Post-tensioned stress ribbon systems in long-span roofs: Case study: Västerås Travel Center

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(1)Post-tensioned stress ribbon systems in long-span roofs - Case study: Västerås Travel Center. Samih Ahmed Guayente Minchot. Master Thesis in Concrete Structures, June 2018 TRITA-ABE-MBT-18309 ISBN 978-91-7729-858-8.

(2) c Ahmed, Minchot 2018 Royal Institute of Technology (KTH) Department of Civil and Architectural Engineering Division of Concrete Structures Stockholm, Sweden, 2018. ii. Post-tensioned stress ribbon systems in long-span roofs.

(3) Abstract The stress ribbon system has numerous advantages, that includes but are not limited to: increasing overall stiffness, control deflections and reduction of materials consumption, which in turn, reduces the load and the cost. Nevertheless, its use is usually limited to bridges, in particular, pedestrian bridges; this can be attributed to the insufficient space that buildings’ usually have for end supports, or/and backstayed cables, that can accommodate the expected high pull-out forces occuring at the cables’ ends. In this work, the roof of Västerås Travel Center, which will become one of the longest cable suspended roofs in the world, was chosen as a case study. The aim was to investigate the optimal technique to model the post-tensioned stress ribbon system for the roof structure using SAP2000, and to assess any possible reduction in the pull-out forces, deflections and concrete stresses. Subsequently, a conventional cable suspended roof was simulated, using SAP2000, and compared to the post-tension stress ribbon system in order to examine the potential of the latter. Moreover, the effects of temperature loads and support movements on the final design loads were examined. Based on the study, a few practical recommendations concerning the construction method and the iterative design process, required to meet the architectural geometrical demands, are stated by the authors. The results showed that the post-tensioned stress ribbon system reduces the concrete stresses, overall deflections, and more importantly, reduces the pull-out forces by up to 16%, which substantially reduces the design forces for the support structures. The magnitude of these reductions was found to be highly correlated to the applied prestressing force, making the size of the prestressing force a key factor in the design. Keywords: cable suspended, stress ribbon, roof structure, post-tensioned concrete, SAP2000.. Post-tensioned stress ribbon systems in long-span roofs. iii.

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(5) Sammanfattning Konstruktioner med spännbandsystem bestående av bärande huvudkablar med pålagda plattor, ofta av betong, har många fördelar. Dessa fördelar inkluderar men begränsas inte till ökad totalt styvhet, kontrollerade nedböjningar och reducerad materialförbrukning, vilket minskar lasten och kostnaden. Deras användning är dock vanligen begränsad till broar, särskilt gång- och cykelbroar, där det finns utrymme för att förankra de höga utdragskrafterna från huvudkablarna. Motsvarande utrymme finns sällan i byggnader. I det föreliggande arbetet har taket till Västerås Resecentrum valts ut som studieobjekt. Taket kommer att bli ett av väldens längsta kabelburna takkonstruktion. Syftet är att undersöka den optimala tekniken för att modellera ett efterspänt spännbandsystem för taket med hjälp av FE-programmet SAP2000 och att bedöma eventuella minskningar på utdragskrafter, nedböjningar och betongspänningar. Därefter modellerades en konventionell kabelburen takkonstruktion med SAP2000, och det jämfördes med det efterspända spännbandsystemet för att undersöka fördelarna av det sistnämnda. Dessutom har effekten av temperaturlasten och upplagsrörelser undersökts på den slutliga modellen. Slutligen ges några praktiska rekommendationer om byggteknik och en iterativ dimensioneringsprocess som är nödvändig för arkitekturgestaltning och dess krav på geometri. Resultaten visar att det efterspända spännbandsystemet gav lägre betongspänningar, mindre totaltnedböjning, och ännu viktigare, mindre utdragskrafter. Krafterna minskade 16%, vilket gav en minskning av konstruktionens horisontella upplagsreaktion. Storleken på reduktionen var direkt proportionell mot spännkraften, så förspänning är en nyckelfaktor vid dimensioneringen.. Post-tensioned stress ribbon systems in long-span roofs. v.

