of determinacy influence the necessity of cross- sectional dimensions and the resulting deflections. An arch rib fixed at both ends requires approximately 0.80 of arch rib depth in a two-hinge arch to achieve the same bending moment from live loads. Deflection of a fixed arch rib is typically 0.75 of the deflection in a two-hinge arch (Nettleton, 1977).

2.5.3.3 Elastic Instability in SABs

In SABs, elastic instability increases due to the out-of-plane loading present in the structure from the start. In general, the stiffness of the arch rib that is exposed to axial compression and bending and torsional moments must be increased. This general rule applies most of all to configurations of SABs where the arch rib diagonally crosses the deck (see Section 2.4.2.5), which results in high lateral moments. The Hulme Arch Bridge in Manchester, UK is an example of this type of bridge (Hussain & Wilson, 1999). However, according to Sarmiento-Comesias et al. (2012), in configurations of ABWIC (see Section 2.4.2.4) that have high flexural and torsional stiffness of the deck and hangers, respectively, and low stiffness of the arch, the arch rib experiences low axial compression, and therefore, the susceptibility to buckling is reduced. Based on the numerical modeling verified by experimental measurements, the angle of arch rib inclination can influence the critical buckling load. Gui et al. (2016) investigated the angle of significance of the arch inclination in the configuration of continuous composite bridges (CCBs). In inclined arch configurations, the slenderness and buckling ratios need to include interactions between the in-plane and out-of-plane buckling. Outward inclined arches are more susceptible to the interaction of in-plane and out-of-plane buckling and to the interaction between local and global buckling. Due to compression loads, both in-plane and out-of-plane buckling can occur in arch ribs. Typically, in-plane buckling is related to a combination of bending moments and compression loads (Dou et al., 2015; Bradford & Pi, 2014). Out-of-plane buckling results from a combination of torsion, biaxial bending, and compression loads (Ziemian, 2010). The interactions are captured with the buckling coefficient method (JRA, 2012).

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Chapter Two: Literature Review

The linear eigenvalue method was used in a case study investigating the stability of a 180m long diagonal arch in China (Qiu, 2010). In this structure, a vertical planar arch on inclined flexible hangers supported a skewed girder deck, which resulted in out-of-plane loads in the arch rib. In this study, several load combinations were used to derive buckling coefficients, and critical loads were considered to investigate the main factors influencing the stability of the structural system. It was found using FE modeling that the main contribution to arch rib stability was provided by cable stays on both sides. Increasing deck stiffness could, up to a certain level, improve the stability of the arch (a stiffer deck experiences lower deflection resulting in a smaller load in the cable stays, which generates an axial load in the arch). It was confirmed that the boundary conditions of the arch significantly affect the stability (a fixed arch is more stable). In addition, the rise-to-span ratio had a relatively large impact on the stability coefficients; a smaller rise resulted in large axial stresses in the arch. The conducted analyses suggested a higher rise-to-span ratio (0.37) to achieve better stability in the arch. It was shown that the lateral bending stiffness of the arch rib had no significant influence on in-plane buckling. A case study on the Ponte della Musica bridge in Rome, Italy conducted by Lacarbonara (2013) also included elastic stability analysis. Numerical models, such as the linear buckling method, used to investigate the ultimate load multiplier in steel arch bridges could not be used for the Ponte della Musica. Selected analyses accounted for pre-critical behaviour and stiffness degradation induced by incremental compressive loads. This assessment was possible with the nonlinear buckling approach based on geometric and material nonlinearities (Komatsu & Sakimoto, 1977). The nonlinear buckling approach investigated the buckling limit of steel arches. The buckling limit was described using 3D analytical model based on elasto-plastic constitutive equations, small-strain finite kinematics, and specific properties of the arch cross-sections (Lacarbonara, 2013). The critical load (load multiplier at which the structure undergoes elastic instability at the divergence bifurcation) was found by applying the vertical load (representing a live load) with increasing magnitude. The conducted analysis showed that, under an increasing load, the inclined arches experienced a nonlinear softening characteristic resulting from a significant reduction in the stiffness of the arches under the developed compression load.

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Chapter Two: Literature Review

This degradation in stiffness occurred at relatively low values (5 and 6 for the considered equilibrium paths) of the load multiplier. Results from the study of the Ponte della Musica bridge concluded that axial compression in SABs affects the elastic instability of the entire structure and represents (due to the spatial alignment of the structure) a significant issue that must be treated accordingly: increasing sectional stiffness, revising boundary conditions, or applying lateral supports in the form of rigid hangers.

2.5.4 Effects of Thermal Loads

The effects of thermal loading on SABs have been studied on a small scale. For instance, Cheng et al. (2010) performed thermal analysis on the Nanning Bridge in China (two unequal inclined arches at different angles support a curved road deck suspended via cables). He considered temperature changes within a 15°C range, which reflected the local conditions. The small temperature fluctuations resulted in relatively small stress changes compared to the effects of dead or live loads. Therefore, even though this case study provided results of a thermal analysis, given the narrow temperature range, the results did not fully contribute to a better understanding of the behaviour of SABs under thermal loading. Sarmiento-Comesias et al. (2011) also conducted a study on SABs that considered thermal effects. Conclusions from her work, which focused on a variation of shapes of superior decks, clarified that deck curvature reduced the axial forces in the deck generated by thermal loads. Hence, the abutments, resisting lateral deformation, could be fully restrained. However, as the axial load was reduced, the bending and torsional moments increased. According to Sarmiento-Comesias et al. (2011), this response was opposite to the response in conventional vertical arch bridges. The structural response of SABs to thermal loading is significantly affected by the geometrical complexity of the system. The magnitude of elongation and contraction of the structural members due to applied thermal loads is a function of material properties, temperature gradient, and, most importantly, dimensions of the structure (Ghali et al., 2009). The force in a member is then dependent on the strain, modulus of elasticity, and the cross-sectional area of the member. In SABs where the arch supports a suspended deck

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Chapter Two: Literature Review using cables of different lengths, thermal loads may be an issue due to the nonlinear character of the cables. Cables of different lengths elongate and contract at different rates under assumed thermal loads and, therefore, have different sag, which results in unequal forces transferred to the arch and deck. The effects of thermal loading on cables stays, particularly in cable-stayed bridges, have been studied by several researchers, for example Sherif (1991) and Vairo & Montassar (2012). According to Sherif (1991), the results of three temperature combinations, representing a maximum temperature drop of – 40°C, for a flexible deck and stiff pylons, indicate that the tensile forces in the cables can reach up to 20% of the maximum highway live load. This magnitude is not concerning when compared to the magnitudes of dead and live loads. As the temperature applied to the cables does not generate extensive axial forces in the cables, the normal forces in the deck can be considered negligible as well. The results of the analysis show that the highest tensile force is generated in the cable closest to the middle pylon and the lowest force occurs in the cable farthest from the pylon. Further, the cable behaviour under thermal loading is also affected by the cable length, its own weight causing sag of the cable, and the location of the cable connection to the deck. Non- linear analysis is required to determine the sag of the cable and the different connection locations. Although thermal loading does not have a significant impact (relatively) on the behaviour of the cables, it has a greater effect on the behaviour of the deck and the main pylon. Due to the linear variation of temperature throughout the deck reflecting a scenario close to reality (assumed to be – 40°C on the bottom surface closer to the water and – 20°C on the top), the magnitude of the generated moments reach the magnitudes of moments from live loads (Sherif, 1991). The main pylon, particularly, the bottom of the shaft (hollow reinforced concrete tower), is significantly affected by thermal loading, and the magnitude of the moment also reaches levels as high as the moments generated by live loads. Despite the fact that cable-stayed bridges represent a different structural concept, the thermal load significantly affects the structure and needs to be taken into account. Other structural components in addition to the cables undergo longitudinal deformation due to temperature change. As stated in the critical analysis of the Hulme Arch Bridge (2009), the diurnal temperature fluctuation affecting the whole structure was a

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Chapter Two: Literature Review critical part in the bridge design. The temperature fluctuation determined in the analysis was as high as 20°C. Temperatures above 0°C, a positive thermal load, reduced the magnitude of pretension in cables by 12% (in the worst case) resulting in the occurrence of high bending moments in the deck. This unfavourable behaviour needs to be addressed by increasing the stiffness of the deck to meet the deflection requirements. Therefore, thermal loads play a significant role, and they need to be analysed and taken into consideration in the design. High temperature differences may occur at the same time on the same structure. Different temperatures of different parts of the same structure are related to the structural orientation in relation to the cardinal directions (north-south, east-west). In a north-south orientation, solar radiation reaches its peak at approximately noon and both sides of the structure are affected equally. However, in an east-west orientation, the south side of the structure is exposed to solar radiation at the maximum level, whereas the northern side remains in the shade. The difference in the temperature is dependent on several variables such as location and orientation of the structure, geometrical and material properties, and environmental conditions (Dilger, 1983). In extreme cases, the temperature difference on the same structure at opposite sides can be as high as 70°C. Dilger (1983) conducted a parametric study on a composite continuous steel-concrete box girder bridge over the Muskwa River near Fort Nelson. He found that the stress induced under the worst thermal combination and for this bridge can reach up to 98 MPa in compression and 42 MPa in tension for steel and 2.9 MPa for tension in concrete (occurring in continuous structures). The results of this study indicated that the geographical orientation of the structure played a significant role and that detailed thermal analyses need to be performed. To conclude, the Canadian environment experiences high differences in temperature throughout the year. For example, the temperature in central Alberta can vary between positive 30°C in summer to negative 45°C in winter, and in addition, temperatures can suddenly change from positive 10°C to negative 10°C within 24 hours. Therefore, the effect of the Canadian climate on the complex configurations of SABs needs to be investigated in detail.

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Chapter Two: Literature Review

2.6 Materials in Spatial Arch Bridges

2.6.1 General

The material composition of any structure can influence the structure’s response under applied loads (Ghali et al., 2009). Specific stiffness (bending, axial, or torsional) can be achieved in any material via a combination of cross-sections and material properties, but the overall response of a structure may differ for different materials. The reason for this difference is the varying specific weight of different materials and their responses to applied loads (creep in concrete or flexural-torsional buckling in steel). In spatial bridge configurations, the responses of different materials exposed to combined out-of-plane loads can play a significant role, and therefore, material selection should represent one of the key design decisions. The sections below describe the materials used in the existing SABs and discuss the application of certain composite systems.

2.6.2 Material Complexity

Material complexity is much lower than geometric complexity in SABs. Most SABs are built from structural steel, particularly from closed steel profiles or steel-concrete composite sections. The shape is typically square, rectangular, or circular. Structural concrete is also used as a common construction material but mainly for decks in the form of box girders (Jorquera, 2007). Other types of materials, such as stainless steel [York Millennium Bridge, UK, Firth, (2002)], stainless steel combined with glass fibres [Sant Fruitos, Spain, Sarmiento-Comesias et al. (2013)], laminated timber [Leonardo da Vinci bridge, Norway, Buelow et al., (2010)], or ultra-high performance concrete [shell arch bridge as research study Strasky, (2008); Terzijski, (2008)], were used only in these particular cases.

2.6.3 Composite Systems

Composite systems, which are structural configurations of two or more different materials, such as steel or fibre reinforced polymer tubes filled with concrete, are commonly used in bridges to achieve optimal performance. Practicality, economics, or aesthetics can be the deciding factors in regards to selecting specific composite systems.

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Chapter Two: Literature Review

Composite systems aim to benefit from the key properties of different materials to resist the demand forces from a variety of load combinations including static, dynamic, and seismic excitation. Because SABs are exposed to out-of-plane loads, the selected materials and assumed structural profile must resist multidirectional loading. Certain materials can withstand the resulting internal forces from out-of-plane loads; however, these materials may have unsatisfactory performance in terms of seismic resistance or durability against cycles of freezing and thawing. This chapter does not list all possible composite materials and describe their function; rather, it discusses composite systems that have been used or may be used in a configuration of SABs. The two sections below describe conventional and advanced composite systems and provide examples related to SABs.

2.6.4 Conventional Composite Systems

Conventional composite systems can be defined as structures where two or more commonly used materials are combined, and the materials must interact. As mentioned in Section 2.6.2, the majority of SABs are constructed from structural steel alone. Even though some bridges comprise two or more materials and are called “composite” bridges, for instance the Ponte della Musica (Lacarbonara (2013)) where a portion of the inclined arch ribs above the deck comprise hollow steel sections and the portion below the deck is made of reinforced concrete, the two materials do not interact. Typically, in composite systems, all materials in the system contribute to resist internal forces at an assumed cross- section (Roberts, 1998). Popular conventional composite systems include steel HSS filled with concrete. The structural profile provides a stay-in-place form and protects the fill. The system of HSS filled with concrete was used, for example, in the inclined arches of a continuous span bridge in Beijing, China (Dou et al., 2015). Due to the bridge’s configuration (15° outward inclination of the arch ribs), the out-of-plane loads are present, and therefore, the bridge is classified as a SAB. Even though the critical buckling load (the main target beyond filling steel tubes with concrete) increased because of the composite system, the total mass of the arch ribs increased as well, which resulted in lower seismic performance. Similarly, a rectangular steel profile filled with concrete was applied in the Hulme Arch Bridge in

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Chapter Two: Literature Review

Manchester, UK where the diagonal arch rib was filled at the crown to resist lateral bending moments and shear forces (Hussain & Wilson, 1999). Applying concrete-filled steel circular shapes (tubes) in long span vertical arch bridges with superior decks can also be combined with tension cables. This innovative hybrid system (currently under Chinese invention patent) was proposed by Liu (2015). Incorporated tension cables placed between deck girders, vertical struts supporting the deck girders, and segments of arch ribs (concrete-filled steel tubes) make the construction more efficient and, in the final structure, the vertical displacements of the arch rib are reduced. This configuration is suitable for extra-long span arch bridges with superior decks. Conventional composite systems function well in many bridge structures and still represent the optimal solution in some SAB configurations. Nevertheless, properties of conventional composite systems, such as their high mass and susceptibility to corrosion, can be disadvantageous in certain environments. Consequently, using advanced composite systems with lightweight, durable materials is increasing (Uddin, 2013).

2.6.5 Advanced Composites Systems

The application of advanced composites systems, in particular advanced composite materials (ACMs), in bridges is rapidly increasing (Uddin, 2013). ACMs, also known as fibre reinforced polymers (FRP), are typically characterized as light, linear-elastic durable materials with a high stiffness-to-weight ratio. ACMs were originally developed in Japan in the 1980’s for aeronautical purposes and slowly expanded to civil engineering applications. Over the last two decades, ACMs were used in thousands of field applications; for example, they were used to reinforce, strengthen, and retrofit systems or used in fully composite systems (El-Hacha, 2000). The material properties of ACMs are a function of the ratios of the combined fibres to the durable matrix that binds the fibres together. The most common fibres are carbon, glass, aramid, and basalt. Matrices are typically made of thermoset (vinylester, polyesters, epoxy, or silicones) or thermoplastic (nylon, polyethylene) synthetic resins. The thermoset resins are typically used in civil engineering applications (El-Hacha, 2000). The material properties of ACMs, considering various types of fibres and matrices, were studied by many researchers. For example, the factors that influence the final properties of the

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Chapter Two: Literature Review composite, such as the volumetric fraction of the fibres and the matrix, the cross-sectional area of the fibres, the method of manufacturing, and the fibre orientation, were studied by Jones (1975). The significance of the interaction between the fibres and matrix that results in the final material properties of the ACM was shown by Tsai & Hahn (1980). Neale & Labossiere (1991) investigated the linear-elastic character of the synthetic fibres up to failure. The significance of the matrix stiffness and volumetric ratio was studied by Rostasy (1993). Extensive research is being conducted to clarify the behaviour of ACMs in structures, and design guidelines are being created so that they can be added to national and international standards, which results in the broader use of composite materials in civil engineering applications. ACMs can be found in the form of reinforcing bars, tendons, cable stays, flexible fabrics, rigid strips, and structural profiles. The application of an individual type of ACM varies depending on the specific purpose. Structural profiles manufactured from ACMs are typically used in newly constructed bridges and can represent a suitable option for fully composite SABs.

2.6.5.1 Application of Structural Profiles made of ACM in SABs

Structural profiles made of ACMs in bridge applications are gaining their popularity not only in deck sections, where the full potential of durable composites can be utilized in relatively simple fashion, but also in more complex all-composite structures. However, there are no SABs constructed with ACMs. Since 1982, when the first all- composite bridge was constructed in China (Potyrala, 2011), our understanding of the application of ACMs in bridge structures is increasing. Therefore, with additional knowledge about the behaviour of structures that are built with ACMs, ACMs may represent a feasible option for SABs. Even though all-composite road bridges exist, for example in 2002, the first all- composite short-span road bridge (with a span of 10m) was opened in Europe near Shrivenham, UK, typically, all-composite systems are more commonly seen in short to mid-span foot bridges (Potyrala, 2011).

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Selecting all-composite systems for a bridge structure is usually driven by a specific requirement, such as chemical durability for bridges in a marine environment as in the Fredrikstad Footbridge, Norway, a bascule bridge with a 56m span, which opened in 2006, or electric non-conductivity for structures in the vicinity of power transmission lines, such as the Lleida Footbridge, Spain, a tied-arch bridge with a 38m span, which opened in 2001. A detailed overview of the majority of hybrid and all-composite structures was done by Potyrala (2011). The main advantages of ACMs, which make them stand out among other conventional materials for bridge structures in Canada, are their corrosion resistance and durability in harsh environmental conditions, such as cycles of freezing and thawing often accompanied with high levels of de-icing salts. The general advantages of ACMs in bridge structures and, therefore, also in SABs, are as follows (Hollaway, 2010): • Durability

o Resistance to atmospheric degradation, de-icing salts, and chemical corrosion

o Reduced requirements for maintenance, and increased life span of the structures • Ability to form custom shapes

o Suitability for curved shapes (aesthetically appealing) o Suitability for complex cross-sections (enhancement of stiffness) • Reduced mass

o Lower transportation costs o More efficient (faster) installation of lighter sections on site o Lower demand forces on the substructure o Lower demand on supporting elements such bearings, cable stays, etc. o Easier operation in movable bridges • Advantageous electrical and thermal properties

o Lower demand on expansion joints due to low thermal expansion (composites including mainly carbon fibres)

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o Low electrical conductivity (composites without carbon fibres) allowing building bridges in the vicinity of transmission power lines, such pedestrian bridges over electrified railroads

In contrast, the disadvantages of ACMs, which are even more perceptible in SABs due to their spatial configuration and the presence of out-of-plane loads, are as follows (Hollawy, 2010): • Deflections

o Larger initial and creep-related deformations due to a low modulus of elasticity and relaxation in the matrices • Susceptibility to vibrations

o The light weight of ACM profiles is an issue because most SABs are pedestrian bridges where human-induced vibration may excite the structure to a point that may exceed the level of comfortable walking or even compromise the stability of the structure

Issues related to particular properties of ACMs in bridge structures need to be addressed via sophisticated analyses and appropriate design decisions.

2.6.5.2 Technology of Suitable Structural Profiles for SABs made of ACMs

Structural profiles made of ACMs used in bridges can be made using several manufacturing technologies. The most common fully-automated techniques for producing profiles for civil engineering structures are pultrusion and filament winding. An advanced technology called “automated fibre placement” that is typically used to manufacture components for aircrafts can also be used to create complex curved components of bridges.

2.6.5.2.1 Pultrusion Pultrusion is a process where several layers of fibres with different orientations are saturated with polymer resin and pulled out at a certain pace through a currying die, forming the final shape (Uddin, 2013). The final product typically consists of four layers: 1) randomly-oriented chopped fibres placed on the surface of the composite, 2) continuous strand mats consisting of continuous randomly-oriented fibres, 3) stitched fabrics with

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Chapter Two: Literature Review different angle orientations, and 4) roving layers that contain continuous unidirectional fibre bundles, which are the main contributor to flexural and axial stiffness of the section (Dalavos, 1996). Even though pultrusion can produce a cross-section of almost any shape from circular or square profiles to more sophisticated honeycomb shapes or “lock-in” profiles (Knippers & Gabler, 2007), these profiles are straight and are constant in the cross-section (not varying with length). This character of pultruded profiles may be limiting for use in SABs where typically curved and, to some extent, profiles of a changing cross-section may be required. However, due to the applicability of certain techniques, pultruded profiles can also be assembled into shapes required by the geometry of SABs configurations. Curved shapes can be achieved by one of two methods. Pultruded profiles can be bent into large radius curvatures, and then a material with a higher stiffness, for example carbon fibres, is glued on the inner radius to secure the curved shape. The other method creates smaller radius curvatures by gluing short, straight segments together. The critical parts of both options are their connections and reliability (Zhou & Keller, 2004; Fibreline Design Manual, 2002). Even though pultruded profiles are typically not manufactured in cross-sections with large dimensions (the standard maximum is around ~300mm) (Fibreline Design Manual, 2002), larger cross-sections and, therefore, greater stiffness is possible by connecting several profiles together typically using glue. This technique was tested in case studies conducted at Lemay Centre for Composites Technology in St. Louis, Missouri and installed at the University of Missouri (Tuakta, 2005).

2.6.5.2.2 Filament Winding Filament winding is a process of winding filaments (rovings or tows) into a geometric pattern, specifically a geodesic stable pattern (Abdel-Hady, 2005), under tension over a rotating mandrel; circular or rectangular shapes can be produced. After curing, the central mandrel is removed from the final product. For straight shapes, the central mandrel is removed simply by pulling it out of the final product; for curved shapes, the central mandrel, which is inflatable, is collapsible to allow it to be removed from inside of the curved, cured profile (Uddin, 2014). Typically, fibres are applied in multiple directions to address the required properties of the final product, and the cross-section of a wound

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Chapter Two: Literature Review profile can vary along its length (Rojas et al., 2014). These two features are highly advantageous for structural profiles of SABs, where multidirectional resistance and changing cross-sections of the profile are desired.

2.6.5.2.3 Automated fibre placement Automated fibre placement (AFP) is based on the principles of manual lay-up technology. AFP is a computerized technology that produces custom shapes on a large scale. A robotic arm places prepreg strip tows, fibres saturated with synthetic resin in a semi-cured state, onto a mold to form a composite layup (Uddin, 2013). The robotic arm can lay the prepared strips in various directions to achieve the desired mechanical properties of the final product. This procedure requires sophisticated programing and coordination of the system. Nevertheless, due to its versatility, almost any shape with custom properties can be manufactured. In certain cases, AFP technology can be combined with a moving mandrel, which results in a faster process. Nonlinear optimization techniques based on artificial neural networks are usually included in the software package. These techniques are applied to predict the material quality as a function of process set points (Heider et al., 2003). Even though this technology has been used for about two decades mainly in aerospace applications, such as aircraft wings (Sloan, 2008), because of its continued development and increasing use, it may be used to produce spatially demanding structural components, such as arch ribs in SABs.

2.6.5.3 Examples and Applications of Suitable ACM Structural Profiles

Pultruded shapes and shapes manufactured via filament winding have gained popularity in bridge applications, especially in combinations with concrete. The following examples describe applications of these composite systems in bridge structures. The main advantages of FRP profiles filled with concrete include a stay-in-place form, a smaller environmental footprint, easy installation, seismic resistance due to increased ductility, corrosion resistance, maintenance free, a much longer life cycle, and blast resistance (Sieble et al., 1996). An FRP profile filled with concrete was developed to increase the seismic performance of vertical columns in buildings. Vertical columns in

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Chapter Two: Literature Review buildings are typically exposed to axial compression and biaxial bending and torsion, which makes them similar to arch ribs in bridges. Self-standing concrete-filled FRP arch ribs in SABs have not been constructed to date; however, the presence of concrete-filled FRP tubes in buried arch bridges is increasing. The Composite Arch Bridge System, known as “Bridge-In-A-BackpackTM,” was developed at the University of Maine, US. In addition to the confining effect of fibres oriented perpendicularly to the longitudinal axis, which results in increased ductility, the developed composite shell contains internal spiral ribs to utilize shear force transfer. The FRP profiles that were used in the innovative bridge system were manufactured via filament winding. Numerous bridges of this kind have been built since 2008, when a pilot bridge, the Neal Bridge (27ft ~ 8.2m long, 23 arch ribs spaced 2ft ~ 0.6m apart) in Pittsfield, ME, was built. Due to the high durability, increasing use, fast construction, low transportation costs, and the low demand of substructure of the FRP tubes filled with concrete, in December 2012, AASHTO approved and published design guidelines for concrete-filled FRP tubes for flexural and axial members (Alberski, 2013). In addition, concrete-filled FRP tubes can be used in bridges with straight horizontal decks. An example of this type of application is the King Channel Bridge, where concrete-filled carbon fibre reinforced polymer (CFRP) tubes were applied in the vertical columns supporting the bent connection and the main girders. In the main girders, flexural reinforcements in the form of reinforcing steel were placed inside the CFRP tubes as in a conventional reinforced concrete beams (Roberts, 1998). In certain cases where increased torsional and bending stiffness is required while the weight is targeted to be low, a hollow FRP profile can be filled with lighter materials such polyurethane foams. Such a system has not been applied in bridge structures; however, it meets code requirements for transmission poles manufactured by Strongwell. The application of pultruded glass fibre reinforced polymer (GFRP) profiles is demonstrated in the Lleida Footbridge, Spain. This double-tied arch bridge with a span of 38m and rise of 6.2m is entirely made of hollow GFRP profiles. The only metal components are angles and bolts that connect individual elements. To achieve the desired stiffness and a sufficient critical buckling load, both the tension and compression members comprise two pultruded profiles (300X90X15mm) glued next to each other and covered

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Chapter Two: Literature Review with flat plates (180X12mm). Longitudinal arch ribs and ties are braced with diagonal GFRP members. Even though the entire structure is very light, the dynamic testing that was conducted proved that the limits imposed on human-induced vibrations were met. A residual deflection due to a minor slip in the bolted connections remained in the structure after removing the loads used during the static test (Potyrala, 2011).

2.6.5.4 Connections of Structural Profiles

In general, ACMs are somewhat difficult to connect due to their fibrous structure and anisotropic character. Nevertheless, connections are one of the key parameters in SAB configurations as they transfer internal and external forces that result from out-of-plane loads. Generally, connections can be divided into to two main categories: material and structural connections.

2.6.5.4.1 Material Connections The assumed structural profiles identified in Section 2.6.5.2 can be made into the desired cross-sections with proper mechanical properties. However, due to manufacturing limitations, the length of the structural profiles is limited, and the prepared parts need to be assembled together. Connecting ACMs can be done either with glued joints or with bolted connections; combinations of both types are also used in industrial applications (Potyrala, 2011).

• Glued joints Glued joints are quickly assembled and are strong; however, they fail in brittle mode (no warning before failure) and are sensitive to the quality of bond between the adhesives and actual ACMs (Tuakta, 2005).

Advantages of glued joints:

o Clean connections o More rigid than traditional bolted connections where slip typically occurs o Outstanding performance under dynamic loading o Fluid and weather tight

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Disadvantages of glued joints:

o Sensitivity to a number of bonding agents in terms of long term loading in certain environmental conditions (humidity and/or elevated temperatures can compromise the structural integrity under sustained loading)

o Brittle failure mode. In general, adhesive bonding can fail in four modes: adhesive failure, cohesive failure, substrate failure, or in a combination of the first two

o Difficult to inspect o High demand on surface preparation o The load bearing capacity of glued joints is not directly proportional to the glued area, i.e., at some point of increasing the connection area, the increase in load bearing capacity stops. This behaviour results from the mechanics of the fracture that happens at the interface of adjacent layers within the composite rather than at the surface of the adhesive (the strength of the bonding agent is larger than strength of the actual composite)

Extensive research is being carried out to overcome the disadvantages of glued joints (Fibreline Design Manual, 2002). A combination of glued and bolted connections is typically applied in critical load bearing connections to avoid a brittle failure mode. Despite certain disadvantages, glued joints are considered the leading connection type for ACMs. The most common bonding agents are two-compound epoxy or single-compound polyurethane resins. Fatigue resistance of glued joints was studied, for example, by Shahverdi et al. (2012).

• Bolted connections Bolted connections are a mechanical connection type for which the capacity is a function of the parameters of the bolts (diameter, arrangement, number), the materials being connected (the direction of the fibres in ACMs in relation to internal forces), and, to some extent, the tension that develops in the bolts. This type of connection is traditional, relatively fast (no curing required), and provides a ductile type of assembly.

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Advantages of bolted connections:

o No specific surface preparation is necessary o Easy to inspect o Disassembly is possible o Quasi-ductile behaviour

Disadvantages of bolted connections:

o Low strength-to-stress concentrations (no stress distribution via yielding as in conventional steel connections)

o Normally require special gaskets or sealants to be fluid and weather tight o Metallic fasteners corrode

Bolted connections are sources of stress concentrations in profiles and in the bolts, and therefore, it is necessary to ensure that adjacent profiles can withstand the loading to avoid the bolts coming out of the profile. Typically, bolted connections in ACMs are similar to those in conventional steel profiles; however, since there is no standard on bolted connections, each manufacturer provides their own guidelines (Fibreline Design Manual, 2002).

2.6.5.4.2 Structural Connections A structural connection can be described as the connection between individual structural components of the bridge. A special type of connection is the joint between the substructure and superstructure. The connection between the cables and deck or an arch rib in SABs follows the principles adopted in conventional steel connections; the connection must allow for the transfer of direct tension from the cables but must also accommodate relevant rotations resulting from the spatial configuration and its displacement. During the design process of the connection, the anisotropic character of ACMs must be taken into account, especially while developing the connections of the cables to the deck or the arch, because at these locations the material may be exposed to a combined loading that results from the deck or the arch and from axial forces in the cables.

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Anchoring the cable stay is an important and complex structural detail; typically, in all-composite bridges, conventional materials are used. An example of such a configuration is the Aberfeldy footbridge, Scotland. In 2002, this bridge was the longest all-composite cable-stayed bridge with a span of 113m; it was constructed from GFRP pultruded profiles. Aramid fibres were used in the cable stays. The cable-to-deck connection was made of aluminum components, and the cable-to-tower connection comprised galvanized steel elements (Cadei & Stratford, 2002). Another example of an all-composite cable-stayed bridge is the foot bridge over a railroad that opened in Kolding, Denmark in 1997. The connections between the superstructure and substructure in conventional (non- spatial) all-composite bridges are not a significant concern as an appropriate bearing pad and the necessary stiffeners at the girder end where mainly shear forces need to be resisted are used. However in configurations of SABs, the connection between the superstructure and substructure must resist not only shear forces but also axial forces and bending and torsional moments. Because no SAB has been constructed from ACMs, there is minimal evidence demonstrating the applicability and feasibility of the connection between the superstructure and substructure. Even though the all-composite Lleida Bridge experiences mainly compression forces, bending moments also develop. Nevertheless, the magnitude of the developed moments is low, and therefore, it is difficult to evaluate the structural response of the all-composite arch bridge in relation to the bending moments in the arch (Sobrino, 2002). Bending moments in an arch in a SAB, particularly at the location of connection of the arch to its abutment, are considered to be a significant issue. A possible alternative may be to use a combination of ACM profiles with concrete. Nevertheless, the structural detail of the connection between a superstructure and substructure for ACMs represents a gap in the research.

2.6.5.5 ACM Cables

Cables manufactured from ACMs represent another key structural component whose properties can influence the structural response in complex configurations of SABs. ACM cables are known for high tensile strength, low weight, and long fatigue life. Depending on the material composition, the issue related to ACM cables is matrix relaxation, which causes loss of tension in the cables, which, consequently, results in

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Chapter Two: Literature Review increasing deck deflection. Anchoring ACM cables can also be an issue. Due to the anisotropic character of ACM cables, anchors need to be more sophisticated, and therefore, cost more, compared to those for steel cables (Wang et al., 2015). Materials used to manufacture cables depend on the purpose and specific configuration of the structure. Typically, carbon, glass, and basalt are used to produce ACM cables; in some cases, fibres are combined within one cable to achieve the desired properties. Cables made of carbon fibre have, on one hand, superior properties such as high strength, chemical stability, and long fatigue life; on the other hand, carbon fibres are the most expensive option and are very light, which makes them more prone to flow-induced vibrations that may result in the aeroelastic instability of the structure (Wang & Wu, 2010). Glass fibres are commonly used in civil engineering applications for their relatively low cost and high strength. However in ACM cables, glass fibres are the heaviest option, and their fatigue life is short compared to carbon fibres. In addition, due to their sensitivity to alkali effects and the high relaxation of the matrix causing large creep, their overall durability is low. Basalt fibres are a relatively new alternative in ACMs; however, they are comparable to glass fibres in terms of their mechanical and physical properties and reasonable cost. The advantage of basalt fibres is that they have better chemical stability and overall durability than do glass or aramid fibres (Zoghi, 2013). Nevertheless, their application in cable stays is limited due to a relatively low modulus of elasticity, which causes large sag in the cables. To overcome the disadvantages of carbon and glass fibres in ACM cables and utilize the advantages of more affordable and heavier basal fibres, hybrid ACM cables, comprised of basalt and carbon fibres, were proposed by Wang & Wu (2009). The proposed hybrid system consists of several layers. The outermost layer is created by combining carbon and basalt strands in a ratio of 1:1, separated with an inner sleeve from viscoelastic material, which provides internal damping to the cable. This outermost layer significantly extends fatigue life (Wang et al., 2016), and the viscoelastic material combined with a heavier basalt fibre core increases the damping characteristics (Wang & Wu, 2009). The core of the hybrid cable consists only of pure basalt fibres.

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ACM cables can play a key role in the long-term performance of bridge structures in general. In SAB configurations, where flexible hangers are one of the main structural components transferring out-of-plane loads, the properties of ACM cables, such as low thermal conductivity and long fatigue life, are highly advantageous.

2.6.5.6 Analysis and Theories Describing the Behaviour of ACMs

ACMs represent an innovative material with superior durability; however, their performance depends on an understanding of their internal structure and their correct application in structures. Due to their fibrous character, ACMs can be classified as an anisotropic material: materials that are not isotropic and where the mechanical and physical properties vary for symmetry planes. The level of anisotropy is defined as the number of symmetry planes in which the properties remain constant. To characterize the constitutive behaviour of fully anisotropic material, 21 independent constants are required. In practical applications, layered composites are assumed to be orthotropic (a special case of anisotropy) where only 9 independent constants fulfill the characterization of the material. Analysis of composite materials can be described using micromechanics (analysis of a single lamina made of matrix and fibre constituents) and macromechanics (analysis of several laminae stacked together in various orientations, which form laminate) (Zoghi, 2013). Structural profiles manufactured using pultrusion or filament winding technology can be generally characterized as layered (laminated) composites. A description of the internal forces in laminated composites can be done using plate theories. In general, there are two groups of plate theories: theories based on an assumption that composite laminates can be described by an equivalent single layer (ESL) and theories based on a 3D elastic continuum. In ESL theories, an appropriate analysis theory should be applied based on the ratio of wall thickness to the representative dimension of the laminate, whose threshold value is 1/20. Laminated composites with a ratio below 1/20 can be analysed using the classical lamination theory, and composites with a higher ratio should be analysed using shear deformation theories (Liewa, 2011). The classical lamination theory (CLT) of shells or plates (proposed by Kirchhoff & Love (1888) that represents an extension of Euler-

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Bernoulli beam theory) assumes that the normals to the midplane remain constant after deformation and neglects transverse normal stresses. In shear deformation theories (corresponding to the Timoshenko bending theory) shear deformation is included as the normals do not remain perpendicular to the midplane after deformation. The first order shear deformation (FSDT) theory assumes that the shear transversal strain is constant throughout the entire thickness of the plate (Reissner, 1945). A shear correction factor is required to correct the strain distribution at the surfaces (the strain at the surface is zero). The magnitude of this factor depends on the material properties, geometry and also on the type of applied load (Liewa, 2011). In higher-order shear deformation theories (HSDT), the transverse shear strain varies as a function of the coordinates within the plate thickness, with zero strain at the surface. An improved higher-order theory proposed by Murthy (1981), and similar to the one proposed by Reissner, uses displacement fields to achieve zero strain at the surfaces. Both FSDT and HSDT are sufficiently efficient (HSDT are typically more demanding in terms of computational time) and highly accurate in the analysis of gross deflections, critical buckling loads, and natural frequencies. The more complex layerwise theory (considered a type of ESL theories) need to be applied to describe interlaminar stresses. The layerwise theory and the 3D continuum theory describe geometric and material discontinuities with a high level of accuracy; however, the demand on computational time significantly increases (Zhang, 2008).

2.6.5.7 Challenges in Structural Profiles made of ACMs

The material properties of profiles made of ACMs present issues that may influence the behaviour of a structure. Some areas for concern in ACM structural profiles include creep, thermal stability under sustained loads, buckling of thin-walled members, and susceptibility to vibrations. These characteristics must be addressed with accurate analyses and careful design. Creep (increasing deformation under sustained load) in structural profiles is a significant issue. Due to the relaxation of the soft polymer matrix over time (creep in fibres typically does not occur to a great extent (ISIS Design Manual 8, 2006)), the relative position between the fibres increases, resulting in higher deformations and loss of internal

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Chapter Two: Literature Review stress (stiffness reduction). Creep in ACM structural profiles is a complex phenomenon and depends on various factors, such as the properties of the fibres and matrices and their volumetric ratio, type of fabrication, and type and direction of loading. The theoretical prediction of creep-related material behaviour can be modeled using the power law developed by Findley (1944). Experimentally, creep behaviour of pultruded GFRP profiles was studied by Sa et al. (2011). In his work, Sa et al. examined the effects of flexural and compressive loading where the load magnitude ranged from 20% to 80% of the maximum strength. The results showed that specimens failed at a level as high as 50% of the maximum strength. Significant deformations occurred after several hours of loading not exceeding 30% of the maximum strength. Short and long-term behaviour of wide flange beams was studied by Bank & Mosallam (1992). The creep testing results showed different rates of creep deflection during the experiment. During the first 2,000 hrs, the rate of creep deflection increased, but after more than 2,000 hrs, the rate remained constant. Creep coefficients for various pultruded profiles made of the most common GFRP materials at service and elevated temperatures were investigated by Smith (2005). Conducted studies and experiments proved that several factors play a key role in the structural behaviour of the selected profiles, and there is still a need to investigate new alternatives such as hybridization, a combination of different fibre types within an assumed profile, in structural profiles. The effect of hybridization in improving creep response was investigated by Barpanda (1998). In recent industrial applications of pultruded profiles in girder road bridges, carbon and glass fibres have been used in hybrid systems to enhance flexural stiffness, lower creep, and reduce cost. An example of this type of hybrid pultruded profile is the double-web, wide flange beam manufactured by Strongwell, where carbon fibres are placed in the flanges and glass fibres are placed in the webs. The performance of ACMs at both high and low temperatures typically decreases. Elevated temperatures cause an accelerated process of matrix relaxation, the composite material to lose its strength, and higher creep effects. Material selection of the matrix can play a significant role. The influence of the resin type (polyester and vinylester) on flexural creep under elevated temperatures (54°C) in GFRP T-shaped sections (with solid and hollow webs) was investigated by Daniali (1991). Profiles with vinylester resin performed better than profiles containing polyester resins at elevated temperatures. Even though the

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Chapter Two: Literature Review coefficient of thermal expansion (CTE) of most ACMs is low and applying these materials in bridges is beneficial (a low demand on the expansion joints), exposure of ACMs to low temperatures may lead to a loss of stiffness. In environments where temperature fluctuations are high and frequent, micro-cracks in the matrix that result from different cycles of expansion and contraction of the individual constitutes may develop. These micro-cracks then affect stiffness and creep related deformations (Cusson, 2002). Creep-rupture is related to the creep of ACMs; it is typically observed in unidirectional reinforcing bars. This phenomenon, sometimes called also “stress- corrosion,” occurs due to the complex interaction of several factors, such as environmental exposure, the presence of cracks in fibres and matrices, the type of loading, and fibre- matrix interface damage. Creep-rupture can be described as a failure of the system after the system was exposed to sustained loads, whose magnitudes exceed a certain stress limit (a limit that differs for different materials) within a certain time period. Both the long-term exposure to low levels of a sustained load and the short-term exposure to high levels of a sustained load may lead to the collapse of the system. Typically, the structures most susceptible to creep-rupture are GFRP. CFRP have the highest resistance to creep-rupture. The limiting values of sustained loads are, for GFRP, ~20% and, for CFRP, ~ 50% of the maximum strength (ISIS Design Manual 8, 2006). Creep-rupture in GFRP pultruded profiles was studied by Batra (2009). Creep and creep-rupture are typically influenced by elevated temperatures, which reduce the load bearing capacity resulting in increased deflections and a reduced level of sustained load (Dutta, 2000). The significance of creep- rupture in BFRP was studied by Banibayat (2015). The creep coefficient of BFRP was determined to be approximately 18% for a 50-year service life and 28% for a 5 year service life. Due to the nature of structural ACM profiles (thin-walled members), buckling is another factor that needs to be addressed in the analysis and design of structures. As in conventional steel profiles, ACM structural profiles are susceptible to some types of buckling such as the global buckling of members and flexural-torsional buckling. The principles of elastic instability (buckling) in SABs are described in Section 2.5.3. The critical buckling load of ACM profiles needs to be found using a more complex method compared to conventional steel profiles. The complex method assumes the number of

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Chapter Two: Literature Review symmetry planes (the level of anisotropy), which results in a number of buckling loads that need to be evaluated. In a typical industrial application, an ACM profile is assumed to be an orthotropic material, and therefore, three buckling loads are evaluated. The lowest obtained load is then taken as the critical buckling load. Formulas to obtain buckling loads depend on whether or not shear deformation is included (Zoghi, 2013). Formulas for calculating the critical buckling load, considering global buckling and flexural-torsional buckling of AMC profiles, can be found in Zureick & Scott (1997), Zureick & Steffen (2000), and Bank (2005). Computing local buckling of flanges and webs or walls in open or closed profiles, respectively, includes moduli of elasticity in two planes. The end conditions of ACM profiles are defined in the same way as they would be defined for conventional steel profiles. Analyses of local buckling in axially loaded thin-walled, closed-section, rectangular beams and open-section I beams are described by Kolar & Springer (2003). Local buckling analyses of a flange or web in compression are specified by Bank (2005). Design guidelines for ACMs in transportation structures are presented by Dalavos (2002). Failure modes of ACM structural profiles are a function of the type of loading and the cross-sectional shape of the profile. Closed profiles are governed by global or local buckling. However, open sections are prone to failure at the connection of the web and flange where stress concentrations occur. In pultruded profiles, this connection may be rich in resin (matrix), and therefore, it is the weakest part of the profile. Stress concentrations are usually treated by specifying an appropriate radius at the connection (Creative Pultrusion Design manual, 2016). In wide-flange beams, critical locations are typically at the support where crippling of the web occurs first (Behzad et al., 2010).