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(7) Preface First, we would like to express our respectful gratitude and sincere appreciation to our supervisor, Fritz King, for his always enthusiastic encouragement and guidance, and to our examiner, Mikael Hallgren, for his invaluable advice in the field. They never spared an opportunity to help us developing personally or academically. We gratefully acknowledge our mentor, Anders Eriksson, for sharing his profound knowledge in our research area. His insightful advice and meticulous guidance were invaluable. He was always, patiently, present throughout the development of this work. Especial thanks to Karl Graah-Hagelbäck, and all the team behind the design of Västerås Travel Center (Daniel, Pontus, Henrik), for letting us play an active role in the project, and for their continuous assistance and constructive discussions. Our time in Tyréns would not have been the same without the continuous sound advice and endless support from Johny Akfidan and the team of K4. Thank you for providing an encouraging working environment and giving us a warm welcome to the team. To all our friends, colleagues and teachers from KTH, and our home universities, thank you for inspiring us to pursue a career in structural engineering and making the last two years in Stockholm truly remarkable. Warmest thanks to our parents for their continuous care and support, and to our siblings for always being there for us, their presence kept us highly motivated throughout our journey. To each of the above and everyone who made this research possible, we extend our deepest appreciation. Stockholm, June 2018 Samih Ahmed and Guayente Minchot. Post-tensioned stress ribbon systems in long-span roofs. vii.

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(9) Contents Contents. ix. List of Figures. xiii. List of Tables. xvii. Nomenclature. xix. 1 Introduction. 1. 1.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.2. Problem statement - case study . . . . . . . . . . . . . . . . . . . . .. 1. 1.3. Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2. 1.4. Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 1.5. Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 1.6. Outline of report . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 2 Theoretical Background 2.1. 2.2. 5. Cable structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. 2.1.1. History review . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. 2.1.2. Examples of existing structures . . . . . . . . . . . . . . . . .. 8. 2.1.3. Categorization of cable roofs . . . . . . . . . . . . . . . . . . . 15. 2.1.4. General structural characteristics . . . . . . . . . . . . . . . . 16. 2.1.5. Stress ribbon structures . . . . . . . . . . . . . . . . . . . . . 19. 2.1.6. Prestress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . 29. Post-tensioned stress ribbon systems in long-span roofs. ix.

(10) CONTENTS 2.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29. 2.2.2. SAP2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30. 2.2.3. Element types . . . . . . . . . . . . . . . . . . . . . . . . . . . 30. 2.2.4. Analysis types . . . . . . . . . . . . . . . . . . . . . . . . . . . 39. 3 Case Study: Västerås Travel Center 3.1. General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43. 3.2. Main drape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47. 3.3. 3.4. 3.2.1. Cable suspended roof with precast concrete panels . . . . . . . 48. 3.2.2. Stress ribbon system . . . . . . . . . . . . . . . . . . . . . . . 48. Materials and components . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3.1. Bearing cables . . . . . . . . . . . . . . . . . . . . . . . . . . . 49. 3.3.2. Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50. 3.3.3. Post-tension tendons . . . . . . . . . . . . . . . . . . . . . . . 50. 3.3.4. Crossbeams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51. Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51. 4 Investigation of modelling options in SAP2000 4.1. 55. Cable vs. frame elements . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.1.1. Cable vs. frame elements applied to a 2D case . . . . . . . . . 56. 4.1.2. Cable vs. frame applied to a simplified version of the case study (3D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57. 4.2. Effect of modelling the concrete panels vs. applying an equivalent uniformly distributed load . . . . . . . . . . . . . . . . . . . . . . . . 59. 4.3. Investigation of support movements . . . . . . . . . . . . . . . . . . . 64. 4.4. 4.5. x. 43. 4.3.1. Movement of the anchoring point throughout the construction stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65. 4.3.2. Quantification of the effect on the main drape . . . . . . . . . 67. Temperature load effects . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.4.1. Temperature effect on a simplified model . . . . . . . . . . . . 69. 4.4.2. Temperature effect on the stress ribbon model . . . . . . . . . 71. Effect of post-tension on the bearing cable forces . . . . . . . . . . . . 71. Post-tensioned stress ribbon systems in long-span roofs.

(11) CONTENTS 5 FE models 5.1. 73. Structural concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.1.1. Common aspects for all models . . . . . . . . . . . . . . . . . 73. 5.1.2. Initial model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75. 5.2. Cable suspended roof, with precast concrete panels . . . . . . . . . . 77. 5.3. Stress ribbon roof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79. 5.4. 5.3.1. Model without post-tension . . . . . . . . . . . . . . . . . . . 80. 5.3.2. Model with post-tension . . . . . . . . . . . . . . . . . . . . . 83. Results reporting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86. 6 Results. 89. 6.1. Cable suspended roof . . . . . . . . . . . . . . . . . . . . . . . . . . . 89. 6.2. Stress ribbon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91. 6.3. 6.2.1. Without post-tension . . . . . . . . . . . . . . . . . . . . . . . 91. 6.2.2. With post-tension . . . . . . . . . . . . . . . . . . . . . . . . . 93. Model comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97. 7 Practical considerations. 101. 7.1. Construction methods . . . . . . . . . . . . . . . . . . . . . . . . . . 101. 7.2. Design process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103. 8 Discussion and conclusions. 107. 8.1. Modelling and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 107. 8.2. Results evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108. 8.3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109. 8.4. Recommendations for future research . . . . . . . . . . . . . . . . . . 109. Bibliography. 111. Appendix. 115. A Analytical calculation for simply supported cable. 115. Post-tensioned stress ribbon systems in long-span roofs. xi.

(12) CONTENTS B Prestress losses. 119. C Initial cable geometry calculation in SAP2000. 127. xii. Post-tensioned stress ribbon systems in long-span roofs.

(13) List of Figures 1.1. The entire station united under one floating roof. Retrieved from [38].. 2. 1.2. Rendering of the travel center. Retrieved from [38]. . . . . . . . . . .. 2. 1.3. Simplification of of the two compared systems. . . . . . . . . . . . . .. 3. 2.1. Sketch of the bridge-type structure developed by D.Jawerth. Retrieved from [25]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. 2.2. Section of the Hovet arena in Stockholm. Retrieved from [32]. . . . .. 6. 2.3. Scandinavium Arena. Retrieved from [2]. . . . . . . . . . . . . . . . .. 7. 2.4. Sacramento River Trail Bridge. Retrieved from [34]. . . . . . . . . . . 10. 2.5. Dulles International Airport after the expansion in 1997. Retrieved from [19]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. 2.6. Portuguese National Pavilion. Retrieved from [29]. . . . . . . . . . . . 12. 2.7. Construction sequence of the Portuguese National Pavilion. Redrawn from [29]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. 2.8. Braga Municipal Stadium. Retrieved from [12]. . . . . . . . . . . . . 14. 2.9. Section diagram of Braga Municipal Stadium. Retrieved from [12]. . . 15. 2.10 Different typologies of cable roofs. Retrieved from [6]. . . . . . . . . . 16 2.11 Effect of different loading patterns in a simply supported cable. . . . 18 2.12 Relation between the ratio of applied loads and the deflection. Redrawn from [24]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.13 Possible alternatives to increase the stiffness. Retrieved from [21]. . . 20 2.14 Effect of different materials, thicknesses and prestressing to the overall stiffness. Retrieved from [21]. . . . . . . . . . . . . . . . . . . . . . . 20 2.15 DS-L Bridge conceptual design. Retrieved from [21]. . . . . . . . . . . 21 2.16 DS-L Bridge under construction. Retrieved from [21]. . . . . . . . . . 22. Post-tensioned stress ribbon systems in long-span roofs. xiii.

(14) LIST OF FIGURES 2.17 Tendon layouts and correspondent prestress equivalent forces. Redrawn from [20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.18 Stress-strain diagram for prestressing steel. Retrieved from [9]. . . . . 25 2.19 Idealized and design stress-strain diagram for prestressing steel. Retrieved from [9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.20 Degrees of freedom for a 3D frame element. . . . . . . . . . . . . . . . 31 2.21 Cable Layout form in SAP2000, including the starting input parameter options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.22 Cable definition in SAP2000. Retrieved from [1]. . . . . . . . . . . . . 33 2.23 Tendon load form in SAP2000. . . . . . . . . . . . . . . . . . . . . . . 36 2.24 Three of the six independent spring hinges in a link/support element. Retrieved from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.25 Different types of link/support elements in SAP2000. Retrieved from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.26 Nonlinear solution controls shown for a one-dimensioned case. . . . . 41 3.1. Rendering of the development area. Retrieved from [37]. . . . . . . . 44. 3.2. Rendering of the roof. Retrieved from [37]. . . . . . . . . . . . . . . . 44. 3.3. Sketch of the roof by BIG Architects. . . . . . . . . . . . . . . . . . . 45. 3.4. Architectural concept for the support structures by BIG Architects. . 46. 3.5. 3D rendering of the roof superstructure by Tyréns. . . . . . . . . . . 46. 3.6. 3D rendering of the main drape, without the concrete panels. . . . . . 47. 3.7. 3D rendering of the main drape, including its main dimensions. . . . 48. 3.8. Simplification of how the tendons (in red) are only anchored in the concrete shell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49. 3.9. Full-locked coil rope. Retrieved from [31]. . . . . . . . . . . . . . . . . 49. 3.10 Arrangement of the bearing cables in 3x3 bundle. . . . . . . . . . . . 50 3.11 The three different structural systems considered. . . . . . . . . . . . 53. xiv. 4.1. Initial geometry and loads of the cable and frame model. . . . . . . . 56. 4.2. Undeformed (initial) shape of the structure. . . . . . . . . . . . . . . 57. 4.3. Deformed (after loading) shape of the structure. . . . . . . . . . . . . 57. 4.4. Relative strain load applied to the crossbeams. . . . . . . . . . . . . . 58. Post-tensioned stress ribbon systems in long-span roofs.