2.6.5.8 FE Modeling of ACMs

Predicting the behaviour of ACMs via FEA can be very accurate and, in modern practice, is often used to analyse composite structures. FEA is a powerful tool whose accuracy is highly dependent on the computational time available for a specific analysis. Typically, as a higher level of accuracy is required, more time (cost) must be available. Formulations of finite elements used in FEA of ACMs are based on analytical theories that describe the material behaviour; in finite elements, the shape function represents the

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Chapter Two: Literature Review relevant theory. As outlined in Section 2.6.5.6, theories describing the material behaviour of laminated composites can be divided into two main categories: (a) equivalent single layer (ESL) theories and (b) continuum-based 3D elastic theories. These theories are used to develop shape functions in particular finite elements. The parameters of individual elements, such as the number of nodes or number of degrees of freedom (DOF) at each node, are derived based on the phenomena the specific element is intended to describe. In general, predicting the material behaviour of ACMs via finite element models can be separated into several categories such as free vibrations, damping and transient dynamic response, buckling and post-buckling, geometric nonlinearity and large deformation analysis, and damage and failure. The complexity of specific finite elements is not described in this section. Rather, the intention is to present the core theme of FEA in predicting the behaviour of ACMs. A detailed review of the developments in FEA, including the specifications of individual elements and particular shape functions, is provided by Zhang & Yang (2008). Even though ACMs are assumed to be linear elastic materials, their behaviour in structural profiles can be nonlinear. For example, pultruded profiles exposed to multiaxial loading experience nonlinear responses due to the interaction of soft the matrix and imperfections (voids) within the material structure (Haj-Ali & Kilic, 2002). Nonlinear material responses can be magnified in structural profiles at material or geometric discontinuities. Finite element (FE) micromechanical constitutive models have been developed and implemented in commercial software to predict nonlinear material responses (Haj-Ali & El-Hajjar, 2003). In practice, coupon testing is used to verify the correctness of the developed FE models. Particular models can utilize viscous layer crack growth in composite materials (Haj-Ali et al, 2006).

2.7 Conclusion to Chapter Two Chapter Two provides a literature review of the topics related to the research in this thesis and identifies the research gaps in the field of SABs. The definition and historical development of spatial structures is presented first with examples that demonstrate the level of understanding of arch construction and materials used in ancient Egypt and Rome followed by a mathematical description of

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Chapter Two: Literature Review antifunicular arches. Modern trends in spatial structures are outlined at the end of the historical review. Subsequently, geometric classifications of complex SABs are described next with detailed descriptions of spatial configurations, geometric ratios, and state-of-the- art form-finding graphical/computational methods. The principles of planar and spatial antifunicular arches are discussed and examples are provided. Then, the structural behaviour of SABs and the response of the system in relation to its geometric and structural configurations is discussed; this section also comments on the effect of flexible and rigid hangers on the structural response. Linear and nonlinear approaches are outlined. The significance of elastic instability and thermal loads in vertical planar arches and spatial arches are discussed. Case studies of individual configurations, including dynamic loading, are presented. Subsequently, material influence on the structural behaviour of SABs is described in terms of conventional and advanced composite systems. Applications of ACMs in structural profiles suitable for SABs are described in depth. A review of the current manufacturing processes, connection details, and the application of ACM cables used in all composite bridges is followed with a description of analytical theories describing the material behaviour of ACMs. The section concludes with an outline of the challenges of using ACMs in bridge structures and numerical methods for predicting material response. A comprehensive literature review, as summarized above, provides an outline of the origins of SABs, their development, and most recent applications. Theoretical works and case studies were presented to show the current state of research and to identify gaps in the understanding of structural behaviour and analysis of SABs. The key gaps in the research were identified as follows: 1) the significance of geometric and stiffness ratios in SABs with an inferior deck; 2) the effect of thermal loads on the structural response of SABs; 3) the configuration of cables and additional cable tensioning in spatial configurations; 4) the significance of material properties on the structural behaviour of SABs; 5) the response of certain geometric configurations to dynamic loading; and 6) the connection of the SAB superstructure constructed from ACMs with the substructure. This thesis focuses on the identified gaps in the current state of research.

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Chapter Three: Development of Models

3.1 Introduction This thesis contributes to our understanding of the structural behaviour of SABs via finite element analyses (FEA) of various configurations. The development of accurate finite element (FE) models plays a significant role. Individual analyses employ models of different complexity. There are linear and nonlinear models. In the nonlinear models, geometric nonlinearity, involving large displacements but small strains, is taken into account. Geometric nonlinearity addresses the displacemnts of the arch and deck and the response of the flexible cables connecting the arch and the deck. The models presented in this chapter provide background for all the types of analysis considered in the thesis. Specific details related to the focus of a specific analysis are described in particular chapters. All models were created assuming basic SI units such as N, m, kg. The present chapter is divided into six sections: an introduction (Section 3.1), a description of the software used (Section 3.2); a description of the structural configurations, differentiating between the linear and nonlinear models (Section 3.3), the methodology for creating the FE models via parametric input files and evaluating the results (Section 3.4), and a listing of the mechanical properties of the materials assumed in particular analyses (Section 3.5). Appendix A is devoted to providing the details related to this chapter and consists of three sections: a justification of the finite elements selected for the arch, deck, and cables in the linear and nonlinear models (Section A.1), a description of the verification and calibration of the models (Section A.2), and, finally, an example of a complete parametric input file (Section A.3).

3.2 Software The analyses conducted and evaluated throughout this thesis have been carried out with the finite element software Abaqus; research version 6.13. Abaqus is used for both the modeling and analysis of the mechanical components and assemblies and also for the visualization of the results.

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Chapter Three: Development of Models

Analyses considering static loads were performed in “Abaqus/Standard”; a general- purpose finite element analyzer that employs traditional implicit integration scheme which is suitable for both linear and nonlinear analysis. A solver which is more suitable for dynamic nonlinear analysis is “Abaqus/Explicit”, which employs an explicit integration scheme to solve highly nonlinear systems with many complex contacts under transient loads. The Abaqus software package was provided and sponsored by the University of Calgary and Compute Canada.

3.3 Structural Configuration of Developed FE Models

The FE models represent pedestrian bridges whose geometric configuration is being changed in order to investigate the significance of the factors considered in this thesis. There are two groups of variables (geometric ratios) reflecting the overall dimensions of the assumed structure: a) the primary variables represented by the ratio of arch rise to span

(f(A)/s) and b) secondary variables represented by: the ratio of deck reach to span (f(D)/s), the angle of arch inclination from a vertical plane (ω), and the angle of arch rotation about a vertical axis bisecting the span length (θ). While the magnitudes of the primary and secondary variables change, the span “s” of the assumed bridges is held constant at 75m.

3.3.1 Character of Models

There are two types of models developed for the analyses carried out in this thesis: a) linear models and b) nonlinear models. The differentiating aspects between the types of model are whether or not geometric nonlinearity is taken into account and the character of cables that connect the arch and deck.

3.3.1.1 Geometric nonlinearity

In the spatial models, it is the geometric nonlinearity, assuming large displacements and small strains, that can affect the overall response of the structure. Due to the arrangement of the SABs in question, it is expected that both the arch and the deck undergo large displacements upon application of load. Nevertheless, in the cables, the developed strains remain small and, therefore, material nonlinearity does not need to be taken into

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Chapter Three: Development of Models account. A schematic sketch showing the principle of large displacements and small strains is presented in Figure 3-1. The figure shows a section trough a SAB at its midspan. The structure consists of a vertical arch and a deck curved in plan. In the initial state, all the structural members, i.e., the arch, deck, and cables, are undeformed, and the cables are assumed taut. Upon the application of load, all the structural members deform. Despite the displacements of the arch and deck being large and the cable developing a sag, the change in distance between the ends of the cable is relatively small and the cable does not develop large strains.

Undeformed shape A ~ B

Deformed shape Cable sag

Figure 3-1: Principle of geometric nonlinearity taking into account large displacements and small strains

Geometric nonlinearity is assumed to have a significant effect only in the models used in Chapter Six where materials with low E are considered. Structures made of materials with low E are expected to undergo larger deformations than structures made of materials with large E. Hence, models assuming geometric nonlinearity become

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Chapter Three: Development of Models advantageous in Chapter Six where ACMs with low E are included in the analyses. Models assuming geometric nonlinearity can predict the response more accurately while taking into account the second order effects. In Abaqus, linear equation solution is used in both linear and nonlinear analysis. In the case of geometric nonlinearity, Abaqus uses the Newton method. In this method, the simulation is broken into a number of time (load) increments and the approximate equilibrium configuration is found at the end of each time increment. Using the Newton method, it often takes Abaqus/Standard several iterations to determine an acceptable solution to each time increment (Abaqus Analysis User’s Guide, Section 7.1.1).

3.3.1.2 Linear Finite Element Model

In the linear FE models, the spatial deformation of the cable sag is assumed to have only a minor contribution to the overall structural response and, therefore, it is ignored. In these models, geometric nonlinearity is not taken into account. The arch and deck were created with the beam elements described in Section A.1.1, while the cables were created with truss elements, described in Section A.1.3. The truss elements can transfer only axial loads and, therefore, the connection between the cables and the arch or the deck behaves as a simple “hinge” transferring translations and allowing for rotations in any directions. Even though a rigid link element could also be used to model the hanger connecting the arch and deck, transferring only axial loads and allowing rotation about any direction, the selected truss elements are more advantageous. The truss elements can model the axial elongation of the cables, which affects the response of the arch and deck. Due to the character of the truss elements (simple elements with a linear interpolation function), the overall procedure is simplified and allows for efficient analysis of a large number of models with a reduced margin of error, avoiding non-convergent solutions. Linear models are employed in Chapters Four and Five.

3.3.1.3 Nonlinear Finite Element Model

The nonlinear models take the geometric nonlinearity into consideration. It is assumed that the displacements of the arch and deck as well as the sag in the cables will affect the overall structural response.

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Chapter Three: Development of Models

The arch and the deck are modeled with the same beam elements as in the linear models. The cables, however, are not modeled with truss elements but with beam elements with reduced bending stiffness as described in Section A.1.4. In the initial state, the cables are assumed to be taut; no existing sag is assigned to the cables. Sag in the cables develops upon application of loads. Therefore, it is the axial stiffness that governs the response of the cables. The connection of the beam elements at adjacent nodes transfers not only the axial loads but also bending moments. Therefore, the connection of the arch or deck to these cables is approached with more sophisticated connector elements (Section A.1.5). Even though the accuracy of the nonlinear models is increased, due to the fact that these models account for the secondary effects, the nonlinear models are more prone to non-convergent solutions. The definition of nonlinear models is sensitive to parameters such as: accurate geometry, the local orientation of connections, and the precise definition the actual nonlinear analysis (represented by the size and number of increments within individual steps). The nonlinear models provide a more accurate response because the second order effects are taken into account. Another advantage of the nonlinear models is their ability to achieve a more realistic response when additional tensioning of the cables is employed as described in Section 3.4.1.4.

3.3.1.4 Comparison of Linear and Nonlinear Models

The correctness of the linear and nonlinear models was verified in Appendix A, Section A.2.Despite a relatively simple arrangement of the linear models, the accuracy of these models is close to the nonlinear models. A comparison of the distribution of the forces in the cables connecting the arch and deck in linear and nonlinear models is shown in Figure 3-2. The material modelled in both models is steel, a material with high E. The configuration consists of a vertical arch and horizontal deck curved in plan. The comparison of linear and nonlinear models is shown for the cables because the cables are the key components. Cables transfer the loads from the deck to the arch and, thus, the response of the cables should be evaluated carefully.

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Chapter Three: Development of Models

Figure 3-2: A comparison of forces in cables in the linear and nonlinear models, assuming material with high E

From the Figure 3-2 is apparent that difference between the cable forces obtained from linear and nonlinear models is less than 10%. A comparison of the cable forces obtained from the linear and nonlinear models assuming materials with low E, such as ACMs, is shown in Figure 3-3.

Figure 3-3: A comparison of forces in cables in the linear and nonlinear models, assuming material with low E

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Chapter Three: Development of Models

From the Figure 3-3 is apparent that the difference between the force in cables obtained from the linear and nonlinear models is larger than 10%. Therefore, it can be said that linear models are sufficiently accurate to be used in configurations assuming materials with relatively high E, such as steel. However, the accuracy of the linear models drops when materials with low E, experiencing larger displacements, are employed. Hence, nonlinear models, taking into account the second order effects, become advantageous in chapter Six, where ACMs with low E are introduced.

3.3.2 Structural Components

The arch, deck, and cables are the main structural components in all the SAB configurations modeled. The configurations assume only SABs with an inferior deck. Attention is given to arrangements with flexible hangers, i.e., cables. An example of a spatial configuration underlying the concept of the SAB with the inferior deck is provided in Figure 3-4.

Figure 3-4: Example of a structural configuration representing a concept of SAB with an inferior deck assumed in the thesis

The arch and the deck comprise hollow structural sections (HSS). A solid section is assumed for the cables, representing locked coil strands. There are 14 equally spaced (5m apart) cables along the span in the models. The outer dimensions of the applied HSS

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Chapter Three: Development of Models profiles and the diameter of the cables vary for a particular type of analysis. Details of the individual analyses are depicted in the particular chapters of this thesis. Material properties of the HSS profiles and cables are provided in Section 3.5.

3.3.3 Structural Connections and Boundary Conditions

The only structural connection in the models is the connection of the cable end to the arch or deck. Connections of individual structural sections (structural profiles connected to each other) within the arch and the deck that are typically done via welding or splicing in the case of steel profiles and via glued of bolted connections in the case of advanced composite materials (see Section 2.6.5.4) are assumed to be “ideal”, and their significance for the structural response is not investigated. The character of the cable connection to the arch or deck differs between the linear and nonlinear models. In the simplified linear FE model the truss element representing the cable can transmit only axial forces and, therefore, it behaves as a “multi-directional” hinge (Section A.1.3). In the more complex nonlinear model, the connection between the cable and the arch or the deck is modeled via an assigned connector element with a spatially oriented connector section (Section A.1.5). In both types of models, the cable to arch or to deck connection represents a typical structural detail where a cable end is prevented from translation in any direction and is allowed to rotate about only the horizontally-oriented axis (relative to the connected member) as illustrated in Figure 3-5.

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Chapter Three: Development of Models

Figure 3-5: Example of cable end to connection to arch or deck

It should be noted that in the spatial configurations, the local orientation of this connection detail plays an important role. Due to the deformations of the arch and the deck, the position of the connection detail may change (translate and rotate in space) and, thus, result in the development of undesired stresses. A definition of the local orientation of the connections of the cables is the same as the definition of the local orientation of the boundary conditions of the deck (section 3.4.1.5). The beam elements used to model the arch and deck are a one-dimensional (1D) representation of the three-dimensional (3D) segment. In the beam elements that are used to model the HSS profiles, nodes are located at an element axis which goes through the centroid of the cross-section of the HSS profile. Therefore, the connection between cables and the arch or deck occurs at the centroidal axis as shown on Figure 3-6.

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Chapter Three: Development of Models

Detail: Cable-to-Arch Connection

Detail: Cable-to-Deck Connection

Cut through the structure at Midspan

Figure 3-6: Detail of cable-to-arch and cable-to-deck connection at centroid axis of structural elements

The connection of a superstructure (arch and deck) to a substructure is done via the definition of boundary conditions (BCs); the linear and the nonlinear models have the same BCs. A procedure for the definition of arch and deck BCs within the FE model is specified in Section 3.4.1.5. In all assumed configurations, the arch has both ends fixed to the adjacent substructure. Such a connection is typically done in practice via anchor bolts as illustrated in Figure 3-7.

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Chapter Three: Development of Models

Bolted connection

Figure 3-7: Example of fixed connection of arch to the substructure

In the FE models, the connection of the deck to the substructure is defined via “rollers” or “pins” representing particular types of bridge bearings. The significance of the different BCs of the deck is examined in Chapter Five.

3.3.4 Discretization

As outlined at the beginning of this section, the main difference between the linear and the nonlinear models is the way the cables are modeled. In both models, the arch and the deck are modeled with same type of beam elements with the same discretization. However, the overall discretization differs between the linear and the nonlinear models due to the different elements used for modeling of cables.

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Chapter Three: Development of Models

As verified in Section A.1.1, the optimal length of beam elements for the assumed structural configuration is 2.5m. Such a dimension results in 30 beam elements for the arch and deck with the span of 75m. In the linear model, one truss element is assigned for each cable and therefore there are 14 truss elements in total. In the nonlinear model, as a result of mesh sensitivity analysis (Section A.2.3), there are 10 beam elements assigned to each cable providing 140 beam elements in total. The total number of nodes in the linear model is 122. This number reflects the fact that the beam and truss elements do not need to be connected via a sophisticated element. At the connection of a beam and a truss element, both elements share the same node. The total number of nodes in the nonlinear model is 472 resulting from the 30 beam elements in the arch and deck (60 in total), 140 elements to model the cables, and 28 connector elements connecting the end nodes of the cables with the nodes in the arch and deck. The larger number of elements results in increase of computational time.

3.3.5 Applied Loads

The loads applied within the models are: permanent load (DL), live load (LL), and global thermal load (GTL). The magnitude of DL reflects the density of selected materials. Design loads assumed within the analyses are determined according to the Canadian Highway Bridge Design Code CHBDC (2014), section 3, Cl. 3.8.9. The magnitude of LL, assuming pedestrian bridges and taking into consideration 75m long span 2m wide deck is 5000N/m. A description of procedures determining a definition of all assumed loads is provided in Section 3.4.1.3. All loads are applied similarly in both the linear and nonlinear models. A local thermal load (LTL) was employed to achieve additional tensioning of the cables in the nonlinear models (Section 3.4.1.4). The sequence of applied loads is as follows: • Undeformed state, no loads applied • Static loads (DL and LL) and GTL are applied first (simultaneously) • LTL applied second to generate additional tensioning in the cables (only in the nonlinear models)

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Chapter Three: Development of Models

3.4 Methodology of FE Modeling and Evaluating of Results

The thesis considers a large number of models whose geometric and material composition change at several levels in order to investigate the sensitivity of the configurations analyzed to the parameters of interest. Therefore, a universal parametric input file was developed to generate the required configurations efficiently. A general scheme of the model was prepared in the graphical interface of Abaqus v. 6.13 utilizing the finite elements described in Section A.1 and verified according to Section A.2. A sophisticated parametric input file was established based on an input file of a model with the general scheme. The internal structure of the input file was modified in order to accommodate the required parameters such as overall structural geometry, the configuration of the cross-sections, material composition, boundary conditions, and loads. Spread sheet based tools and particular keywords (commands) were combined to develop the parametric input file.

3.4.1 Pre-processing and Development of Universal Parametric Model

The parameters of interest are: • Superstructure geometry • Cross-sectional parameters and material properties definition • Load definition • Boundary condition definition

Typically, an input file consisted of a “model data” section defining nodes, elements, materials, assembly, or initial conditions and a “history data” section defining analysis type, loading, or output requests. Model and history data were separated by definition of a step. A step is described in the history data portion of the input file. Everything appearing before the first step definition is model data, and everything appearing within and following the first step definition is history data. The step defines the analysis procedure such as static, dynamic, heat transfer, etc. Only one analysis procedure was allowed per step and therefore multiple steps may have been required, as per Abaqus Analysis User’s Manual (2014).

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Chapter Three: Development of Models

Creating of FE models directly via a parametric input file is a powerful method whose main advantage is high efficiency while generating a large number of models. Nevertheless, this method requires a detailed approach and knowledge of particular keywords (commands). Essential parts of the model and history data of the universal parametric input file developed for the purposes of this thesis are described in the sections below. The entire parametric input file is provided in the Section A.3.

3.4.1.1 Superstructure Geometry

The spatial geometry of the superstructure was defined via coordinates of nodes in the arch, deck, and cables. The shape of the curved arch and curved deck was determined via a combination of trigonometric functions with a parabolic curve in general form expressed via Equation 3-1. Where f is the rise of the parabola, s is the span of the parabola, and x is the coordinate along span of the parabola. 4 = ( ) 𝑓𝑓 𝑦𝑦 2 ∗ 𝑠𝑠 − 𝑥𝑥 ∗ 𝑥𝑥 Equation 3-1 𝑠𝑠

The geometric variables incorporated in the spread sheet in order to define a particular shape and position of the arch and deck are specified below.

Arch related variables: • Span: s

• Arch rise: f(A)

• Arch rise to span ratio: f(A)/s • Angle of arch inclination from vertical plane: ω • Angle of rotation about vertical axis at located at midspan: θ

Deck related variables • Span: s

• Deck reach: f(D) (applicable only in configurations with curved deck)

• Deck reach to span ratio: f(D)/s (applicable only in configurations with curved deck)

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Chapter Three: Development of Models

• Rise of the camber of the deck in vertical direction: f(VC)

• Span of the deck assuming deck camber in vertical direction: s(VC) = s

• Ratio of the rise of the deck camber in vertical direction and the span: f/s(VC)

• Vertical offset of the arch and deck abutments: g(V)

• Horizontal offset of the arch and deck abutments: g(H)

Formulae that include the geometric variables of the arch and deck, following the equation of a parabola combined with specific trigonometric functions, are specified below. The X, Y, and Z coordinates correspond to the longitudinal, vertical and transversal directions, respectively. Node coordinates of the arch in the longitudinal, vertical and transversal directions are expressed via Equations 3-2, 3-3, and 3-3, respectively The parameter ( ) represents a reference coordinate that is increasing linearly from zero to total span in𝑋𝑋 𝑖𝑖the0 longitudinal direction between the abutments of the arch. 𝑠𝑠

= ( ) ( ) 2 2 ( ) 𝑠𝑠 𝑠𝑠 𝑋𝑋𝑖𝑖 𝐴𝐴 − 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 ∗ � − 𝑋𝑋𝑖𝑖 0 � Equation 3-2

4 ( ) ( ) = ( ) ( ) cos ( ) 𝑓𝑓 𝐴𝐴 𝑌𝑌𝑖𝑖 𝐴𝐴 � 2 ∗ �𝑠𝑠 − 𝑋𝑋𝑖𝑖 0 � ∗ 𝑋𝑋𝑖𝑖 0 � ∗ 𝜔𝜔 𝑠𝑠 Equation 3-3

( )

𝑍𝑍𝑖𝑖 𝐴𝐴 = 0, ( ) ( ) + ( ) ( ) ( ) , ( ) 1 ( ) = 𝜋𝜋 ⎡ 𝑖𝑖 0 𝑖𝑖 0 𝑤𝑤 𝐴𝐴𝐴𝐴 ⎤ 𝜃𝜃 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔 ∗ 𝑋𝑋 �𝑠𝑠𝑠𝑠𝑠𝑠 �𝑋𝑋 ∗ 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝐴𝐴 𝑖𝑖 ∗ 𝑘𝑘 � ∗ 𝑓𝑓 � ⎢ sin ( ) + ( ) �𝑁𝑁𝑁𝑁 − � ∗ 𝐿𝐿 ( ) ( ) ⎥ 𝐼𝐼𝐼𝐼 2 1 ⎢ 𝑠𝑠 ( )𝜋𝜋 ( ) ⎥ ⎢ 0 𝑖𝑖 0 𝑤𝑤 𝐴𝐴𝐴𝐴 ⎥ � − 𝑋𝑋 � ∗ 𝜃𝜃 �𝑠𝑠𝑠𝑠𝑠𝑠 �𝑋𝑋 ∗ 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝐴𝐴 𝑖𝑖 ∗ 𝑘𝑘 � ∗ 𝑓𝑓 � ⎣ �𝑁𝑁𝑁𝑁 − � ∗ 𝐿𝐿 Equation ⎦3-4

In the Equation 3-4, there is a condition that takes into consideration the angle , which is the angle of rotation of the arch about a vertical axis at located at the midspan 𝜃𝜃of

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Chapter Three: Development of Models the arch. When angle = 0, the equation assumes an arch only inclined from the vertical plane. Vertical planar 𝜃𝜃arches can be also defined; in such cases ( ) is zero. In the second part of the condition, i.e., the case when 0, the inclination of𝑍𝑍 𝑖𝑖the𝐴𝐴 arch from the vertical plane (angle ω) is zero and the arch is rotated𝜃𝜃 ≠ only about the vertical axis at midspan. It should be noted that the coordinate ( ) establishes whether or not the arch is contained in a plane. The necessity for a nonplanar𝑍𝑍𝑖𝑖 𝐴𝐴 arch may arise in situations where a spatial configuration requires an antifunicular arch. Equation 3-4 allows for the definition of a nonplanar arch via a sinusoidal goniometric function. Arguments of the function, the

( ) (the number of sinusoidal half-waves) and ( ) (the arch reach in the z-direction), can

𝑘𝑘be𝑤𝑤 varied in a form-finding process. Variables 𝑓𝑓 𝐴𝐴𝐴𝐴( ) and ( ) stand for the number of nodes in the arch and the projected length of the𝑁𝑁𝑁𝑁 arch𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 between𝐿𝐿 individual𝑖𝑖 nodes whose z- direction coordinate is being sought, respectively. To achieve a planar arch, the variable

( ) is set to zero. The shape of a nonplanar antifunicular arch is the result of a complex

𝑓𝑓form𝐴𝐴𝐴𝐴 -finding procedure. The details of such a procedure were described in Section 2.4.4. The coordinates of the deck nodes in the longitudinal, vertical and transversal directions are expressed via Equations 3-5, 3-6, and 3-7, respectively.

( ) = ( )

𝑋𝑋𝑖𝑖 𝐷𝐷 𝑋𝑋𝑖𝑖 0 Equation 3-5

4 ( ) ( ) = ( ) ( ) + ( ) 𝑓𝑓 𝑉𝑉𝑉𝑉 𝑌𝑌𝑖𝑖 𝐷𝐷 � 2 ∗ �𝑠𝑠 − 𝑋𝑋𝑖𝑖 0 � ∗ 𝑋𝑋𝑖𝑖 0 � 𝑔𝑔 𝑉𝑉 𝑠𝑠 Equation 3-6

4 ( ) ( ) = ( ) ( ) + ( ) 𝑓𝑓 𝐷𝐷 𝑍𝑍𝑖𝑖 𝐷𝐷 � 2 ∗ �𝑠𝑠 − 𝑋𝑋𝑖𝑖 0 � ∗ 𝑋𝑋𝑖𝑖 0 � 𝑔𝑔 𝐻𝐻 𝑠𝑠 Equation 3-7

The coordinates of the cable nodes in the longitudinal, vertical and transversal directions are expressed via Equations 3-8, 3-9, and 3-10, respectively. ( ) represents the location of a cable along a span. In a general configuration, the structure𝑋𝑋𝑖𝑖 𝐶𝐶 −comprises0 14 equally spaced cables located 5m apart in vertical planes perpendicular to the longitudinal line connecting the arch abutments. The parameters ( ) and ( ) stand for total

𝑁𝑁𝑖𝑖 𝐼𝐼𝐼𝐼 𝑁𝑁𝑖𝑖 𝑁𝑁𝑁𝑁 - 77 -

Chapter Three: Development of Models number of nodes within one cable and the node order in the cable, respectively. The inclination of the cables is a function of the mutual positions of the arch and deck.

( ) = ( )

𝑋𝑋𝑖𝑖 𝐶𝐶 𝑋𝑋𝑖𝑖 𝐶𝐶−0 Equation 3-8

( ) ( ) = + 1 ( ) ( ) ( ) 1 𝑌𝑌𝑖𝑖 𝐴𝐴( −) 𝑌𝑌𝑖𝑖 𝐷𝐷 𝑌𝑌𝑖𝑖 𝐶𝐶 𝑌𝑌𝑖𝑖 𝐷𝐷 �𝑁𝑁𝑖𝑖 𝐼𝐼𝐼𝐼 − � ∗ � � 𝑁𝑁𝑖𝑖 𝑁𝑁𝑁𝑁 − Equation 3-9

( ) ( ) = + 1 ( ) ( ) ( ) 1 𝑍𝑍𝑖𝑖 𝐴𝐴( −) 𝑍𝑍𝑖𝑖 𝐷𝐷 𝑍𝑍𝑖𝑖 𝐶𝐶 𝑍𝑍𝑖𝑖 𝐷𝐷 �𝑁𝑁𝑖𝑖 𝐼𝐼𝐼𝐼 − � ∗ � � 𝑁𝑁𝑖𝑖 𝑁𝑁𝑁𝑁 − Equation 3-10

It should be noted that in a linear model, the cables are created via truss elements that have only two nodes and location of these two nodes matches the coordinates of nodes in arch and deck. Therefore, Equations 3-8, 3-9, and 3-10 are not relevant in the linear model.

3.4.1.2 Definition of Cross-sectional Parameters and Material Properties

The cross-section of each structural component, i.e., arch, deck and cables, is defined via an overall dimension, a thickness of the assumed profile, and a location of section points. Section points are determined in order to read stress at the desired location. The definition of material properties is directly related to the assumed material. An example of cross-sectional parameters and material properties for an arch made of steel in the parametric input file is presented in Figure 3-8.

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Chapter Three: Development of Models

*PARAMETER **ARCH **arch material properties (STEEL) E_arch = 2.00E+11 G_arch = 7.70E+10 nu_arch = 0.3 Density_arch = 7800 CTE_arch = 1.20E-05 **arch outside dimensions and wall thickness r_arch = 0.375 t_arch = 0.025 **Section points arch SP_01_X1_A = 0.000000 SP_01_X2_A = 0.362500 SP_02_X1_A = 0.000000 SP_02_X2_A = -0.362500 SP_03_X1_A = 0.362500 SP_03_X2_A = 0.000000 SP_04_X1_A = -0.362500 SP_04_X2_A = 0.000000 *Part, name=Part01_Arch ** Section: Section01_Arch Profile: Profile01_Arch *Beam General Section, elset=_PickedSet131, poisson = , density=, section=PIPE , 0.,0.,-1. , , *Section Points , , , , , , , *End Part Figure 3-8: Example of a parametric definition of cross-sectional parameters and material properties of a steel arch

3.4.1.3 Definition of Loads

Load in the parametric input file is applied in two individual steps. Static and thermal loads are defined in the first step, while the second step is defined to determine additional tensioning of cables.

3.4.1.3.1 Definition of Static Load Static load is taken as gravity load and a live load. The gravity load is applied to all components and the live load is applied as a line load on the deck. The gravity load reflects specific density of the materials defined in Table 3-1. The magnitude of the live load is specified in Section 3.3.5.

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Chapter Three: Development of Models

An additional weight resulting from a non-structural mass that represents a walking surface is determined via a line load as well as the live load, i.e., there are two loads defined as a line load. The magnitude of the additional weight is 972N/m, which is kept constant in all models. Both the gravity and the live load were defined in the first loading step. An example of a gravity and live load definition in the parametric input files is presented in Figure 3-9.

*PARAMETER **Magnitude of LL Ped_Load=-5000 Walking_Surf=-972 LL=Ped_Load+Walking_Surf ** LOADS ** Name: GravityLoad Type: Gravity *Dload _PickedSet270, GRAV, 9.81, 0., -1., 0. ** Name: Load_02_LL Type: Line load *Dload Deck_Segments, PY, ** Figure 3-9: Example of parametric definition of gravity and live loads

3.4.1.3.2 Definition Global Thermal Load A global thermal load (GTL) was defined in the first loading step as a predefined field. The magnitude of the GTL varied for particular configurations. An example of global thermal load definition in the parametric script is presented Figure 3-10.

*PARAMETER **Global_Thermal_Loading G_Th_L =-50 ** LOADS ** PREDEFINED FIELDS ** Name: Predefined Field_Th_Load Type: Temperature *Temperature _PickedSet272, ** Figure 3-10: Example of parametric definition of a global thermal load

3.4.1.4 Additional Tensioning of the Cables

Even though other methods exist, for example through definition of an initial stain in the material, additional tensioning of the cables, applied to the models considered in this

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Chapter Three: Development of Models thesis, is implemented through application of localized thermal load (LTL). It is the negative thermal load that is applied to the cables that generates additional tensioning in those cables. Negative thermal load results in a desired contraction of the cables that will then develop additional tension because the cable lengths are fixed between the arch and the deck. In practical applications, additional tensioning of the cables would be achieved with a jack. Due to the symmetrical arrangement of the models assumed in this thesis, the LTL is defined to two cables at a time because the lengths of the cables that are located at the same distance from the vertical centre line of the bridge are identical. Additional tensioning of cables is employed only in Chapter Six. Chapters Four and Five do not examine the effect of additional tensioning of cables. A magnitude of the LTL determines the additional force that is developed in the cables. There are several ways to determine an “optimal” level of additional cable tensioning. An analysis that searches for a required deck shape is typically involved. It is the DL that is being considered when determining the shape of the deck (Nettleton, 1997). The magnitude of the total force on a cable (including additional cable tensioning) in practical applications should not exceed 40% of the ultimate strength of the cables under permanent load condition (Chen et al., 2000). In this thesis, the magnitude of the –ve LTL (the temperature) that develops the required additional tensile force in individual cables is determined via Equation 3-11. Equation 3-11 is based on equation of parabolic curved, captured in Equation 3-1.

( ) ( ) ( ) = + 4 × 1 × 𝐶𝐶 𝑖𝑖 𝐶𝐶 𝑖𝑖 𝐿𝐿𝑇𝑇𝑇𝑇 𝑖𝑖 𝑇𝑇𝑇𝑇 𝑚𝑚 � − � 𝑛𝑛 𝑛𝑛 Equation 3-11

Therefore, the parabolic distribution was selected in order to account for the factors listed above via three different parameters: (1) ; the base temperature that is applied to all cables regardless of their length, (2) ; 𝑇𝑇𝑇𝑇the peak value added to at midspan (representing a “rise” of the parabolic curve),𝑚𝑚 and (3) ; the parameter determining𝑇𝑇𝑇𝑇 the increase in curvature of the parabolic distribution. The default𝑛𝑛 magnitude of the parameter

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Chapter Three: Development of Models

is 15 which reflects the 75m span of the bridge and the 5 m spacing of the cables in the

𝑛𝑛reference structural configuration assumed in this thesis. Values of n below 15 are beneficial because the contribution of both the CF(V) and

CF(H) components of the CF are less and reduce the deck deflection at midspan; therefore, less additional cable tensioning is required. The parameter ( ) indicates a coordinate of individual cables. In the reference structural configuration, there𝐶𝐶 𝑖𝑖 are 14 cables organized symmetrically, hence, “i” ranges from 1 to 7. An example of LTL definition in the parametric input file that assumes parameters = 200° , = 200° , and = 15 is presented Figure 3-11. 𝑇𝑇𝑇𝑇 − 𝐶𝐶 𝑚𝑚 − 𝐶𝐶 𝑛𝑛

*PARAMETER **Cables_Additional_tensioning_temperatures T_Clbs_01_14 = -249.8 T_Clbs_02_13 = -292.4 T_Clbs_03_12 = -328.0 T_Clbs_04_11 = -356.4 T_Clbs_05_10 = -377.8 T_Clbs_06_09 = -392.0 T_Clbs_07_08 = -399.1 ** LOADS ** PREDEFINED FIELDS ** Name: PT01 Type: Temperature *Temperature Part03_Cables-1.Set03_Cables01, ** Name: PT02 Type: Temperature *Temperature Part03_Cables-1.Set03_Cables02, ** Name: PT03 Type: Temperature *Temperature Part03_Cables-1.Set03_Cables03, ** Name: PT04 Type: Temperature *Temperature Part03_Cables-1.Set03_Cables04, ** Name: PT05 Type: Temperature *Temperature Part03_Cables-1.Set03_Cables05, ** Name: PT06 Type: Temperature *Temperature Part03_Cables-1.Set03_Cables06, ** Name: PT07 Type: Temperature *Temperature Part03_Cables-1.Set03_Cables07, ** Figure 3-11: Example of parametric definition of LTL utilized for additional cable tensioning

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Chapter Three: Development of Models

3.4.1.5 Definition of Boundary Conditions

The BCs of the arch and the deck represent conditions that would be present at the ends of the arch and deck in practice. The end nodes of the arch and deck represent the springings and abutments of the arch and the deck, respectively. For the arch, both ends were taken to be fixed. In the deck, not all DOFs were restrained and, therefore, the deck end nodes were allowed to translate or rotate in certain directions as a function of the selected configuration of SABs. In the parametric input file, the availability of a certain DOF in a specific direction is simply represented via a listing of the DOF. Omission of a specific translation or rotation means that DOF is allowed. An example is presented on Figure 3-12. In the figure, it is shown that DOFs 1, 2, 3, and 4 are listed and, therefore, restrained. The DOFs 5 and 6 not listed and, therefore, free, allowing rotation about the Y and Z axes.

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Chapter Three: Development of Models

*PARAMETER **Orientation_Of_Deck_BC_as_function_of_deck_rise_for_Datum_CSYS Deck_f_to_s_ratio=0.25 span=75 deck_reach=Deck_f_to_s_ratio*span X=1 **"Z" coordinated for Datum_CSYS Z=(4*deck_reach/span**2)*(span-X)*X ** ASSEMBLY *Assembly, name=Assembly *Nset, nset=Deck_LHS_node, internal, instance=Part02_Deck-1 1, *Nset, nset=Deck_RHS_node, internal, instance=Part02_Deck-1 17, *Nset, nset="_T-Datum csys_01", internal Deck_LHS_node, *Transform, nset="_T-Datum csys_01" ,0.,,0,1,0 *Nset, nset="_T-Datum csys_02", internal Deck_RHS_node, *Transform, nset="_T-Datum csys_02" -,0.,,0,1,0 *Orientation, name="Datum csys-1" 1.,0.,0., 0., 1., 0. 1, 0. *End Assembly ** ** BOUNDARY CONDITIONS ** Name: BC-2_DeckBC_LHS Type: Displacement/Rotation *Boundary Deck_LHS_node, 1, 1 Deck_LHS_node, 2, 2 Deck_LHS_node, 3, 3 Deck_LHS_node, 4, 4 ** Name: BC-2_DeckBC_RHS Type: Displacement/Rotation *Boundary Deck_RHS_node, 1, 1 Deck_RHS_node, 2, 2 Deck_RHS_node, 3, 3 Deck_RHS_node, 4, 4 **

Figure 3-12: Example of parametric definition of local direction for deck boundary condition

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Chapter Three: Development of Models

A schematic representation of the general six DOFs (three translational and three rotational) is shown in Figure 3-13.

Global directions: X, Y, Z 3

Y X

1 4 3 5 6

6 5

Deck

2 4 1

Figure 3-13: A schematic representation of general directions of DOFs shown on a curved deck

In Chapters Four and Five, the orientation of deck BCs follows the global orientation as shown in Figure 3-13. However, in Chapter Six, in order to achieve BCs that are more feasible in industrial applications, the BCs of the deck were altered; a local orientation is selected. In such case, the direction “1” follows the longitudinal axis of the deck. Because the local orientation of the deck BCs, in Chapter Six is a function of the deck shape, in particular the deck reach f(D) in curved decks, the local orientation of the deck boundary conditions is defined as a parameter. In configurations with a straight deck, the local orientation follows the global directions. However, in configurations with a curved deck, the local orientation of the deck BCs parameter plays a key role. An example of definition of the local orientation of the deck BCs is presented in Figure 3-12. As outlined in Section A.2, in order to avoid numerical singularities resulting from under-defined BCs, the deck must have another constraint. This constraint is a “vertical

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Chapter Three: Development of Models roller” located at the mid-span of the deck, constraining translation in the longitudinal direction (X global direction). All the other translational and rotational DOFs are available. By constraining the translations only in the longitudinal direction, the FE model becomes spatially stable, and susceptibility to numerical singularities decreases dramatically. The significance of variability in the BCs of the deck is studied in Chapter Five.

3.5 Material Properties used in FE Models

The material properties used in the analyses are presented in Table 3-1. The properties were taken from the Handbook of steel construction for steel, Dalavos (1996) for advanced composite materials (ACMs), and Wang & Wu (2010) for the cables. Table 3-1: Material properties of assumed materials STEEL GFRP CFRP Material property Arch Arch Arch Cables Cables Cables &Deck &Deck &Deck E [GPa] 200 165 54 80 147 230 G [GPa] 77.00 63.30 3.77 3.77 3.84 3.84 CTE [m/m/°C] 12E-06 12E-06 8E-06 8E-06 2.8E-06 2.8E-06 nu [-] 0.30 0.30 0.27 0.27 0.24 0.24 ρ [kg/m^3] 7,800 7,800 1,982 2,600 1,535 1,600 F(t-max) [MPa] 500 1,630 1,800 1,500 3,250 3,400 Stress Limit [%] N/A 40% 20% 20% 50% 50% Stress Limit [MPa] 400 652 360 300 1,625 1,700

In the case of ACMs, the intention here was not to develop material models that would reflect any specific product. The objective was to develop models with generic material properties that would represent two types of ACMs used for pultruded profiles. It should be noted that although the properties of pultruded profiles are not strictly isotropic, isotropic material models listed for the Glass Fibre Reinforced Polymer (GFRP) and Carbon Fibre Reinforced Polymer (CFRP) were assumed because the main contribution to member stiffness is provided via longitudinally oriented fibres having isotropic character.

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Chapter Four: Significance of Geometry

4.1 Introduction In Chapter Four the significance of geometric variables on the overall structural response of the three assumed configurations of SABs is investigated, under changing global thermal load (GTL) and live load (LL). The GTL is applied equally over the entire structure at three distinctive levels, and is considered to be the main load in question. The LL is applied in two load cases to identify its effect on the bending moment in the arch. In conventional structures, thermal loads can result in extensive stresses and deformations, and therefore, in the present chapter, attention is mainly given to this kind of loading. Linear FE parametric models are employed to perform the proposed analyses (details on linear parametric models are provided in Section 3.3.1.2). Chapter Four is divided into five sections. Section 4.2 outlines the main objectives of this chapter. The specifications of the linear FE models and the methodology are provided in Section 4.3, with details of the geometry of the three assumed configurations in relation to the developed FE models. The range of individual variables, magnitudes of assumed loads, and combinations of the variables are specified. Section 4.4 focuses on a description and discussion of the results. The results are discussed taking into account the significance of the individual variables and the type of structural response in question. Several types of structural responses are evaluated, such as axial forces, bending moments in the arch, bending moments in the deck, and tensile forces in the cables. The key understandings obtained from the results are discussed, and details of the investigation are provided in Appendix B. Section 4.5 summarizes the main contributions of the present chapter. It focuses on the trends in the structural response of individual components (the arch, deck, and cables) taking into account the assumed geometric variables and two types of load (LL and GTL). Critical configurations are identified, and the effects of the key parameters are discussed. The obtained understanding of the structural response provides the basis for the subsequent chapters of this thesis.

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Chapter Four: Significance of Geometry

4.2 Objectives

The main goal of Chapter Four is to investigate the significance of geometric variables on structural response in the arch, deck, and cables in the three structural configurations, while taking into account the effect of GTL and LL. The individual objectives of this Chapter are:

• To determine if the effect of GTL is a significant issue in the assumed configurations of SABs compared to the effect of LL;

• To investigate which of the defined geometric variables has the most influence on the structural response under GTL and LL;

• To identify which combination of primary and secondary variables represents the most sensitive configuration to provide a base configuration for following chapters; and

• To describe the structural response of the three main configurations under GTL and LL and to provide a guideline for the analysis and design of configurations that are similar in nature to those presented in Chapter Four.

4.3 Model Specifications

4.3.1 Characteristics of the Models and Applied Materials

Linear FE models are used to evaluate the structural response of the assumed configurations. The developed configurations assume steel as the main structural material of the arch and deck. The cables comprise high tensile strength steel tendons. Specific material properties are provided in Section 3.5. Additional tensioning of the cables is not considered.