(15) LIST OF FIGURES 4.5. Plan view of the models. . . . . . . . . . . . . . . . . . . . . . . . . . 60. 4.6. Different options to apply area loads in SAP2000. . . . . . . . . . . . 61. 4.7. The two options to apply the load. . . . . . . . . . . . . . . . . . . . 63. 4.8. 3D rendering of the support structures and expected movements. . . . 64. 4.9. 3D and plan view of the support structure frame. . . . . . . . . . . . 65. 4.10 Proposed construction sequence. . . . . . . . . . . . . . . . . . . . . . 66 4.11 Force-displacement graph for the corner support.. . . . . . . . . . . . 67. 4.12 Spring support conditions. . . . . . . . . . . . . . . . . . . . . . . . . 68 4.13 Two simplified models to apply the temperature load. . . . . . . . . . 69 4.14 Deflections under different loads: dead load (blue) and temperature load (orange). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.15 Relation between the bearing cable force and the prestress force. . . . 72 5.1. Plan view and geometry of the final models. . . . . . . . . . . . . . . 74. 5.2. Example of the curved frame geometry generator in SAP2000. . . . . 76. 5.3. Connection between the crossbeams and the cables. . . . . . . . . . . 77. 5.4. Main structural elements of the cable suspended roof model. . . . . . 78. 5.5. Example of how the shell elements are divided in between crossbeams for the cable suspended roof. . . . . . . . . . . . . . . . . . . . . . . . 78. 5.6. Main structural elements of the stress ribbon without post-tension. . 79. 5.7. Main structural elements of the stress ribbon with post-tension. . . . 79. 5.8. Example of how the shell are connected to the adjacent elements for the stress ribbon system. . . . . . . . . . . . . . . . . . . . . . . . . . 80. 5.9. Properties of the gap link used in SAP2000. . . . . . . . . . . . . . . 81. 5.10 Sketch of how the required nodes for the gap distance were achieved in SAP2000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.11 The model after drawing the concrete panels. Detail of the physical space between the panels. . . . . . . . . . . . . . . . . . . . . . . . . 82 5.12 Gap links (in red) between the panels divided in 10x10 elements. . . . 82 5.13 Sketch of the equal constraints (in red) applied to nodes between the shells and the cables (in blue). . . . . . . . . . . . . . . . . . . . . . . 83 5.14 Extrapolation and averaging of stresses in the shells. . . . . . . . . . 87. Post-tensioned stress ribbon systems in long-span roofs. xv.

(16) LIST OF FIGURES 6.1. Location of the output results. . . . . . . . . . . . . . . . . . . . . . . 89. 6.2. Axial stresses in the bearing cables, for the cable suspended roof model. 90. 6.3. Deflection in the bearing cables, for the cable suspended roof model. Note: similar deflections, some colours don’t show . . . . . . . . . . . 90. 6.4. Stresses in the concrete along path SC, for the cable suspended roof model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91. 6.5. Axial stresses in the bearing cables, for the stress ribbon model without post-tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92. 6.6. Deflection in the bearing cables, for the stress ribbon model without post-tension. Note: similar deflections, some colours don’t show . . . 92. 6.7. Stresses in the concrete along path SC, for the stress ribbon model without post-tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . 92. 6.8. Axial stresses along cable B, for the stress ribbon model. . . . . . . . 94. 6.9. Deflection along cable B, for the stress ribbon model. . . . . . . . . . 94. 6.10 Stresses in the concrete along path SC, for the stress ribbon model. . 95 6.11 Stresses trendline during post-tension.. . . . . . . . . . . . . . . . . . 96. 6.12 Stresses trendline for snow load case. . . . . . . . . . . . . . . . . . . 96 6.13 Comparison of axial stresses along cable B for the different models. . 97 6.14 Comparison of deflections along cable B for the different models. . . . 98 6.15 Comparison of stresses in the shell top face along path SC. . . . . . . 99 6.16 Comparison of stresses in the shell bottom face along path SC. . . . . 99. xvi. 7.1. Construction sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . 102. 7.2. Flowchart for the iteration process. . . . . . . . . . . . . . . . . . . . 103. 8.1. Local bending effect due to the deformed shape. . . . . . . . . . . . . 109. Post-tensioned stress ribbon systems in long-span roofs.