4.3.2 Reference Configuration

The structural response from the spatial configurations assumed in the present chapter is being compared to a reference configuration. The reference configuration is a vertical planar arch supporting a straight deck. The BCs of the arch and deck are the same as in the spatial configurations. The primary variable (the f(A)/s ratio) of the reference

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Chapter Four: Significance of Geometry configuration varies in the same fashion as in the spatial configurations in order to provide comparable geometries.

4.3.3 Boundary Conditions

The boundary conditions (BCs) of the arch and deck are the same in the three configurations. A schematic sketch of the assumed BCs is shown in Figure 4-1.

Arch

UR3 UR3

U1 U1

Deck

U1 = Translation Reference line UR3 = Rotation

Deck

Figure 4-1: A schematic sketch depicting the boundary conditions of the arch and deck

The arch is fixed at both ends. BCs of the deck allow for the following: 1) a longitudinal displacement in the direction parallel to the line (the “reference line”) that connects the springings of arch and 2) a rotation about the horizontal axis that is

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Chapter Four: Significance of Geometry perpendicular to the reference line at the ends of the deck. The BCs in the FE models are described in Section 3.4.1.5. The reason for the same BCs in all three configurations (regardless of their geometric uniqueness) is to investigate the influence of the assumed concept and determine the factors that may result in a more efficient design. The significance of the various BCs of the deck for the overall structural response of the three configurations is investigated in Chapter Five. The abutments of the arch and deck lie on the same reference line, and therefore, there is no vertical or lateral difference between the starting and ending coordinates of the arch and deck.

4.3.4 Geometry of Spatial Configurations

The structural responses are investigated in the three spatial configurations to cover most cases of possible out-of-plane loads. The assumed configurations are as follows: Configuration C01 has a vertical arch supporting a curved deck, Configuration C02 has an arch inclined from a vertical plane supporting a straight deck, and Configuration C03 has a vertical arch rotated about its vertical axis at the midspan supporting a straight deck. In all configurations, the arch is planar, symmetric in plan-view, and its shape follows a parabolic curve (described by Equation A-16). The shape of the curved deck in Configuration C01 is also a parabolic curve. The curved deck is contained in a horizontal plane. Structural arrangements and a visualization of the primary and secondary variables in Configuration C01, C02, and C03 are presented in Figure 4-2, Figure 4-3, and Figure 4-4, respectively. The primary and secondary variables relevant to the three assumed configurations are presented Table 4-1. While the primary and secondary variables change, the assumed span of 75m remains constant in all conducted analyses. Table 4-1: Primary and secondary variables of the assumed spatial configurations Configuration Primary Variable Secondary Variable

C01 f(D)/s: Deck reach over span ratio f(A)/s: Arch rise C02 ω: Angle of arch inclination from a vertical plane over span ratio C03 θ: Angle of arch rotation about a vertical axis

A range of the primary and secondary variables in the configurations is presented in Table 4-2.

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Chapter Four: Significance of Geometry

Table 4-2: Range of primary and secondary variables Configuration C01 Configuration C02 Configuration C03 Parameter f(A)/s ratio f(A)/s ratio f(A)/s ratio 0.15 0.20 0.25 0.15 0.2 0.25 0.15 0.2 0.25 Span; s = [m] 75 75 75 75 75 75 75 75 75 Arch Rise; 11.25 15 18.75 11.25 15 18.75 11.25 15 18.75 f(A) = [m] Deck Shape Curved Straight Straight 0 0 0 Deck Reach; 11.25 11.25 11.25 N/A f(D) [m] 15 15 15 18.75 18.75 18.75 Angle of 0 0 0 Arch 15 15 15 N/A N/A Inclination; 30 30 30 ω = [°] 45 45 45 Angle of 0 0 0 Arch 15 15 15 N/A Rotation; 30 30 30 θ = [°] 45 45 45

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Chapter Four: Significance of Geometry

f(A)

s f(D)

f(D)

Figure 4-2: Configuration C01: a graphical representation of the relevant geometric variables and a 3D perspective view

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Chapter Four: Significance of Geometry

f(A) f(A)

Figure 4-3: Configuration C02: a graphical representation of the relevant geometric variables and a 3D perspective view

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Chapter Four: Significance of Geometry

f(A)

Figure 4-4: Configuration C03: a graphical representation of the relevant geometric variables and a 3D perspective view

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Chapter Four: Significance of Geometry

4.3.5 Cross-sectional Properties

In the models, the structural members, such as the arch and deck, comprise hollow structural sections (HSS). A solid circular section is assumed for the cables. The specific parameters of the assumed cross-sections are listed in Table 4-3. In the table, I22 represents the moment of inertia about a vertical axis and I11 represents the moment of inertia about a horizontal axis.

Table 4-3: Cross-sectional properties of the arch, deck, and cables Parameter Arch Deck Cables Shape

Diameter [m] 0.750 N/A 0.015 Height [m] N/A 0.400 N/A Width [m] N/A 2.000 N/A Wall thickness [m] 0.025 0.020 N/A Cross-sectional Area [m2] 4.59E-02 5.94E-02 1.77E-04 Moment of inertia: I22 [m4] 3.06E-03 5.04E-02 2.49E-09 Moment of inertia: I11 [m4] 3.06E-03 3.70E-03 2.49E-09 Torsional constant [m4] 6.12E-03 1.17E-02 4.97E-09 Weight per unit length [N/m] 3,860 8,990 14

The selected cross-sectional properties remain unchanged in the analyses. These cross-sectional properties were selected based on case studies of a similar nature (for example Sarmiento-Comesias et al., 2011). It should be noted that the idea was not to design a specific structural member. The intention was to select sections that have reasonable dimensions and allow for the evaluation of the structural response in question.

4.3.6 Applied Loads

Three types of load are applied to the FE models: DL, LL, and GTL. The magnitude of the loads was determined according to Section 3.3.5.

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Chapter Four: Significance of Geometry

The distribution of LL over the deck can significantly influence the distribution and magnitude of the internal forces in the arch and deck. In vertical planar arches, LL applied only to one half of the deck results in larger bending moments in the arch (Nettleton, 1977). In contrast, in particular configurations of SABs, larger internal moments in the arch resulted from LL applied uniformly over the entire deck (Sarmiento-Comesias, 2011). Therefore, in the configurations considered here, the LL is applied using two arrangements: 1) LL applied over the entire deck (LL100) and 2) LL applied over one half of the deck (LL50). LL50 starts at the abutment and ends at the midspan of the deck. GTL, applied equally to all structural components, is considered to be the main load because, in conventional structures, this type of load can cause extensive stresses (Dilger, 1983 and Sherif, 1991). Therefore, in spatially complex structures, such as SABs, this concern also arises. The magnitude of the GTL is a function of the assumed temperature change. There are three temperature levels considered in this study. The reference level is 0°C. The other two levels represent positive (+ve) and negative (–ve) thermal loads with magnitudes of -50°C and +50°C, respectively. The magnitude of GTL was selected to reflect the Canadian climate to examine the effect of possible extreme GTL. The +ve or –ve GTL applied simultaneously with DL represents two loading cases. The LL50 or LL100 applied simultaneously with DL represents other two load cases. DL applied individually also represents one loading case. The DL is always applied to the structure. In total, there are five loading cases that are being applied and the responses from these cases compared. The abbreviations of the five load cases are: a) DL, b) +ve GTL, c) –ve GTL, d) LL50, and e) LL100.

4.3.7 Structural Response

The thirteen outputs investigated are summarized in Table 4-4. From the list, the key outputs such as the axial force (SF1) and combined moment (SM(COMB)) in the arch, combined bending moment (SM(COMB)) in the deck, and tensile axial force in the cables (CF) are discussed in detail in Section 4.4. An evaluation of the remaining outputs is implemented in the complete result tables, describing the trends in the structural behaviour, and is provided in Appendix B.3.

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Chapter Four: Significance of Geometry

Table 4-4 lists single outputs, such as SF1 and CF, and combined outputs, such as

SM(COMB). The approach of the combined outputs was selected to account for the effect of out-of-plane loads to which the arch and deck are exposed. The combined outputs are shear force (SF(COMB)), section bending moment (SM(COMB)), and displacement (U(COMB)), and the combined quantities are calculated using Equation 4-1, Equation 4-2, and Equation 4-3, respectively.

( ) = 2 + 3 2 2 𝑆𝑆𝑆𝑆 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 �𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 Equation 4-1

( ) = 1 + 2 2 2 𝑆𝑆𝑆𝑆 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 �𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 Equation 4-2

( ) = 2 + 3 2 2 𝑈𝑈 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 �𝑈𝑈 𝑈𝑈 Equation 4-3 Table 4-4: List of structural responses evaluated in Chapter Four Symbol Description SF1 Axial force in the direction of the longitudinal axis of the structural component SF2 Shear force in the vertical direction SF3 Shear force in the horizontal transversal direction

SF(COMB) Combined shear force taking into account the effect of both SF2 and SF3 SM1 Section bending moment about the local axis in the horizontal transversal direction (vertical bending moment) SM2 Section bending moment about the local axis in the vertical direction (transversal bending moment)

SM(COMB) Combined bending moment taking into account the effect of both SM1 and SM2 SM3 Torsional moment U2 Displacement in the vertical direction U3 Displacement in the horizontal transversal direction

U(COMB) Combined displacement taking into account the effect of both U2 and U3 CF Axial force in a cable

CF(H) Horizontal component of the cable force

CF(V) Vertical component of the cable force

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Chapter Four: Significance of Geometry

4.4 Results and Discussion

This section is organized as follows: Section 4.4.1 determines the critical pattern of the LL distribution that governs the magnitude of the bending moment in the arch. The critical case of LL is compared to GTL in Section 4.4.2. Furthermore, Section 4.4.2 determines the combination of the primary and secondary variables that results in a configuration that is the most sensitive to the applied GTL and LL. A geometric variable (either primary or secondary) that controls the arch behaviour is identified. Section 4.4.3 discusses the mechanism of the structural response and underlines the new achieved understanding. In order to avoid confusion, two terms should be clarified. The configuration whose combination of primary and secondary variables results in the largest change in the structural response is called the most “significant” configuration. It can be also said that the configuration is the most “sensitive” to the applied load. However, the most significant (the most sensitive) configuration does not always correlate with a configuration in which the maximum magnitude (in the particular type of the structural response) occurs. Hence, the configurations that are the most significant (most sensitive) and configurations that result in maximum magnitude are differentiated within the text. In the text that follows two terms are being used: a) “directly proportional” and b) “inversely proportional”. These terms were adopted in order to describe the character of the relationship between the change in geometry and the change in the structural response. When the term directly proportional is used, for example: “a change in bending moment in the arch is directly proportional to change in the f(A)/s ratio”, the intention is to express that an increase in the geometric variable, the f(A)/s, results in an increase in the bending moment in the arch. When the term inversely proportional is used, for example: “a change in axial load in the arch is inversely proportional to the f(A)/s ratio”, the intention is to express that when the geometric variable decreases, the axial load in the arch increases. These two terms are often use to avoid lengthy repetitive descriptions.

4.4.1 Critical Live Load Distribution

In the three spatial configurations, all possible combinations of primary and secondary variables were compared to determine the critical pattern of applied LL. Details

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Chapter Four: Significance of Geometry of the comparison of the two patterns of applied LL are provided in Appendix B, Table B-1, Table B-2, and Table B-3 for the C01, C02 and C03, respectively.

The pattern of LL that results in a larger SM(COMB) in the arch is presented in Table 4-5. A comparison of the bending moment distributions for the two LL patterns in the reference vertical planar arch, C01, C02, and C03 are presented in Figure 4-5, Figure 4-6, Figure 4-7, and Figure 4-8, respectively.

Table 4-5: Critical patterns of LL distribution in the three spatial configurations Configuration Critical live load distribution C01 LL100: Live load distributed equally over the entire length of the deck C02 LL50: Live load distributed only over one half of the deck starting at the deck abutment and ending at the midspan C03 LL100: Live load distributed equally over the entire length of the deck

From Figure 4-5, the reference arch behaves as anticipated; LL50 results in a significantly larger bending moment in the arch than the bending moment for LL100. The difference between the bending moments resulting from LL50 and LL100 is ~750% (see

Table B-1), and it occurs in the configuration with f(A)/s = 0.25 (the largest in the range).

Figure 4-5: A comparison of the bending moment distributions in the reference

vertical planar arch f(A)/s = 0.25

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Chapter Four: Significance of Geometry

In C01, the larger SM(COMB) results from LL100, as shown in Figure 4-6. The difference between LL50 and L100 is ~15% (see Table B-1) in a combination of f(A)/s =

0.15 and f(D)/s = 0.25; this result is an opposite trend to the reference arch. Therefore, the difference between moments is inversely proportional to arch rise f(A) and directly proportional to deck reach f(D).

Figure 4-6: A comparison of the bending moment distributions in the arch of C01

with f(A)/s = 0.15 and f(D)/s = 0.25

A comparison shown in Figure 4-7 indicates that the inclined arch in C02 follows the same behavioural pattern as the reference arch: LL50 causes a larger SM(COMB) in the arch. The largest difference in magnitude, ~38% (see Table B-2), due to LL100 and LL50, occurs in the configuration with f(A)/s = 0.15 and ω = 15°. The trend in sensitivity to the type of LL pattern is inversely proportional to the arch rise and to the angle of arch inclination from the vertical plane. Further, even though the difference between moments is smaller in the combination with f(A)/s = 0.25 and ω = 45° (~6%), the overall magnitude of SM(COMB) in this configuration is larger than in the combination with f(A)/s = 0.15 and ω = 15°.

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Chapter Four: Significance of Geometry

Figure 4-7: A comparison of the bending moment distributions in the arch of C02

with f(A)/s = 0.15 and ω = 15°

From Figure 4-8, SM(COMB) resulting from LL100 is larger than that from LL50 (see

Table B-3). The only exception is the combination with f(A)/s = 0.25 and θ = 15° where

SM(COMB) from LL50 is larger than that from LL100. Nevertheless, the difference in the combination is only ~1.5% and, therefore, LL100 is the critical case that is used in subsequent analyses. In C03, the largest difference between LL100 and LL50 is ~6.5%, which occurs in the combination with f(A)/s = 0.15 and θ = 30°. The smallest difference of

~0.25% is present in the combination with f(A)/s = 0.20 and θ = 15°. Consequently, in C03, the difference between the bending moments that result from LL100 and LL50 is highly dependent on a particular combination of f(A) and angle θ, and therefore, this case may require additional attention during analysis.

Figure 4-8: A comparison of the bending moment distributions in the arch of C03

with f(A)/s = 0.15 and θ = 30°

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4.4.2 Comparison of Global Thermal Load with Critical Case of Live Load

The structural response in the critical case of the applied LL is compared with the structural response in the case with the applied GTL. In particular, the change in the structural response resulting from the addition of the GTL or the LL to the initial state, when the structure is exposed only to DL, is compared. DL is the case that is always applied on the structure. The comparison is carried out for all three configurations under all the assumed geometric combinations. Both the positive and negative GTL are included in the comparison. The geometrical variables that control a structural response are identified for all components of the structure. The sections below discuss the crucial types of response such as SF1, SM(COMB), SM3, and CF. Details related to the other types of responses such as SF(COMB) and U(COMB) are provided in Appendix B. A comparison of the structural response in all three structural components (arch, deck and cables) is performed in a tabular form accompanied with charts showing the distribution of a particular response. Values of the internal forces, provided in the tables, represent maximum magnitudes achieved in specific load cases. Each configuration and structural component is assigned a set of two tables. These two tables show the magnitude of the particular response in question resulting from application of DL, DL and LL, DL and +ve GTL, and DL and –ve GTL. The first table from the set lists the magnitudes of the internal forces and the second compares the change due to the applied loads. The difference between individual cases of LL and GTL is indicated in percentage (%) to identify the variation in the response due to +ve and –ve GTL. The last column in the second table states whether or not the applied GTL has a larger effect than the critical case of LL. It should be noted that particular values of a change in a specific type of structural response, presented in the body of text, are taken from the two detailed result tables. Direct cross-reference to these result tables is provided; these result tables are placed in a relevant appendix section, due to their large size.

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4.4.2.1 Configuration C01

4.4.2.1.1 Expected Behaviour In C01, the effect of GTL is compared with the effect of LL100. The arch in C01 acts as a curved cantilever beam that resists the out-of-plane load transferred from the deck via the cables (see Figure 4-2).

The horizontal component of the force in the cable, CF(H), develops the out-of-plane bending moments in the arch, and the vertical component of the force in the cable, CF(V), causes the in-plane bending moments and axial forces in the arch. Even though the deck in C01 takes a curved shape, the deck does not act as an arch because the BCs of the deck (see Section 4.3.3 for details) does not allow for the development of an axial thrust in the deck. Therefore, the deck acts rather as a curved beam (resisting bending) than an arch (resisting compression). The arrangement of the BCs of the deck ensures that the bending stiffness (EI) of the deck controls the structural response in the arch. The significance of EI of the deck is studied in Chapter Five. As a consequence of the structural combination of the arch, deck, and BCs of the deck, it is expected that the arch will be exposed to large bending moments and relatively small axial loads compared to the reference arch.

4.4.2.1.2 Axial Force in the Arch GTL has only a minor effect on axial force (SF1) in the arch compared to LL100. The largest increase in SF1 due to GTL is only ~3% (see Tables B-4 and B-5). An increase in the axial force due to LL100 is approximately ten times larger than the increase due to GTL. A comparison of the effects of GTL and LL100 in the most sensitive combination with f(D)/s = 0.25, taking into account the variability in the f(A)/s ratio, is shown in Figure

4-9. The figure shows that, with an increasing f(A)/s ratio, the significance of the applied GTL decreases. The most variation in the distribution of SF1 occurs at the midspan, and the least variation occurs at the abutments of the arch. The primary variable (f(A)/s) has a more significant impact on the SF1 distribution than the secondary variable (f(D)/s). A change in SF1 due to f(A)/s is ~48% and due to f(D)/s is ~3% (see Table B-60). Therefore, the primary variable controls SF1 in the arch when GTL is considered.

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Figure 4-9: Configuration C01 — Distribution of axial forces in the arch comparing the critical cases of LL100, +ve and –ve GTL, and the unloaded state for three levels

of f(A)/s and the most sensitive ratio, f(D)/s = 0.25

The significance of the changing primary and secondary variables under LL100 is compared in Figure 4-10. As may be seen, the sensitivity of SF1 in the arch is directly proportional to f(A)/s and inversely proportional to f(D)/s. The largest increase (~32%) occurs when f(A)/s = 0.25 and f(D)/s = 0.15. The highest SF1 is exerted on the arch when f(A)/s = 0.15 and f(D)/s = 0.15. In this configuration, the influence of the cantilevered, curved deck on the out-of-plane deflection of the arch is the lowest (the highest occurs in configuration with f(D)/s = 0.25), and therefore, the arch experiences a high SF1 compared to the other levels of f(D)/s. As f(A) increases, the influence of deck reach f(D) increases, i.e., SF1 in the arch drops. Even though the horizontal component of the cable force is the smallest in the configuration with a large f(A), the flexibility of a tall arch is large, and therefore, the arch is more prone to out-of-plane displacement, which results in the drop of

SF1 in the arch. The secondary variable, f(D)/s, has a significant influence on the magnitude and distribution of SF1 in the arch. However, a higher change in SF1 occurs due to the change in the primary variable, the f(A)/s ratio. Thus, the analysis or design of the arch for C01 should consider the magnitude of the primary variable first.

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Figure 4-10: Configuration C01 — Distribution of axial forces in the arch considering

the effect of LL100 for three levels of f(A)/s (0.15, 0.20, and 0.25) and four levels of

f(D)/s (0.00, 0.15, 0.20, and 0.25)

4.4.2.1.3 Combined Bending Moment in the Arch

The effect of GTL on the magnitude and distribution of SM(COMB) in the arch of C01 is minor compared to the effect of LL100. The most sensitive combination with the highest difference (~1.2%) due to –ve GTL is f(A)/s = 0.15 and f(D)/s = 0.15 (see Table B-9).

The significance of GTL on the distribution of SM(COMB) is inversely proportional to both f(A)/s and f(D)/s. This trend is similar to all assumed combinations of the primary and secondary variables.

A comparison of the effect of GTL and LL on the distributions of SM(COMB) in the arch in the combination with the smallest f(D)/s ratio (f(D)/s = 0.15) and three levels of f(A)/s is presented in Figure 4-11. A distribution of SM(COMB) in the arch that results from LL100 is presented in Figure 4-12.

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Figure 4-11: Configuration C01 — Distribution of combined moments in the arch comparing the critical cases of LL100, +ve and –ve GTL, and the unloaded state for

three levels of f(A)/s and the most sensitive ratio, f(D)/s = 0.15

Figure 4-12: Configuration C01 — Distribution of combined bending moments in the arch considering the effect of LL100 for three levels of f(A)/s (0.15, .020, 0.25) and four

levels of f(D)/s (0.00, 0.15, 0.20, and 0.25)

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The following can be concluded from the figures: • LL100 represents a governing load compared to the effect of GTL.

• For LL100, SM(COMB) increases with an increase in both the primary and secondary

variables; the increase in SM(COMB) due to the primary variable is greater than that due to the secondary variable.

• The primary variable controls the magnitude and distribution of SM(COMB) in the arch.

4.4.2.1.4 Torsional Moment in the Arch The effects of GTL and LL100 on the torsional moment (SM3) in the arch are compared in Figure 4-13. As shown in the figure, LL100 governs the magnitude of SM3.

Figure 4-13: Configuration C01 — Distribution of torsional moments in the arch comparing the critical cases of LL100, +ve and –ve GTL, and the unloaded state for

three levels of f(A)/s and the most sensitive ratio, f(D)/s = 0.25

LL100 results in a maximum change of 39% when f(A)/s = 0.25 and f(D)/s = 0.25, i.e., the combination with the largest deck reach and arch rise. The change in SM3 for the same combination of primary and secondary variables due to GTL is only 3% (see Table B-11). The sensitivity to GTL is directly proportional to both the primary and secondary variable; in other words, the largest change due to GTL in SM3 occurs in a combination with a large deck reach and a large arch rise.

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The effect of changing geometry on SM3 in the arch under the governing LL100 case is presented in Figure 4-14. In addition to the sensitivity of SM3 to a change in geometry, the maximum magnitude of SM3 is directly proportional to f(A)/s and f(D)/s; in other words, the maximum torsion is exerted on the arch when both the deck reach and arch rise are large (see Table B-10). This behaviour is expected because the magnitude of the torsional moment is a function of both the deck reach and arch rise. The effect of arch rise is larger than the effect of deck reach. Therefore, the arch rise should be given the most attention during the design process to control the torsion in the arch efficiently.

Figure 4-14: Configuration C01 — Distribution of torsional moments in the arch

under LL100 three levels of f(A)/s (0.15, 0.20, 0.25) and for four levels of f(D)/s (0.00, 0.15, 0.20, and 0.25)

4.4.2.1.5 Combined Bending Moment in the Deck The effect of GTL is negligible, especially compared to the effect of LL100 on the magnitude and distribution of SM(COMB) in the deck of C01. The change in SM(COMB) due to GTL is only ~0.5% (see Table B-19).

The most sensitive combination is f(A)/s = 0.15 and f(D)/s = 0.15. The least sensitive combination is f(A)/s = 0.25 and f(D)/s = 0.25. Therefore, the less rigid a structural system is, the less sensitive the SM(COMB) in the deck is to changes in GTL. Further, sensitivity to

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GTL in the deck is inversely proportional to both f(A)/s and f(D)/s. The distribution for the most sensitive configuration (f(D)/s = 0.15) that compares the effect of GTL and LL100 is shown in Figure 4-15.

Figure 4-15: Configuration C01 — Distribution of combined moments in the deck comparing the critical cases of LL100, +ve and –ve GTL, and the unloaded state for

three levels of f(A)/s and the most sensitive ratio, f(D)/s = 0.15

Figure 4-16 shows the distribution of SM(COMB) due to the governing case of LL100. When considering the effect of LL, the highest difference is apparent at midspan of the deck. The effect of LL100 is larger than effect of GTL along the entire span. It can be noted that the f(A)/s represents a more significant parameter. When taking into account the effect of LL100, the figure confirms that the trend in distribution and sensitivity to change in the primary and secondary variables due to LL is the same as due to GTL. The most sensitive configuration consists of f(A)/s = 0.15 and f(D)/s

= 0.15. The highest magnitude of SM(COMB) is exerted on the deck in combination that comprise of f(A)/s = 0.15 and f(D)/s = 0.25.

It can be said that as higher the SM(COMB) in the deck is, as less sensitive the structural system is to change in geometry. The effect of f(A)/s and f(D)/s on the SM(COMB) in the deck under applied LL100 is almost identical.

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Figure 4-16: Configuration C01 — Distribution of combined moments in the deck

considering the effect of LL100 for three levels of f(A)/s (0.15, .020, 0.25) and four

levels of f(D)/s (0.00, 0.15, 0.20, and 0.25)

4.4.2.1.6 Tensile Force in the Cables The effect of LL100 is larger than the effect of GTL on the structural response of

CF. The significance of GTL is inversely proportional to f(A)/s and directly proportional f(D)/s. The most sensitive combination with a difference of ~3.5% due to –ve GTL is f(A)/s =

0.20 and f(D)/s 0.15. The maximum axial force is exerted on the cables when f(A)/s = 0.15 and f(D)/s = 0.25 (see Table B-22 and Table B-23). A typical distribution of CF that compares the effect of GTL and LL for the most sensitive ratio, f(D)/s = 0.15, and three levels of f(A)/s is shown in Figure 4-17. Due to the symmetry, the figure shows only half of the cables (there are fourteen cables in total).

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Figure 4-17: Configuration C01 — Distribution of axial forces in the cables comparing the critical cases of LL100, +ve and –ve GTL, and the unloaded state for

three levels of f(A)/s and the most sensitive ratio, f(D)/s = 0.15

As shown in Figure 4-17, the effect of GTL is significantly less than the effect of LL100 in the cables along the span. The highest change in force in a cable due to applied LL100 occurs in the first cable (the cable next to the abutment of the arch) and the effect gradually decreases toward the fourth cable. At the fifth cable the magnitude of CF starts to increase under all assumed loads. The distribution of CFs for the governing case of LL100 is presented in Figure 4-18. As shown in the figure, in all combinations, the magnitude of CF in the first cable is the highest.

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Figure 4-18: Configuration C01 — Distribution of axial forces in the cables under

LL100 for three levels of f(A)/s (0.15, 0.20, and 0.25) and four levels of f(D)/s (0.00, 0.15, 0.20, and 0.25)

There are several factors that cause the large magnitude of CF in the first cable. First, the actual magnitude of LL is carried by the first cable. In total, there are 14 cables evenly spaced 5m apart. Therefore, each inner cable (the cables between the first and last cables) carries a portion of the load from two adjacent deck segments of even length (2.5m and 2.5m). However, the outer cables (the first and last cable) carry a load from one inner deck segment with a length of 2.5m and a load from one outer deck segment with a length of 5m. Hence, the total load carried by the outer cables covers deck segments with a total projected length of 7.5m. Second, the actual length of the deck segments causes a large CF in the first cable. The cables are equally spaced 5m apart. However, the curved shape of the deck, which is defined by the equation of a parabola, has segments of uneven length. The cable length is longest in the cables at the midspan. When a load is applied, these cables undergo the largest elongation, which is a function of the cable length. The

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Chapter Four: Significance of Geometry displacement of the arch is also the largest at the midspan. The combined effect of the largest cable elongation and largest arch displacement causes the deck to descend the most at the midspan. The large displacement of the deck at the midspan results in a large rotation of the deck at the abutment; this rotation is resisted by the shorter outer cables. Therefore, the force in the first and last cable is the largest. At the midspan, the spatial displacement of the arch and deck is the largest. The displacement of the arch and deck in the transversal horizontal direction is a function of the horizontal component of the cable force and arch rise f(A). The arch with the largest f(A) is the most flexible and is the most prone to transversal horizontal deflection; however, the arch is exposed to the smallest CF(H). At the midspan, the angle α is the smallest in configurations with a small arch rise f(A), and therefore, CF(H) is the largest, which results in a large displacement of the arch and deck. When the arch and the deck experience displacements in opposite directions, this situation causes the angle α to increase, which reduces CF in the configurations with a small f(A). In the configurations with a large f(A), the arch is more flexible than in the configurations with a small f(A); however, the CF(H) is the smallest in the configurations with a large f(A), which causes the smallest combined displacement of the arch and deck in opposite directions and, therefore, the smallest increase in α. This scenario results in a larger cable force than in the configurations with a small f(A).

The key parameter is the angle α; as the angle, which is function of arch rise f(A) and deck reach f(D), increases, the force in the cables decreases. Consequently, the distribution and magnitude of the forces are directly related to the distribution and magnitude of the internal forces in the arch and deck.

4.4.2.2 Configuration C02

4.4.2.2.1 Expected Behaviour In C02, the effect of GTL is compared with the effect of LL50. In the unstressed state, i.e., before DL is taken into consideration, the cables in C02 are contained in the same plane as the arch (see Figure 4-3). However, the applied loads, DL, LL, and GTL, cause the deck to deflect. The deflected deck results in a change of the position of the

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cables, which in turn causes an increase in CF(H), and therefore, out-of-plane loads develop and are exerted on the arch. Further, the arch also experiences in-plane and out-of-plane displacements, and therefore, the magnitude of the out-of-plane loads may vary as a function of the overall geometry and the rigidity of the entire system; in other words, the arch with a large rise is more prone to deflection and, therefore, more susceptible to the out-of-plane loads that develop. Since the planes in which the cables and the arch are contained are identical in the unstressed state, the arch in C02 may be exposed to larger axial loads and smaller bending moments compared to C01. Nevertheless, the inclination of the arch combined with a deflection of the deck, which is a function of the BCs of the deck and the EI of the deck, results in out-of-plane bending moments in the arch.

4.4.2.2.2 Axial Force in the Arch A comparison of SF1 in the arch of C02 under LL50 and GTL is presented in Table B-24 and Table B-25. As shown in tables, the effect of GTL on SF1 in the arch is smaller than the effect of LL50. The largest change in SF1 is ~3% due to GTL and ~24% due to LL50 using the same combination of primary and secondary variables. The sensitivity of

SF1 in the arch to GTL is inversely proportional to f(A)/s and directly proportional to ω. The largest increase and the largest magnitude resulting from +ve GTL occurs when f(A)/s = 0.15 and ω = 45°.

The distribution of SF1 under the four assumed loads, three levels of f(A)/s, and the most sensitive angle of arch inclination ω (45°) is presented in Figure 4-19. The figure indicates that the change in SF1 is the largest at the midspan and the lowest at the abutments of the arch. This behaviour holds for all types of applied loads (DL, LL, and GTL). For GTL, a change in SF1 due to a change in the primary variable is ~51.5%, and due to the secondary variable, it is only ~0.4% (see Table B-63). The parameter that controls the magnitude and distribution of SF1 under GTL is the primary variable, f(A)/s.

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Figure 4-19: Configuration C02 — Distribution of axial forces in the arch comparing the critical cases of LL50, +ve and –ve GTL, and the unloaded state in the arch under

three levels of f(A)/s and the most sensitive angle of arch inclination, ω = 45°

A distribution of SF1 in the arch under LL, considering three levels of f(A)/s and four levels of ω, is presented in Figure 4-20. From the figure is apparent that the difference in SF1 due to a change in the angle ω is uniform along the entire arch length. However, the increase in SF1 due to a uniform increase of the angle ω (from 15° to 45°) is not uniform. The increase due to change from ω = 30° to ω = 45° is significantly larger than the increase due to change from ω = 15° to ω = 30°. The highest SF1 is exerted on the arch when f(A)/s = 0.15 and ω = 45°. The trend of sensitivity to a change in geometry is inversely proportional to f(A)/s and directly proportional to ω. The reason for a larger SF1 in the arch in a configuration with a small f(A)/s and a large angle of arch inclination ω is the transformation of angle α. Due to the increase in ω, angle α is reduced (α = 90° – ω), and therefore, a larger CF is generated. The secondary variable controls the magnitude and distribution of SF1 in the arch in C02.

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Figure 4-20: Configuration C02 — Distribution of axial forces in the arch under LL50

for three levels of (A)/s (0.15, 0.20, and 0.25) and four levels of arch inclination ω (0°, 15°, 30°, and 45°)

4.4.2.2.3 Combined Bending Moment in the Arch In the reference configuration, which has a vertical arch, the effect of LL50 is larger than the effect of GTL. In contrast, in C02, which has an arch inclination that starts at ω =

30°, the effect of GTL is larger than the effect of LL50. Further, starting at f(A)/s = 0.15 and

ω = 45°, LL50 reduces the magnitude of SM(COMB) in the arch. A larger effect results from

–ve GTL than from +ve GTL. The sensitivity of GTL is inversely proportional to both f(A)/s and ω. When GTL is applied, the largest change in SM(COMB) is ~25%; this change occurs when f(A)/s = 0.15 and ω = 15° due to –ve GTL (see Table B-24 and Table B-25).

The distribution of SM(COMB) in the arch that takes into account the most sensitive combination (where effect of GTL is larger than the effect of LL50, i.e., ω = 30°) and compares the effect of GTL and LL50 is shown in Figure 4-21. From the figure is apparent that the effect of the GTL is larger than the effect of the LL50 mainly at midspan. At the abutments the effect of the GTL is still larger than that of LL50; however, the difference between GTL and LL50 is small.

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Chapter Four: Significance of Geometry

Figure 4-21: Configuration C02 — Distribution of combined moments in the arch comparing the critical cases of LL50, +ve and –ve GTL, and the unloaded state under

three levels of f(A)/s and the most sensitive angle or arch inclination, ω = 30°

Figure 4-22: Configuration C02 — Distribution of combined bending moments in the

arch under LL50 for three levels of f(A)/s (0.15, 0.20, and 0.25) and four levels of arch inclination ω (0°, 15°, 30°, and 45°)

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The distribution of SM(COMB) due to LL50, comparing the effect of changing geometry, is shown in Figure 4-22. The figure indicates that the largest effect of LL50 results from a change of the secondary variable, the angle ω. This effect is most apparent at the midspan. Therefore, the secondary variable controls the magnitude and distribution of

SM(COMB) in the arch.

The largest magnitude of SM(COMB), resulting from the applied LL50, is exerted on the arch in combination of f(A)/s = 0.25 and ω = 45°. The trend of SM(COMB) is directly proportional to both the f(A)/s and to ω.

A larger SM(COMB) in the vicinity of the midspan of the arch is a result of a larger deflection of the arch at the midspan. The enlarged deflection of the arch increases the influence of the out-of-plane load that is transferred from the deck via the cables.

When the arch is tilted more from the vertical plane, i.e., ω increases, CF(H) also increases, which causes an increase in the magnitude of the out-of-plane bending moment.

When LL50 is applied, the significance of a change in f(A)/s is minor (~28%) compared to the effect of the angle of arch inclination ω (~87%). Therefore, ω controls the magnitude and distribution of SM(COMB) in the arch.

4.4.2.2.4 Torsional Moment in the Arch A comparison of the effect of GTL and LL50 on SM3 in the arch is presented in Figure 4-23. As shown in the figure, GTL has almost no effect on SM3. The difference due to GTL is less than 1% (see Table B-31). In the same configuration, the effect of LL50 is – 42%. The negative sign indicates that LL50 applied on the deck reduces the magnitude of SM3 in the arch, which can be utilized in efficient designs of C02. The effect of changing geometry on SM3 under only LL50 is presented in Figure 4-24. As shown in the figure, the maximum magnitude of SM3 is exerted on the arch when f(A)/s = 0.25 and ω = 45° (see Table B-30). The table also indicates that the arch inclination controls the magnitude of SM3. The sensitivity of SM3 to a change in geometry is directly proportional to both the primary and secondary variables; in other words, the largest drop in SM3 is achieved in a combination with a large arch rise and large arch inclination (f(A)/s = 0.25 and ω = 45°). Therefore, as the dimensions of the bridge increase, the sensitivity of SM3 to a change in geometry increases.

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Figure 4-23: Configuration C02 — Distribution of torsional moments in the arch comparing the critical cases of LL50, +ve and –ve GTL, and the unloaded state for

three levels of f(A)/s and the most sensitive inclination of the arch, ω = 45°

Figure 4-24: Configuration C02 — Distribution of torsional moments in the arch

under LL50 for three levels of f(A)/s (0.15, .020, 0.25) and four levels of arch inclination (0°, 15°, 30°, 45°)

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4.4.2.2.5 Combined Bending Moment in the Deck

The effect of LL50 on SM(COMB) in the deck is larger than the effect of GTL. However, the difference between GTL and LL50 is only minor, particularly in configurations with a large angle ω (see Table B-36 and Table B-37). When f(A)/s = 0.15 and ω = 45°, the increase in SM(COMB) is 0.99% and 0.94% due to LL50 and +ve GTL, respectively. Based on this observation, the effect of +ve GTL is almost identical to the effect of LL50.

The sensitivity of SM(COMB) under +ve GTL to changing geometry is inversely proportional to f(A)/s and directly proportional to ω. The most sensitive combination is f(A)/s = 0.15 and ω = 45°. The effect of ω represents the governing parameter under both LL50 and GTL. For

LL50, the change in f(A)/s leads to only a minor change in SM(COMB) and, therefore, the f(A)/s does not present a concern.

The distribution of SM(COMB) in the most sensitive configuration is presented in

Figure 4-25. The figure shows that SM(COMB) reaches almost the same magnitude under all four types of applied load.

Figure 4-25: Configuration C02 — Distribution of combined moments in the deck comparing the critical cases of LL50, +ve and –ve GTL, and the unloaded state for

f(A)/s = 0.15, ω = 45°

Nevertheless, if only a bending moment in the vertical plane, i.e., a bending moment that results in vertical displacements of the deck, is taken into consideration, the

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Chapter Four: Significance of Geometry difference in the distribution of internal forces is more perceptible as shown in Figure 4-26. As shown in the figure, even though the difference between the response (resulting from the application of GTL and LL) is minor, the actual magnitude of the bending moment in the vertical plane (SM1) is significantly larger due to LL and GTL than the response due to only DL.

Figure 4-26: Configuration C02 — Distribution of the moments vertical plane in the deck comparing the critical cases of LL50, +ve and –ve GTL, and the unloaded state

for f(A)/s = 0.15, ω = 45°

In a situation when LL100 is applied, the effect of GTL is greater than the effect of LL100. The distribution of SM1, comparing GTL and LL100, is presented in Figure 4-27. As shown, the effect of GTL is almost twice as large as the effect resulting from LL100 at the midspan of the deck.

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Chapter Four: Significance of Geometry

Figure 4-27: Configuration C02 (f(A)/s = 0.15, ω = 45°) — Distribution of moments vertical plane in the deck comparing critical case of LL100, +ve and –ve GTL and unloaded state

The significance of ω on SM1 of the deck (considering f(A)/s = 0.15 and comparing the GTL with LL100) is demonstrated in Figure 4-28.

Figure 4-28: Configuration C02 — Distribution of vertical moments in the deck comparing LL100, +ve and –ve GTL, and the unloaded state in the arch for four

levels of ω (0°, 15°, 30°, and 45°) and the most sensitive ratio, f(A)/s = 0.15

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The Figure 4-28 shows that the effect of –ve GTL is larger than effect of LL100 at all levels of arch inclination. As ω increases, the significance of GTL increases as well. Nevertheless, when the LL50 is compared with the GTL as shown on Figure 4-29, the effect of GTL is close to the effect of LL50 only in a configuration with the highest arch inclination, i.e. ω = 45°. In the two combinations that have lower arch inclinations (ω = 30° and 15°), the effect of LL50 is larger than the effect of GTL. The effect of LL50 is the largest with the smallest arch inclination, ω = 15°. This situation occurs because the cables contract under –ve GTL. As ω increases, CF(H) also increases, and therefore, a higher load is exerted on the deck, which results in a larger SM2 and, subsequently, SM1 in the deck.

Figure 4-29: Configuration C02 — Distribution of vertical moments in the deck comparing LL50, +ve and –ve GTL, and the unloaded state in the arch for four levels

of ω (0°, 15°, 30°, and 45°) and the most sensitive ratio, f(A)/s = 0.15

The described behaviour holds for LL50 and for –ve GTL. However, in the case of –ve GTL, the contraction in the cables, which is not present in the case of LL50, increases the tension in the cables and ultimately reduces the magnitude of the load that is supported by the deck.

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When the arch becomes more vertical, the effect of the contracting cables is combined with the effect of the decreasing magnitude of CF, and therefore, for –ve GTL, SM1 in the deck is smaller compared to SM1 resulting from LL50. As shown in Figure 4-30, the moment distribution in the horizontal plane of the deck, SM2, is almost insensitive to the type of load that is applied to the structure. It should be noted that the EI of the deck is larger in the horizontal direction (section properties of the deck are presented in Table 4-3) and, therefore, the EI of the deck influences the moment distribution. The other factor that results in a significant difference in the moment distribution in the vertical and horizontal directions is the BCs of the deck. The significance of the EI and BCs of the deck is studied in Chapter Five.

Figure 4-30: Configuration C02 — Distribution of moments in the horizontal plane of the deck comparing the critical cases of LL50, +ve and –ve GTL, and the unloaded

state for f(A)/s = 0.15, ω = 45°

4.4.2.2.6 Tensile Force in the Cables A comparison of the effect of GTL and LL50 on CF in C02 is presented in Table B-40 and Table B-41. As shown in the tables, in all assumed combinations, the effect of LL50 is larger than the effect of GTL. The significance of –ve GTL under the changing geometry is inversely proportional to both f(A)/s and ω. The most sensitive combination is f(A)/s = 0.15 and ω = 15°. The maximum change in CF due to –ve GTL reaches a magnitude of ~37%. The effect of LL50 causes an increase in CF by ~85%, which makes the applied LL50 the governing load case.

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Chapter Four: Significance of Geometry

The distribution of CF for the most sensitive configuration (ω = 15°) and a comparison of the effect of GTL and LL50 is presented in Figure 4-31. As shown, the effect of LL50 is more significant at all levels of f(A)/s, and the significance of –ve GTL is larger than the +ve GTL only at the first two cables. At the third cable, the significance of +ve GTL is larger than the significance of –ve GTL.

Figure 4-31: Configuration C02 — Distribution of axial forces in the cables comparing the critical cases of LL50, +ve and –ve GTL, and the unloaded state for

three levels of f(A)/s (0.15, 0.20, 0.25) and the most sensitive arch inclination, ω = 15°

The distribution of CF under the changing geometry, while considering the effect of the LL50 alone, is shown in Figure 4-32. As shown in the figure, the change in ω has a higher impact on the magnitude of CF than a change in the ratio f(A)/s. This trend holds for all cables except the first cable. At location of the first cable, f(A)/s has a higher significance than ω. As ω increases, the magnitude of CF also increases because the angle α is reduced. In general, in configurations with a large angle ω (and a small angle α = 90° – ω), both CF and CF(H) are large.

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Figure 4-32: Configuration C02 — Distribution of axial forces in the cables under

LL50 for three levels of f(A)/s (0.15, 0.20, and 0.25) and four levels of ω (0°, 15°, 30°, and 45°)

4.4.2.3 Configuration C03

4.4.2.3.1 Expected Behaviour In C03, the effect of GTL is compared with LL100. C03 has an arch that is rotated diagonally above a straight deck. The arrangement of the cables, whose inclination varies along the span, with the largest inclination at the springings of the arch and the smallest at the midspan, results in a CF(H) that loads each half of the symmetric arch in an opposite direction (see Figure 4-4). Therefore, it the arch will be exposed to large out-of-plane bending moments. The magnitude of the out-of-plane bending moments will be directly proportional to the magnitude of arch rotation, the angle θ. Due to the arch rotation, the length of the cables along the span change. In the reference arch, the outer cables (the cables in the vicinity of the springings of the arch) are the shortest. The length of the cables gradually increases towards the midspan. However, in C03, the trend of the cable length is different than in the reference arch. The length of the outer cables increases proportionally with the magnitude of angle θ. In a configuration with a large angle θ, the outer cables are longer than the cables located in the vicinity of the

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Chapter Four: Significance of Geometry midspan. The increased length of the cables, when combined with a small angle α at location of the outer cables (a small α causes a large CF and CF(H)) affects the structural response in the arch and deck under GTL.

4.4.2.3.2 Axial Force in the Arch A comparison of SF1 in arch of C03 under LL100 and GTL is presented in Table B-42 and Table B-43. The tables show that the effect of LL100 is significantly larger than the effect of GTL. The significance of GTL under the changing geometry is inversely proportional to f(A)/s and is directly proportional to the angle θ. The highest magnitude of

SF1 in the arch due to +ve GTL is exerted on the arch when f(A)/s = 0.15 and θ = 45°. This trend in SF1 is inversely proportional to f(A)/s and directly proportional to θ. The change in

SF1 due to the variability of f(A)/s can be as high as ~40%, whereas the change in SF1 due to the angle θ is ~3%. Therefore, f(A)/s controls SF1 when GTL is taken into account. The distribution of SF1 in the arch that compares the effects of LL100 and GTL and considers the most sensitive configuration with θ = 15° is presented in Figure 4-33.

Figure 4-33: Configuration C03 — Distribution of axial forces in the arch comparing the critical cases of LL100, +ve and –ve GTL, and the unloaded state for three levels

of f(A)/s and the most sensitive angle of arch rotation, θ = 15°

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As shown in the Figure 4-33, the effect of LL100 is significantly greater along the entire length of the arch. Further, because GTL is applied to the whole structure and LL100 is applied only on the deck, the distribution of SF1 is different for the two loads. In the case of LL100, the difference in the magnitude of SF1 between the arch springings and the midspan is larger than that in the case of GTL. The comparison presented in Figure 4-33 shows that LL100 represents a governing case. The significance of LL100 is inversely proportional to f(A)/s and directly proportional to θ. The highest difference in magnitude (~64%), due to LL100 being exerted on the arch, occurs when f(A)/s = 0.15 and θ = 45°. The maximum magnitude of SF1 occurs in the same configuration. In case of LL100, the angle θ controls the magnitude and distribution of

SF1. The change in SF1 when θ changes is ~24%, whereas the change in SF1 when f(A)/s changes is ~2.5%. This trend is opposite to the trend under GTL. A comparison of the effect of changing geometry, for only LL100, is presented in Figure 4-34.

Figure 4-34: Configuration C03 — Distribution of axial forces in the arch under

LL100 for three levels of f(A)/s (0.15, 0.20, and 0.25) and four levels of arch rotation θ (0°, 15°, 30°, and 45°)

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As shown in the Figure 4-34, the arch rotation has a significant impact on the magnitude of SF1 in the arch.

In the reference arch, a large SF1 is achieved in an arch with a small f(A)/s ratio and a small SF1 is exerted on the arch with a high f(A)/s ratio, as expected. Further, in the reference arch, SF1 is significantly larger in magnitude at the springings of the arch than the magnitude of SF1 at the midspan. However, in C03, as θ increases, the difference in SF1 between the springings and midspan diminishes. In the configuration with maximum arch rotation, i.e., θ = 45°, there is almost no difference in SF1 between the springings and midspan of the arch. This behaviour is inversely proportional to f(A)/s; in other words, as f(A)/s increases, the difference in SF1 between the midspan and springings declines. In the combination with the smallest f(A)/s, the magnitude of SF1 at the midspan is even larger than at the springings. Such behaviour is opposite to a conventional vertical planar arch supporting a straight deck. From Figure 4-35, comparing the effects of LL and GTL in the critical combination of arch rotation θ = 45°, the trend of change in the magnitude of SF1 (as it is larger at the midspan of the arch compared to at the springings of the arch) is a function of the magnitude of the load applied on the deck. In a case where only GTL or DL is considered, the magnitude of SF1 in the arch is smaller than in a case that considers only LL100. The reason for a low magnitude of SF1 at the springings is the inclination of the cables that transfer the load from the deck to the arch. In configurations with a large θ, the inclination of the cables from the vertical plane is the largest (the angle α is small) at the arch springings. Cables with a small angle α experience a large CF(H). Hence, the portion of the arch rib that is located in the vicinity of the springings is exposed to large lateral forces, and the centroid axis of the arch tends to displace from its original position significantly. The displaced axis of the arch results in a transformation of the angle α, and therefore, out- of-plane bending moments develop in the arch. Thus, the magnitude of SF1 declines at the springings of the arch. The effect of a large CF(H) decreases in the direction away from the springings, and therefore, the out-of-plane bending moments are low and SF1 is large at the midspan. Hence, in a combination with a large angle θ, the difference in SF1 between the springings and the midspan is small. This behaviour is exaggerated in arches with low rise.

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In a combination with f(A)/s = 0.15 and θ = 45°, SF1 is larger at the midspan than at the springings, which is unusual in an arch-like structure. Therefore, analyses and design of the arch in C03 should consider this behaviour.

Figure 4-35: Configuration C03 — Distribution of axial forces in the arch comparing the critical cases of LL100, +ve and –ve GTL, and the unloaded state in the arch for

three levels of f(A)/s and the most sensitive angle of arch rotation, θ = 45°

4.4.2.3.3 Combined Bending Moment in the Arch

Comparisons of SM(COMB) in the arch of C03 are presented in Table B-46 and Table B-47. As is shown, the effect of LL100 is larger than the effect of GTL in all assumed geometric combinations. In general, the sensitivity of the arch to GTL is inversely proportional to f(A)/s and directly proportional to θ. The most sensitive combination consists of f(A)/s = 0.15 and θ = 45° and results in a change of ~13% due to +ve GTL. In comparison, the LL100 result in a change of ~40% for the same combination.

The increase in SM(COMB) due to +ve GTL in C03 is opposite to the trend in C01 and C02; in most cases for C01 and C02, –ve GTL caused the internal forces to increase.

The distribution of SM(COMB) in the arch, which compares the effect of GTL and

LL100 at three levels of f(A)/s and the most sensitive arch inclination with θ = 45°, is presented in Figure 4-36. As shown in the figure, the highest magnitude of SM(COMB) is

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Chapter Four: Significance of Geometry located at the springings of the arch where a critical difference of ~13% occurs. However, at the midspan of the arch, the difference between GTL and the reference DL is the largest.

Figure 4-36: Configuration C02 — Distribution of the combined moments in the arch comparing the critical cases of LL100, +ve and –ve GTL, and the unloaded state for

three levels of f(A)/s and the most sensitive angle or arch inclination, θ = 45°

A comparison of the distribution of SM(COMB) that considers only LL100 is presented in Figure 4-37. From the figure, the angle θ is more significant than the effect of f(A)/s. The largest difference in SM(COMB) among the individual configurations occurs at the arch midspan. The most sensitive configuration occurs when f(A)/s = 0.15 and θ = 15°. The trend is inversely proportional to both f(A)/s and θ. From Figure 4-37, SM(COMB) in the arch of C03 is significantly larger than a bending moment in the reference arch.

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Chapter Four: Significance of Geometry

Figure 4-37: Configuration C03 — Distribution of the combined bending moments in

the arch under LL100 for three levels of f(A)/s (0.15, 0.20, and 0.25) and four levels of arch rotation θ (0°, 15°, 30°, and 45°)

The largest magnitude of SM(COMB) due to LL100 is exerted on the arch when f(A)/s = 0.15 and θ = 45° and, therefore, as the internal moment increases, the sensitivity to a change in geometry decreases. It is the large CF(H) that causes a large SM(COMB) in configurations with a low f(A)/s. In configurations with a large ω, the cables become less vertical, and hence, CF increases. When a low f(A)/s is combined with a large angle θ, CF(H) increases even more, which causes an increase in SM(COMB).

4.4.2.3.4 Torsional Moment in the Arch A comparison of the effects of GTL and LL100 on SM3 in the arch is presented in Figure 4-38. The figure indicates that GTL has a larger effect than LL100 on SM3. However, the difference in SM3 due to GTL is minor when compared to the difference in SM3 due to LL; for GTL, the difference is only ~3% at most (see Table B-49). For the same combination of primary and secondary variables, the effect of LL100 is –13%. The presence of LL100 reduces the magnitude of SM3 in the arch.

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Chapter Four: Significance of Geometry

Figure 4-38: Configuration C03 — Distribution of torsional moments in the arch comparing the critical cases of LL100, +ve and –ve GTL, and the unloaded state for

three levels of f(A)/s and the most sensitive angle of arch rotation, θ = 45°

The effect of changing geometry under only LL100 is presented in Figure 4-39. The figure shows that the arch rotation has a more significant influence on SM3 than arch rise (see Table B-66). The combination with a large arch rotation (θ = 45°) results in the smallest SM3 at all levels of f(A)/s. The smallest SM3 is exerted on the arch when f(A)/s = 0.15 and θ = 45°. The magnitude of SM3 increases with an increase in arch rise, as expected. However, the maximum magnitude of SM3 occurs when f(A)/s = 0.25 and θ = 30°. Therefore, the arch rise should be low and the arch rotation should be at a moderate level to control the torsion in the arch.

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Chapter Four: Significance of Geometry

Figure 4-39: Configuration C03 — Distribution of torsional moments in the arch

under LL100 for three levels of f(A)/s (0.15, 0.20, 0.25) and four levels of arch rotation (θ = 0°, 15°, 30°, 45°)

4.4.2.3.5 Combined Bending Moment in the Deck

Comparisons of SM(COMB) in the decks are presented in Table B-54 and Table B-55. Similarly to the arch behaviour, the effect of LL100 in the deck represents the governing load case. In general, the significance of GTL on the deck is inversely proportional to both the f(A)/s ratio and the angle of arch rotation θ. The most sensitive combination occurs when f(A)/s = 0.15 and θ = 15°, which results in a difference of 12% due to –ve GTL. Further, in a case of LL100, the highest change of the internal forces (48%) occurs in a combination of f(A)/s = 0.25 and θ = 30°, which is opposite to the results for GTL.

The distribution of SM(COMB), which compares the effect of GTL and LL100 for the most sensitive combination of arch rotation (θ = 15°), is presented in Figure 4-40. The figure shows that at midspan the effect of +ve GTL is slightly larger than the effect of

LL100. However, at the abutments of the deck where SM(COMB) reaches its maximum value, the effect of LL100 is larger than the effect of GTL. This trend happens for all three levels of the f(A)/s. The +ve GTL applied equally to the entire structure simultaneously causes an elongation of all components. An elongated arch is more prone to out-of-plane

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Chapter Four: Significance of Geometry displacement. The elongation in the cables results in a change in the angle α. The angle α increases which changes the ratio of CF(H) and CF(V). Therefore, CF is reduced, which results in larger displacements and moments in the deck. Based on a comparison of the data provided in Table B-54 and the distribution of internal forces shown in Figure 4-40, it can be concluded that the effect of the applied GTL can result in a change in the internal forces as high as that due to LL100. Therefore, the analysis and/or design of the deck in C03 may require more attention.

Figure 4-40: Configuration C03 — Distribution of the combined moments in the deck comparing the critical cases of LL100, +ve and –ve GTL, and the unloaded state for

three levels of f(A)/s and the most sensitive angle of arch rotation, θ = 15°

The distribution of SM(COMB) under changing geometry for only LL100 is presented in Figure 4-41. From the figure is apparent that the highest magnitude of SM(COMB) is exerted on the deck when f(A)/s = 0.15 and θ = 45°; this trend is, therefore, inversely proportional to f(A)/s and directly proportional to θ. However, the significance of LL100 on the change in SM(COMB) is opposite to the trend of the maximum magnitude. The significance of LL100 is directly proportional to f(A)/s and inversely proportional to θ. The highest change in SM(COMB) due to a change in geometry under LL100 reaches 48% when

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Chapter Four: Significance of Geometry

f(A)/s = 0.25 and θ = 30°. The sensitivity to a change in the arch rotation is larger than the sensitivity to a change in arch rise. Hence, the angle θ controls SM(COMB) in the deck of C03 (see Figure 4-41 and Table B-54).

Figure 4-41: Configuration C03 — Distribution of the combined bending moments in

the deck under LL100 for three levels of f(A)/s (0.15, 0.20, and 0.25) and four levels of arch rotation θ (0°, 15°, 30°, and 45°)

4.4.2.3.6 Tensile Force in the Cables In the cables of C03, the overall effect of LL100 is larger than the effect of GTL. A comparison of CF, resulting from individual cases, is presented in Table B-58 and Table

B-59. The significance of the applied GTL is inversely proportional to both f(A)/s and θ.

The highest difference of 3% occurs when f(A)/s = 0.15 and θ = 15° due to +ve GTL. The effect of LL100 in the same configuration is 38%, which indicates that the significance of GTL on the cables is minor.

In C03, the length of the cables is a function of the arch rise f(A), arch rotation θ, and the shape of the arch. In the reference arch, the longest cables are located at the midspan and the shortest in the vicinity of the springings. However, in a rotated arch, the length of the cables in the vicinity of the springings is not necessarily the shortest, and therefore, a change in structural behaviour occurs, which is a function of the cable length.

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Chapter Four: Significance of Geometry

The distribution of CF under only LL100 is presented in Figure 4-42. As shown, the largest CF occurs at the midspan in all assumed combinations. The reference arch experiences the largest CF at the vicinity of the springings. The maximum magnitude of CF occurs when f(A)/s = 0.15 and θ = 45°. The trend is inversely proportional to f(A)/s and directly proportional to θ. Even though the effects of the primary and secondary variables are similar (the effect of primary variable is almost the same as the effect of secondary variable), the angle θ controls the magnitude of CF in C03 (see Table B-66).

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4.4.3 Principles of the Structural Behaviour in the Assumed Configurations

The structural responses of the arches and decks in the three configurations have one common aspect: the angle, α, between the horizontal plane of the deck and the cables. Even though the distribution of the angle α differs in each configuration, the principle shown in Figure 4-43 is the same for all the three structural arrangements. As shown in the figure, in a case when the deck reach f(D) remains constant and arch rise f(A) increases, α also increases.

A change in α results in a change in both CF(H) and CF. CF(V) remains constant for all configurations because it represents a vertical reaction to the loads exerted on the deck, and it is independent of a change in geometry of the arch and the inclination of the cables.

It should be noted that CF(V) remains constant in the general case: however, due to the fact that the SABs in question are indeterminate systems, the actual magnitude of CF(V) may vary as a function of the relative stiffness of all structural members. Nevertheless, in the general case it can be assumed that the magnitude of CF(V) does not change significantly and, therefore, due to a change in α, only CF(H) and CF vary. The type of change in CF(H) and CF is inversely proportional to α. In other words, as α increases, the magnitudes of

CF(H) and CF decline. This description covers a general mechanism that applies to a structure in an undeformed state as presented in Figure 4-43. However, in a spatial configuration that experiences a deformed state, the distribution of α becomes more complex. Due to the uneven displacement of the arch and deck along the span, the magnitude of which is also a function of EI of the arch and deck, α undergoes a transformation. A transformation of α (a change in magnitude) in a deformed state is uneven along the span, and, therefore it affects the structural response of the arch and deck in an uneven fashion. Examples of a change in α along the span in the three spatial configurations are presented in Figure 4-44. Figure 4-44 shows sections cuts, which are made at the midspan of the bridge structures, to illustrate where the uneven magnitude of α is the most perceptible. Further, the inclination of each cable in the deformed state holds a different magnitude, which results from a different spatial displacement of the arch and deck along span. In addition, the arch and deck are most sensitive to a displacement at the midspan and the least sensitive in the vicinity of the abutments. A response of the arch and the deck depends on

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Chapter Four: Significance of Geometry the response of the cables, which is a function of cable length. In SABs, the long cables tend to have a larger elongation and sag compared to short cables. The magnitude of the cable elongation and sag is a function of the load applied to a cable end. When a load is applied on the deck, a response in the arch depends on the length of the cables. The long cables transfer a smaller load compared to short cables because, for long cables, the load is transformed to reduce the sag of the cables and to lengthen the cables instead of transferring the load onto the arch. It should be noted that this behaviour of cables can be modeled using FE models employed in Chapter Six. In the deformed state, the transformation of α is a function of EI of the arch and deck and also of a combination of the primary and secondary variables.

f(A)1 < f(A)2 < f(A)3  α1 < α2 < α3  CF1 > CF2 > CF3

CF(H)1 > CF(H)2 > CF(H)3 CF(H)1 α1 CF(V)1 = CF(V)2 = CF(V)3 CF(V)1 CF1

CF(H)2 α2

3 ) A

( CF(V)2 f CF2

2 )

A ( f

1 CF(H)3 )

A ( α3 α3 f α2 CF(V)3 α1 CF3

f(D) = Constant

Figure 4-43: Schematic sketch showing the change in cable forces as α changes

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Chapter Four: Significance of Geometry

a) Configuration C01

b) Configuration C02

c) Configuration C03

Figure 4-44: Distribution of the uneven changes in magnitude of α along the span, shown on section cuts made at the midspan of the bridge structure in a deformed state for a) C01, b) C02, and c) C03

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Chapter Four: Significance of Geometry

Due to the different magnitudes of the deformed arch and deck (a change in α), the angle γ becomes either “positive” or “negative.” The positive change results in an increase and the negative change causes a drop in the magnitude of the transformed angle α, α(TRAN). The positive and negative changes in α then result in a drop or an increase in CF as indicated in Figure 4-43. The distribution of the cable forces then influences the structural response in the arch and deck. An example of a positive transformation of α in C02 is presented in Figure 4-45. The figure compares two cases with the same angle of inclination

ω (ω = 45°) but different f(A)/s ratios.

α1 = α2 Case #01 Case #02 EI(A)1 = EI(A)2 U3(A)2 (A) f /s = 0.15 EI(D)1 = EI(D)2 f(A)/s = 0.25

U3(A)1 U2(A)2

f(A)2 U2(A)1

f(A)1 γ2

γ1

α1 α2

U2(D)1 α1(TRAN) α2(TRAN) U2(D)2

U3(D)1 U3(D)2

f(A)1 < f(A)2 Cable length 1 < Cable length 2  U3(A)1 < U3(A)2 & U2(A)1 < U2(A)2  γ1 > γ2:  α1(TRAN) > α2(TRAN) U3(D)1 < U3(D)2 & U2(D)1 < U2(D)2 CF1 < CF2

Figure 4-45: Transformation of angle α and γ resulting in larger cable forces in the

configuration with a larger arch rise f(A) as a function of arch and deck stiffness

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Chapter Four: Significance of Geometry

Cross-sectional properties of the arch and deck in the two cases presented in Figure

4-45 do not differ. The bending stiffness of the arch EI(A) and deck EI(D) are the same in both cases. Nevertheless, in Case #02, the larger f(A)/s indicates that the lengths of the arch and the cables are longer compared to Case #01, and therefore, the response of the deck is also affected. The deck in Case #02 will experience large deflections, which will also influence distribution of bending moments. A relationship between the displacement of the arch and deck influences the magnitude of the angle α(TRAN). The change in α, presented in Figure 4-45, is denoted by γ, which is the angle between the cable inclination in an undeformed and a deformed state; hence, α(TRAN) = α + γ. As shown in the figure, based on a larger displacement of the arch and deck in Case #02, the magnitude of γ is smaller, which results in a smaller increase of the angle α(TRAN). Therefore, a drop in CF(H) and CF is smaller compared to Case #01. A graphical comparison of Case #01 and Case #02 in a 3D view is presented in Figure 4-46. As shown, both the displacements of the arch and deck are larger in Case #02.

Further, U3 of the deck is larger than U2 of the deck as a result of a larger EI(H) of the deck. This behaviour confirms that EI of the structural components plays a key role in the transformation of α, and as a direct consequence it influences the response in the arch and deck. The principle of the transformation of α was explained in C02; however, the nature of the mechanism is the same for the three different configurations of SABs. It was observed that the EI of the arch and deck, the magnitude of the primary and secondary variables, and BC of the deck influence the magnitude of the transformed angle

α(TRAN). Hence, a significance of these factors is studied in Chapter Five and Chapter Six in order to provide a clearer understanding of the structural behaviour of SABs.

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Chapter Four: Significance of Geometry

Case # 1 f(A)/s = 0.15

Case # 2 f(A)/s = 0.25

Figure 4-46: Comparison of spatial out-of-plane displacement of the arch two cases

with different f(A)/s ratios in configuration C02 resulting transformation of angle α and γ. Scale factor = 5.0

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Chapter Four: Significance of Geometry

4.5 Conclusion to Chapter Four

Chapter Four provides a detailed evaluation of the structural behaviour of three different spatial configurations: C01, C02, and C03. The following sections summarize the conclusions that can be made about the structural behaviour of the individual configurations.

4.5.1 Configuration C01

The assumed GTL does not represent a significant concern in C01. The effect of GTL is less than the effect of the governing LL100 in all structural components. The increase in the internal forces due to LL100 ranges from ~31% to ~39%, whereas the effect of GTL ranges between ~0.5% and ~5%. The response in the arch is more sensitive to changes in the f(A)/s ratio, and the response in the deck is more sensitive to changes in f(D)/s.

The response in the cables is also significantly affected due to changes in f(D)/s.

In the arch, the sensitivity to GTL is inversely proportional to the f(A)/s ratio; that is, as f(A)/s increases, the sensitivity to the applied GTL decreases. The sensitivity in the deck is also inversely proportional to the f(D)/s ratio, i.e., with an increase in the f(D)/s ratio, the sensitivity to the applied GTL decreases.

The maximum magnitude in a structural response (SF1, SM(COMB), CF, etc.) is insensitive to the type of load. In other words, the effect of geometry is greater than the effect of different loads (LL is only applied on the deck and GTL is applied to the entire structure). A combination of the primary and secondary variables that results in the maximum axial load in the arch under LL results in the maximum axial load when GTL is applied. The structural response is inversely proportional to f(A)/s and directly proportional to f(D)/s: therefore, the largest forces are exerted on the arch, deck, and cables when the arch rise is small (f(A)/s = 0.15) and the deck reach is large (f(D)/s = 0.25). The overall structural response in C01 is a function of the out-of-plane displacements of the arch and deck that influence the transformation of angle α between the horizontal plane of the deck and the inclined cables, which transfer the loads between the arch and deck. The combination in which the out-of-plane displacements of the arch and deck are the largest is f(A)/s = 0.15 and f(D)/s = 0.25.

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Chapter Four: Significance of Geometry

The following can be concluded about the effect of changing geometry:

• In C01, the primary variable (f(A)/s) controls the structural response in the arch. In

general, the magnitude of the internal forces in the arch increases as f(A)/s decreases, as

expected. When f(A)/s changes from 0.25 to 0.15, the following is observed:

o When the deck reach is small (f(D)/s = 0.15) and the arch rise is small (f(A)/s

= 0.15), SM(COMB) in the arch increases by 58%, and SF1 increases by only

17%. When the deck reach is large (f(D)/s = 0.25), the SF1 increases by only

6% and SM(COMB) by 46%. The torsion in the deck decreases when f(A)/s decreases by 17% on average.

o Therefore, SM(COMB) in the arch represents the key concern and should be considered first during the design process. • In the deck, the effect of changing the primary variable from 0.25 to 0.15 mostly influences the axial load.

o When the deck reach is small (f(D)/s = 0.15) and the arch rise is small (f(A)/s = 0.15), SF1 in the deck increases by 82%, SM(COMB) increases by 58%, and CF increases by 59%.

o Compared to the arch, the axial load in the deck is more sensitive to changes in arch geometry. Therefore, a rigorous analysis should be carried out when designing the deck in C01.

4.5.2 Configuration C02

The analyses showed that in C02, the assumed GTL can be a significant concern. The bending moment and displacement of the deck are larger due to GTL than due to LL50. In the arch, the effect of GTL is almost identical to the effect of LL50. The overall change in the structural response due to GTL ranges from ~2.5% in SF1 of the arch to ~ 67% in U(COMB) of the deck. The range of response due to LL50 starts at

~25% in SF1 of arch to ~87% in U(COMB) of the arch. The structural response is more sensitive to a variation in the angle of arch inclination from the vertical plane, ω, rather than to f(A)/s. The f(A)/s ratio has greater significance only in SF1 in the arch and SM(COMB) in the deck under GTL. In the cables, f(A)/s greatly influences the magnitude of CF under both types of assumed load. The greatest load in the cables occurs when there is a large rise

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Chapter Four: Significance of Geometry and a large inclination of the arch. The magnitude of load in the cables is also a function of deck stiffness, which is further investigated in Chapter Five. Overall, the response in the arch under GTL and LL50 is inversely proportional to f(A)/s; as f(A)/s increases, the sensitivity to the applied GTL and LL50 decreases. Therefore, the arch rise should be large to reduce the sensitivity to GTL in C02.

In general, the response in the deck is inversely proportional to both f(A)/s and ω; as f(A)/s or ω increases, the sensitivity under the applied loads decreases, with the exception of

SM(COMB) in the deck under GTL. In this case, the response is inversely proportional to f(A)/s but directly proportional to ω. In other words, SM(COMB) in the deck is large when the arch rise is small and the inclination of the arch is large. In the cables, the trend under GTL and LL50 is the same and correlates with the trend in the arch: the sensitivity in the cables is inversely proportional to both f(A)/s and ω. The maximum magnitudes in the assumed structural responses are insensitive to the type of load except U(COMB) in the deck. The maximum magnitude of U(COMB) in the deck is inversely proportional to f(A)/s and directly proportional to ω under GTL and directly proportional to both f(A)/s and ω under LL50. The response in the arch and deck is inversely proportional to f(A)/s and directly proportional to ω except SM(COMB) and U(COMB), which have a maximum magnitude that is directly proportional to both f(A)/s and ω. The maximum forces in the deck and cables under both types of applied load are inversely proportional to f(A)/s and directly proportional to ω. The greatest internal forces occur when f(A)/s = 0.15 and ω = 45°, and this combination is the most sensitive to applied loads. The overall structural response in C02 is a function of the out-of-plane displacement of the arch and deck. The configuration with the largest out-of-plane displacement of the entire structure is when f(A)/s = 0.25 and ω = 45°, under LL50. The following can be concluded about the effect of changing geometry:

• As f(A)/s decreases, only SF1 and CF increase in magnitude; the bending moment and displacement of the arch decrease.

o At all levels of arch inclination, the trend is similar. When the arch rise is

small (f(A)/s = 0.15), SF1 in the arch increases by ~36% on average and

SM(COMB) drops by only ~9%.

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Chapter Four: Significance of Geometry

o LL applied on the deck reduces the torsion in the arch. The torsion in the arch decreases by 40% compared to the effect of DL alone.

• The response in the deck is almost insensitive to a change in f(A)/s.

o SM(COMB) in the deck, at all levels of arch inclination, increases by less than 5%.

o The displacement of the deck is the largest when the arch inclination is the smallest (ω = 15°). However, the change is only 9% when f(A)/s changes and, hence, the displacement is not a significant concern. • The axial forces in the arch and the tensile forces in the cables must be given a high level of attention while designing a SAB in C02.

4.5.3 Configuration C03

The effect of LL100 is larger than the effect of GTL only on some types of structural response. In contrast, a change in U(COMB) of the arch and deck is larger due to the effect of GTL. In general, the effect of applying GTL results in changes ranging from ~2.5% in CF to ~57% in U(COMB) in the arch. The effect of applying LL100 causes changes between

~39% in CF to ~63% in SF1 in the arch. In the arch, the f(A)/s ratio has a higher influence than angle θ and in the deck it is the angle θ that has a higher influence than the f(A)/s ratio.

In the cables, f(A)/s controls the magnitude of CF under GTL, and the angle θ controls the magnitude of CF under LL100. The sensitivity of all structural components to the applied GTL and LL100 is inversely proportional to f(A)/s; in other words, the smaller f(A)/s is, the higher the difference in the structural response. Nevertheless, the structural responses of the individual components (arch, deck, and cables) experience different trends in sensitivity. In general, the sensitivity decreases with an increase in arch rotation. However, C03 is the least predictable in terms of the structural behaviour compared to the other two spatial configurations. The trends of the maximum magnitudes in the structural responses under both GTL and LL is the same: the structural response is inversely proportional to f(A)/s and directly

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Chapter Four: Significance of Geometry proportional to θ. The maximum responses in the arch, deck, and cables occur when there is a small arch rise and a large arch rotation, f(A)/s = 0.15 and θ = 45°. The overall structural response in C03 is highly dependent on the magnitudes of the displacement of the arch and the deck, which affect the transformation of the angles between the cables and arch and deck. In this configuration, the angle of inclination of each cable (α) is different, and therefore, it results in a complex response. The configuration with the largest out-of-plane displacement of the entire structure is with f(A)/s = 0.15 and θ = 45°. The following can concluded about the effect of changing geometry:

• In the C03, the effect of changing f(A)/s ratio has a similar effect as in the C02, the axial load in the arch changes at a greater scale than the bending moment.

o When the arch rotation is small (θ = 15°), SF1 in the arch increases by 45%, whereas SM(COMB) increases by only 21%.

o When the arch rotation is large (θ = 45°), SF1 in the arch increases by 63%, and SM(COMB) increases by only 7%.

o The torsion is not significantly affected: at most, the torsion drops by 27% when the arch rotation is small.

o The cables experience only minor changes. Due to a drop in f(A)/s, CF increases by 33% when the arch rotation is large.

• In the deck, the most significant change in SM(COMB) occurs when the arch rotation is small.

o When the arch rotation is small (θ = 15°), SM(COMB) increases by 30%, and U(COMB) increases by 29%.

o The arch rotation has a significant influence. When the arch rotation is large (θ = 45°), SM(COMB) increases by only 6%, whereas U(COMB) increases by 47%. • SF1 in the arch is most influenced by a change in geometry and should be given a high level of attention during the design process. The effect of arch rotation magnifies the effect of arch rise.

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Chapter Four: Significance of Geometry

The critical configurations (combinations of primary and secondary variables) identified in this chapter are the structural arrangements used for analyses in the following chapters.

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

5.1 Introduction The significance of changing boundary conditions (BCs) of the deck and the variability in bending stiffness (EI) of the arch and deck on the susceptibility of the arch to buckling are investigated in this chapter. The distribution and magnitude of internal forces in the arch is studied, and a critical distribution of a live load (LL) on the deck is determined. The critical combination of the primary and secondary variables, identified in Chapter Four, is employed in the present chapter. The magnitude of the axial force in a compression member plays a key role because exceeding the member’s critical value may lead to a loss of structural stability, i.e., to global buckling. Because global buckling represents a state of elastic instability - occurring at loads lower than the yield stress - this phenomenon is highly important. Current knowledge in the field of elastic instability in SABs was discussed in Section 2.5.3. In the selected configurations of SABs considered in this thesis, the arch (carrying loads either from a straight or a curved deck) is unsupported along its entire length, and therefore, it is more susceptible to in-plane and out-of-plane buckling compared to conventional vertical planar arches (carrying loads from a straight deck). In conventional planar arches, both types of buckling (in- and out-of-plane) may occur as well. However, in SABs, due to their spatial definition, both types of buckling may occur simultaneously (Sarmiento-Comesias, 2015). The susceptibility of the arch to buckling is a function of parameters such as the overall geometry of the structure, sectional properties (in- and out-of-plane bending stiffness), the distribution and magnitude of internal forces (particularly axial loads and bending moments), and BCs of the structural member that is exposed to compression loads. In SABs, these parameters also include sectional properties and BCs of the deck. The other aspect that influences the susceptibility of SABs to buckling due to their spatial definition is the direction of the load exerted on the arch. In the assumed configurations, loads from the deck are transferred to the arch via cables, which are organized in a certain alignment. Due to the spatial arrangement of the arch and deck (which differs for individual configurations) in the assumed SABs, the alignment of the

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness cables changes when load is applied and thus influences the direction the forces the cables transfer to the arch. This aspect affects the magnitude and distribution of internal forces in the arch, and therefore, it needs to be considered while evaluating the susceptibility to buckling. The effect of geometrical imperfections (also affecting the susceptibility to buckling) that may result from manufacturing or assembly deviations (studied for example by Sarmiento-Comesias (2015)) are not be included in the result evaluation of the present chapter. Investigating the susceptibility to buckling does not consider any nonlinear effects, which may result, for instance, from the imperfections. Hence, linear FE parametric models are employed to perform the proposed analyses (details on the linear FE parametric models are provided in Section 3.3.1.2). In the assumed SABs, the arch is exposed to high axial loads and bending moments, resulting in a large out-of-plane displacement of the arch, as expected. Therefore, it is important to find an optimal combination of these parameters because due to their interaction, the susceptibility of the arch to buckling increases. In general, the magnitude of the axial load exerted on the arch in SABs (resulting from a load applied on the deck) is smaller compared to the magnitude of axial loads in conventional arches because of out-of-plane bending moments in the arch of a SAB; therefore, SABs may be less susceptible to buckling. However, SABs may buckle at low axial loads because the out-of-plane bending moments result in second order effects that enlarge the deflection of the arch, which then affects the susceptibility of the arch to buckling. The Canadian Highway Bridge Design Code CHBDC S6-14 in Section 10, Cl. 10.9.4 provides an equation that establishes a proportion of axial and bending resistance of a straight member exposed to axial compression and bending. However, more particular approaches to indicate such proportion in curved members exposed to out-of-plane loads are not indicated. Therefore, an investigation of the proportions of the internal forces such as axial loads and bending moments in the arch of spatial configurations is required. This chapter is organized into five sections. The objectives of the present work are discussed in Section 5.2. The specifications of the proposed FE models, such as variability

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness in the deck BCs, EI of the arch and deck, and the applied loads, are described in Section 5.3. The results of the analyses are discussed in Section 5.4 and an optimal combination of the arch and deck profiles with a particular ratio of the EI proposed; the influence of the assumed BC of the deck is considered, and critical combinations are identified. The conclusions and a chapter summary are provided in Section 5.5. Relevant details, such as large tables or detailed charts, are provided in Appendix C.

5.2 Objectives

The main goal of Chapter Five is to investigate the effect of changes in BCs of the deck and a variability in EI of the arch and deck on the magnitude SF1 and SM(COMB) in the arch under a critical pattern of LL. The particular objectives are:

• To determine a critical distribution of LL that results in the highest SF1 in the arch in the three spatial configurations (critical values of primary and secondary variables were identified in Chapter Four);

• To identify a combination of EI of the arch and deck that results in the smallest and the

largest SF1 and SM(COMB) in the arch for a particular arrangement of BCs of the deck in the three spatial configurations; and

• To establish ratios of vertical EI(V) and horizontal EI(H) bending stiffness of the arch

and deck that result in the smallest SF1 and SM(COMB) in the arch simultaneously, representing a combination that is (for the assumed BCs of the deck) the least susceptible to buckling.

5.3 Model Specification

5.3.1 The Geometry of Models

Geometries of the spatial configurations used in the analyses represent the critical combinations of the primary and secondary variables, identified in Chapter Four. In these configurations, the magnitude of SF1 in the arch reaches the highest level. Particular combinations of the primary and secondary variables are listed in Table 5-1.

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

Table 5-1: Combinations of primary and secondary variables in individual spatial configurations Configuration Description Primary Variable Secondary Variable C01

A vertical arch supporting a f(A)/s = 0.15 f(D)/s = 0.15 curved deck

C02

An arch inclined from a vertical f(A)/s = 0.15 ω = 45° plane supporting a straight deck

C03 A vertical arch rotated about its vertical axis at f(A)/s = 0.15 θ = 45° midspan supporting a straight deck

5.3.2 Bending Stiffness of the Arch and Deck

The bending stiffness of the structural profiles assumed for the arch and deck is varied, as shown in Table 5-2, to investigate the effect of bending stiffness on the magnitude and distribution of SF1 and SM(COMB) in the arch. A generalized profile of the assumed beam elements is used to achieve a constant cross-sectional area, which results in constant weight per unit length of the arch and deck in the structural profiles. The details of the generalized profile of beam elements is provided in Section A.1.

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

Table 5-2: Geometric properties of the assumed structural sections for the arch and deck Geometric Stiffness level for the Arch(A) and the Deck(D) Cables Property S01A(S01D) S02A(S02D) S03A(S03D) S04A(S04D) Equivalent Shape

Diameter [m] N/A N/A N/A N/A 0.050 Height [m] ~0.400 ~2.000 ~0.400 ~2.000 N/A Width [m] ~0.400 ~0.400 ~2.000 ~2.000 N/A Cross-sectional 1.18E-01 1.18E-01 1.18E-01 1.18E-01 1.96E-03 Area [m2] Moment of 3.70E-03 3.70E-02 3.70E-03 3.70E-02 3.07E-07 inertia: I22 [m4] Moment of 3.70E-03 3.70E-03 3.70E-02 3.70E-02 3.07E-07 inertia: I11 [m4] Torsional 6.26E-03 1.18E-02 1.18E-02 6.26E-02 6.14E-07 constant [m4] Weight per 8,990 8,990 8,990 8,990 150 unit length [N/m]

5.3.3 Boundary Conditions

The BCs of the deck are varied to investigate their influence on the structural response. The BCs of the arch remain constant throughout the analyses. The arch rib is fixed at both ends. In total, there are four types of deck BCs. Definitions of the BCs, which are a function of allowed (indicated by “1”) or constrained (indicated by “0”) degrees of freedom (DOF), are summarized in Table 5-3. The integers 1, 2, and 3 represent a longitudinal, a vertical, and a transversal direction, respectively, within the global coordinate system of the developed FE models.

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

Table 5-3: Assumed types of deck boundary conditions Type Description Config. U1 U2 U3 UR1 UR2 UR3 Roller and one pin at each end of the deck C01, (R_1P): Allows for BC#01 C02, 1 0 0 0 0 1 translation in direction C03 1 and rotation about direction 3 Pinned at both ends (P_P): Allows only BC#02 rotation in all three C01 0 0 0 1 1 1 directions (no translations possible) Fixed at both ends BC#03 (F_F): No translations C01 0 0 0 0 0 0 or rotations allowed Roller and two pins (R_2P): Allows for C02, BC#04 translation in direction 1 0 0 0 1 1 C03 1 and rotation about direction 2 and 3

Graphical representations of the four types of deck BCs, indicating allowed DOFs, are presented in Figure 5-1 and Figure 5-2.

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

UR3 UR3

U1 U1

UR3 a) BC#01: U1

UR3

U1 UR3 UR3

UR2

UR1 UR3 b) BC#02:

UR2

UR3

UR1

c) BC#03:

Figure 5-1: The types of BCs of the deck that apply to C01: a) BC#01: Roller and one pin, b) BC#02: Two pins, c) BC#03: Fixed both ends.

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

UR3 UR3

U1 U1

Deck U1 UR3 a) BC#01:

UR3

U1 UR2 b) BC#04: U1 UR3

UR2

UR3

Figure 5-2: The types of BCs of the deck that apply to C02 and C03: a) BC#01: Roller and one pin, b) BC#04: Roller and two pins.

5.3.4 Loads and Materials

There are two types of assumed loads: a live load (LL) and a dead load (DL). The LL is applied simultaneously with the DL to create only one load case. As in Chapters Four and Five, the magnitude of LL reflects the purpose of the bridges in question, i.e., pedestrian bridges. The magnitude of LL for pedestrian bridges, determined according to the CHBDC S6-14, with a span of 75m and a deck width of 2m, is 5000N/m (see details in Section 3.3.5). There are two types of LL distributions assumed in Chapter Five: LL100 (LL distributed equally over the entire length of the deck) and LL50 (LL distributed equally only over one half of the deck, starting at a deck abutment and ending at the midspan). Both types of LL distribution are compared to determine a critical pattern that results in a larger SF1 in the arch. This comparison of the LL distribution is different from

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness the comparison conducted in Chapter Four. In Chapter Four, the focus was given only to the magnitude and distribution of bending moments whereas in the present chapter attention is given to the axial loads, bending moments, and the displacements of the arch. It is a common practice to evaluate these three types of responses while addressing susceptibility to buckling. The material properties of structural steel, the main material assumed in the entire structure, are presented in Section 3.5.

5.3.5 Achieving the Objectives of Chapter Five

To address the first objective: • All combinations of the primary and secondary variables assumed in Chapter Four are considered here to determine the critical pattern of LL. • The effects of LL50 and LL100 are compared, and the type of critical LL that causes a

larger SF1 in the arch is identified. Other types of responses such as SM(COMB), U2 and U3 of the arch are also evaluated. • A graphical comparison of the distributions of the SF1 is provided. • The identified type of the critical pattern of the LL is used as a governing load case in the analyses.

To address the second objective: • The three spatial configurations with particular combinations of the primary and secondary variables (identified in Table 5-1) are analyzed, while the critical pattern of LL is considered. • In each configuration, the EI of the arch and deck and deck BCs are varied according to Table 5-2 and Table 5-3, respectively. • The resulting structural responses are compared, the combinations of EI of the arch and deck and arrangements of the deck BCs are evaluated, and an optimal combination is identified for each spatial configuration. • The magnitude of the bending moments in the arch considers the effect of the in- and

out-of-plane load, and therefore, a combined bending moment SM(COMB), calculated according to Equation 4-2, is evaluated.

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

• For the spatial configurations, which are exposed to out-of-plane loads, the magnitude

of SF1 and SM(COMB) may be inversely proportional, i.e., an increasing trend in the

SF1 may result in a decreasing trend in the SM(COMB) and vice versa.

To address the third objective: • Combinations of assumed structural profiles of the arch and deck with a particular EI and BCs of the deck, identified in objective # 02, that identify either the lowest SF1 or

SM(COMB), in the arch, are compared and a combination that results in the lowest SF1

and SM(COMB) simultaneously is determined. • In total, there are four levels of EI of the arch and deck and, thus, there are sixteen

combinations per one type of deck BCs. The magnitude of SF1 and SM(COMB) for all combinations are ranked. The combination with the smallest magnitude of internal forces is assigned number 1, and the combination with the largest magnitude is ranked number 16. The combination of EI of the arch and deck that results in the smallest SF1

and SM(COMB) simultaneously is identified according to the ranking: the combination of

the smallest rankings for both SF1 and SM(COMB) is determined. • For the combination of EI of the arch and deck that results in the smallest SF1 and

SM(COMB) for a particular type of deck BCs, a ratio of EI(V) and EI(H) of the arch and deck (considering the assigned ranks) is established. • A tabular and graphical comparison is provided.

5.4 Results and Discussion

The main findings and conclusions regarding the structural responses in the individual spatial configurations are described and discussed in this section. Large tables and complex charts are provided in Appendix C. It should be noted that particular values of a change in a specific type of structural response, presented in the body of text, are taken from the two detailed result tables developed for each combination. Direct cross-reference to these result tables is provided: these result tables are placed in a relevant appendix section due to their large size.

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

5.4.1 Determining of a Critical Live Load Distribution

Comparisons of the effects of the two selected patterns of LL distribution (LL100 and LL50) on the magnitude of SF1 in the arch of C01, C02, and C03 are provided in Table C-1, Table C-2, and Table C-3, respectively. The effects of the two LL patterns on distributions of SF1, SM(COMB), U2, and U3 in the arches of the three spatial configurations are shown in Figure 5-3, Figure 5-4, Figure 5-5, and Figure 5-6, respectively. The following can be concluded from the figures: • In all three assumed configurations, considering all levels of the geometric variables, the effect of LL100 is larger than the effect of LL50. • The combination of the primary and secondary variables that results in the largest SF1 in the arch is also the combination that results in the largest increase in SF1, due to the applied LL100. • Sensitivity to a change in the LL distribution is inversely proportional to both the primary and secondary variables in C01; in other words, in a combination with a large

f(A)/s and f(D)/s, the difference between the effect of LL50 and LL100 is small. In C02 and C03, sensitivity to a change is inversely proportional to the primary variable and directly proportional to the secondary variable. • C01 experiences the smallest SF1, and C03 experiences the largest SF1 in the arch (see Figure 5-3). This observation indicates that C03 is more prone to buckling, considering large in- and out-of-plane moments (see Figure 5-4). However, the spatial displacements of the arch in C03 are small compared to C01 (see Figure 5-5 and Figure 5-6). Hence, while considering the large bending moments (see Figure 5-4) combined with a large U2 and U3 of the arch in C01, C01 is the most susceptible to buckling.

• The smallest magnitude of SM(COMB) is exerted on the arch in C02; when combined with a moderate level of SF1, this finding indicates that C02 is the least susceptible to out-of-plane buckling. • C03 is the most sensitive to a change in the LL pattern as a change results in the largest difference in magnitude of SF1. This characteristic may affect the fatigue life of C03 compared to the other two configurations.

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

These conclusions are based on a geometry employed in Chapter Four (one type of deck BCs, and arch and deck stiffness) and, therefore, a more rigorous approach is necessary to verify these trends while taking into account the effects of EI and BCs of the deck (see Section 5.4.2 and 5.4.3).

Figure 5-3: A comparison of a distribution of SF1 in the arch in C01, C02, and C03 for LL100 and LL50

Figure 5-4: A comparison of a distribution of SM(COMB) in the arch in C01, C02, and C03 for LL100 and LL50

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

Figure 5-5: A comparison of a distribution of U2 in the arch in C01, C02, and C03 for LL100 and LL50

Figure 5-6: A comparison of a distribution of U3 in the arch in C01, C02, and C03 for LL100 and LL50

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

5.4.2 Structural Behaviour under changing Stiffness

C01, C02, and C03 are approached separately to study the structural behaviour of the individual configurations, taking into account the changing EI of the arch and deck and BCs of the deck. The effect of a changing EI is evaluated for a particular arrangement of the BCs of the deck. A ratio of the EI of the arch and deck that results in the lowest magnitude of SF1 and SM(COMB) identifies the combination that is least susceptible to buckling; this ratio is established for each configuration and type of BCs. Key findings are concluded and discussed.

5.4.2.1 Configuration C01

The three types of deck BCs for C01 are defined in Table 5-3. Each type of deck BCs is evaluated individually; tabular and graphical comparisons are provided in Appendix C. Structural sketches that represent combinations of EI of the arch and deck, from which

SF1 and SM(COMB) reach their minimal or maximal magnitudes (as concluded in sections below), are provided in Tables C-10 and C-11, respectively.

5.4.2.1.1 Configuration C01 — Deck Boundary Conditions: BC#01 The effects of a changing EI of the arch and deck for deck BC#01 on SF1 and

SM(COMB) are summarized in Tables C-12 and C-13, respectively. As shown in these tables, the smallest SF1 and SM(COMB) occur in the same stiffness configuration, S04D and S01A (a vertically and horizontally stiff deck and a vertically and horizontally flexible arch). The largest SF1 and SM(COMB) also occur in the same stiffness configuration, S03D and S03A (a vertically stiff and horizontally flexible arch and deck). A distribution of SF1 and

SM(COMB), capturing the sixteen combinations for EI, is presented in Figure 5-7 and Figure 5-8, respectively. I can be said that there is a combination of arch and deck EI that causes the same trend in the distribution and magnitude of SF1 and SM(COMB). Specifically, a combination of arch and deck EI that results in a small SF1 also results in a small SM(COMB).

A general trend in structural behaviour is that the EI(V) of the deck and EI(H) of the arch control the magnitude of SF1 and SM(COMB). A large EI(V) of the deck and a large EI(H) of the arch prevent a large U2 in the deck and a large U3 in the arch, which results in a

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

small transformation in the angle α. Hence, the CF and CF(H) do not significantly change as a load is applied; the consistency in CF and CF(H) results in a small change in SF1 and

SM(COMB).

Vertically stiff decks (S02D and S04D) result in a small SF1 and SM(COMB), whereas vertically flexible decks (S01D and S03D) result in a large SF1 and SM(COMB). The deck essentially acts as a curved beam rather than a horizontal arch because there is no horizontal reaction imposed at the abutments to generate axial thrust (U1 is allowed). The UR1 at the deck abutments is not allowed and, therefore, the deck acts as a cantilever beam, in which its EI(V) represents a parameter that controls the U2 of the deck. The magnitude of the deck U2 affects the following: a) the amount of force transferred to the arch via the cables and b) the magnitude of the transformed angle α – as the deck U2 increases, the α also increases; this mechanism results in a reduction of the CF and CF(H).

The EI(H) of the deck controls U3 of the deck. Decks with a small EI(H) tend to deflect more with vertical loads applied on the deck; thus, this deflection affects the transformation of α, the low EI(H) results in a large transformation of α. However, the EI(H) plays a less significant role than the EI(V) of the deck, i.e., when EI(V) is low, α undergoes a larger transformation than when EI(H) is low.

In the configuration with arch stiffness S01A and S04A (where the ratio of EI(V) to

EI(H) is the same), the arch tends to equally deflect in the U2 and U3 directions. This characteristic of the arch deformation does not significantly affect the transformation of α, and hence, the resulting SF1 and SM(COMB) are smaller compared to other levels of arch stiffness with an uneven ratio of EI(V) to EI(H).

Selecting a structural profile with the same ratio of EI(V) to EI(H) can be advantageous and can reduce the susceptibility to buckling for BC#01.

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

Figure 5-7: Configuration C01: A comparison of the distributions of SF1 in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#01

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

Figure 5-8: Configuration C01: A comparison of the distributions of SM(COMB) in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#01

5.4.2.1.2 Configuration C01 — Deck Boundary Conditions: BC#02 Compared to BC#01, BC#02 does not allow translation in the U1 direction; further, there are no translational DOF allowed in any direction. Hence, the curved deck no longer acts as a curved beam but rather as a two pinned arches. Rotations about the axes in all three directions are allowed. Hence, the deck no longer acts as a cantilever when the vertical displacements are considered. BC#02 influences the use of the CF: the cables directly transfer the loads from the deck without any contribution of the deck EI as is the

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

case in BC#01. Thus, the distribution of SF1 and SM(COMB) in the arch changes significantly compared to BC#01

For BC#02, the effects of a changing EI of the arch and deck on SF1 and SM(COMB) are summarized in Tables C-14 and C-15, respectively. As shown in the tables, the minimum SF1 is exerted on the arch in a combination of S04D and S04A (a vertically and horizontally stiff arch and deck). The minimum SM(COMB) occurs in a combination of S01D and S04A (a vertically and horizontally flexible deck and a vertically and horizontally stiff arch). Distributions of SF1 and SM(COMB), which capture the sixteen combinations, are presented in Figure 5-9 and 5-10, respectively. In C01, assuming BC#02, the out-of-plane displacement of the arch plays a key role. Nevertheless, the deck stiffness controls the arch displacement, and therefore, it governs the distribution and magnitude of SF1 and SM(COMB) in the arch. The deck with BC#02 behaves as a two-hinged arch exposed to a load in the direction which is perpendicular to its plane. A reaction force to this type of load is directly transferred to the arch, because the rotation at the deck ends is released; therefore, the load applied on the deck directly influences the arch displacement. A transformation of the angle α is a function of deck displacement and, therefore, a function of deck stiffness. Decks with the same value of EI(V) to EI(H) (S01D and S04D) represent the governing cases.

A deck with a low EI(V) and EI(H) (S01D) deflects more with an applied load. Hence, the arch does not significantly deflect, and the resulting transformation of α is small. A small change in α leads to a small drop in CF and CF(H), while keeping SF1 and

SM(COMB) large.

When a deck with a large EI(V) and EI(H) (S04D) is employed, the deck does not significantly deflect. Therefore, the arch, acting as a curved cantilever beam with fixed ends, deflects more in return to the loads transferred from the deck. Thus, when S04D is used, the transformation of α is large, CF and CF(H) drop more significantly, and the resulting SF1 and SM(COMB) are reduced.

Arches with a low EI(H) results in increase of α, and therefore, the drop in SF1 and

SM(COMB) is even more perceptible in arches S02A and S01A. The decks with an uneven value of EI(V) and EI(H) result in moderate SF1 and SM(COMB) in the arch. The effects of the

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deck EI (described above) are the same at all levels of the arch EI. Both SF1 and SM(COMB) are related proportionally: the arch and deck stiffness that result in a small SF1 also result in a small SM(COMB).

Figure 5-9: Configuration C01: A comparison of the distributions of SF1 in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#02

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Figure 5-10: Configuration C01: A comparison of the distributions of SM(COMB) in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#02

5.4.2.1.3 Configuration C01 — Deck Boundary Conditions: BC#03 The effects of a changing EI of the arch and deck for the deck BC#03 on SF1 and

SM(COMB) are summarized in Tables C-16 and C-17, respectively. The minimum SF1 is exerted on the arch in a combination of S04D and S02A. The minimum SM(COMB) is exerted on the arch in a combination of S04D and S01A. In both cases, the vertically and horizontally stiff deck (S04) represents the governing case. In addition, in both cases, the arch is flexible in the horizontal direction. Distributions of SF1 and SM(COMB) capturing the sixteen combinations are presented in Figure 5-11 and Figure 5-12, respectively.

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It can be said that that the governing parameter is the EI(V) of the deck. Because the deck is fixed at the ends, it behaves essentially as a curved cantilever beam. Hence, the

EI(V) of the deck controls the U2 of the deck, and consequently, the amount of load transferred to the arch.

Vertically stiff decks (S02D and S04D) result in a small SF1 and SM(COMB). A low

EI(H) in S02D causes uneven U3 and U2 of the deck. Therefore, the transformation of α is not as significant as in the case with deck S04D. Hence, the vertically stiff but horizontally flexible arch in S02A combined with the deck in S04D results in a smaller SF1 and

SM(COMB) than for the deck S02D. Vertically flexible decks (S01D and S03D) are capable of carrying less vertical load (the U2 is larger than in vertically stiff decks) and, thus, the S01D and S03D decks result in a large SF1 and SM(COMB) in the arch. This trend is amplified in the arches with a low EI(V), such as S01A or S03A. The vertically flexible arch in S03A combined with decks S01D or

S03D result in the largest SF1 and SM(COMB). In this combination, the EI(H) of the deck does not play a significant role. Further, there is no difference between the structural response if S01D or S03D is employed. Due to a large U2 of the arch and deck in S03D combined with either S01A or S03A, the angle α undergoes only a minimal transformation.

Therefore, CF and CF(H) are slightly reduced, compared to combinations with stiff decks, which results in the largest SF1 and SM(COMB).

The significance of a high and a low EI(V) of the deck holds for all levels of arch stiffness. A vertically and horizontally stiff deck (S04D) represents the optimal deck stiffness because it results in the smallest SF1 and SM(COMB). To achieve the smallest SF1 and SM(COMB), the arch needs to be vertically stiff but horizontally flexible to allow for a larger transformation of α, which results in a more significant drop in CF and CF(H); this drop directly affects the magnitude of SF1 and SM(COMB) in the arch.

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Figure 5-11: Configuration C01: A comparison of the distributions of SF1 in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#03

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

Figure 5-12: Configuration C01: A comparison of the distributions of SM(COMB) in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#03

5.4.2.2 Configuration C02

The two types of deck BCs in C02, BC#01 and BC#04, are defined in Table 5-3. Each type of deck BCs is evaluated individually; tabular and graphical comparisons are provided. Structural sketches that represent a combination of EI of the arch and deck, from which SF1 and SM(COMB) reach their minimal or maximal magnitude (as concluded in sections below), are provided in Tables C-22 and C-23, respectively

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5.4.2.2.1 Configuration C02 — Deck Boundary Conditions: BC#01 The effects of a changing EI of the arch and deck for the deck BC#01 on SF1 and

SM(COMB) are summarized in Tables C-24 and C-25, respectively. The tables show that the lowest SF1 is exerted on the arch in a combination of S02D and S02A (a vertically stiff and horizontally flexible deck and arch). The lowest SM(COMB) is achieved in a combination of S03D and S01A (a vertically flexible and horizontally stiff deck and a vertically and horizontally flexible arch). Figure 5-13 and Figure 5-14 compare the distributions of SF1 and SM(COMB), respectively. A change in the EI of the arch and deck results in an opposite effect on the magnitude of SF1 and SM(COMB), i.e., a stiffness combination that causes a small SF1 results in a large SM(COMB) and vice versa. However, a low EI(H) of the arch achieves both a small SF1 and SM(COMB). The deck EI controls the structural response of the arch; in other words, the effect of a change in the EI of the deck overcomes the effect of a change of the EI of the arch at all levels of arch EI. The deck S02D results in the smallest SF1 and the largest SM(COMB) for all levels of arch stiffness (see Section 4.4.3 for an explanation of this behaviour). The deck S02D (vertically stiff but horizontally flexible) is more prone to inward deflection U3 (toward the inclined arch). The inward U3 of the deck causes an increase in α and results in a drop in CF(H) and the actual CF. Therefore, the magnitude of SF1 in the arch is reduced.

The reduced CF(H) can no longer counter-act the CF(V), and therefore, the bending moment in the arch increases.

The effect of EI of the arch plays a secondary role. When the EI(V) of the arch is low, the arch U2 is large. The large U2 of the arch results in a small increase in α and the

CF is not significantly reduced. Therefore, the SF1 in the arch is large, and the SM(COMB) is small. When the EI(H) of the arch is low, the U3 of the arch is large (toward the deck). The large U3 of the arch causes α to increase, which reduces CF. Therefore, the SF1 in the arch is small. However, due to the large U3 of the arch, the magnitude of SM(COMB) increases.

The EI(H) of the arch and deck play a key role in the transformation of the angle α, and consequently, in the distribution and magnitude of SF1 in the arch. Thus, the smallest

SF1 is exerted on the arch when both the arch and the deck have a low EI(H). The profiles of the arch and deck that have the same ratio of EI(V) and EI(H) (S01A, S01D, S04A, and

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

S04D) experience approximately the same U2 and U3. Therefore, the magnitude of the transformation of α remains at a moderate level, which maintains a large SF1 and SM(COMB) as loads are applied.

Figure 5-13: Configuration C02: A comparison of the distributions of SF1 in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#01

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Figure 5-14: Configuration C02: A comparison of the distributions of SM(COMB) in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#01

5.4.2.2.2 Configuration C02 — Deck Boundary Conditions: BC#04 The deck arrangement BC#04 allows for UR2 at the deck abutments, and therefore, the deck is more prone to U3 toward the arch. This behaviour influences the magnitude of CF, and subsequently, the SF1 in the arch. The effects of a changing EI of the arch and deck for the deck BC#04 on SF1 and SM(COMB) are summarized in Tables C-26 and C-27, respectively. As shown in the tables, the smallest SF1 occurs in a stiffness combination of

S02D and S02A, and the smallest SM(COMB) occurs in S02D and S01A. Distributions of

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SF1 and SM(COMB), which capture the sixteen combinations, are presented in Figure 5-15 and Figure 5-16, respectively. The trends in the structural behaviour of the configurations for BC#01 and BC#04 are similar. For both BC#01 and BC#04, the EI of the deck controls the response, i.e., at all levels of arch stiffness, the vertically stiff and horizontally flexible deck (S02D) results in the smallest SF1. The horizontally stiff and vertically flexible deck (S03D) results in the largest SF1 at all levels of arch stiffness. The arches with the same ratio of EI(V) and EI(H) (S01A and S04A) result in an almost identical magnitude and distribution of SF1 in the arch. For BC#04, the variability in the deck EI does not significantly influence the magnitude of SM(COMB). There is almost no change in SM(COMB) due to a change in the EI of the deck. This behaviour holds for all levels of the EI of the arch. The largest SM(COMB) occurs in a vertically stiff and horizontally flexible arch (S02A). The smallest SM(COMB) is achieved in arches S01A and S04A where the EI(V) and EI(H) is kept at the same ratio. This trend holds for all levels of deck stiffness. A vertically flexible and horizontally stiff arch

(S03A) results in an SM(COMB) of a moderate magnitude. The EI of the deck controls the magnitude and distribution of SF1, and the EI of the arch controls the magnitude and distribution of SM(COMB) in the arch. Due to the released rotation UR2 at the deck abutments, the deck is more prone to displacement toward the inclined arch with a minor effect of the EI(H). Therefore, the ratio of EI(V) and EI(H) of the arch controls the transformation of α, and consequently, the magnitude of CF controls the magnitude of SF1 and SM(COMB) in the arch.

The ratio of arch stiffness EI(V) and EI(H) is crucial. If the ratio of the arch EI(V) and

EI(H) is the same (S01A or S04A), the arch deflects at the same magnitude in the vertical and horizontal directions. The approximately equal U2 and U3 of the arch cause only a minor change in the angle α, and therefore, the CF(H) remains high and can counter-act the

CF(V), resulting in a small SM(COMB). If the ratio of the arch EI(V) and EI(H) is not the same (S02A or S03A), the transformation of α is large. For instance, in the case of S02A, the arch is horizontally flexible and its U3 is large. The large U3 of the arch results in an increase of α, and therefore, CF(H) drops. The reduced CF(H) can no longer counter-act the magnitude of CF(V), and therefore, the magnitude of SM(COMB) increases.

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Figure 5-15: Configuration C02: A comparison of the distributions of SF1 in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#04

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

Figure 5-16: Configuration C02: A comparison of the distributions of SM(COMB) in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#04

5.4.2.3 Configuration C03

The two types of deck BCs in C03 are BC#01 and BC#04, defined in Table 5-3. Each type of deck BCs is evaluated individually; tabular and graphical comparisons are provided. Structural sketches that represent a combination of EI of the arch and deck, from which the SF1 and SM(COMB) reach their minimal or maximal magnitude (as concluded in sections below), are provided in Tables C-32 and C-33, respectively.

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

5.4.2.3.1 Configuration C03 — Deck Boundary Conditions: BC#01 The effects of a changing EI of the arch and deck for deck BC#01 on SF1 and

SM(COMB) are summarized in Tables C-34, and C-35, respectively. The tables indicate that the smallest SF1 and SM(COMB) occur in a combination of S04D and S03A and S01D and S04A, respectively. Figure 5-17 and Figure 5-18 depict the distributions of SF1 and

SM(COMB), respectively, in the arch. In horizontally flexible arches (S01A and S02A), the particular EI of the deck that results in a small SF1 also results in a small SM(COMB). However, when the EI(H) of the arch increases, the trend reverses: the particular EI of the deck that results in a small SF1 generates a large SM(COMB) in the arch. The vertically stiff decks (S02D and S04D) result in a small SF1 and large

SM(COMB) when combined with horizontally flexible arches (S01A and S02A). However, when the EI(H) of the arch increases (S03A and S04A), the vertically stiff decks are no longer necessary to achieve a low SF1. The reason for this response is the transformation of α, which is different for each cable along the span (taking into consideration a symmetry in plan-view). In horizontally flexible arches (S01A and S02A), when loads are applied, the arch experiences a larger U3 compared to S03A and S04A, which in return causes an increase in

α. A larger angle α results in a smaller CF and smaller CF(H), and therefore, a smaller SF1 and SM(COMB) are achieved. Further, decks with a low EI(H) (S01D and S02D) cause an even larger increase in α.

When a large EI(H) of the arch is employed (S03A and S04A), the transformation of

α is not as high, resulting in a larger CF and CF(H), and therefore, SF1 in the arch increases (particularly in the vicinity of the abutments where α is very small). Due to a large starting magnitude of α at the midspan (the cables are almost vertical), the arch and deck experience the largest U2 and U3 at this location. Hence, the transformation of α is large.

This behaviour decreases CF, and therefore, SF1 in the arch at the midspan drops. Further, horizontally stiff decks (S03D and S04D) have a small U3. U3 of the arch is significantly influenced by a “locked” UR2 at the deck abutments in BC#01. The significance of an allowed UR2 for the deck BC#04 is described in the following section.

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

Figure 5-17: Configuration C03: A comparison of the distributions of SF1 in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#01

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

Figure 5-18: Configuration C03: A comparison of the distributions of SM(COMB) in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#01

5.4.2.3.2 Configuration C03 — Deck Boundary Conditions: BC#04 Deck BC#04 allows for UR2 at the deck abutments, and therefore, the deck is more prone to the out-of-plane displacement U3. The magnitude of U3 of the deck significantly influences the transformation of α, which affects the distribution of SF1 and SM(COMB) in the arch. In C03, the magnitude of α is different for each cable. Hence, the magnitude of

CF(H) and CF in each cable differs. The CF(V) is constant along the entire deck length because it represents only the reaction to the vertical load exerted on the deck. In the vicinity of the abutments, α is small; thus, α results in a large CF and CF(H), generating

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness large out-of-plane bending moments in both the arch and the deck. In the vicinity of the midspan, α is larger, and therefore, CF(H) is smaller. Nevertheless, due to the allowed UR2 at the deck abutments, even a low CF(H) can initiate U3, which influences the transformation of α, which in turn affects SF1 and SM(COMB) in the arch. The effects of a changing EI of the arch and deck for the deck BC#04 on SF1 and

SM(COMB) are summarized in Tables C-36 and C-37, respectively. The smallest SF1 results from a combination of S04D and S03A, and the smallest SM(COMB) is exerted on the arch in a combination of S02D and S03A. Distributions of SF1 and SM(COMB) capturing the sixteen combinations are presented in Figure 5-19 and Figure 5-20, respectively.

The trend in SF1 and SM(COMB) in BC#04 is very similar to the trend in BC#01. For

BC#04, the vertically stiff deck results in a small SF1 and a small SM(COMB) at all levels of arch stiffness.

For the case with a horizontally stiff arch (S03A and S04A) and S02D, SM(COMB) increases at the midspan because of the low horizontal stiffness of S02D; this combination results in a large U3 of the deck. Due to the large U3 of S02D, the capacity of the deck to carry vertical loads decreases (most perceptible at midspan), and thus, more load is transferred via the cables to the arch, which results in a large, mainly vertical, bending moment in the arch.

When a combination of S02D and S03A is employed, a low SM(COMB) for BC#04 is caused due to the low CF(H) that results from a large transformation of α. A vertically stiff deck (S02D) does not experience a large U2 (compared to vertically flexible decks).

However, due to the allowed UR2 at the deck abutments and low EI(H), the deck S02D experiences a large U3 to accommodate CF(H). Consequently, α increases, which results in a subsequent reduction of CF(H) and a smaller SM(COMB) in the arch. Overall, an employment of BC#04 represents an advantageous option in C03.

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

Figure 5-19: Configuration C03: A comparison of the distributions of SF1 in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#04

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

Figure 5-20: Configuration C03: A comparison of the distributions of SM(COMB) in the arch for LL100 and the 16 combinations of arch and deck stiffness. Deck BC type: BC#04

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

5.4.3 Susceptibility to Buckling

Susceptibility to buckling is based on the structural response described in Section 5.4.2. Each configuration is approached separately to determine the susceptibility to buckling. Section 10.9.4 in CHBDC S6-14 states that the in- and out-of-plane EI of the arch should be considered when evaluating a ratio of the axial forces and bending moments in a compression member. Hence, the structural response of the arch is evaluated in a similar fashion; in other words, the magnitude and distribution of SF1, SM(COMB), U2, and U3 are analyzed for all types of assumed deck BCs. Combinations of the arch and deck profiles, with a ratio of EI(V) and EI(H) that results in the smallest SF1 and SM(COMB) in the arch are summarized in Table 5-4. The provided ratios of EI(V) and EI(H) represent the optimal combination for a particular arrangement of the deck BCs. The proposed optimal combinations are the least susceptible to buckling. The details, on which Table 5-4 is based, are provided in Table C-38 to C-44. The effects of the selected arrangements of the deck BCs on the level of susceptibility to buckling are described separately for individual configurations in sections that follow.

Table 5-4: Proposed optimal mechanical ratios of deck and arch stiffness resulting in a combination, which is the least susceptible to buckling Confi- Deck BC Deck Arch Ratio of Deck and Arch guration Type: Stiffness Stiffness Stiffness Vertical 10 : 1 BC#01 S04D S01A Horizontal 10 : 1 Vertical 1 : 10 C01 BC#02 S03D S04A Horizontal 10 : 10 Vertical 10 : 10 BC#03 S04D S02A Horizontal 1 : 10 Vertical 10 : 10 BC#01 S02D S04A Horizontal 1 : 10 C02 Vertical 10 : 10 BC#04 S02D S04A Horizontal 1 : 10 Vertical 10 : 1 BC#01 S02D S03A Horizontal 1 : 10 C03 Vertical 1 : 1 BC#04 S03D S03A Horizontal 10 : 10

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

5.4.3.1 Configuration C01

The magnitude and distribution of SF1, SM(COMB), U2, and U3 of the arch at determined ratios of EI(V) and EI(H) of the arch and deck, showing the significance of a particular type of deck BC, are presented in Figures 5-21, 5-22, 5-23, and 5-24, respectively. The following can be concluded from the figures: • For a combination of S04D and S01A, which are identified as optimal for BC#01, the change in the type of BCs of the deck has a significant impact, as expected.

• The fixed ends of the deck in BC#03 reduce the magnitude of SF1, SM(COMB), U2, and U3. In contrast, the released rotational DOFs at the deck ends in BC#02 cause an increase in the structural responses. • The large EI of the deck in both directions controls the distribution of internal forces in the arch for all three arrangements of the BCs of the deck. Its effect is most perceptible for BC#03 because the deck acts as a cantilever beam in this case. For this combination, the structural response of the deck overcomes the effect of the arch, and therefore, the stiffness of the arch can be low. • The stiffness of the deck must be large in both directions to reduce the susceptibility to buckling in C01. • A combination of S03D and S04A, which are identified as optimal for BC#02, indicates that the released rotational DOFs for BC#02 allow a large portion of the load

to be transferred to the arch. Therefore, the EI(V) and EI(H) of the arch must be large

enough to accommodate such loads and prevent large U2 and U3. The EI(V) of the deck does not influence the distribution of the internal forces in the arch due to the released

rotational DOFs at the deck ends. However, a large EI(H) of the deck can reduce the

internal forces in the arch because the deck acts an arch loaded by CF(H). Hence, the

EI(H) of the deck plays a significant role in BC#02.

• When the EI(V) and EI(H) of the arch is large, the effects of deck BCs are negligible. • For a combination of S04D and S02A (identified as optimal for BC#03), it is proven that the effect of deck stiffness controls the arch action in C01. • In C01, all the combinations of arch EI with stiff decks are less susceptible to buckling than combinations with flexible decks. The effect of the arch EI cannot overcome the effect of deck stiffness in BC#03.

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

• Employing BC#03 can significantly reduce the susceptibility to buckling. BC#02 is the least suitable arrangement of deck ends in regards to buckling of the arch.

Figure 5-21: Configuration C01 — A comparison of the effects of different deck BCs on the distribution of SF1 in the arch for the combination of arch and deck stiffness

that results in the smallest SF1 and SM(COMB)

Figure 5-22: Configuration C01 — A comparison of the effects of different deck BCs

on the distribution of SM(COMB) in the arch for the combination of arch and deck

stiffness that results in the smallest SF1 and SM(COMB)

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

Figure 5-23: Configuration C01 — A comparison of the effects of different deck BCs on the distribution of U2 in the arch for the combination of arch and deck stiffness

that results in the smallest SF1 and SM(COMB)

Figure 5-24: Configuration C01 — A comparison of the effects of different deck BCs on the distribution of U3 in the arch for the combination of arch and deck stiffness

that results in the smallest SF1 and SM(COMB)

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

5.4.3.2 Configuration C02

As shown in Table 5-4, the ratio of the arch and deck stiffness that results in the smallest SF1 and SM(COMB) in the arch is the same for both BC#01 and BC#04. The effects of the released UR2 rotation in BC#04 on SF1, SM(COMB), U2, and U3 are presented in Figures 5-25, 5-26, 5-27, and 5-28, respectively. The following can be concluded from the figures:

• BC#04 results in a smaller SF1 and SM(COMB) in the arch compared to BC#01. • The effect of a different type of deck BCs is more perceptible in the distribution of the

SF1 than distribution of the SM(COMB). BC#04 reduces the drop in SF1, localized at the position of the first cable. The released rotation UR2 for BC#04 results in a more gradual distribution of SF1 in the arch. • The deck stiffness controls the magnitude of SF1 in the arch for both arrangements of deck BCs. A vertically stiff and horizontally flexible deck (S02D) results in the smallest SF1 at all levels of arch stiffness for both types of deck BCs; this is the optimal combination of the arch and deck stiffness (Section 5.4.2.2). • The released UR2 for BC#04 results in a larger U2 of the arch. For BC#04, due to the

low EI(H) of S02D, the deck has greater displacement in the U3 direction than for BC#01. Thus, for BC#04, the ability of the deck to carry vertical loads decreases and a larger amount of load is transferred to the arch, resulting in a larger U2 of the arch. Nevertheless, the U3 of the arch is reduced for BC#04 because the released UR2, making the deck more prone to U3 deflection, causes α to increase, which in turn

reduces CF and CF(H). • BC#04 positively reduces the susceptibility to buckling in C02. • A ratio of S02D and S04A combined with BC#04 is the least susceptible to buckling in C02.

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

Figure 5-25: Configuration C02 — A comparison of the effects of different deck BCs on the distribution of SF1 in the arch for the combination of arch and deck stiffness

that results in the smallest SF1 and SM(COMB)

Figure 5-26: Configuration C02 — A comparison of the effects of different deck BCs

on the distribution of SM(COMB) in the arch for the combination of arch and deck

stiffness that results in the smallest SF1 and SM(COMB)

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

Figure 5-27: Configuration C02 — A comparison of the effects of different deck BCs on the distribution of U2 in the arch for the combination of arch and deck stiffness

that results in the smallest SF1 and SM(COMB)

Figure 5-28: Configuration C02 — A comparison of the effects of different deck BCs on the distribution of U3 in the arch for the combination of arch and deck stiffness

that results in the smallest SF1 and SM(COMB)

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

5.4.3.3 Configuration C03

In C03, the deck stiffness controls the susceptibility to buckling for the two types of the deck BCs, as indicated in Table 5-4. In both arrangements, the smallest SF1 and

SM(COMB) occur in the arch S03A (a vertically flexible and horizontally stiff arch). The effects of the released UR2 for BC#04 on SF1, SM(COMB), U2, and U3 are presented in Figures 5-29, 5-30, 5-31, and 5-32, respectively. The following is concluded from the figures: • For the two cases that are being compared, the effect of the arch and deck profiles

(with a particular ratio of EI(V) and EI(H)) is larger than the effect of the deck BCs. Further, the effect of the deck BCs is only minor. • For BC#01, a combination of S02D and S03A is the optimal case. For BC#04, the

released UR2 results in a larger SF1, SM(COMB), and U2 and a smaller U3 of the arch. The combination of S03D and S03A, the optimal case, experiences a larger change in terms of the structural response for BC#04 response than for BC#01. • Due to the released UR2 in BC#04, the deck is more prone to U3 deflection, which

increases α. Consequently, an increase in α causes CF(H) to be low. However, due to the large U3 of the deck, more load is transferred to the arch. Hence, the susceptibility to buckling increases in BC#04. • BC#01 represents a more efficient arrangement when vertically stiff decks are selected. BC#04 reduces the susceptibility to bucking when vertically flexible decks are used.

• The magnitude of EI(H) of the arch and deck controls the susceptibility to buckling in C03. • BC#01 with vertically stiff decks is the least susceptible to buckling.

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

Figure 5-29: Configuration C03 — A comparison of the effects of different deck BCs on the distribution of SF1 in the arch for the combination of arch and deck stiffness

that results in the smallest SF1 and SM(COMB)

Figure 5-30: Configuration C03 — A comparison of the effects of different deck BCs

on the distribution of SM(COMB) in the arch for the combination of arch and deck

stiffness that results in the smallest SF1 and SM(COMB)

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness

Figure 5-31: Configuration C03 — A comparison of the effects of different deck BCs on the distribution of U2 in the arch for the combination of arch and deck stiffness

that results in the smallest SF1 and SM(COMB)

Figure 5-32: Configuration C03 — A comparison of the effects of different deck BCs on the distribution of U3 in the arch for the combination of arch and deck stiffness

that results in the smallest SF1 and SM(COMB)

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5.5 Conclusion to Chapter Five

The following can be concluded about the general response of each configuration to a change in EI of the arch and deck as a function of deck BCs.

5.5.1 Configuration C01

The response in C01 with deck BC#01 is controlled by EI(H) of the arch and EI(V) of the deck. When the arch is horizontally stiff and the deck vertically stiff, the axial forces in the arch are large and the bending moments in the arch are small. Such a configuration is the least susceptible to buckling. When considering the deck with BC#02, the stiffness of the deck governs the response in the arch. A vertically and horizontally flexible deck results in large axial forces and bending moments in the arch and, therefore, increases the susceptibility of the arch to buckling. In contrast, a deck that is vertically and horizontally stiff deforms less and causes a larger displacement of the arch than the flexible deck. The larger displacements of the arch cause large bending moments in the arch; nevertheless, the axial loads in the arch are reduced due to out-of-plane arch displacements. In other words, due to the out of plane displacements of the inclined arch, the arch is no longer capable of carrying as high axial loads as a conventional vertical arch. Overall, stiff deck reduces the susceptibility of the arch to buckling. Fixed ends of the deck, i.e., BC#03, represent an option that results in a configuration that is the least susceptible to buckling from all the three deck BCs. When the deck is vertically and horizontally stiff, most of the load is carried by the deck and the out- of-plane displacements of the arch are small, which results in small susceptibility to buckling. Low horizontal stiffness of the arch improves the described behaviour.

5.5.2 Configuration C02

In C02, a combination of the arch and deck stiffness that results in small axial load in the arch typically results in large bending moment in the arch. It can be said that the deck stiffness governs the response in the arch regardless of the arch stiffness. The vertically stiff and horizontally flexible deck results in the smallest SF1 and the largest

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SM(COMB) in all levels of arch stiffness when compared to the effect of the other levels of deck stiffness. The difference between the responses in the arch resulting from BC#01 and BC#04 in C02 is that for the BC#04 the variability in deck bending stiffness does not influence the magnitude of SM(COMB). Due to the released to released rotation about vertical axis at the deck ends, there is almost no change in SM(COMB) due to change in deck stiffness for the particular arch stiffness. Such behaviour holds for all levels of arch bending stiffness. When BC#04 is selected, the deck bending stiffness controls the magnitude and distribution of SF1 in the arch and the arch bending stiffness controls the magnitude and distribution of SM(COMB) in the arch. The BC#04 reduces the susceptibility of the arch to buckling because the released rotation about vertical axis at the deck ends reduces both the axial loads and bending moments.

5.5.3 Configuration C03

The main difference between C03 and C01 or C02 is that in C03 the arch crosses over the straight deck and, therefore, the inclination of each cable, from a vertical plane, is different (symmetry midspan applies). Another difference is that in comparison to C01, C02, or conventional vertical planar arch, in C03, the cables are the shortest at midspan and longest at the arch abutments. The response in the cables is critical; it is a function of both the cable inclination and cable length. The highest cable forces occur in cables located in the vicinity of arch abutments because the inclination of these cables, from a vertical plane, is large. Nevertheless, the large length of these cables, resulting in large elongation, reduces the effect of high tensile forces in the cables on the response in the arch. Whereas at midspan, despite the small inclination of the cables from a vertical plane (resulting in relatively smaller forces in the cables in comparison to the cables with a large inclination from a vertical plane), the effect of the cables on the response in the arch is large. Also, the arch is prone to larger displacements at midspan than in vicinity of the abutments. Therefore, in C03 it is the horizontal stiffness of the arch that controls the response in the arch. Deck horizontal stiffness can also affect the response in the arch; however, a change in the deck

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Chapter Five: The Significance of Boundary Conditions and Bending Stiffness horizontal stiffness has a smaller impact on the response in the arch than a change in the arch horizontal stiffness. The end conditions of the deck can significantly influence the general type of response described above. The BC#04 increases the susceptibility of the arch to buckling because the released rotation about vertical axis at the deck ends reduces the effect of horizontal deck stiffness and the deck is more prone to horizontal displacements. Displacement of the deck in horizontal direction causes an increase mainly in bending moments in the arch, which, in C03, represents the critical type of response. The response of the arch can be improved by selecting BC#01. A combination of BC#01, a deck of a small horizontal bending stiffness, and an arch with large horizontal stiffness reduces the susceptibility of the arch to buckling.

The result summary tables, which provide organized data as a function of arch and deck EI and deck BCs, accompanied with structural sketches (provided in Appendix C), when combined with the conclusions of this chapter, can be used as a guideline. Chapter Five establishes a base for the spatial configurations that are studied in Chapter Six.

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6.1 Introduction Any structure exposed to weather conditions undergoes a process of deterioration. In Canadian climates, the effect of weather conditions is elevated due to the frequent presence of de-icing salts and cycles of freezing and thawing that accelerate the deterioration process, particularly in structures such as parking garages and bridges. Hence, a desire for durable materials, such as advanced composite materials (ACMs), arises. ACMs are increasingly used in civil engineering applications for their superior durability and high stiffness and strength to weight ratios. The mechanical properties of ACMs can be utilized to reinforce and strengthen structures or construct all-composite assemblies. ACMs also represent an alternative to traditional materials that have been used in the construction of SABs. Nevertheless, the fibrous character of ACMs accompanied with our limited understanding of the response of structures exposed to out-of-plane loads result in a rare occurrence of all-composite SABs. Therefore, several factors in the structural response of SABs that are entirely constructed from ACMs are now investigated. The challenges in SABs entirely constructed from ACMs are related primarily to the material properties of ACMs. Typically, ACMs have a low modulus of elasticity (E) and are susceptible to creep, which leads to large deformations. Creep-rupture can cause a collapse of the entire structure. The low weight of ACM may be advantageous in terms of designing a substructure. However, the low mass of the superstructure may influence the damping properties and the structure may become more prone to undesired vibrations. ACMs have a low coefficient of thermal expansion (CTE), which results in small demands on the deck expansion joints and in small fluctuations of stresses due to the reduced contraction and expansion of the cables that support the deck. These factors influence the structural response, and therefore, they need to be addressed via detailed analyses. In this chapter not all the possible cases are covered, but rather focus is placed on the key factors identified as an outcome of the literature review (Section 2.6.5) and the findings in Chapters Four and Five. Therefore, the main focus is investigating the effect of the following: 1) creep and creep-rupture, 2) CTE, and 3) changing deck stiffness upon applied LL and GTL.

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There are two types of ACMs assumed for the proposed analyses: 1) glass fibre reinforced polymers (GFRP) and 2) carbon fibre reinforced polymers (CFRP). Both material types are employed in the form of HSS profiles. The response of the cables, connecting the arch and deck, directly influences the structural behaviour of these two components. The nonlinear character of the cables can skew the accuracy of the results. Hence, a detailed approach, employing the developed nonlinear FE models, is established to take into account the effect of sag and the additional tensioning of the cables. In contrast to Chapters Four and Five, only one configuration is considered here. The spatial configuration C01B consists of an in-plan curved deck and a parabolic arch inclined from a vertical plane. The chapter is divided into the following five sections. The objectives are presented in Section 6.2. Section 6.3 describes the specifications of the developed models including the geometrical and material properties in relation to the analysis types. Section 6.4 presents and discusses the achieved results of the specific analyses as follows: Section 6.4.1 describes the effect of susceptibility to creep, creep-rupture, and effect of additional tensioning of cables; Section 6.4.2 discusses the sensitivity of the assumed structural configuration to different CTE; and Section 6.4.3 depicts the details related to the magnitude of deck deflection and the distribution of stresses in the deck while taking into account the variability in deck stiffness. Section 6.5 underlines the most significant findings. Details of the analyses, such as large tables or complex charts, are provided in Appendix D.

6.2 Objectives The main goal of Chapter Six is to investigate the significance of material influence on the structural behaviour of a particular spatial configuration. The specific objectives are:

• To determine whether or not the spatial configuration C01B, using GFRP and CFRP in the structure, is susceptible to creep-rupture due to out-of-plane loads when a particular deck deflection limit (L/500) is imposed;

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• To investigate the significance of creep on the distribution and magnitude of displacements and stresses in the arch, deck, and cables made of GFRP;

• To determine changes in stress distributions in structural profiles made of GFRP (employed in the arch and deck) when the EI of the arch is reduced by 50% (compared to a model configuration investigated in the previous objective). Additional tensioning of the cables is employed to achieve the imposed deck deflection limit of L/500;

• To determine the level of a stress change in the structural profiles and cables made of the two types of AMCs, GFRP and CFRP, with a low CTE (compared to steel) when exposed to –50°C and +50°C, assuming that a principle of constant-bending-stiffness is employed in the arch and deck and a principle of constant-axial-stiffness is employed in the cables;

• To determine an optimal ratio of EI(V) and EI(H) of the deck that results in the smallest deck deflection at the midspan and the smallest stress fluctuation in the deck, assuming

only GFRP within the entire structure, while taking into account three levels of f(D)/s (0.25, 0.20, and 0.15) and nine different deck cross-sections; and

• To examine the effect that GFRP, CFRP, and steel have on the distribution and magnitude of stress in two critical deck profiles, while assuming the most sensitive

ratio f(D)/s (0.25) and three levels of applied GTL (0°C, +50°C, and -50°C).

6.3 Model Specifications The common aspects of the models related to all analysis types are provided in the sections below. Chapter Six assumes three main analysis types: • Analysis AN#01: The effect of creep • Analysis AN#02: The significance of CTE • Analysis AN#03: The influence of the variability in deck stiffness

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6.3.1 Nonlinear Model

Nonlinear FE models (developed according to Section 3.3.1.3) are employed to account for the nonlinear character of the cables, whose response directly impacts the behaviour of the arch and deck. In Chapters Four and Five, the cables were modeled with truss elements to ignore the effect of sag in the cables. This assumption is acceptable for an analysis that clarifies the overall structural response, while considering a change in geometry or BCs. However, in terms of investigating the significance of material properties on the stress distribution, the sag of the cables cannot be ignored because the level of resulting stresses (in the stress sensitive ACM profiles) can change significantly. It should be noted that the even though the effect of the cable sag is taken into account, in these models, the length of the cables was established at the un-deformed state, i.e., the cables assume no initial sag.

6.3.2 Selection of a Critical Configuration

The structural configuration consists of an in-plan curved deck (identified as critical in Section 5.4.3.1) and an arch, inclined from a vertical plane by angle ω = 30° (identified as critical in Section 4.4.2.2). The combined configuration is designated as “C01B.” C01B is based on the findings presented in the previous chapters and on a stress analysis that takes into account the three main spatial configurations (C01, C02, and C03) and the three assumed materials (GFRP, CFRP, and steel). The detailed results from these analyses are provided in Table D-1 and Table D-2. The magnitude and type of stress (tensile or compression) exerted on individual structural components in the assumed configurations is compared. The results show that in C01 and C02, the structural components experience high stresses more frequently compared to C03. Therefore, it seems logical to combine the geometry of C01 and C02 into the proposed configuration C01B. The ranges of the primary and secondary variables of C01B are presented in . A graphical representation of C01B (depicting the primary and secondary variables), including sections cuts through the arch and deck with varying dimensions, is presented in Figure 6-1.

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Table 6-1: Range of primary and secondary variables assumed in Chapter Six Analysis Geometric variable Value

f(A)/s 0.15

AN#01 & AN#02 f(D)/s 0.25 ω 30°

f(A)/s 0.15

AN#03 f(D)/s 0.15/0.20/0.25 ω 30°

Cable Diam. D: Arch Diam.

H: Deck Height

W: Deck Width

f(A): Arch rise

s: Span ff(D)(D);: DeckDeck rReaceach h

Figure 6-1: Spatial Configuration C01B

6.3.3 Boundary Conditions

The BCs of the arch remain unchanged; in other words, the arch is fixed at both ends. However, the BCs of the deck are altered to provide a more realistic arrangement. The characteristics of the deck BCs are classified in Chapter Five as BC#02 (see Table 5-3). The altered deck BCs is designated as BC#02B. At the BC#02B, the rotational DOF about the longitudinal deck axis (UR1) is not allowed. The local orientation at the deck abutments is altered to follow the longitudinal axis of the deck, which is a function of the f(D)/s. A definition of the BC#02B in terms of the DOF is provided in Table 6-2. A schematic representation of BC#02B is shown in Figure 6-2. The selected deck BC#02B remains unchanged throughout all analyses in Chapter Six.

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Table 6-2: Definition of deck boundary condition BC#02B Type Description U1 U2 U3 UR1 UR2 UR3 Restricted Pin-connection (P_P_R) Allows for rotation about local direction 2 and 3. BC#02B 1 1 1 1 0 0 Rotation about the deck longitudinal axis is not permitted. No translations are allowed.

Figure 6-2: Schematic representation of deck boundary conditions BC#02B (Restricted Pin-connection) in C01B

6.3.4 Cross-sections

Structural profiles are modeled via beam elements with a rectangular cross-section (C-S), representing closed HSS. The employment of HSS sections assumes the ACMs are in the form of pultruded profiles (see Section 3.5 for reference). A reason to select a rectangular box section is the ability to determine the stresses as a function of the following: a) particular location within the assumed C-S and b) actual magnitude of the area of the C-S, which changes as the outer dimensions of the HSS are varied. The selection of rectangular sections also reflects standard structural profiles and, hence, the

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Chapter Six: Application of Advanced Composite Materials in SABs magnitude of stresses can be evaluated against particular stress limits that need to be met in the individual types of ACM. In the present chapter, all analyses take into account profiles made of the three assumed materials (see Section 6.3.6), whose overall dimensions change to address the particular objectives. While the overall dimensions of the assumed structural profiles are varied, the thickness of the HSS wall remains unchanged throughout all analyses. The thickness of the HSS wall in the arch and the deck is 25mm and 20mm, respectively, following the assumed parameters determined in Table 4-3.

6.3.4.1 Principle of a Variation in the Bending and Axial Stiffness

The analyses AN#02 and AN#03 are based on the principle of constant-bending- stiffness (CBS), described as follows: • In AN#02, the dimensions of the arch, deck, and cables are based on results of AN#01C, which considers GFRP structural profiles. • The EI of the arch and deck (found via AN#01C) is set as a reference to other materials in AN#02. To achieve the same EI, while taking into account materials with different E, the outer dimensions of the structural profiles must change. • The principle of CBS is graphically demonstrated in Figure 6-3. • Similarly to the principle of CBS applied to profiles assumed for the arch and deck, a principle of constant-axial-stiffness (CAS) is applied to cables. A cable made of material with large E will have small A and vice versa.

E(GFRP) < E(STEEL) I(GFRP) > I(STEEL) EI(GFRP) = EI(STEEL)

GFRP Profile STEEL Profile Figure 6-3: Principle of constant-bending-stiffness (CBS)

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It should be realized that even though the principle of CBS is considered, the structural profiles have different axial stiffness. The different axial stiffness of these profiles is unavoidable because to achieve the constant bending stiffness simultaneously with the constant axial stiffness is not feasible.

6.3.4.2 Section Points

The significance of the three materials, whose performance is stress sensitive (creep-rupture in ACM), is investigated and, therefore, the actual magnitude of the stresses becomes an important factor. In C01B, four locations of section points (SPs) were determined to capture the stresses within the C-S of the arch and deck. Individual SPs are defined at the middle of the wall of the HSS profile. The locations of the SPs in the arch and deck are presented in Figure 6-4.

SP#01: TOP

SP#01: LHS SP#03: RHS

SP#01: TOP SP#01: BOTTOM

SP#01: LHS SP#03: RHS SP#01: BOTTOM

Figure 6-4: Locations of section points in the cross-section of arch and deck

6.3.5 Loads

There are three different types of loads exerted on the structure: 1) DL, 2) LL, and 3) GTL. Because the properties of the assumed materials vary, the magnitude of DL varies as well. As shown in Chapters Four and Five, the LL100 represents the critical pattern, and hence, the LL100 is selected. The magnitude of the LL is determined in accordance with Section 3.3.5. The magnitude of GTL covers three levels: –50°C, 0°C, and +50°C. In

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Chapter Four, the effects of the LL and GTL were examined separately; in this chapter, the LL and GTL are applied at the same time to represent a more realistic case to which the assumed configuration may be exposed. It should be noted that all three types of loads, i.e., DL, LL, and GTL are applied simultaneously in the first loading step. When taken into account, additional tensioning of the cables is applied in a separate second loading step.

6.3.6 Materials

The assumed HSS profiles take into account three material types: 1) GFRP, 2) CFRP, and 3) structural steel. Structural steel is employed in the models to serve as a reference material. The mechanical properties of the selected materials are presented in Section 3.5.

The magnitude of the effective modulus of elasticity (E(EFF)), which is used to take creep into account, was established according to Smith (2005). In the GFRP pultruded profiles and cables, due to the effect of creep, the initial modulus of elasticity (E(IN)) drops to a range of 70% to 90%, as a function of temperature and length of service life. Based on the assumed temperature range (–50°C to +50°C) and service life of 75 years, it was determined that the E(EFF) equals 75% of E(IN). The magnitude of E(EFF) for the GFRP pultruded profiles and cables is presented in Table 6-3.

Table 6-3: E(EFF) for the GFRP profiles and cables

Component E(IN) [GPa] E(EFF) [GPa] GFRP pultruded profile 54 40.5 GFRP Cable 80 60

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6.4 Results and Discussion

6.4.1 Creep Analysis

Creep analysis, AN#01, is subdivided into three sections to investigate the particular parameters affected by the creep, such as the effect of E, EI, or weight per unit length (WPUL), which are discussed below.

6.4.1.1 Susceptibility to Creep-rupture

The proposed analysis with focus on the creep related effects taking into account the susceptibility to creep-rupture is designated “AN#01A”. The factors to consider are as follows: • Developed spatial configurations assume one material type to create the entire structure. GFRP and CFRP are employed; steel is assumed as a reference material. • EI of the arch is varied to meet the imposed deck deflection limit of L/500 (in accordance with AASHTO (2009) Section 5 Deflections). The arch diameter is varied until a satisfactory value of deck U2 is achieved. At this point, the contribution of the EI of the deck and the cables is only minor and, therefore, the C-S properties of the deck and cables remain unchanged. • The stresses achieved in the configurations, assuming GFRP and CFRP, are compared to the stress limits (listed in Table 3-1), which are imposed to avoid creep-rupture. • Properties of the assumed C-S in AN#01A are provided in Table D-3.

Results of the AN#01A are presented in Table 6-4. The table shows the diameter of the arch rib, required to archive imposed deck deflection limit. In order to meet the limit of L/500, the bending stiffness of the arch needs to be enlarged to such a level where the stresses that develop do not exceed the stress limits imposed to avoid creep-rupture. The results show that the structural response of the arch and the EI of the arch control the vertical deflection of the deck in C01B. Due to the serviceability limit, creep-rupture in C01B (entirely made of ACM) is not a significant concern. Nevertheless, due to the low E in the GFRP profiles, the arch diameter needs to be significantly larger than if made from CFRP or steel.

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Table 6-4: Results of analysis AN#01A Stress [MPa] Creep- Arch: Deck: Vert. L/500 rup. Mat. Diam. W*H deck Satisfie Arch Deck Cables limit [m] [m] deflection d Y/N [MPa] GFRP 4.50 2.0*0.4 L/501 Y 19 13 65 360 CFRP 2.75 2.0*0.4 L/502 Y 47 13 68 1,700 STEEL 2.70 2.0*0.4 L/527 Y 73 24 125 N/A

6.4.1.2 Significance of Weight per Unit Length and Modulus of Elasticity

The proposed analysis investigating the effect of WPUL and E is designated “AN#01B-I”. In AN#01B-I, the same configurations as in AN#01A are studied in order to establish the significance of different WPUL of the three materials in question. The distribution of displacements in the arch and deck are compared and the effect of WPUL described. A comparison of the arch EI that results in satisfactory deflection of the deck is shown in Table 6-5. This table shows that even though the GFRP profile must have the largest diameter to result in satisfactory deflection of the deck and the WPUL differs among the three materials, the final EI of the GFRP arch is very similar to those of the arches made of CFRP and steel. It should be noted that the principle of CBS was not employed in this analysis.

Table 6-5: EI of the arch resulting in satisfactory deck deflections Arch EI Vertical Deck Material E [N/m2] I [m4] [N*m2] deflection GFRP 5.40E+10 4.12E-01 2.22E+10 L/501 CFRP 1.47E+11 1.49E-01 2.19E+10 L/502 STEEL 2.00E+11 1.16E-01 2.31E+10 L/527

A comparison of the U2 and U3 of the deck is presented on Figures 6-5 and 6-6, respectively. These figures consider identical C-S of the deck: W = 2.0m, H = 0.4m in all materials.

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Figure 6-5: U2 of the deck, considering deck: W = 2.0m, H = 0.4m in all materials

Figure 6-6: U3 of the deck, considering deck: W = 2.0m, H = 0.4m in all materials

From the Figure 6-5 it is apparent that despite the same C-S of the deck, the different E and WPUL, in the three configurations, do not affect the final magnitude or distribution of U2 significantly. Such behaviour results from the ratio of EI(H) and EI(V) of the deck (EI(H) being larger than EI(V)) combined with the geometric configuration assumed for the C01B. In the configuration in question, the deck reach is large (f(D)/s = 0.25) and the arch rise small (f(A)/s = 0.15). Hence, the CF(H) is large and the EI(H) of the deck is utilized rather than EI(V) of the deck.

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While the effect of low EI(V) in GFRP deck profiles does not influence the U2 of the deck significantly, the effect of low EI(H) of the GFRP deck on the U3 of this deck is more evident. EI(V) and U2 of the deck made of the three materials are presented in Table 6-6 and

EI(H) and U3 in Table 6-7.

Table 6-6: EI(V) and U2 of the deck

Material E [N/m2] I [m4] EI(V) [N*m2] U2 GFRP 5.40E+10 3.05E-03 1.64E+08 L/501 CFRP 1.47E+11 3.05E-03 4.48E+08 L/502 STEEL 2.00E+11 3.05E-03 6.09E+08 L/527

Table 6-7: EI(H) and U3 of the deck

Material E [N/m2] I [m4] EI(H) [N*m2] U3 GFRP 5.40E+10 4.08E-02 2.20E+09 L/7,843 CFRP 1.47E+11 4.08E-02 5.99E+09 L/18,475 STEEL 2.00E+11 4.08E-02 8.16E+09 L/79,447

As expected, U3 of the deck is largest in the GFRP deck, the deck with the smallest

EI(H). Hence, it can be concluded that in C01B, the EI of the arch controls U2 of the deck and EI(H) of the deck controls U3.

The distributions of U2 and U3 of the arch, affecting the deck displacement, are presented in Figure 6-7 and Figure 6-8, respectively. From these figures it is apparent that U2 and U3 of the arch are the largest in the CFRP arch, not in the GFRP arch, as may have been expected. Such behaviour is a result of the interaction between the EI of the arch and deck.

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Figure 6-7: U2 of the arch, considering deck: W = 2.0m, H = 0.4m in all materials

Figure 6-8: U3 of the arch, considering deck: W = 2.0m, H = 0.4m in all materials

As pointed out earlier, the arch is the structural component that governs U2 of the deck. And in order to achieve the same U2 of the deck, the EI of the arch needs to be the same regardless of the material used. However, because EI(H) of the CFRP deck is larger than EI(H) of the GFRP deck, the CFRP deck (carrying the same amount of LL, generating similar CF(H) as in the case of the GFRP deck) will tend to deflect less in the horizontal direction, due to a larger deformation of the CFRP arch than the arch made of GFRP. In

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other words, because the EI(H) of the CFRP deck is larger than EI(H) of the GFRP deck, the CFRP arch (with the same stiffness as the GFRP arch) will deflect more to accommodate the loads transferred via the cables from the horizontally stiff deck. It can be noted that in the configuration assuming steel, the U2 and U3 of the arch are smaller than in the configuration assuming CFRP despite the EI(V) and EI(H) of the steel deck being larger than that of the CFRP deck. The reason for such behaviour is not the WPUL of the steel deck but the WPUL of the steel arch. The WPUL of the steel arch is significantly larger than the WPUL of the CFRP and, therefore, the weight of the steel arch counteracts the CF(H). Hence, the resulting U2 and U3 of the steel arch is smaller in comparison to the configuration made of CFRP. The outcome of the large WPUL of the steel arch and deck is compensated by large stress in the cables as shown in Figure 6-9. The CFRP arch and deck, with low WPUL, then result in the smallest stress in the cables, except the first cable which exposed to the largest amount of force as explained in Section 4.4.2.1.6.

Figure 6-9: Stresses in cables (Cable Ø = 50mm in all materials)

The distribution of stresses, at the four designated section points, in the arch and deck are shown in Figure 6-10 and Figure 6-11, respectively.

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a) Stress at SP#01: Top

b) Stress at SP#02: Bottom

c) Stress at SP#03: RHS

d) Stress at SP#04: LHS

Figure 6-10: Distribution of stresses in the arch comparing the effect of GFRP, CFRP, and steel at four section points

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a) Stress at SP#01: Top

b) Stress at SP#02: Bottom

c) Stress at SP#03: RHS

d) Stress at SP#04: LHS

Figure 6-11 Distribution of stresses in the deck comparing the effect of GFRP, CFRP, and steel at four section points

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The figures show that the stresses in the steel arch and deck are the highest due to the highest WPUL of steel. Also, the difference between the maximum and minimum stress is the highest in the steel profiles. The profiles made of GFRP and CFRP with low WPUL have similar magnitude and distribution of stresses, particularly in the deck. However, the difference between the maximum and minimum stress in the GFRP and CFRP profiles is much smaller in comparison to the steel profiles. Such an attribute of commonly used GFRP profiles represents a promising feature for these profiles to be used in spatial structures. Details of the stress fluctuations in the arch, deck, and cables are provided in Appendix D, Tables D-4, D-5, and D-6. The smallest stress fluctuation was identified in the GFRP profiles; the profiles with the smallest E and moderate WPUL among the three materials. However, the stress distribution is time dependent in the GFRP profiles and the effect of creep plays a significant role. Hence, the significance of creep on the stress distribution in the arch and deck, assuming only GFRP profiles, is studied next.

6.4.1.3 Significance of Creep on the Stress Distribution

The proposed analysis, investigating the effect of E(EFF), is designated “AN#01B- II”. In this analysis, only GFRP is modeled for the arch and deck because among the three materials, GFRP is most susceptible to creep. The cables assume all three materials in order to examine the effect of creep in the cables. Five different models were analyzed. The designation of the models, assignment of the materials, and level of E(EFF) in the members and cables are listed in in the Table 6-8. Table 6-8: Parameters of models in analysis AN#01B-II

Model Material of E(EFF) of arch Material of E(EFF) of cables designation arch and deck and deck [%] cables [%] M(REF) GFRP 100 GFRP 100 M001 GFRP 75 GFRP 75 M002 GFRP 75 GFRP 100 M003 GFRP 75 CFRP 100 M004 GFRP 75 STEEL 100

The cross-sectional properties of the arch, deck, and cables were the same as for the AN#01A, as specified in Section 6.4.1.1. The magnitude and distribution of U2 and U3 of

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Chapter Six: Application of Advanced Composite Materials in SABs the arch and deck are compared and checked against the imposed limit of L/500 in order to verify whether or not the reduced stiffness ( EI(EFF)) in the arch may contribute to a reduction of vertical deck displacement. The resulting distributions and magnitudes of U2 and U3 of the deck are presented in Figure 6-12 and Figure 6-13, respectively.

Figure 6-12: U2 of the deck taking into account creep in the arch and deck and five levels of axial stiffness of the cables

Figure 6-13: U3 of the deck taking into account creep in the arch and deck and five levels of axial stiffness of the cables

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A comparison of U2 of the deck as a function of cable axial stiffness, taking into account E(EFF), WPUL, and the ability of cables to creep is provided in Table 6-9. Table 6-9: Deck vertical displacement as a function of changing parameters of cables Cable Cable Creep Difference Material Model EA(EFF) WPUL U2 Deck in in vert. of cables [N*m2] [N/m] Cables displ. [%] M(REF) GFRP 1.57E+08 38 L/501 No N/A M001 GFRP 1.18E+08 38 L/365 Yes 27% M002 GFRP 1.57E+08 38 L/400 No 20% M003 CFRP 3.24E+08 20 L/491 No 2% M004 STEEL 3.24E+08 150 L/477 No 5%

From the table is apparent that only the model M(REF) (the model assuming no creep at all) meets the deflection limit L/500. A combination of Figure 6-12 and Table 6-9 indicates that the effect of creep in the arch and the deck is more significant than the effect of creep in cables. Model M002 (assuming creep in the arch, deck, and cables) and model M003 (assuming creep in the arch and deck only) experience increases in deck deflection U2 by 27% and 20%, respectively. Table 6-10 provides a comparison of the differences in U2 of the deck, and axial stiffness, WPUL, and EA of the cables. Table 6-10: The significance of changes of the cable properties on deck vertical displacement Cable Difference Difference Difference Material Model EA(EFF) in Deck in EA(EFF) in WPUL Deck U2 of cables [N*m2] U2 [%] [%] [%] M(REF) GFRP 1.57E+08 N/A N/A N/A L/501 M001 GFRP 1.18E+08 27% -33% 0% L/365 M002 GFRP 1.57E+08 20% 0% 0% L/400 M003 CFRP 3.24E+08 2% 52% -90% L/491 M004 STEEL 3.24E+08 5% 52% 75% L/477

From the table is apparent the effective axial stiffness of the cable (EA(EFF)) governs the U2 of the deck. When CFRP (M003) and steel (M004) cables (assuming no creep and the same EA(EFF) (which is increased by 52% in comparison to the reference GFRP cable in

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M001, assuming creep in the cables) are employed, the deck vertical displacement improves by 25% and 22%, respectively.

The effect of the different WPUL of the cables is present, but very minor. This is apparent from comparing models M003 and M004 in Table 6-10, which have identical

EA(EFF) of the cables. The large increase in WPUL (by 75%) in the steel cables (M004) results in a change of the deck deflection by 5% in a comparison to M(REF). The light CFRP cables, with a 90% drop in WPUL, result in a change in the deck deflection of only by 2% in comparison to M(REF). Hence, it can be concluded that the effect of light-weight ACM cables (M003) on the vertical displacement of the deck is very minor and it is not effective to use high performance CFRP cables. Nevertheless, CFRP cables have much better response to fatigue loading than steel cables and, therefore, employment of CFRP cables can be advantageous. The effects of creep (resulting in a drop in the bending stiffness) in the arch and changing EA(EFF) of the cables on U2 and U3 of the arch are presented in Figure 6-14 and Figure 6-15, respectively.

Figure 6-14: U2 of the arch taking into account creep in the arch and deck and five levels of axial stiffness of the cables

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Figure 6-15: U3 of the arch taking into account creep in the arch and deck and five levels of axial stiffness of the cables

From the figures is apparent that the effect of creep in the arch and deck results in almost identical U2 and U3 of the arch. The effect of variability in EA(EFF) of the cables influences the arch displacement insignificantly. The large WPUL of steel cables causes an increase in both the U2 and the U3 of the arch. The effect of creep in the cables does not affect the arch displacements. A comparison of the forces in the cables is presented in Figure 6-16. In models M003 and M004, where the axial stiffness of cable is high, the stress at the first cable is larger than in the other models. The large WPUL of the steel cables (M004) causes an increase of the stresses in the cables, particularly at midspan where the cable length and, therefore, the total weight of the cables is the largest. It can be concluded that creep in the arch and deck has only a minor influence on stress changes in the cables.

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Figure 6-16: Stress in the cables taking into account creep in the arch and the deck and five arrangements of cables axial stiffness and susceptibility to creep

6.4.1.4 Effect of Additional Tensioning of Cables

The proposed analysis, AN#01C, focuses on creep related effects and considers the additional tensioning of cables.

Model M001 from AN#01B-II, which assumes creep with E(EFF) = 75% in all structural components (arch, deck, and cables), is applied. Properties of M001 are adapted into model M001B. M001B also assumes E(EFF) = 75% in all structural components. The materials and C-S properties of the deck and cables remain the same as in M001. However, the C-S properties of the arch change. The EI(EFF) of the arch (governing the deck vertical displacement in C01B) is reduced by 50%. The outer diameter of the arch is reduced from

4.5m, in M001, to 3.5m in M001B to achieve a reduction of the arch EI(EFF). Additional tensioning of the cables is applied in accordance with the principles in Section 3.4.1.4 to obtain a satisfactory U2 of the deck (L/500). The temperature resulting in additional tensile force in individual cables is determined via Equation 3-11. Parameters of Equation 3-11: , , and , are determined in the present section via an iterative analysis to achieve satisfactory𝑇𝑇𝑇𝑇 𝑚𝑚 deck𝑛𝑛 deflection and a gradually deflected deck. For convenience, Equation 3-11 is presented again:

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( ) ( ) ( ) = + 4 × 1 × 𝐶𝐶 𝑖𝑖 𝐶𝐶 𝑖𝑖 𝐿𝐿𝐿𝐿𝐿𝐿 𝑖𝑖 𝑇𝑇𝑇𝑇 𝑚𝑚 � − � Equation 3-11 𝑛𝑛 𝑛𝑛

The results from M001B (including the developed distribution of additional tensile forces in the cables) are compared to the two models addressed in Section 6.4.1.3: M001 and M(REF) assuming E(EFF) = 75% and 100% in all components, respectively. First, a comparison of the deflected shapes of the deck and the arch are conducted, and then the magnitude and distribution of the stresses at the four designated SPs is evaluated to determine whether or not the applied additional tensioning of the cables can result in stresses that are close to the creep-rupture limit. A comparison of the magnitudes and distributions of cable forces along the span is also performed to check whether or not the load in the cables exceeds 40% of the ultimate cable strength, which is the threshold magnitude typically used in additional cable tensioning (Chen 2000). The resulting parameters, Tb, m, and n, obtained for Equation 3-11 are listed in Table 11.

Table 6-11: Resulting parameters of Equation 3-10 providing an optimal distribution of additional tensile forces in the cables Parameter Value Tb -200°C m -66°C n 15

A comparison of the U2 of the deck, taking into account the EI(EFF) of the arch that varies in all three models, is provided in Table 6-12.

Table 6-12: The EI of the arch and U2 in the deck achieved from models M(REF), M001, and M001B Arch Cable Vertical Arch EI(EFF) Creep in Model WPUL EA(EFF) deck [N*m2] Cables [N/m] [N*m2] deflection

M(REF) 4.77E+10 6,834 1.57E+08 N L/501 M001 3.58E+10 6,834 1.18E+08 Y L/365 M001B 1.68E+10 5,307 1.18E+08 Y L/504

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Comparison of the resulting magnitude and distribution of U2 and U3 of the deck are presented in Figure 6-17 and Figure 6-18, respectively.

Figure 6-17: U2 of the deck taking into account creep in the structural components and additional tensioning of cables

Figure 6-18: U3 of the deck taking into account creep in the structural components and additional tensioning of cables

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Displacements U2 and U3 of the arch are compared in Figure 6-19 and Figure 6-20, respectively.

Figure 6-19: U2 of the arch taking into account creep in the structural components and additional tensioning of cables

Figure 6-20: U3 of the arch taking into account creep in the structural components and additional tensioning of cables

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The distribution of the stresses in the cables is shown in Figure 6-19.

Figure 6-21: Stress distribution in the cables, taking into account additional tensioning of the cables and creep in the arch, deck, and cables

The following can be concluded from the tables and figures for AN#01C: • AN#01C proves that the additional tensioning of cables can be a very efficient way to meet the deflection limit of the deck while reducing material usage and keeping the stresses below the imposed limits. The reduction of the arch EI by 50% results in a material savings of ~22% while meeting the L/500 deflection limit and the creep rupture limit. • Due to the employment of additional tensioning of the cables, the distribution of stresses in the deck is not influenced by creep in the arch. The magnitudes and distribution of the stresses in the deck of M001B (assuming creep and additional

tensioning of the cables) closely correlates with the distribution of stresses of M(REF) (assuming no creep and reduced EI of the arch). Therefore, as long as the deflected shape of the deck meets the required profile (achieved via additional tensioning of the cables), large stress redistribution in the deck (apparent in M001 where the deck deflection L/365 is below the imposed limit L/500) can be avoided.

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• The distribution and magnitude of stresses in the arch (the main component affecting the magnitude of deck displacement) is significantly affected by additional cable

tensioning and the effect of creep. Due to the low EI(EFF) of the arch, the arch experiences larger deflections and increased stresses. Nevertheless, the stresses in the arch at all four designated SPs are below the creep-rupture limit, and therefore, the increased stress level is not of concern. • The stresses in the cables are affected insignificantly by additional tensioning of the cables because the low EI of the arch, in the arch assuming creep, cannot provide enough resistance to hold the additional stresses that develop in the cables; consequently, the arch bends instead of overstressing the cables.

Stress fluctuations at the designated SPs in the C-S of the deck are summarized in Appendix D, see Tables D-7 and D-8, and Figure D-3. Stress fluctuations in the arch are shown in and Tables D-9 and D-10, and Figure D-4. To conclude, the proposed procedure to provide an optimal magnitude and distribution of the additional tensile forces in the cables proved that additional cable tensioning can overcome the negative aspects of creep in the GFRP profiles employed in the deck and yet result in less material use in the arch.

6.4.2 The Significance of Variability in the Coefficient of Thermal Expansion

AN#02 explores the effect of the CTE of the three materials on stress distribution in the arch, deck, and cables. The analysis assumes the principle of CBS and CAS as depicted in Section 6.3.4.1. C-S properties of the arch, deck, and cables used in the present analyses are listed in Appendix D: Table D-11, Table D-12, and Table D-13, respectively. Dimensions of the structural profiles of the reference model, assuming GFRP, are based on the results of AN#01C. WPUL of individual profiles, assuming the three different materials, varies. Therefore, even though CBS is employed, the structural response is skewed due to the different WPUL values. This skewing cannot be ignored because the WPUL is significantly different among the three materials, and it influences the distribution of stresses in the structural profiles. However, due to the complexity of the process to model

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Chapter Six: Application of Advanced Composite Materials in SABs the section with CBS, CAS, and constant WPUL, while considering industry used structural profiles, the effect of the skewed response is taken into consideration and discussed appropriately. The CTE of the ACMs is significantly lower than the CTE of the steel. Therefore, the objective is to determine whether or not an employment of commonly used GFRP profiles can be advantageous in C01B. A configuration assuming CFRP is also considered; however, due to the much higher cost of CFRP profiles, this option is assumed only for comparison. CFRP in structural profiles is typically present in “hybrid” sections that utilize the high tensile strength of the CFRP only in particular locations to reduce cost and maximize efficiency (see Section 2.6.5.7 for state of the art in this field). The application of hybrid profiles is not within the scope of AN#02. The resulting stress changes in the individual structural components (arch, deck, and cables) due to the +ve and –ve GTL are presented in Figure 6-22 and Figure 6-23, respectively.

+50°C Applied to entire structure

Figure 6-22: Stress change in all three materials due to the application of a +ve thermal load in the arch, deck, and cables

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– 50°C Applied to entire structure

Figure 6-23: Stress change in all three materials due to the application of a –ve thermal load in the arch, deck, and cables

Ratios of a stress change, due to the applied +ve and –ve GTL, are summarized in Table 6-13.

Table 6-13: Stress change and ratio of stress change due to +ve and –ve GTL in the individual structural components for GFRP, CFRP, and steel Stress change Stress change Structural Ratio of stress Material due to due to Component change +ve GTL -ve GTL GFRP -17.2% 12.8% 1.4 : 1 Arch CFRP -6.1% 5.4% 1.2 : 1 STEEL -58.8% 27.2% 2.2 : 1 GFRP -4.3% 4.4% 1 : 1 Deck CFRP -1.3% 1.3% 1.1 : 1 STEEL -8.3% 7.2% 1.2 : 1 GFRP -0.7% 0.8% 1 : 1 Cables CFRP -0.4% 0.4% 1 : 1 STEEL -0.8% 0.9% 1 : 1

The distributions of the U2 and U3 of the arch are captured in Figure 6-24 and Figure 6-25, respectively.

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Figure 6-24: Distribution U2 of the arch for GFRP, CFRP, and Steel at three levels of GTL

Figure 6-25: Distribution U3 of the arch for GFRP, CFRP, and Steel at three levels of GTL

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Table 6-14 compares the stress reduction in the arch, deck, and cables upon the applied GTL. The magnitude of the U2 of the deck at midspan, assuming the three materials types, is summarized in Table 6-15. Table 6-14: Stress reduction in individual structural components when GFRP and CFRP is employed instead of steel in in C01B Arch Deck Cables Difference in +ve -ve +ve -ve +ve -ve Material due to GTL GTL GTL GTL GTL GTL GFRP vs. Steel 41.62% 14.37% 4.06% 2.82% 0.09% 0.11% CFRP vs. Steel 52.78% 21.79% 7.04% 5.92% 0.47% 0.50%

Table 6-15: Vertical deck displacement at the midspan for GFRP, CFRP, and steel at three levels of applied GTL Deck deflection in vertical direction at midspan Material 0°C +50°C –50°C GFRP L/393 L/2362 L/210 CFRP L/410 L/665 L/326 STEEL L/408 L/-645 L/194

The distributions of U2 and U3 of the deck are presented in Figure 6-26 and Figure 6-27, respectively.

Figure 6-26: Distribution U2 of the deck for GFRP, CFRP, and Steel at three levels of GTL

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Figure 6-27: Distribution U3 of the deck for GFRP, CFRP, and Steel at three levels of GTL

The following can be concluded from the tables and figures for AN#01B: • In the deck, the presence of ACMs (both types) results in a uniform stress fluctuation along the entire deck length due to +ve and –ve GTL. However, in the steel deck, the stress significantly varies. Such behaviour is affected by the high CTE in the steel cables, which affects the response in the deck and the arch. • In the arch and deck at all SPs, assumed in the cross section, the smallest stress fluctuation is present in the profiles that assume GFRP, the material with the smallest E. In contrast, steel, material with the highest E, experiences the highest stress fluctuation in the deck. • In the arch, the application of CFRP and steel results in the highest stress fluctuation at particular SPs as a function of the location of the SP and the applied level of GTL. The CFRP profiles experience the highest stress fluctuation at 0°C and +ve GTL. The steel profiles experience the highest stress fluctuation at –ve GTL. • The arch, acting as the main structural component that carries the loads from the deck in addition to the weight of cables, is exposed to the highest stress fluctuation and the highest change in the stress magnitude overall. This trend is common to all assumed

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materials. In steel, due to the highest CTE, the change in the stress magnitude is the largest. • Even though the +ve and –ve GTL are applied to the entire structure, the structural response of individual components is not even. The absolute magnitude of the stress drop due to the +ve GTL is larger than the absolute magnitude of the stress increase due to –ve GTL. The ratios of a change in the stresses due to +ve and –ve GTL in the individual structural components were established. In the arch, where the difference in the stress change due to the applied GTL is the most evident, the ratio of the stress change due to +ve to –ve GTL is approximately 2:1 (see Table 6-13 for reference). • The fluctuation of stresses in the deck, assuming ACMs, is low. Nevertheless, the fluctuation in the U2 of the deck is significant despite the low CTE of the ACMs. In particular, the deck displacement in a configuration assuming the GFRP, which has a low CTE, is similar to the deflection resulting from a configuration assuming steel which has a high CTE. Hence, a high level of attention should be given to the factors influencing the deflection in GFRP decks, such as 1) the EI of the arch and EA of the cables, taking into account creep effects, and 2) the accurate definition of additional cable tensioning as depicted in AN#01C.

For details, see Appendix D; stress fluctuation at designated SPs in the deck at the three levels of applied GTL is presented in Tables D-14, D-15, and D-16, and Figures D-5 and D-6. Tables D-17, D-18, and D-19, and Figures D-7 and D-8 present the stress fluctuation in the arch. A stress distribution in the cables along the span is shown in Figure D-9. To concluded, the analysis AN#02 clarifies the effect of low CTE in ACM. However, the present analysis also clarified important factors relevant to the structural response of C01B, assuming conventional steel.

6.4.3 The Significance of Variability in Deck Stiffness

As identified in Section 6.4.1.2, the magnitude of deck U2 is governed by the EI of the arch and EA of the cables regardless of the deck EI(V). However, the magnitude of deck

U3, which is a function of the deck EI(H), can significantly affect the U2 of the deck.

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Therefore, in the present section, a rigorous approach investigates the significance of a ratio of EI(V) and EI(H) of the deck, taking into account the effect of material properties, secondary geometric variable f(D)/s, and GTL.

6.4.3.1 Effect of Variability in Deck Stiffness and Deck Reach

6.4.3.1.1 Displacement of the Deck Since C01B with GFRP profiles experiences the smallest stress fluctuation in both the arch and the deck (as identified in AN#02, see Section 6.4.2), AN#03A-I (discussed in the present section) considers the GFRP profiles as the main structural material. AN#03A-I refines the results achieved in AN#02, and therefore, the C-S properties of the arch and cables remain the same as in AN#02 (see Table D-11 and Table D-13). However, the C-S properties of the deck are varied. Hence, a new set of structural profiles is defined. Nine different deck profiles are developed to examine the effect of changing stiffness on the distribution of displacements and stresses in the deck. The circumferences and thickness of the wall of all nine profiles are kept the same to avoid skewing of the results due to a different WPUL. Such an arrangement results in the same axial stiffness of the assumed deck profiles. The dimensions and particular deck stiffness ratio (DSR) of the developed deck profiles are summarized in Table D-20.

AN#03A examines the effects of different DSRs at three levels of f(D)/s as depicted in Table D-20. The resulting magnitudes of the U2 at the midspan of the deck is presented in Figure 6-28.

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a) Ratio f(D)/s = 0.25 Deck Stiffness Ratio (EI(H) : EI(V))

b) Ratio f(D)/s = 0.20 Deck Stiffness Ratio (EI(H) : EI(V))

c) Ratio f(D)/s = 0.15 Deck Stiffness Ratio (EI(H) : EI(V))

Figure 6-28: C01B assuming only GFRP; a comparison of the deck displacement in the vertical direction at three levels of deck reach to span ratio

Due to a change in the f(D)/s, the actual length of the deck changes as well; therefore, the employment of the “L/value” term finds a more practical application to provide an accurate comparison. Detailed comparisons of the distributions of U2 and U3 of the deck are shown in Figures D-10 and D-11, respectively. The distribution of stresses in the cables is presented in Figure D-12.

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Particular DSRs that result in the minimum and maximum U2 of the deck at the midspan are different for individual f(D)/s ratios. The critical DSRs, specific for the assumed f(D)/s, are summarized in Table 6-16. A comparison of a change in the U2 of the deck and stresses in all structural components, taking into account the critical DSRs, is presented in Table 6-17.

Table 6-16: Critical DSRs that result in the maximum and minimum vertical displacement of the deck at the midspan at three levels of deck reach

Ratio f(D)/s Status Vertical Displacement of the Deck DSR Min L/462 15:1 0.25 Max L/375 1:3 Min L/522 15:1 0.20 Max L/495 1:3 Min L/652 1:15 0.15 Max L/645 15:1

Table 6-17: Changes in deflections and stresses at critical DSRs Difference in: Difference in Stress Combination U2 of the deck ARCH CBL 02 CBL07 DECK f/s(D) = 0.25: 18.8% -0.6% -13.7% 8.1% -29.2% [(15:1) vs. (1:3)] f/s(D) = 0.20: 5.2% -0.5% -6.9% 4.1% -8.0% [(15:1) vs. (1:3)] f/s(D) = 0.15: -1.0% -0.2% -8.7% 2.0% 25.1% [(15:1) vs. (1:15)]

As shown in Table 6-16, a vertically flexible deck experiences a smaller displacement than a vertically stiff deck, with the exception of the configuration assuming f(D)/s = 0.15. This unexpected behaviour of C01B with a curved deck is opposite to conventional bridges with straight decks. Such behaviour, resulting from a transformation of the out-of-plane loads, is a function of the EI(V) and EI(H) of the deck, which is affected by the sag in the cables. The mechanism of the out-of-plane load transformation in configurations that assumes deck profiles with DSR = 15: 1 and DSR = 1:3 are presented schematically in Figure 6-29 and

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Figure 6-30, respectively. In both figures, the same loads and the same combination of the primary and secondary variables were applied.

Inclination Angle (Smaller than in DSR = 1:3)

Original Cable Sag

New Cable Sag (Larger than in DSR = 1:3)

2) Smaller Vertical Displacement 1) Smaller Horizontal Displacement

Figure 6-29: Schematic sketch capturing the mechanism of deck displacement in C01B assuming the deck profile with DSR = 15:1 (vertically flexible and horizontally stiff deck)

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In the configuration assuming the deck with a DSR = 15:1, shown in Figure 6-29, the vertically flexible but horizontally stiff deck does not deflect inwards as significantly as the deck with DSR = 1:3. Therefore, the “original” distance between the cable ends does not enlarge as significantly as in the configuration assuming the deck with a DSR = 1:3. Hence, the resulting U2 of the deck is smaller for DSR = 15:1.

Inclination Angle (Larger than in DSR = 15:1)

Original Cable Sag

New Cable Sag (Smaller than in DSR = 15:1)

2) Larger Vertical Displacement

1) Larger Horizontal Displacement

Figure 6-30: Schematic sketch capturing the mechanism of deck displacement in C01B assuming the deck profile with DSR = 1:3 (vertically stiff and horizontally flexible deck)

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The cable sag in the configuration that assumes a deck with DSR 15:1 is larger than in the configuration that assumes a deck with DSR 1:3; therefore, the forces in the cables are smaller in the configuration assuming deck with DSR = 15:1 (see Figure D-12 for reference). The following can be concluded from the tables and figures for AN#03A-I.

• Both the U2 and U3 of the deck are functions of the EI(V) and EI(H) of the deck and

f(D)/s, as expected. The most significant changes in the U2 and U3, as deck stiffness

varies, are present at f(D)/s = 0.25.

• In configurations with f(D)/s = 0.25, the deck is exposed to the largest CF(H). Therefore,

even the deck with a large EI(H) (DSR = 15:1) experiences a large U3. The increase in the U3 of the deck influences the magnitude of the U2 of the deck. This trend is

directly proportional to the f(D)/s ratio; in other words, the smallest fluctuation in the U2 and U3 of the deck, due to variability in deck stiffness, is present in configurations

with f(D)/s = 0.15. • The DSR that results in the smallest U2 in the deck at the midspan is DSR = 15:1 (a

deck with a high EI(H)). However, a deck with this DSR is exposed to higher stresses

compared to decks with a low EI(H). The deck with DSR = 1:3 experiences a larger U2, but the shape of the deflected deck is more gradual, and the stresses are significantly smaller compared to the deck with DSR = 15:1. Hence, the deck with a DSR = 1:3 is optimal in terms of the distribution of deck stresses and U2. • The principles of deck displacement in C01B are described as a function of a particular

DSR, ratio f(D)/s, and proportion of sag in cables.

o The curved deck in C01B, where the ratio of the secondary variable (f(D)/s)

does not exceed the ratio of the primary variable (f(A)/s), behaves similarly to the deck in conventional arch bridges with straight decks.

o When the f(D)/s is higher than f(A)/s, the vertically flexible deck deflects less than the vertically stiff deck. This behaviour results from the presence of out-of-plane loads, whose transformation takes into account the magnitude

of cable sag. The transformation is a function of the ratio EI(V) and EI(H) of the deck.

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6.4.3.1.2 Internal Forces in the Deck and Arch

The maximum magnitudes of the SM(COMB) in the arch and deck are compared in Table 6-18; the negative and positive signs in the table indicate a drop and a rise, respectively, in the SM(COMB) when a deck with DSR 1:3 is employed. The significant distribution of the SM(COMB) in the deck is shown in Figure 6-31. The following can be concluded from the table and figure for AN#03A-II.

• The distribution and magnitude of the SF1 in the arch and deck and the SM(COMB) in the arch are not significantly influenced by a variation in the deck stiffness. However,

the distribution and magnitude of the SM(COMB) in the deck is affected significantly by the selection of the deck profile.

• Comparison of the SM(COMB), summarized in Table 6-18, indicates that a change in the deck profile can result in a change of the bending moment that is twice as high. In the

configuration with f(D)/s = 0.25, the difference between models M001A and M006A is

–104.20%; in other words, the SM(COMB) in the M006A is more than two times smaller

than the SM(COMB) in M001A.

• With a decreasing f(D)/s, the difference in the SM(COMB) drops. The difference between

M001B and M006B, assuming f(D)/s = 0.020, is –27.09%. Further, in the configuration

with the smallest f(D)/s ratio, f(D)/s = 0.15, the difference is +2.59%, which indicates a change in the structural behaviour as a function of the selected DSR. • The selection of a deck profile with a particular value of DSR can significantly influence the magnitude and distribution of the bending moments in the deck. In general, vertically stiff decks with a DSR not lower than 1:3 experience low bending moments compared to other levels of DSR.

Appendix D provides details showing the distributions of SM(COMB) in the arch, considering the deck profiles in models M001 and M006 at three levels of f(D)/s, is shown in Figures D-14, D-15, and D-16, respectively. A comparison of the maximum magnitude of the SF1 in the deck and arch is shown in Table D-22.

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Chapter Six: Application of Advanced Composite Materials in SABs

Table 6-18: The magnitude and differences in change of SM(COMB) in the arch and deck for M001 and M006

SM(COMB) SM(COMB) Difference Difference Model DSR [N*m] [N*m] M001 vs. M006 M001 vs. M006 Deck Arch in the Deck in the Arch M001A 15:1 2.18E+05 6.38E+06 -104.20% -0.44% M006A 1:3 1.07E+05 6.35E+06 M001B 15:1 1.08E+05 4.68E+06 -27.09% -0.43% M006B 1:3 8.48E+04 4.66E+06 M001C 15:1 6.56E+04 3.11E+06 2.59% -0.43% M006C 1:3 6.73E+04 3.10E+06

Figure 6-31: Distribution of SM(COMB) in the deck for M001 and M006 at three levels

of ratio: f(D)/s = 0.25, 0.20, and 0.15

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Chapter Six: Application of Advanced Composite Materials in SABs

6.4.3.2 Effect of Material Properties and Level of Applied GTL

In AN#03A, it was found that the distribution and magnitude of the bending moment in the deck is highly dependent on the assumed deck profile with a particular deck

DSR. In the configuration with only GFRP and f(D)/s = 0.25, the difference in the SM(COMB) between the models M001A and M006A is more than100%. Therefore, in AN#03B, the significance of the material properties and GTL on a distribution of the SM(COMB) and stresses in the deck was investigated, while considering the critical DSR resulting from M001A and M006A. The C-S of the assumed profiles in M001A and M006A, taking into account the principle of CBS (see Section 6.3.4.1 for reference), are listed in Table D-23 and Table D-24, respectively. The resulting magnitude of the stress fluctuations, which shows the effect of the variability in material composition and applied GTL on the stress fluctuation, is summarized in Table 6-19.

Table 6-19: Stress fluctuations in the decks assuming deck profiles of M001 and M006 for all three materials and three levels of applied GTL GTL Model GFRP CFRP STEEL Min in Max in M001 28.1 51.1 93.0 GFRP STEEL 0°C M006 14.7 28.5 38.3 GFRP STEEL Difference [%] 91.0% 79.2% 142.9% CFRP STEEL M001 31.1 53.2 112.4 GFRP STEEL +50°C M006 13.7 28.0 35.0 GFRP STEEL Difference [%] 127.7% 89.8% 220.8% CFRP STEEL M001 25.3 49.3 74.7 GFRP STEEL -50°C M006 15.6 29.0 44.4 GFRP STEEL Difference [%] 61.6% 70.0% 68.2% GFRP CFRP

The following can be concluded for AN#03B. • The trend identified in C01B, assuming GFRP, holds for CFRP and steel as well. The deck with a DSR = 15:1 experiences significantly larger moments than a deck with a DSR = 1:3. Overall, the configuration assuming GFRP, CFRP, and steel experiences a

difference in the SM(COMB) at moderate, low, and high levels, respectively. • The effect of the different CTE at the three levels of GTL changes the distribution and

magnitude of the SM(COMB) along the span as a function of the material, selected deck

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Chapter Six: Application of Advanced Composite Materials in SABs

profile, and level of applied GTL. Applying the –ve GTL in steel decks with a DSR =

15:1 and with a high CTE causes the magnitude of SM(COMB) to drop to a similar level as in the GFRP and CFRP decks with a low CTE. This trend is mainly an outcome of the different WPUL (steel is the heaviest, GFRP is moderate, and CFRP is the lightest). • The effects of CTE and different WPUL values on stress in the cables are only minor. • C01B, taking into account the GFRP profiles with a small E, experiences the smallest stresses compared to the CFRP and steel profiles. The maximum stress is exerted at the middle of the outer surface of the deck, i.e., at the LHS section point (see Figure 6-4 for reference). • The stress fluctuation along the deck span (length) is also the smallest in the GFRP profiles at all three levels of assumed GTL. • In C01B, taking into consideration both types of decks with the critical DSR (M001 and M006) and assuming GFRP, the stress fluctuation is the smallest and represents the advantageous character of commonly used GFRP profiles.

Appendix D provides the details as follows. A summary of the resulting SM(COMB) in the deck is presented in Table D-25. A comparison of the distribution of the SM(COMB), taking into account the parameters of M001A and M006A, is shown in Figure D-17. Table D-26 and Figure D-18 compare the magnitude and distribution of stresses in the cables. A comparison of the maximum stresses in the deck at temperature of 0°C, +50°C, and –50°C is presented in Tables D-27, D-28, and D-28, respectively. Figures D-19, D-20, and D-21 compare stress fluctuations at a critical SP of the deck.

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Chapter Six: Application of Advanced Composite Materials in SABs

6.5 Conclusion to Chapter Six

The results showed that the idea of constructing SABs entirely from ACMs is feasible under certain conditions. In cases where serviceability limits are to be met, creep effects from sustained load appear not to be a significant concern as the section profiles need to be enlarged to achieve the desired stiffness, which reduces the magnitude of the stresses. The procedure employing additional tensioning of the cables showed that as long as the required deck profile is achieved, stress redistribution due to the creep in the deck does not represent a significant concern. The low CTE of ACMs reduces the stress fluctuations due to temperature effects in the cables and deck of the SAB configurations considered, compared to steel. However, the spatially inclined arch did appear sensitive to even low levels of temperature variation. Analysis of varying deck stiffness considering the non-linear effect of the cables clarified the significance of lateral and vertical deck stiffness for three levels of deck out- of-plane geometry and three levels of thermal loading. It was found that by selecting a vertically stiff deck the overall deck deflection increases. However, the maximum stress can be reduced as well as the stress fluctuation within the deck due to thermal loading. In the case of a vertically stiff deck, the bending moment in the deck can be reduced by more than 100%, in a comparison to a vertically flexible deck.

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Chapter Seven: Design Criteria and Directions of Future Work

The present chapter proposes design criteria that result in the efficient design of the three spatial configurations studied in this thesis. Furthermore, directions of future work are outlined.

7.1 Design Criteria The proposed design criteria are based on summary tables provided in sections B.3.1, B.3.2, B.3.3, C.2.1, C.2.5, and C.2.8,. These complex tables and the design criteria, provided below, are intended to serve as guidelines during analysis and design of SABs studied in this thesis.

7.1.1 Configuration C01

Loads • A live load distributed over the entire deck represents the governing load case. • A thermal load results in an effect smaller than 10% in comparison to the effect of LL, and therefore, it is not a significant concern.

Arch rise • Arch rise is the governing variable that controls the response in the arch and deck. • A low arch rise increases the axial load and reduces the bending moments. Therefore, the arch rise should be established first and kept low. • Bending moments in the arch are the governing type of response; the design of the arch needs to consider these bending moments. • Inclination of the arch, starting at 30° (C01B), reduces the bending moments in the arch and introduces large axial loads in the deck.

Deck reach • As deck reach increases, the bending moment in the arch increases. • The optimum ratio of arch rise to deck reach is 1:1.

o When the ratio is smaller than 1 (the deck reach is larger than arch rise), the arch is exposed to larger bending moments. Also, in this combination, the

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Chapter Seven: Design Criteria and Directions of Future Work

contribution of deck stiffness to deflection of the deck changes. Vertically stiff decks deflect more than vertically flexible decks; this finding is opposite to conventional planar arches with a straight deck. The achieve smaller stresses in the deck, the deck should be vertically stiff with ratio of horizontal to vertical stiffness 1:3.

Boundary conditions of the deck • Boundary conditions of the deck play a significant role in susceptibility of the arch to buckling and should be established before selecting the stiffness of the deck and arch. • Fixed ends of the deck result in small loads in the cables and the arch. However, the abutments of the deck must be designed for large moments.

Ratio of arch and deck bending stiffness • An optimum ratio of arch and deck bending stiffness, assuming fixed ends of the deck, is as follows:

o 10:10 in the vertical direction (both the arch and deck must be equally stiff in the vertical direction); and

o 1:10 in the horizontal direction (the deck stiffness should be significantly larger in the horizontal direction than the stiffness of the arch).

7.1.2 Configuration C02

Loads • A live load distributed over one half of the deck represents the governing load case. • A thermal load can result in an effect that is larger than the effect of live loads, particularly in the deck. • The effect of thermal load in the arch can be as high as the effect of live loads. • The arch rise should be large to reduce the sensitivity to thermal loads.

Arch rise • A change in the arch rise does not influence the structural response of the arch as significantly as a change in the arch inclination.

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Chapter Seven: Design Criteria and Directions of Future Work

• Reducing the arch rise increases the axial loads and decreases the bending moments in the arch.

Arch inclination • Arch inclination represents the governing variable (it needs to be established first).

o When large, the sensitivity to GTL is large and axial loads in the arch increase less significantly than bending moments. • Arch inclination should be kept below 30° from the vertical plane to keep the axial loads and bending moments in the arch low. • When combined with a curved deck of a small deck reach (C01B), the structural system becomes more efficient.

Boundary conditions of the deck • Rotation allowed about the vertical axis at the deck ends reduces the axial loads and bending moments in the arch and, subsequently, reduces the susceptibility to buckling.

o The bending moment in the arch is less sensitive to a change in the arch inclination than the axial loads in the arch.

Ratio of arch and deck stiffness • Bending stiffness of the deck in the horizontal direction controls the arch behaviour.

o When large, the arch is exposed to high axial loads and small bending moments. • A combination of a stiff deck and flexible arch results in small axial loads and small bending moment in the arch. • When the vertical and horizontal bending stiffness of the arch is the same, the bending moment in the arch is low.

7.1.3 Configuration C03

Loads • A live load distributed over the entire deck represents the governing load case. • A live load governs the magnitude of internal forces.

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Chapter Seven: Design Criteria and Directions of Future Work

• A thermal load governs the magnitude of displacements (the displacements resulting from thermal loads are larger than the displacements resulting from live loads).

o The arch rise should be large and arch rotation small to reduce the sensitivity to thermal loads.

Arch rise • A change in the arch rise controls the response in the arch.

o The arch rise should be large to keep the axial loads and bending moments in the arch low.

Arch rotation • A change in the arch rotation controls the response in the deck and cables. • A small arch rise and a large arch rotation result in large axial loads at the midspan of the arch (the axial load at the midspan of the arch is larger than at the arch springings, which is opposite to conventional vertical arches). Therefore, this combination should be avoided. • Large axial forces are exerted on the arch when the arch rise is low. The rotation of the arch should be low to reduce the large axial loads.

Boundary conditions of the deck • Rotation allowed about the vertical axis at the deck ends increases the susceptibility of the arch to buckling and, therefore, it should be avoided.

Ratio of arch and deck stiffness • A large horizontal bending stiffness of the arch reduces the susceptibility to buckling.

o Both the axial load and bending moment in the arch are low when the horizontal stiffness of the arch is large. • A large horizontal bending stiffness of the deck is required to keep the susceptibility to buckling low.

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Chapter Seven: Design Criteria and Directions of Future Work

7.2 Directions of Future Work

7.2.1 Arrangement of the Cables

In the spatial configurations studied in this thesis, the arrangement of the cables was kept the same in all analyses: the cables were located 5m apart in a vertical plane. However, a different arrangement of the cables may result in smaller fluctuation of stresses. Large stress fluctuations are apparent in C01, in which the first cable is overstressed. In a special form of vertical planar arches, called the “network arches,” cables are inclined in the plane of the arch and some cables may cross other cables more than once. In these bridges, the efficiency of the arch configuration is increased, resulting in smaller cross-sections of the arch rib, which in turn leads to a lighter and more attractive appearance (Tveit, 2002). Future studies should focus on several different patterns of the cable arrangement to propose more efficient design criteria and reduce the overstressed elements. The effect of the angle of cable inclination, crossing of cables, and total number of cables on the response of the arch and deck should be evaluated.

7.2.2 Additional Tensioning of the Cables

In Chapter Six, the effect of additional tensioning of the cables was introduced on a small scale to investigate its influence in configurations assuming GFRP profiles and tendons exposed to creep. Nevertheless, due to the complexity of additional tensioning of cables in spatial configurations and the objectives of the proposed research, the topic of additional tensioning of cables was not developed further. Future studies should focus on developing practical curves that provide parameters for Equation 3-11, which proposes an optimal level of additional tensioning of cables, resulting in the desired deck profiles. Developing the proposed curves should consider the effect of arch and deck bending and axial stiffness, weight per unit length, and required final shape of the deck.

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Chapter Seven: Design Criteria and Directions of Future Work

7.2.3 Response to Dynamic Loading

Most SABs are pedestrian bridges, which may be susceptible to human induced vibrations. The natural frequency of the structure is the first parameter that should be checked according to national and international standards (Bachmann, 1997). To meet the limits imposed by these standards, the first (lowest) natural frequency of the bridge must be higher than the specified threshold value. Investigating the natural frequencies and accelerations in complex configurations of SABs is a sophisticated procedure. The natural frequency is a function of the stiffness and mass of the structure (Ghali, 2009). In SABs, the stiffness of the entire system is influenced by its proportions and, therefore, the natural frequencies of these structures may be low. Structures with low natural frequencies need to be evaluated using a more detailed analysis to verify whether or not the acceleration of the structure would disturb the comfort of pedestrian walking (Sarmiento-Comesias, 2015). Future studies should focus on modal and dynamic analyses that consider the parameters in this thesis, such as the primary and secondary variables, boundary conditions of the deck, variability in bending stiffness of the arch and the deck, and material diversity. The results of the proposed analyses can be used to establish guidelines for analysis and design that provide information about advantageous distributions of mass and stiffness of lightweight bridges susceptible to vibrations. Gusting winds are a different type of dynamic loading, which may result in flow induced vibrations. The majority of SABs are short or medium span bridges, and therefore, flow induced vibrations are not a significant concern. Nevertheless, particular arrangements of cables or slender decks may be prone to gusting winds. Hence, flow induced vibrations in cables, vortex shedding, and galloping (in presence of ice) may also be possible directions of future work.

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Chapter Eight: Summary and Final Remarks

This work describes the structural behaviour of SABs. SABs represent state-of-the- art structures designed to be unique pieces of architecture. These structures are built to celebrate innovation in technology, prosperity, or important jubilees. Nevertheless, due to the high demand for aesthetic appearances, SABs are exposed to out-of-plane loads, which cause challenges during analysis and design. The geometry of SABs can be very complex. This thesis focuses on SABs with an inferior deck. The hangers are assumed to be flexible cables. Parametric 3D linear and nonlinear models are developed and the responses of SABs to the assumed loads are studied using FEA. Three main configurations are investigated to determine the responses to a wide spectrum of out-of-plane loads exerted on the arch and the deck. Relevant variables, such as f(A)/s, f(D)/s, ω, and θ, are considered while examining the effect of changing EI of the arch and deck, EA of the cables, additional tensioning of the cables, and material properties of ACMs. The described structural behaviours represent new knowledge about SABs with an inferior deck, and this new knowledge directly contributes to the field of structural engineering and applied science.

The analyses that investigated the effects of changing geometric variables, such as arch rise, deck reach, arch inclination, and arch rotation, showed that only in C01, the arch rise (the primary variable) controlled the response in the arch and deck. In C02 and C03, the arch inclination and arch rotation (the secondary variables), respectively, have a more significant effect on the structural response of the arch and deck. As anticipated, in all three configurations, a low arch rise results in large axial loads in the arch. This response is identical to conventional vertical planar arches. The spatial configurations studied assume an arch rib unsupported along its entire length, and therefore, high axial loads are an issue. In these combinations, susceptibility to buckling increases, particularly when high axial loads are combined with out-of-plane loads exerted on the arch via the inclined cables. An adequate selection of secondary variables can efficiently control the magnitude of the axial loads in the arch. In C01, the axial loads in the arch are reduced when the deck reach is

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Chapter Eight: Summary and Final Remarks large. However, to achieve low axial loads in the arch of C02 and C03, the arch inclination and arch rotation need to be small. The analyses showed that only C02 behaved similarly to conventional vertical planar arches in terms of the distribution of LL on the deck, which caused larger bending moments in the arch. In C02, the LL distributed over a half of the deck represented the governing load case. In C01 and C03, the LL distributed over the entire deck resulted in larger bending moments in the arch. The effects of the identified critical patterns of LL were compared with the effects from an applied GTL. The analyses showed that in C02, starting at an arch inclination of

30°, the resulting SM(COMB) in the arch, under –ve GTL, reached magnitudes approximately 10% higher than under LL. In the deck of C02, the magnitude of the bending moment resulting from the applied GTL was not higher than the one resulting from LL. However, the difference between GTL and LL was only minor. In other words, the effect of GTL is almost the same as the effect of LL, which meant that the effect of GTL cannot be ignored in C02. The significance of the applied GTL was most perceptible in a combination with a low f(A)/s ratio (0.15). In C01 and C03, the significance of GTL was also the most apparent in combinations with low f(A)/s. However, the effect of GTL on the internal forces in the arch and deck was smaller than the effect of the governing LL. Due to the large out-of- plane loads in C01, resulting from the arrangement of the deck, which acted as a curved cantilever beam supported by cables, the effect of GTL was the smallest. Applied GTL can result in larger out-of-plane displacement of the arch and deck than applied LL in C03.

Investigating the effects of a changing EI of the arch and deck as a function of the

BCs of the deck on the magnitude and distribution of SF1, SM(COMB), U2, and U3 of the arch determined the level of susceptibility of the arch to global buckling. Specific ratios of

EI(V) and EI(H) of the arch and deck, resulting either in the smallest SF1 or SM(COMB), were established for each configuration as a function of the BCs of the deck. These ratios were presented with structural sketches of the proposed arrangements. It was found that the type of BCs of the deck significantly affected the susceptibility of the arch to buckling in the individual configurations.

In C01, for deck arrangement BC#01, EI(V) and EI(H) of the deck need to be large and EI(V) and EI(H) of the arch need to be small to achieve small SF1 and small SM(COMB) in

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Chapter Eight: Summary and Final Remarks the arch and, thus, reducing the susceptibility to buckling. For deck arrangement BC#02,

EI(V) and EI(H) of the deck need to be large and small, respectively, and both EI(V) and EI(H) of the arch need to be large to achieve small susceptibility to buckling. For BC#03, the smallest SF1 and SM(COMB) in the arch result from a combination of large EI(V) and EI(H) in the deck and a large EI(V) and low EI(H) in the arch.

The following can be concluded about C02: For both BC#01and BC#04, EI(V) of the deck needs to be large, EI(H) of the deck needs to be small, and both EI(V) and EI(H) of the arch need to be large so that the susceptibility to buckling is small. Evaluating arch displacements indicates that BC#04 results in conditions that make the arch in C02 less susceptible to global buckling. Further, deck stiffness governs the magnitude of SF1 in the arch for both arrangements of deck BCs. The characteristics of BC#04 also significantly affect the distribution of SF1 in the arch compared to BC#01. For BC#04, the allowed rotation UR2 at the deck ends causes the more gradual distribution of SF1 in the arch, which reduces the susceptibility to local buckling.

The structural behaviour of C03 can be summarized as follows: For BC#01, EI(V) and EI(H) of the deck need to be large and small, respectively, and EI(V) and EI(H) of the arch need to be small and large, respectively, to result in a low SF1 and SM(COMB) in the arch.

For BC#04, EI(V) and EI(H) of the arch need to be the same as in BC#01 (the EI(V) and EI(H) need to be small and large, respectively); however, due to the released rotation UR2, EI(V) of the deck can be small, and EI(H) of the deck needs to be large to achieve small SF1 and

SM(COMB) in the arch. The released rotation UR2 in BC#04 causes an increase in susceptibility of the arch to buckling.

This thesis also investigates the effect of material significance, in particular, the effect of the materials properties of ACMs. The responses of the HSS profiles made of GFRP and CFRP employed in all-composite assemblies are compared to configurations made of steel. The principles of CBS and CAS are introduced to compare the effects of the different mechanical properties of the three main materials. The effect of the material properties of ACMs is investigated in configuration C01B, which is a hybrid combination of C01 and C02. The geometric arrangement of C01B comprises a parabolic arch inclined from a vertical plane and a curved deck with altered boundary conditions BC#02B.

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Chapter Eight: Summary and Final Remarks

A detailed approach is used to determine the effect of the nonlinear character of the cables that experience sag, whose magnitude affects the magnitude of the forces at the ends of the cables. The effect of additional cable tensioning is described, and the parameters of the proposed equation, which determine the level of additional tensioning in the individual cables that influence the final deck profile, are established as a function of the bending stiffness of the arch. When evaluating the effect of creep and susceptibility to creep-rupture, creep- rupture is not a significant concern in C01B if a certain serviceability limit on deck deflection in U2 is imposed. It was confirmed that the governing element affecting deck displacement is not the deck but the arch. EI(V) of the deck only minimally contributes to the deck deflection. The creep in the arch and deck only minorly influences the stress change in the cables; the stress in the cables reaches similar magnitudes in all assumed models. Due to the use of additional tensioning of the cables, the distribution of stresses in the deck is not influenced by creep in the arch. The magnitude and distribution of the stresses in the deck, in the configuration that experiences creep and is exposed to additional tensioning of the cables, is very close to the distribution and magnitude of the stresses in the deck of the configuration where no creep and reduced bending stiffness are present. Therefore, as long as the deck deflection in C01B meets the required profile and the magnitude of the required displacement, the large stress redistribution, which results from the effect of creep and the absence of additional cable tensioning, can be mitigated. Even though the assumed GFRP profiles experience large deflections and may undergo significant stress distributions (particularly in the deck due to the effect of creep if no additional cable tensioning is employed), the overall stress fluctuation is low compared to the profiles with conventional steel. The analysis of the effect of CTE on the models with the three different materials found that for low CTE, the models with ACMs resulted in smaller changes in the stresses under GTL compared to steel, as expected. In addition to this trend, the stress fluctuations in particular structural components were the smallest when GFRP profiles with the smallest modulus of elasticity were employed. Nevertheless, the effect of low CTE did not influence all structural components the same; the cables were affected the least and the arch the most. Applying the three materials causes the same trend: the effect of +ve GTL resulted in a

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Chapter Eight: Summary and Final Remarks larger stress change than the effect of –ve GTL. The ratio of change due to +ve and –ve GTL was ~2.2:1 for steel and 1.2:1 for CFRP. The fluctuation of stresses in the deck assuming ACMs was low under GTL; however, the fluctuation in the vertical displacement of the deck was significant despite the low CTE of ACMs. Evaluating the effect of varying deck stiffness found that the most significant changes in U2 and U3 of the deck occurred when the deck reach was large (f(D)/s = 0.25).

When f(D)/s = 0.25, the deck was exposed to the largest forces transferred from the cables (resulting from a small angle α). The variability in deck stiffness had only a minor effect on the stresses in the arch and the cables. Based on the analysis of the effect of varying deck stiffness, two critical ratios, DSR = 15:1 (a vertically flexible and horizontally stiff deck) and DSR = 1:3 (a vertically stiff and horizontally flexible deck), were established. The responses in the configurations that employ the selected DSR ratios show that in C01B the vertically flexible deck (DSR = 15:1) deflects less than the vertically stiff deck (DSR = 1:3). This trend is opposite to conventional arch bridges with straight decks, and it results from the proportions of the assumed deck profiles, where EI(H) of the deck plays a significant role when combined with the nonlinear response of the cables. In the deck with DSR = 15:1, the displacements are small; however, the stresses are large compared to the deck with DSR = 1:3. The effect of varying deck bending stiffness is most apparent in the distribution and magnitude of the combined bending moments (SM(COMB)) in the deck; the magnitude of SM(COMB) can be reduced to half when the deck with DSR = 1:3 is employed. This trend holds for all three assumed materials regardless of the level of applied GTL.

This work also proposes design guidelines for the three configurations and establishes several straightforward rules. These rules take into consideration the geometrical proportions of the arch and deck, ratios of bending stiffness, and the end conditions of the deck. Researching the effect of additional variables, such as arrangement of cables, additional tensioning of cables, and mass and stiffness distribution, affecting the structural response to dynamic loads that may contribute to the presented design guidelines is a potential direction of future work.

To conclude, this thesis establishes geometric ratios of primary and secondary variables and identifies the effect of varying bending stiffness of the arch and deck and the

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Chapter Eight: Summary and Final Remarks end conditions of the deck on the susceptibility of the arch to buckling. The properties of ACMs in the assumed structural profiles were considered, and the significance of creep, thermal load, and additional tensioning of cables was investigated. Optimal ratios of deck stiffness that result in a reduction of deck displacement and bending moments in the deck were established in relation to the shape of the deck profile and to the type of material used. Design guidelines were proposed and directions of future work were discussed.

The results of this research can be used as guidelines and criteria. However, due to the high complexity of SABs, the proposed guidelines and criteria were not intended to cover all the particular details but rather to cover a wide spectrum of cases that can be considered in the preliminary stage of the design. The efficient establishment of the overall geometric ratios, the end boundary conditions, and the ratios of bending and axial stiffness can accelerate the design process and result in cost saving decisions.

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APPENDIX A: DETAILS TO CHAPTER THREE A. Appendix A:

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Appendix A: Details to Chapter Three

A.1. Selection of Finite Elements

The selection of a proper finite element and mesh is a key step. The requirement is that the selected finite element is able (through its response) to reflect correctly the effect of the applied loads in the assumed configuration. The response of an element to applied loads is influenced by five aspects: 1) the element family; beam, shell, continuum elements, etc.; 2) available degrees of freedom (DOF); calculated translations and rotations directly related to the element family; 3) the number of nodes and interpolation function (the displacements and rotations are calculated at the nodes and interpolated via specific functions along the element); 4) formulation; Lagrangian formulation – element deforms with the material (typical for structural problems), or Eulerian formulation – elements are fixed in space and material flows through them; (typical for fluid mechanics problems); and 5) integration type; typical is Gaussian quadrature for most elements, Abaqus 6.13 reference manual (2014). One of the significant aspects is the interpolation function; sometimes also called the shape or approximating function. The selection of an appropriate interpolation function and, therefore, a selection of an appropriate finite element depends on the required level of accuracy. Typically, polynomial or trigonometric functions are used for most of finite elements in order to describe the behaviour of the field variable within the element, Desai (2011). Use of large numbers of lower order elements can provide as satisfactory an answer as fewer higher order elements. The elements selected for the analyses in this thesis are structural elements; particularly, truss and beam elements. Structural finite elements are commonly used in industrial practice as well as in research case studies. Several examples are described in the following paragraph. The investigation of long span truss arches done by Farreyre (2005) utilize truss elements for diagonal members and beam elements for top and bottom chords in spatial roofs. Critical buckling loads in bridges with inclined parabolic arch ribs were examined by Gui (2016). Arches and hangers were modeled via beam elements and truss elements, respectively. A structural response of the Gateshead Millennium Bridge was evaluated with two-node bar (truss) elements in hangers and with engineering thick beam elements (elements accounting for shear deformation) in an arch and a deck as per a research case

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Appendix A: Details to Chapter Three study, (Lucas, 2002). An investigation of a structural response of a diagonal arch employed spatial beam elements for the deck girders and the arch rib. Hangers were modeled via flexible cable elements (Qiu, 2010). Specifications of the structural finite elements that are used for analyses conducted in this thesis are described next.

A.1.1. Finite Elements for Arch and Deck

Arch and a deck in both the linear and the nonlinear models are modeled with B32 beam elements. Both the arch and the deck are components that are exposed to axial and shear forces as well as bending and torsional moments. Therefore, the selected finite element must respond to such loads. From the library of structural elements, Abaqus v. 6.13 offers a large group of suitable beam elements with specific attributes appropriate for planar and spatial problems. In order to model a spatial configuration resisting the out-of-plane loads, a three- dimensional B32 beam element was selected for both the arch and the deck. This three node (two integration points) beam element is shear flexible (Timoshenko definition) and has a quadratic shape function providing relatively simple but accurate interpolation within the length of the element. Timoshenko beams can be subjected to large axial strains. The axial strains due to torsion are assumed to be small. In a combined axial-torsion loading case, the torsional shear strains are calculated accurately only when the axial strain is not large. Abaqus assumes that the transverse shear behavior of Timoshenko beams is linear elastic with a fixed modulus and, thus, independent of the response of the beam section to axial stretch and bending, Abaqus Analysis User’s Guide (2014), Section 29.3.3. At each node, each beam element has 6 degrees of freedom (DOF) that allow for obtaining translations and rotations in all three directions (X, Y, Z). Each node can transfer shear and axial forces and moments. The versatility of beam elements in spatial structures is represented by the availability of various cross-sections that may be assigned to the beam element. Cross- sections for beam elements that are available in Abaqus v. 6.13 can be solid or thin-walled.

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Appendix A: Details to Chapter Three

If thin-walled cross-sections are used, then they can be open or closed. Specific properties of beam cross-section can be defined via: • Selecting of a particular shape of the cross-section from provided Abaqus library whose dimensions are to be defined. In the Abaqus library there are standard open and closed sections such as I-beams, T-beams, circular or rectangular hollow structural shapes, but also hexagonal shapes or L-shapes. • Entering of specific geometric quantities such as area, moments of inertia, and torsional constant into a general beam cross-section. A general beam cross-section distributes the geometric quantities in a solid section. The advantage of a general beam section is that the individual geometric quantities can be defined separately. Such a property can be beneficial in the definition of cable-like components having high axial but low bending stiffness (definition of flexible, cable-like components is described in Section A.1.4) • The meshed cross-section where an arbitrary shape can be created (drawn) and specific two-dimensional (2D) elements (warping elements) are used to calculate the geometric quantities numerically. An employment of the meshed cross-section can beneficial particularly in modeling of complex cross-sections (inclined internal stiffeners, curved surfaces, or combination of multiple materials) when it is difficult to obtain geometric quantities otherwise. Nevertheless, in the warping elements there is only one DOF; the out-of-plane warping displacement. Therefore, the meshed beam cross-section is not suitable for thermal analysis as per Abaqus Analysis User’s Guide (2014), Section 10.6.1. In order to take the advantage of the versatility of the meshed cross-section and overcome its disability to model thermal loads certain semi-procedure exists. The geometric quantities of complex cross-section are acquired from the meshed cross- section and then entered into the general beam cross-section whose available degrees of freedom take into account the coefficient of thermal expansion that is necessary for thermal analysis.

A definition of section points (the assumed points within the assumed section at which output variables such as the axial stresses or strains are collected) is allowed for all beam cross-sections described above.

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Appendix A: Details to Chapter Three

Since the nonlinear material behaviour is not considered during the assumed analyses, the numerical integration of section properties is set to default, i.e., to “integration before analysis”. In this option Abaqus precomputes the beam cross-section quantities and performs all section computations during the analysis in terms of the precomputed values. This method combines the functions of beam section and material descriptions (stress strain material relation is not required), Abaqus Analysis User’s Guide (2014), Section 29.3.5.

A.1.2. Finite Elements for Hangers

In the FE models, two types of hanger are employed. The sections below specify hanger for the linear and nonlinear model.

A.1.3. Hangers for the Linear Model

Hangers in the linear model are created with the T3D2 truss element. Only one element is used to connect the arch and deck. T3D2 truss element is a two node element applicable in 3D space with linear interpolation function. In general, truss elements are long slender members that can transmit only axial load. No moments or forces perpendicular to the centreline are supported. An application of the truss elements as cables in structural applications represents a certain simplification because the truss elements cannot account for deformation of flexible cables under self-weight; the sag. Nevertheless, truss elements are simple elements (with low demand on computational time) and can provide reasonably accurate results valuable in preliminary analyses where neither a detailed definition nor a response of a deflected cable is required. In certain cases, the application of truss elements in cable-like configurations is recommended. For example, the 3-node truss element available in Abaqus/Standard is often useful for modeling of curved reinforcing cables in structures such as prestressed tendons in reinforced concrete or long slender pipelines used in the offshore industry. Modeling of the flexible cable-like members with multiple truss elements in space is possible. However, the procedure is sensitive to the alignment of the elements, requires numerous sub-steps, and may be less versatile in spatial configurations. Details of a

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Appendix A: Details to Chapter Three procedure describing how to model cables with multiple truss elements are provided in, for example, Davies (1988).

Figure A-1: Selection of the “tension-only” material behaviour assigned to the truss elements utilized for modeling of hangers in the linear models.

In order to improve the cable-like behaviour modeled with truss elements, the truss elements are defined as “tension-only” members. Such members cannot transfer compression forces. Abaqus v. 6.13 offers the “tension-only” option as a property of linear elastic material. An example of assignment of “tension-only” property is shown in Figure A-1. The tension-only attribute of cables modeled with truss elements is advantageous in

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Appendix A: Details to Chapter Three thermal analyses where the compression axial stress (generated under positive thermal loads) is omitted, Abaqus Analysis User’s Guide (2014), Section 29.2.1.

A.1.4. Hangers for the Nonlinear Model

The hangers for the nonlinear model are created with B32 beam elements with a particular definition of bending stiffness. The particular definition of bending stiffness is related to the basic concept of flexible cables. In general, the purpose of structural cables is to resist tension forces and, therefore, the cables must have high axial stiffness. However, the bending stiffness of the cables in an unstressed state is low. Abaqus v. 6.13 does not offer any cable elements, and therefore, other alternatives were sought. The approach presented by Davies (1988), described in Section A.1.3 above, suggests modeling cables via a system of truss elements. This option, however, is not the most suitable for the spatial configurations in this thesis. Another different approach was proposed by Todisco (2014) who used structural elements connected with hinges that allow for rotation at assumed points to achieve cable-like behaviour. Nevertheless, in this work, the modeling of flexible cables is performed with structural beam elements that have a bending stiffness reduced to 1% of its original value. This approach was suggested and applied by several authors, for example, Davies (1988) and Steiner (2011). Even though the specified threshold value of 1% represents an optimum, the magnitude of the actual moment of inertia influences the convergence level of the solution. When the magnitude of the bending stiffness of the cable is high, the analysis reaches a convergent solution at a faster rate. In addition, the large difference between axial and bending stiffness may lead to ill-conditioning (Davies, 1988).

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Appendix A: Details to Chapter Three

Figure A-2: Definition of geometric quantities in a generalized profile of a beam element cross-section

The proposed method of reduced bending stiffness utilizes the advantage of the generalized beam profile (Section A.1.1) where the geometric quantities can be defined directly with arbitrary magnitudes. In order achieve a cable-like behaviour via beam elements, the axial stiffness “EA” is kept as per the original material. However, the bending stiffness “EI” must be reduced. The modulus of elasticity, E, remains unchanged because, as a material constant, it is used to calculate the other geometric quantities. Nevertheless, the second moment of area (moment of inertia), I, can be reduced in both directions to achieve the required low bending stiffness of the cables. A definition of the particular moment of inertia is shown in Figure A-2 above. A verification of the suitability of the beam elements with reduced bending stiffness to model flexible cables is provided in Section A.2.2.

A.1.5. Finite Elements for Connections

A sophisticated type of finite element is required to connect the cables to the arch and deck only in the nonlinear models. In the linear model, an application of the specific connection elements is not necessary because the truss elements used to model the cables

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Appendix A: Details to Chapter Three can only transfer axial loads, and therefore, any connection of truss elements with other elements behaves as a joint free to rotate in any direction. The finite elements used in the nonlinear model to connect the cables to the arch and deck are connector elements CONN3D2. The connector elements are applied at both ends of the cable. CONN3D2 is a two-node element applicable in 3D space. In Abaqus v. 6.13, a connector element is modeled as a connector wire (zero length is also applicable) to which a connector section is assigned. A connector section determines the attributes of the modeled connection. A universal joint (U-joint), which combines universal and joint connector sections, is selected to achieve an accurate and convergent solution in the nonlinear models. The joint connector section transfers all translational DOFs, and the universal connector section allows for two rotational DOFs. The two allowed rotational DOF are “UR1,” which allows for a rotation at an element end about a horizontal axis that is perpendicular to the longitudinal direction of the element, and “UR3,” which allows for a rotation at an element end about a vertical axis that is perpendicular to the longitudinal direction of the element. The third rotational DOF, “UR2” (rotation about its own longitudinal axis), is constrained to ensure stability (avoid singularities) of the systems. The definition of a fixed UR2 also reduces the numerical size of the problem. A local orientation of a connector section becomes imperative in spatial configurations of SABs where the cables are connected to a curved arch or deck. Due to sag and the spatial displacements of the cables, misleading bending and torsional moments can occur in the modeled cables. These bending or torsional moments in the cables are not correct and, therefore, undesirable. A correct local orientation of the connector section must be defined to avoid a deviation in the cable response. A definition of the local orientation of the connector sections as a function of the shape of the arch and deck is provided in Section 3.4.1. A graphical representation of a universal and a joint connector section presenting specified degrees of freedom is shown in Figure A-3.

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Appendix A: Details to Chapter Three

Adopted from Abaqus Analysis User’s Guide, Section 31.1.5 (2014). Figure A-3: A graphical interpretation of a connector section “U-joint” used to model the connection of the cables to the arch and deck

A connection of two nodes at an identical location (a connector element of zero length) is possible given that the two nodes belong to different parts. Placing a connector element at a conjunction of two components of the same part that meet at the same node is not possible (two nodes must be present). An assignment of a connector element and its section is in Abaqus v. 6.13 can be done in a part module or an assembly module.

A.2. Verification and Calibration Certain steps need to be carried out to ensure correctness and accuracy of the results obtained from an FE model. Typically, a comparison of the results obtained from an analytical model (AM) and an FE model is conducted. Further, the AM can only reach a certain level of complexity. Developing an AM that is too complex would lead to loss of efficiency, and the AM itself could be vitiated due to an error. Calibrating models involves a mesh sensitivity analysis and a selection of the most suitable elements from possible options. In general, a mesh sensitivity analysis investigates the influence of a finite mesh on the response of the model. Typically, an acceptable level of a refined mesh is the level at which the difference between two subsequent sets of results (the model with original and refined mesh) does not exceed 10%. An optimal mesh quality, in models using structural elements, is determined by a procedure that increases the number of elements.

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Appendix A: Details to Chapter Three

A.2.1. Verification of Selected Structural Elements for Arch and Deck

The response of the entire structure depends on the response of the individual elements. Therefore, a verification of the particular elements is considered first. Elements that are selected for modeling the spatial configurations of SABs assumed in this thesis must provide a correct response to applied axial and shear forces, bending and torsional moments, and thermal loads. The simple cases provided below demonstrate the suitability of the selected elements. For three different beam elements, the results obtained from the reference AM are compared with results from the FE models to demonstrate the suitability of the B32 beam element that was selected for the arch and deck. All compared beam elements are spatial beams with a Timoshenko definition (shear flexible). The specific elements are as follows: • B31: two-node beam element with a linear interpolation function • B32: three-node beam element with a quadratic interpolation function • B33: two-node beam element with a cubic interpolation function

The structural configuration that was used to compare the three types of beam elements consists of a 10m long steel cantilever beam, which had a downward force of 100kN and a torsional moment of 10kN*m applied at the free end. The assumed beam section is an HSS steel profile that is 2.0m wide and 0.4m high with a wall thickness of 0.025m (deck section assumed to verify the entire structure). Details of the cross-sectional properties are provided in Table A-4. The material properties of steel are taken from Section 3.5. The assumed configuration is exposed to shear force and bending and torsional moments. A schema of such configuration is shown in Figure A-4 below.

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Appendix A: Details to Chapter Three

L = 10m A = 5.94E-02m2 F = 100kN T = 100kN*m

Figure A-4: A spatial configuration assumed to verify beam elements that resist the combined shear forces and bending and torsional moments

Results of a mesh sensitivity analysis comparing a magnitude of lateral displacements in all three assumed beam elements are shown in Figure A-5. As shown in the figure, B31 achieves consistent results after having assigned seven elements. This number, i.e., 7 elements necessary with B31 element type, compared to the other elements, is too high, and therefore, B31 is rejected from further consideration. The responses of B32 and B33 are very similar in nature and, in fact, B33 achieves consistent results in terms of displacements even with a coarse mesh. Nevertheless, the application of B33 in the spatial configuration results in non-convergent solutions because B33 does not have a fully nonlinear formulation, which is necessary to represent the geometric nonlinearity introduced by the cables within structural configurations (Abaqus Documentation, 2014).

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Appendix A: Details to Chapter Three

Mesh sensitivity: Comparison of finitite elements: B31/B32/B33 4.90E-02

4.80E-02

4.70E-02

4.60E-02 B31

4.50E-02 B32 B33

Displacement [m] 4.40E-02 AM 4.30E-02

4.20E-02 0 1 2 3 4 5 6 7 8 9 10 11 No of elements

Figure A-5: A comparison of the displacements of a free end cantilever beam

Therefore, for the FE analyses assumed in this thesis, the spatially stable beam element B32 with a quadratic interpolation function was selected. The difference between the results of the AM model and FE model with B32 elements is 4.13%. Hence, an application of B32 is acceptable. Further, the assumed AM does not take into consideration a shear deformation, and therefore, the results of the FE model represent a higher magnitude of final displacement. The mesh sensitivity analysis indicating an adequate number of elements is shown in Figure A-6. As shown in the figure, the optimum accuracy is reached with four beam elements over the 10m long cantilever beam. Hence, the length of one element is established to be 2.5m. This length is, therefore, implemented into the arch and deck in the linear and nonlinear models of the entire structure as presented in Section A.2.7.

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Appendix A: Details to Chapter Three

Mesh sensitivity analysis: Beam element B32 4.323985E-02 4.323980E-02 4.323975E-02 4.323970E-02 4.323965E-02 4.323960E-02 AM 4.323955E-02 B32 4.323950E-02 Displace me nt [m] 4.323945E-02 4.323940E-02 4.323935E-02 0 1 2 3 4 5 6 7 8 9 10 11 No of elements

Figure A-6: A comparison of number of elements required to achieve a consistent level of displacement

The capability of B32 to carry thermal and axial loads was verified via a comparison with the AM, which took into account a 10m long beam (with the same parameters as used for the case shown in Figure A-4) that is fixed at both ends. This configuration is presented in Figure A-7. A principle of the compatibility of displacements is described in Equation A-1, Equation A-2, and Equation A-3. is the displacement due to thermal load, is the displacement due to axial load, is the𝛿𝛿𝑇𝑇𝑇𝑇 cross-sectional area, is the modulus of 𝛿𝛿𝐴𝐴elasticity, is the difference in applied𝐴𝐴 temperature, is 𝐸𝐸the coefficient of thermal expansion,∆𝑇𝑇 and is the axial load resulting from 𝐶𝐶the𝐶𝐶𝐶𝐶 applied temperature. A positive thermal load of +100°C𝐹𝐹 was applied, and the reactions at the beam ends in the horizontal direction were compared. Both the AM and FE models had the same horizontal reaction with a magnitude of 7,128kN at each end causing a compression stress of 120MPa.

𝛿𝛿𝑇𝑇𝑇𝑇 𝛿𝛿𝑉𝑉𝑉𝑉 Equation A-1

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Appendix A: Details to Chapter Three

× × × = × 𝐹𝐹 𝐿𝐿 𝐶𝐶𝐶𝐶𝐶𝐶 𝐿𝐿 ∆𝑇𝑇 Equation A-2 𝐴𝐴 𝐸𝐸

= × × ×

𝐹𝐹 𝐴𝐴 𝐸𝐸 ∆𝑇𝑇 𝐶𝐶𝐶𝐶𝐶𝐶 Equation A-3

L = 10m A = 5.94E-02m2 ΔT = 100°C

Figure A-7: A configuration assumed to verify beam elements that resist thermal loads

A.2.2. Verification of Flexible Cables Modeled with Beam Elements

As described in Section A.1.4, flexible hangers (cables) are modeled with B32 beam elements with a reduced bending stiffness. During the verification process of the proposed approach, the following parameters were considered: • Element type and number of elements • Level of convergence • Boundary conditions • Vertical and horizontal reactions • Cable sag (magnitude and shape of deformation) • Ability to carry tensile stress • Ability to carry negative and positive thermal loads • Ability not to produce any compressive stress under positive thermal loads

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Appendix A: Details to Chapter Three

A.2.3. Mesh Sensitivity Analysis of the Suspended Cable

The suitability of B32 is presented in Section A.2.2. The results of a mesh sensitivity analysis, which considers a 10m long steel cable with a diameter of 50mm that is exposed only to a gravity load, are presented in Figure A-8. There is no additional tension applied to the cable. In the unstressed state, the cable is straight with no initial sag.

5.650E-02

5.648E-02

5.646E-02

5.644E-02

5.642E-02

5.640E-02 B32

Displace me nt [m] 5.638E-02

5.636E-02

5.634E-02 0 5 10 15 20 25 30 35 40 45 50 55 No of elements

Figure A-8: Mesh sensitivity analysis of B32 indicating the suitable number of elements for flexible cables

From Figure A-8, it can be seen that a consistent level of displacement at the middle of the cable length is achieved with 10 elements. This value is, therefore, implemented into the FE models of the entire structure. The determined number of beam elements is assigned to each cable in the bridge configuration regardless of the cable length. A constant number of elements in each cable (1) maintains the simplicity of the model, while developing the parametric input file and a spreadsheet-based tool that provides the coordinates of individual nodes applicable to an arbitrary configuration, (2) maintains a constant number of nodes (elements) in all models, and (3) allows for the same curvature of each cable. The element type and boundary conditions are directly related to the level of convergence. An analysis of boundary conditions indicated that number of rotational degrees of freedom needs to be reduced from six to four to achieve a convergent solution in the assumed configuration. A critical degree of freedom is a rotation about the longitudinal

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Appendix A: Details to Chapter Three axis of the element. The relevant number of degrees of freedom at the connection of the cable to the arch or deck in the assembled structure is controlled by a selected connector element section as described in Section A.1.5. In general, the convergence level is sensitive to the size and number of assumed increments within loading steps. It was found that the optimum number of increments was 1,000, and the initial increment size was 0.01 as shown in Figure A-9. Further, in certain spatial configurations, the number of increments was significantly increased to achieve a convergent solution (sensitive to the local orientation of the connector elements).

Figure A-9: An example of a setting of the incremental steps for the nonlinear analysis

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Appendix A: Details to Chapter Three

A.2.4. Magnitude of Cable Displacement

The verification of cable deformation after a load is applied assumes a 10m long, weightless, steel cable with a diameter of 50mm, which is loaded with concentrated force at the midspan. The magnitude of the force that reflects the total weight of the steel cable is 1502N. The configuration of the AM is schematically presented in Figure A-10.

Adopted from Ghali (2009); Chapter 23.3.

Figure A-10: Configuration of a weightless cable loaded by a concentrated force at its midspan assumed to verify the cable modeled with beam elements with reduced bending stiffness

A geometric nonlinear analysis that considers large displacements, but small strains, needs to be introduced to account for the effect of the deformed shape of the cable, which influences the magnitude of the tensile force in the cable. In geometrically nonlinear problems, the magnitude of the suspended cable displacement is a function of the tension applied to the cable. Therefore, determining the internal forces must consider equilibrium about points on the deformed shape, which is unlike linear analysis where the equilibrium of forces assumes an undeformed initial state (Ghali, 2009). In Abaqus, a linear equation solution is used in the linear and nonlinear analyses. In the nonlinear analysis, Abaqus/Standard uses the Newton method or a variant of it, such as the Riks method. The Riks method is used to solve a set of linear equations at each iteration. Sparse solver, which applies the linear equation solution to a “multifront” technique, is suitable for a physical model that is made from several parts or branches

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Appendix A: Details to Chapter Three connected together such as configurations of SABs where the arch, deck, and set of cables are the only structural components. However, an iterative linear equation solver in Abaqus/Standard is more suitable for single “blocky” (solid) three dimensional structures, as per Chapter 6.1.4 and 6.1.5 (Abaqus Documentation, 2014). An iterative procedure utilizing Equation A-4 and Equation A-5 is used to find displacements and reaction forces of the assumed configuration shown in Figure A-10. is the total tensile force in the cable, is the increment in tensile force due to the applied𝑁𝑁 vertical load , is the applied vertical∆𝑁𝑁 load (representing the entire mass of the assumed cable) at point𝑄𝑄 B𝑄𝑄, E is the modulus of elasticity, A is the cross-sectional area of the assumed cable, ( ) is the cable elongation due to developed tensile force, b is the cable length, and

D is the𝛿𝛿 vertical𝐶𝐶 displacement due to the applied vertical load.

( ) ( ) ( ) = + = + × ( ) = + + 2 0 0 𝐸𝐸𝐸𝐸 0 2𝐸𝐸𝐸𝐸 𝑏𝑏 2 𝑏𝑏 𝐶𝐶 𝑁𝑁 𝑁𝑁 ∆𝑁𝑁 𝑁𝑁 𝐿𝐿 𝛿𝛿 𝑁𝑁 � 𝑏𝑏 � ��2� 𝐷𝐷 −Equation2� A-4

2 = 𝑁𝑁𝑁𝑁+ 𝑄𝑄 2 2 𝑏𝑏 2 �� 𝐷𝐷 Equation A-5

The proposed nonlinear iterative procedure assumes the vertical displacement D and then calculates the tensile force N. The out of balance load is calculated next according to Equation A-5. 𝑄𝑄𝐵𝐵

Table A-1: A comparison of the analytical and FE models for nonlinear cable behaviour Parameter AM FE Difference Displacement; D [m] 0.078 0.077 1.13% Vertical reaction [N] 751.446 751.500 0.01% Horizontal reaction [N] 48,041.808 47,489.400 1.15%

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Appendix A: Details to Chapter Three

The solution is found via a spreadsheet-based tool after several iterations when the assumed displacement D provides an out of balance load equal to the applied vertical load . A comparison indicating a close match of obtained𝑄𝑄 results𝐵𝐵 from the analytical and FE models𝑄𝑄 is shown in Table A-1. The close match indicates that beam elements with reduced bending stiffness represent the behaviour of flexible cables well.

A.2.5. Deformed Shape of the Cable

The shape of a cable deformed under its own weight is another important parameter of the verification. Cables resisting only gravity loads (self-weight) deform in the shape of a catenary. A catenary can be described by the hyperbolic cosine function as shown in Equation A-6 where y is the vertical coordinate, x is the horizontal coordinate, and is the ratio of / in which is the linear density (the weight of a cable per unit length𝑎𝑎 along cable’s longitudinal𝑇𝑇0 𝑊𝑊0 axis),𝑊𝑊 and0 is the horizontal component of the tensile force, which is constant at any point on the𝑇𝑇 0curve; the ratio is, therefore, constant for a given configuration. 𝑎𝑎 In practical applications, a second degree parabolic curve represented by Equation A-7 (same notations as per Equation A-6 apply) is typically used instead of the catenary curve because both curves have similarly deformed shapes, and the definition of a parabolic curve is much simpler.

= 𝑥𝑥 𝑦𝑦 𝑎𝑎 ∗ 𝑐𝑐𝑐𝑐𝑐𝑐ℎ Equation A-6 𝑎𝑎

= 2 2 𝑥𝑥 𝑦𝑦 𝑎𝑎 ∗ Equation A-7 𝑎𝑎

The difference between the two curves becomes more apparent in cables with larger sag (Sherif, 1991). However, in cables with a small sag, as assumed for the configurations of SABs in this thesis, the difference is minor. A comparison of both curves is presented in Figure A-11; as shown in this figure, the maximum difference is less than 5%, and therefore, the utilization of the parabolic equation to model funicular and antifunicular

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Appendix A: Details to Chapter Three shapes of suspended cables and erected arches, respectively, is considered acceptable for purposes of this thesis.

Location along span [m]

-5 -4 -3 -2 -1 0 1 2 3 4 5 1.20 2.00%

1.00 1.50% 0.80

0.60 1.00%

0.40 Displace me nt [m] 0.50% [%] Difference 0.20

0.00 0.00%

Difference CATENARY PARABOLA

Figure A-11: A comparison of the deformed shapes of the cables using the parabola and catenary equations

Displacements of individual nodes obtained from the FE model is plotted against the parabolic curve as shown in Figure A-12. The low level of difference (no more than 1%) that is apparent on the secondary axis indicates a close correlation of both curves concluding that B32 with reduced bending stiffness is suitable for modeling flexible cables.

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Appendix A: Details to Chapter Three

Location along span [m]

0 1 2 3 4 5 6 7 8 9 10 0.000E+00 0.000%

-1.000E-02 -0.150% -2.000E-02 -0.300% -3.000E-02 -0.450% -4.000E-02 -0.600% Difference [%] Difference Displace me nt [m] -5.000E-02 -0.750% -6.000E-02 -0.900% Difference Equation of parabola FE Model

Figure A-12: A comparison of the deformed shape of a cable modeled using B32 elements with reduced bending stiffness and AM using a parabolic curve

A.2.6. Capability to Carry Thermal Loads

Cables that are modeled using B32 with reduced bending stiffness must be able to carry thermal loads to correctly respond to temperature variation and not to generate any compressive stresses.

RV TR

L θ RH

q f ds dy

T0 dx

Figure A-13: Cable deformed under self-weight: related symbols

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Appendix A: Details to Chapter Three

A configuration, shown in Figure A-13, is exposed to a temperature change of ΔT = +/-100°C to verify the response to thermal loads. The configuration assumes a steel cable, 50mm in diameter, spanning a distance of L = 10m with a sag f = 2m. In the configuration, the sag increases, and the tensile force in the cable decreases as a result of the thermal expansion that occurs after a positive thermal load is applied. The increased physical length of the cable affects the angle at the connection of the cable to its support and, therefore, the magnitude of the tensile force in the cable. Applying a negative thermal load has the opposite effect: the cable sag decreases, and the tension force increases. The relations for AM are expressed by the equations below and are adapted from Ghali (2009). The length of the parabolic curve is obtained using Equation A-8 where is the straight distance between cable ends and 𝐿𝐿 is𝐶𝐶 the cable sag. 𝐿𝐿 𝑓𝑓 1 4 + + 16 = + 16 + × 2 82 2 2 2 2 𝐿𝐿 𝑓𝑓 �𝐿𝐿 𝑓𝑓 𝐿𝐿𝐶𝐶 �𝐿𝐿 𝑓𝑓 𝑙𝑙𝑙𝑙 � � 𝑓𝑓 𝐿𝐿 Equation A-8

Equation A-9 describes the linear thermal deformation where is the coefficient of thermal expansion, and is the temperature difference. 𝐶𝐶𝐶𝐶𝐶𝐶 ∆𝑇𝑇 = × × 𝛿𝛿𝑇𝑇𝑇𝑇 𝐶𝐶𝐶𝐶𝐶𝐶 𝐿𝐿𝐶𝐶 ∆𝑇𝑇 Equation A-9

The vertical and horizontal reactions at the cable anchors are computed with Equation A-10 and Equation A-11, respectively, where is the weight of the cable per unit length. 𝑞𝑞𝐶𝐶 = × /2

𝑅𝑅𝑉𝑉 𝑞𝑞𝐶𝐶 𝐿𝐿𝐶𝐶 Equation A-10

× = 8 2 𝑞𝑞𝐶𝐶 𝐿𝐿 𝑅𝑅𝐻𝐻 𝑓𝑓 Equation A-11

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Appendix A: Details to Chapter Three

The magnitude of the reaction force in the tangential direction at the cable anchors is obtained using Equation A-12.

= + 2 2 𝑇𝑇𝑅𝑅 �𝑅𝑅𝐻𝐻 𝑅𝑅𝑉𝑉 Equation A-12

The tensile force at an arbitrary point on the cable is expressed with Equation A-13 where and represent differential increments in the vertical and horizontal directions, respectively.𝑑𝑑𝑑𝑑 The𝑑𝑑𝑑𝑑 magnitude of is a function of the assumed curve; Equation A-6 and Equation A-7 express catenary and𝑑𝑑𝑑𝑑 parabola curves, respectively.

= 2 + 1 𝑑𝑑𝑑𝑑 𝑇𝑇𝐶𝐶 𝑅𝑅𝐻𝐻�� 𝑑𝑑𝑑𝑑 Equation A-13

The horizontal and vertical components of the tensile force in a cable are expressed with Equation A-14 and Equation A-15.

( 𝑇𝑇𝐻𝐻 𝑅𝑅𝐻𝐻 ) 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑡𝑡 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 Equation A-14

= × 𝑑𝑑𝑑𝑑 𝑇𝑇𝑉𝑉 𝑅𝑅𝐻𝐻 � � Equation A-15 𝑑𝑑𝑑𝑑

A comparison of the AM and FE models is summarized in Table A-2. A graphical representation of the displacement and tensile force distribution under three levels of thermal loads is shown in Figure A-14 and Figure A-15, respectively. As shown in Table A-2, the FE model is close to the reference values obtained using AM. As expected, applying a positive thermal load causes the sag to increase and the tensile forces to decrease. The highest difference, 12.55%, occurs in the horizontal reaction due to ΔT = –100°C. The reason for such a high difference is that at ΔT = 0.00°C the AM does not assume any deformation of the cable whereas the FE model does. In other words,

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Appendix A: Details to Chapter Three for the FE model, the displacement at the centre is less than the given sag of 2m (the distributed cable weight “pushes” all segments of the cable downwards resulting in an uplift of the node at the centre of the curve). Even though the difference exceeds the threshold value of 10%, the results obtained from the FE models are acceptable. Table A-2: A comparison of the AM and FE models for suspended cables under thermal loads Level of thermal Reaction AM Model FE Model Difference load ΔT = -100°C 824.01 825.25 0.15%

RV [N] ΔT = 0.00°C 825.01 825.22 0.03% ΔT = +100°C 826.00 825.21 -0.10% ΔT = -100°C 945.88 1081.58 12.55%

RH [N] ΔT = 0.00°C 939.02 993.30 5.46% ΔT = +100°C 932.30 920.09 -1.33% ΔT = -100°C 1254.47 1360.46 7.79%

TR [N] ΔT = 0.00°C 1249.95 1291.37 3.21% ΔT = +100°C 1245.57 1235.93 -0.78% ΔT = -100°C 1.985 1.827 -8.70% Displacement ΔT = 0.00°C 2.000 1.969 -1.56% at centre [m] ΔT = +100°C 2.014 2.106 4.33%

Location along span [m]

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 0.000

-0.500

-1.000

-1.500

Displace me nt [m] -2.000

-2.500 ΔT = -100°C ΔT = 0.00°C ΔT = +100°C

Figure A-14: A comparison of the displacements of a cable, modeled using B32 elements, exposed to three levels of thermal load

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Appendix A: Details to Chapter Three

The displacements of the nodes in a cable under the assumed thermal loads are shown in Figure A-14. As shown in the figure, the developed FE models using B32 can correctly represent flexible cables because, under the applied negative or positive thermal load, the vertical displacements fluctuate as anticipated. From Figure A-15, the tensile force distribution under the assumed thermal loads also correctly reflects the expected behaviour. The tensile force is reduced under positive thermal loads and is increased under negative thermal loads. Further, the magnitude of the tensile force at the centre, shown in Figure A-15, matches the magnitude of the horizontal reaction presented in Table A-2, which also confirms the correctness of the developed model. In addition, no compressive forces are generated upon applying positive thermal loads. This finding confirms that modeling flexible cables using B32 with reduced bending stiffness provides correct results.

Location along span [m]

0 1 2 3 4 5 6 7 8 9 10 1,400

1,300

1,200

1,100

Tensile Force [N] Force Tensile 1,000

900

800

ΔT = -100°C ΔT = 0.00°C ΔT = +100°C

Figure A-15: A comparison of the distributions of the tensile force in a cable, modeled using B32, exposed to three levels of thermal load

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Appendix A: Details to Chapter Three

A.2.7. Verification of the Entire Structure

As described in Sections A.2.1 and A.2.2, the individual components of the structure, the arch, deck, and cables, modeled with assumed finite elements, can correctly respond to the applied static and thermal loads. Therefore, a final verification step of the developed FE models is a verification of the entire structure. The verification of the entire structure needs to take into account several factors such as the complexity of possible spatial configurations of SABs, the reliability of the analytical model (AM), and the response of the individual structural components modeled with the selected finite elements. Since the geometry of SABs can be very complex and developing an accurate AM, which reflects complex configurations, can be vitiated due an error, a verification of the entire structure consists of a vertical planar arch with a straight deck. In this configuration, a reliable AM can validate the response of the FE model that comprises finite elements already established for the static and thermal loads. The assumed configuration is a pedestrian bridge with an arch span and rise of 75m and 15m, respectively. The deck is suspended on vertical cables from the arch. The cables are not additionally tensioned. This configuration is graphically presented in Figure A-16.

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Appendix A: Details to Chapter Three

f(A)

Figure A-16: A configuration used to verify the entire structure

An applied LL is determined according to CHBDC (2014), and it is equally distributed along the span. The magnitude of the thermal load is assumed arbitrarily to impose extreme conditions. The arch is fixed at both ends (hinge-less arch). The deck rests on horizontal rollers located at each end of the deck allowing for movement in a longitudinal direction and a rotation about the transversal axis. The centre of the deck is secured with a vertical roller that allows for vertical displacement. This constraint is imposed to avoid a singularity. The parameters of the assumed configuration are listed in Table A-3.

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Appendix A: Details to Chapter Three

Table A-3: Parameters of the assumed configuration used to verify the entire structure Parameter Magnitude Span; s [m] 75

Arch rise; f(A) [m] 15

Rise to span ratio; f(A)/s [ - ] 0.20 Live load applied on deck [N/m] 5000 Level of thermal load [°C] -50/0/+50 Number of cables [ - ] 14 BCs of the arch Fixed at both ends Horizontal rollers at both ends, vertical BCs of the deck roller at midspan

Equation A-16 describes the shape of the arch using a parabolic curve where ( ) is the arch rise, is the span of the arch, and ( ) is the horizontal coordinate along the 𝑓𝑓span.𝐴𝐴 4 𝑠𝑠 ( ) 𝑋𝑋𝑖𝑖 0 = ( ) ( ) 𝑓𝑓 𝐴𝐴 𝑦𝑦 2 ∗ �𝑠𝑠 − 𝑋𝑋𝑖𝑖 0 � ∗ 𝑋𝑋𝑖𝑖 0 Equation A-16 𝑠𝑠

Cross-sectional properties of the individual structural components used in the verification analysis are listed in Table A-4. The verification of the entire structure compares global displacements and reaction forces at the arch abutments. The displacements of interests are as follows: (1) the vertical displacement at the arch crown at the midspan, U2A.CR. and (2) the vertical displacement of the deck at the midspan, U2D.MS.. A combined effect of an axial shortening is employed to calculate

U2A.CR.. The axial thrust described in Equation A-17 where is the axial shortening, is the axial thrust in the arch obtained using Equation A-18, 𝛿𝛿𝐴𝐴 is the length of the parabolic𝑁𝑁𝐴𝐴 arch obtained via Equation A-8, and is the cross-sectional𝐿𝐿𝐴𝐴 area of the arch. The length of the parabolic arch is taken into 𝐴𝐴consideration𝐴𝐴 while obtaining the deformation due to the applied thermal loads𝐿𝐿𝐴𝐴 obtained via Equation A-9.

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Table A-4: Cross-sectional properties of the main structural components Parameter Arch Deck Cables Shape

Diameter [m] 0.750 N/A 0.050 Height [m] N/A 0.400 N/A Width [m] N/A 2.000 N/A Wall thickness [m] 0.020 0.025 N/A Cross-sectional Area [m2] 4.59E-02 5.94E-02 1.96E-03 Moment of inertia: I22 [m4] 3.06E-03 5.04E-02 3.07E-09 Moment of inertia: I11 [m4] 3.06E-03 3.70E-03 3.07E-09 Torsional constant: J [m4] 6.12E-03 1.17E-02 6.14E-09 Weight per unit length [N/m] 3,860 8,990 150

× = × 𝑁𝑁𝐴𝐴 𝐿𝐿𝐴𝐴 𝛿𝛿𝐴𝐴 𝐴𝐴𝐴𝐴 𝐸𝐸 Equation A-17

In the Equation A-18, refers to a horizontal reaction at the arch abutment. The terms and represent differential𝑅𝑅𝐻𝐻𝐻𝐻 increments in the vertical and horizontal direction, respectively.𝑑𝑑𝑑𝑑 The𝑑𝑑𝑑𝑑 magnitude of is a function of the parabolic shape of the arch rib expressed in Equation A-16. The𝑑𝑑 horizontal𝑑𝑑 reaction at the arch abutment is obtained using Equation A-21. 𝑅𝑅𝐻𝐻𝐻𝐻

= 2 + 1 𝑑𝑑𝑑𝑑 𝑁𝑁𝐴𝐴 𝑅𝑅𝐻𝐻𝐻𝐻�� 𝑑𝑑𝑑𝑑 Equation A-18

The total displacement of the deck at the midspan results in the arch displacement (described below), cable elongation (Equation A-9 and Equation A-17), and the displacement of the deck due to the self-weight of the deck and the live load applied on the deck.

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Appendix A: Details to Chapter Three

The procedure that combines axial shortening due to axial thrust and the applied thermal load covers following steps. First, the axial thrust, which is obtained from Equation A-18, is applied at the arch ends to calculate the axial shortening . The reduced length of the arch is then used to find the first new (smaller) arch rise. The𝛿𝛿𝐴𝐴 difference between the original and the first new arch rise represents the vertical arch drop at the arch crown. In the second step, the reduced arch length (due to the axial load) is set as the starting parameter to calculate the axial deformation due to the applied thermal load . Based on the character of the thermal load (positive or negative), the length of the 𝛿𝛿arch𝑇𝑇𝑇𝑇 changes (increases in the case of a positive thermal load and decreases in the case of a negative thermal load). The length of the arch after applying the thermal load is then used to obtain the second new rise of the arch. The difference between the first and the second new arch rise represents the arch deflection at the crown of the arch due to the applied thermal load. A summation of the vertical arch displacement due to the axial and thermal loads provides the total vertical arch displacement at the crown. The reaction forces at the arch abutments are calculated using the equations specified below. There are several methods to calculate the reaction forces in the fixed ends of the arch such as graphic methods and methods using sophisticated software programs to perform the required integration. Nevertheless, for the purpose of verifying the entire structure, the method of elastic centroid that utilizes spreadsheets was adapted from Shrive (2000). The moment at the elastic centroid is calculated according to Equation A-19 where is the moment effect at a general point P on the arch profile due to the loading between𝑀𝑀 abutment𝑝𝑝 A and the point P, is the modulus of elasticity of the material assumed for the arch, is the second moment of𝐸𝐸 area of the arch cross-section. 1 𝐼𝐼 = 𝑠𝑠 1 ∫0 𝑀𝑀𝑝𝑝 𝑑𝑑𝑑𝑑 𝑀𝑀 𝑠𝑠 𝐸𝐸𝐸𝐸 0 Equation A-19 ∫ 𝐸𝐸𝐸𝐸 𝑑𝑑𝑑𝑑

The vertical reaction at the arch abutment A is obtained using Equation A-20 where corresponds to the coordinate𝑅𝑅𝑉𝑉𝑉𝑉 along the span. 𝑥𝑥

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Appendix A: Details to Chapter Three

1 = 1 ∫ 𝑀𝑀𝑝𝑝 𝑑𝑑𝑑𝑑 𝑉𝑉𝑉𝑉 𝐸𝐸𝐸𝐸 𝑅𝑅 2 𝑥𝑥 Equation A-20 ∫ 𝐸𝐸𝐸𝐸 𝑑𝑑𝑑𝑑

Equation A-21 is used to calculate the horizontal reaction at the arch abutment

A taking into consideration the axial shortening due to the axial𝑅𝑅 𝐻𝐻thrust𝐻𝐻 in the arch. The parameter corresponds to the vertical coordinate of the arch as a function of . 1 𝑦𝑦 𝑥𝑥 = 𝑠𝑠 1 ∫0 𝑀𝑀𝑝𝑝 𝑑𝑑+𝑑𝑑 𝑅𝑅𝐻𝐻𝐻𝐻 𝑠𝑠 𝐸𝐸𝐸𝐸 2 𝐿𝐿𝐴𝐴 0 �𝑦𝑦 𝐴𝐴 ∫ 𝐸𝐸𝐸𝐸 𝑑𝑑𝑑𝑑� 𝐸𝐸𝐴𝐴 Equation A-21

The moment effect at a general point P is obtained via Equation A-22 where is the uniformly distributed𝑀𝑀 live𝑝𝑝 load, is the weight of an individual arch segment, and𝑞𝑞 𝐿𝐿𝐿𝐿 is the number of assumed segments𝑊𝑊 into𝑠𝑠 which the arch was divided. The difference𝑛𝑛 between and is the distance between the centroid of the entire arch and the centroid𝑥𝑥𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐of 𝑐𝑐𝑐𝑐𝑐𝑐an 𝑐𝑐individual𝑥𝑥𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 segment.𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 ( 1) + 2 2 = + 𝑛𝑛 [( )( )] 2 ∆𝑥𝑥 𝑞𝑞𝐿𝐿𝐿𝐿 � 𝑛𝑛 − ∆𝑥𝑥 � 𝑀𝑀𝑃𝑃 � 𝑊𝑊𝑠𝑠 𝑥𝑥𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝑥𝑥𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖=1 Equation A-22

The moment at any point on the arch due to the applied load can be obtained via Equation A-23. ( ) = +

𝑀𝑀 𝑥𝑥𝑖𝑖 𝑀𝑀 𝑅𝑅𝑉𝑉𝑥𝑥𝑖𝑖 − 𝑅𝑅𝐻𝐻𝑦𝑦𝑖𝑖 − 𝑀𝑀𝑃𝑃𝑃𝑃 Equation A-23

The reaction forces were calculated using spreadsheets to conduct the required numerical integration (presented in the equations). A comparison of the developed AM to the FE model at three levels of assumed thermal loads plus gravity and a live load is summarized in Table A-5. The most accurate match is achieved in the vertical reactions, and the least accurate match in the reaction moments. The reason for the large difference in the reaction moments is the number of

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Appendix A: Details to Chapter Three segments into which the arch is divided. Further, the displacements in the FE models are generally larger than the ones obtained from AM because the finite elements applied in the FE model consider not only axial deformation but also shear deformation. From Table A-5, the developed FE model closely correlates with a reliable AM. Therefore, the assumed settings of individual finite elements that represent the entire structure provide correct results, and the developed FE model can be used under specific alterations to investigate the spatial configurations in this thesis.

Table A-5: Summary of verification of entire structure ΔT = -50°C AM FE Difference Arch Vert. Displ. at crown [m] -1.17E-01 -1.26E-01 6.78% Deck Vert. Displ. at midspan [m] -7.68E-03 -8.21E-03 6.40%

RV [N] 694,134 694,102 0.00%

RH [N] -870,480 -910,665 4.41%

M(A) [N*m] 49,487 55,321 10.55% ΔT = 0°C AM FE Difference Arch Vert. Displ. at crown [m] -8.91E-03 -9.33E-03 4.46% Deck Vert. Displ. at midspan [m] -4.04E-03 -4.27E-03 5.28%

RV [N] 694,214 694,195 0.00%

RH [N] -867,441 -908,991 4.57%

M(A) [N*m] 49,835 56,035 11.06% ΔT = +50°C AM FE Difference Arch Vert. Displ. at crown [m] 9.94E-02 9.51E-02 -4.46% Deck Vert. Displ. at midspan [m] -1.30E-02 -1.39E-02 6.44%

RV [N] 694,293 694,305 0.00%

RH [N] -864,432 905,874 4.57%

M(A) [N*m] 50,183 58,698 14.51%

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Appendix A: Details to Chapter Three

A.3. Parametric Input File

This section provides a complete input file as an example of such file that has been employed in the nonlinear analysis. The input file presented below serves as a reference for studying purposes, and when copied to Abaqus, it can be used for actual analysis. Parameters in the input file define the configuration C01B (see Chapter Six for details) that takes into consideration steel as the main structural component and GTL at level of –50°C. It should be noted that the input file presented below is a combination of two steps. In the first step, cross-sectional dimensions and node coordinates are generated via developed spread-sheet tools. In the second step, the generated data are inserted from the spread-sheet into the input file whose structure contains the commands and parameters.

The geometric parameters of the assumed structure defined within the input file are:

• f(A)/s = 0.15

• f(D)/s = 0.25 • ω = 30° • Diameter of the arch profile = 1.5m • Wall thickness of the arch profile = 0.025m • Dimensions of the deck cross-section: Height = 0.4m. Width = 2.0m • Wall thickness of the deck profile = 0.020m • Cable diameter = 0.050m

Parametric input file:

*Heading ** Job name: Job_M001_S_C01B_fA015_fD025_omg30_-50I.inp ** Generated by: Abaqus/CAE 6.13-3 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** **All units basic SI *PARAMETER ** **Magnitude of LL; “1” or “0” indicates “on” or “off” in LL case selection MLTP_LL_050=1 MLTP_LL_100=0 Ped_Load=-5000 Walking_Surf=-972 LL_050=(Ped_Load+Walking_Surf)*MLTP_LL_050 LL_100=(Ped_Load+Walking_Surf)*MLTP_LL_100

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Appendix A: Details to Chapter Three

** **Orientation_Of_Deck_BC_as_function_of_deck_rise_for_Datum_CSYS [-] Deck_f_to_s_ratio=0.25 **Span, the distance between abutments of the arch [m] span=75 deck_rise=Deck_f_to_s_ratio*span X=1 **"Z" coordinated for Datum_CSYS Z=(4*deck_rise/span**2)*(span-X)*X ** **Global_Thermal_Loading [°C] G_Th_L =-50 **Cables: Temperature of develop Additional tensioning in cables [°C] ** Use Equation 3-33 to determine the “temperature profile” T_Clbs_01_14 = 0 T_Clbs_02_13 = 0 T_Clbs_03_12 = 0 T_Clbs_04_11 = 0 T_Clbs_05_10 = 0 T_Clbs_06_09 = 0 T_Clbs_07_08 = 0 **DECK **deck material properties STEEL E_deck = 2.00E+11 G_deck = 7.70E+10 nu_deck = 0.3 Density_deck = 7800 CTE_deck = 1.20E-05 **deck outside dimensions and wall thickness a = 2.000 b = 0.400 t1 = 0.020 t2 = 0.020 t3 = 0.020 t4 = 0.020 **Section points deck SP_01_X1_D = 0.000000 SP_01_X2_D = 0.190000 SP_02_X1_D = 0.000000 SP_02_X2_D = -0.190000 SP_03_X1_D = 0.990000 SP_03_X2_D = 0.000000 SP_04_X1_D = -0.990000 SP_04_X2_D = 0.000000 **ARCH **arch material properties STEEL E_arch = 2.00E+11 G_arch = 7.70E+10 nu_arch = 0.3 Density_arch = 7800 CTE_arch = 1.20E-05 **arch outside dimensions and wall thickness r_arch = 0.750 t_arch = 0.025 **Section points arch SP_01_X1_A = 0.000000 SP_01_X2_A = 0.737500

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Appendix A: Details to Chapter Three

SP_02_X1_A = 0.000000 SP_02_X2_A = -0.737500 SP_03_X1_A = 0.737500 SP_03_X2_A = 0.000000 SP_04_X1_A = -0.737500 SP_04_X2_A = 0.000000 **CABLES **cable material properties STEEL (from tab Cable Prop_Copied) E_cable = 1.65E+11 G_cable = 6.33E+08 nu_cable = 0.3 Density_cable = 7800 CTE_cable = 1.20E-05 **cable sectional properties STEEL (from tab Cable Prop_Copied) Cable_CS_Area = 1.963E-03 Cable_I11 = 3.07E-09 Cable_I12 = 0 Cable_I22 = 3.07E-09 Cable_J = 6.14E-09 **Section points cable SP_01_X1_C = 0.000000 SP_01_X2_C = 0.016000 SP_02_X1_C = 0.000000 SP_02_X2_C = -0.016000 SP_03_X1_C = 0.016000 SP_03_X2_C = 0.000000 SP_04_X1_C = -0.016000 SP_04_X2_C = 0.000000 ** ** PARTS ** *Part, name=Part01_Arch *Node 1 , 0 , 0 , 0 2 , 5 , 2.424871131 , -1.212435565 3 , 10 , 4.5033321 , -2.25166605 4 , 15 , 6.235382907 , -3.117691454 5 , 20 , 7.621023553 , -3.810511777 6 , 25 , 8.660254038 , -4.330127019 7 , 30 , 9.353074361 , -4.67653718 8 , 35 , 9.699484522 , -4.849742261 9 , 40 , 9.699484522 , -4.849742261 10 , 45 , 9.353074361 , -4.67653718 11 , 50 , 8.660254038 , -4.330127019 12 , 55 , 7.621023553 , -3.810511777 13 , 60 , 6.235382907 , -3.117691454 14 , 65 , 4.5033321 , -2.25166605 15 , 70 , 2.424871131 , -1.212435565 16 , 75 , 0 , 0 17 , 2.5 , 1.255736835 , -0.627868418 18 , 7.5 , 3.507402885 , -1.753701443 19 , 12.5 , 5.412658774 , -2.706329387 20 , 17.5 , 6.9715045 , -3.48575225 21 , 22.5 , 8.183940066 , -4.091970033 22 , 27.5 , 9.04996547 , -4.524982735 23 , 32.5 , 9.569580712 , -4.784790356 24 , 37.5 , 9.742785793 , -4.871392896

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25 , 42.5 , 9.569580712 , -4.784790356 26 , 47.5 , 9.04996547 , -4.524982735 27 , 52.5 , 8.183940066 , -4.091970033 28 , 57.5 , 6.9715045 , -3.48575225 29 , 62.5 , 5.412658774 , -2.706329387 30 , 67.5 , 3.507402885 , -1.753701443 31 , 72.5 , 1.255736835 , -0.627868418 32 , 1.25 , 0.638693735 , -0.319346868 33 , 3.75 , 1.851129301 , -0.92556465 34 , 6.25 , 2.976962326 , -1.488481163 35 , 8.75 , 4.01619281 , -2.008096405 36 , 11.25 , 4.968820754 , -2.484410377 37 , 13.75 , 5.834846158 , -2.917423079 38 , 16.25 , 6.614269021 , -3.307134511 39 , 18.75 , 7.307089344 , -3.653544672 40 , 21.25 , 7.913307127 , -3.956653564 41 , 23.75 , 8.432922369 , -4.216461185 42 , 26.25 , 8.865935071 , -4.432967536 43 , 28.75 , 9.212345233 , -4.606172616 44 , 31.25 , 9.472152854 , -4.736076427 45 , 33.75 , 9.645357935 , -4.822678967 46 , 36.25 , 9.731960475 , -4.865980238 47 , 38.75 , 9.731960475 , -4.865980238 48 , 41.25 , 9.645357935 , -4.822678967 49 , 43.75 , 9.472152854 , -4.736076427 50 , 46.25 , 9.212345233 , -4.606172616 51 , 48.75 , 8.865935071 , -4.432967536 52 , 51.25 , 8.432922369 , -4.216461185 53 , 53.75 , 7.913307127 , -3.956653564 54 , 56.25 , 7.307089344 , -3.653544672 55 , 58.75 , 6.614269021 , -3.307134511 56 , 61.25 , 5.834846158 , -2.917423079 57 , 63.75 , 4.968820754 , -2.484410377 58 , 66.25 , 4.01619281 , -2.008096405 59 , 68.75 , 2.976962326 , -1.488481163 60 , 71.25 , 1.851129301 , -0.92556465 61 , 73.75 , 0.638693735 , -0.319346868 *Element, type=B32 1, 1, 32, 17 2, 17, 33, 2 3, 2, 34, 18 4, 18, 35, 3 5, 3, 36, 19 6, 19, 37, 4 7, 4, 38, 20 8, 20, 39, 5 9, 5, 40, 21 10, 21, 41, 6 11, 6, 42, 22 12, 22, 43, 7 13, 7, 44, 23 14, 23, 45, 8 15, 8, 46, 24 16, 24, 47, 9 17, 9, 48, 25 18, 25, 49, 10 19, 10, 50, 26

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20, 26, 51, 11 21, 11, 52, 27 22, 27, 53, 12 23, 12, 54, 28 24, 28, 55, 13 25, 13, 56, 29 26, 29, 57, 14 27, 14, 58, 30 28, 30, 59, 15 29, 15, 60, 31 30, 31, 61, 16 *Nset, nset=_PickedSet131, internal, generate 1, 61, 1 *Elset, elset=_PickedSet131, internal, generate 1, 30, 1 *Nset, nset=Set01_Arch, generate 1, 61, 1 *Elset, elset=Set01_Arch, generate 1, 30, 1 ** Section: Section01_Arch Profile: Profile01_Arch *Beam General Section, elset=_PickedSet131, poisson = , density=, section=PIPE , 0.,0.,-1. , , *Section Points , , , , , , , *End Part ** *Part, name=Part02_Deck *Node 1 , 0.1 , 0 , 0 2 , 5 , 0 , 4.666666667 3 , 10 , 0 , 8.666666667 4 , 15 , 0 , 12 5 , 20 , 0 , 14.66666667 6 , 25 , 0 , 16.66666667 7 , 30 , 0 , 18 8 , 35 , 0 , 18.66666667 9 , 37.5 , 0 , 18.75 10 , 40 , 0 , 18.66666667 11 , 45 , 0 , 18 12 , 50 , 0 , 16.66666667 13 , 55 , 0 , 14.66666667 14 , 60 , 0 , 12 15 , 65 , 0 , 8.666666667 16 , 70 , 0 , 4.666666667 17 , 74.9 , 0 , 0 18 , 2.5 , 0 , 2.416666667 19 , 7.5 , 0 , 6.75 20 , 12.5 , 0 , 10.41666667 21 , 17.5 , 0 , 13.41666667 22 , 22.5 , 0 , 15.75 23 , 27.5 , 0 , 17.41666667 24 , 32.5 , 0 , 18.41666667 25 , 42.5 , 0 , 18.41666667

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26 , 47.5 , 0 , 17.41666667 27 , 52.5 , 0 , 15.75 28 , 57.5 , 0 , 13.41666667 29 , 62.5 , 0 , 10.41666667 30 , 67.5 , 0 , 6.75 31 , 72.5 , 0 , 2.416666667 32 , 1.25 , 0 , 1.229166667 33 , 3.75 , 0 , 3.5625 34 , 6.25 , 0 , 5.729166667 35 , 8.75 , 0 , 7.729166667 36 , 11.25 , 0 , 9.5625 37 , 13.75 , 0 , 11.22916667 38 , 16.25 , 0 , 12.72916667 39 , 18.75 , 0 , 14.0625 40 , 21.25 , 0 , 15.22916667 41 , 23.75 , 0 , 16.22916667 42 , 26.25 , 0 , 17.0625 43 , 28.75 , 0 , 17.72916667 44 , 31.25 , 0 , 18.22916667 45 , 33.75 , 0 , 18.5625 46 , 36.25 , 0 , 18.72916667 47 , 38.75 , 0 , 18.72916667 48 , 41.25 , 0 , 18.5625 49 , 43.75 , 0 , 18.22916667 50 , 46.25 , 0 , 17.72916667 51 , 48.75 , 0 , 17.0625 52 , 51.25 , 0 , 16.22916667 53 , 53.75 , 0 , 15.22916667 54 , 56.25 , 0 , 14.0625 55 , 58.75 , 0 , 12.72916667 56 , 61.25 , 0 , 11.22916667 57 , 63.75 , 0 , 9.5625 58 , 66.25 , 0 , 7.729166667 59 , 68.75 , 0 , 5.729166667 60 , 71.25 , 0 , 3.5625 61 , 73.75 , 0 , 1.229166667 *Element, type=B32 1, 1, 32, 18 2, 18, 33, 2 3, 2, 34, 19 4, 19, 35, 3 5, 3, 36, 20 6, 20, 37, 4 7, 4, 38, 21 8, 21, 39, 5 9, 5, 40, 22 10, 22, 41, 6 11, 6, 42, 23 12, 23, 43, 7 13, 7, 44, 24 14, 24, 45, 8 15, 8, 46, 9 16, 9, 47, 10 17, 10, 48, 25 18, 25, 49, 11 19, 11, 50, 26 20, 26, 51, 12

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21, 12, 52, 27 22, 27, 53, 13 23, 13, 54, 28 24, 28, 55, 14 25, 14, 56, 29 26, 29, 57, 15 27, 15, 58, 30 28, 30, 59, 16 29, 16, 60, 31 30, 31, 61, 17 *Nset, nset=_PickedSet128, internal, generate 1, 61, 1 *Elset, elset=_PickedSet128, internal, generate 1, 30, 1 *Nset, nset=Set02_Deck, generate 1, 61, 1 *Elset, elset=Set02_Deck, generate 1, 30, 1 ** Section: Section02_Deck Profile: Profile02_Deck *Beam General Section, elset=_PickedSet128, poisson=, density=, section=BOX , , , , , 0.,0.,-1. , , *Section Points , , , , , , , *End Part ** *Part, name=Part03_Cables *Node 1 , 65 , 4.5033321 , -2.25166605 2 , 65 , 0 , 8.666666667 3 , 55 , 7.621023553 , -3.810511777 4 , 55 , 0 , 14.66666667 5 , 45 , 9.353074361 , -4.67653718 6 , 45 , 0 , 18 7 , 35 , 9.699484522 , -4.849742261 8 , 35 , 0 , 18.66666667 9 , 25 , 8.660254038 , -4.330127019 10 , 25 , 0 , 16.66666667 11 , 15 , 6.235382907 , -3.117691454 12 , 15 , 0 , 12 13 , 5 , 2.424871131 , -1.212435565 14 , 5 , 0 , 4.666666667 15 , 10 , 4.5033321 , -2.25166605 16 , 10 , 0 , 8.666666667 17 , 20 , 7.621023553 , -3.810511777 18 , 20 , 0 , 14.66666667 19 , 30 , 9.353074361 , -4.67653718 20 , 30 , 0 , 18 21 , 40 , 9.699484522 , -4.849742261 22 , 40 , 0 , 18.66666667 23 , 50 , 8.660254038 , -4.330127019 24 , 50 , 0 , 16.66666667 25 , 60 , 6.235382907 , -3.117691454 26 , 60 , 0 , 12