(17) List of Tables 2.1. Main facts about the Sacramento River Trail Bridge. . . . . . . . . . 10. 2.2. Main facts about the Dulles International Airport. . . . . . . . . . . . 11. 2.3. Main facts about the Portuguese National Pavilion. . . . . . . . . . . 13. 2.4. Main facts about the Braga Municipal Stadium. . . . . . . . . . . . . 15. 2.5. Types of analysis in SAP2000. . . . . . . . . . . . . . . . . . . . . . . 40. 2.6. Differences between linear and nonlinear analyses. . . . . . . . . . . . 40. 3.1. Cable material and section properties.. 3.2. Concrete properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50. 3.3. Tendon properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51. 4.1. Results from the investigation. . . . . . . . . . . . . . . . . . . . . . . 56. 4.2. Material properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58. 4.3. Comparison between cable and frame elements. . . . . . . . . . . . . 59. 4.4. Comparison between the panel’s model and UDL model. . . . . . . . 62. 4.5. Displacement at the cable anchorage point for the different construction stages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66. 4.6. Comparison between pinned and spring support conditions. . . . . . . 68. 4.7. Support movements and deflection due to temperature load in the bearing cables, simplified models. . . . . . . . . . . . . . . . . . . . . 70. 4.8. Axial load in the cables due to temperature load in the bearing cables, simplified models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70. 4.9. Deflection and axial forces in the bearing cables from the temperature load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71. 5.1. Steel material properties. . . . . . . . . . . . . . . . . . . . . . . . . . 74. . . . . . . . . . . . . . . . . . 49. Post-tensioned stress ribbon systems in long-span roofs. xvii.

(18) LIST OF TABLES 5.2. Concrete material properties. . . . . . . . . . . . . . . . . . . . . . . 74. 5.3. Section properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74. 5.4. Convergence parameters. . . . . . . . . . . . . . . . . . . . . . . . . . 75. 5.5. Tendon parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84. 5.6. Tendon section modification. . . . . . . . . . . . . . . . . . . . . . . . 85. 5.7. Prestressing force applied. . . . . . . . . . . . . . . . . . . . . . . . . 86. 6.1. Final results for the cable suspended roof model. . . . . . . . . . . . . 91. 6.2. Final results for the stress ribbon model without post-tension. . . . . 93. 6.3. Variation of the deflection during the different load cases, for the stress ribbon model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95. 6.4. Final results for the post-tensioned stress ribbon model. . . . . . . . . 96. 6.5. Comparison of maximum axial stress along cable B for the different models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97. 6.6. Comparison of deflections at midspan for the different models. . . . . 98. 6.7. Comparison of minimum stresses in the top face for the different models. 99. 6.8. Comparison of minimum stresses in the bottom face for the different models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100. xviii. Post-tensioned stress ribbon systems in long-span roofs.

(19) Nomenclature Acronyms 2D. 2 Dimension. 3D. 3 Dimension. FEM Finite Element Method GUI. Graphical User Interface. SLS. Service-ability Limit State. UDL Uniformly Distributed Load ULS Ultimate Limit State Greek Symbols α. Coefficient of thermal expansion. ∆. Variation of a quantity. δ. Deflection, displacement. ε. Strain. γ. Safety factor. µ. Coefficient of friction between the tendons and their ducts. µi. Snow load shape coefficient. ϕ. Creep factor. ρ. Material density. τ. Shear stress. σ. Normal stress. θ. Angle. Subscripts. Post-tensioned stress ribbon systems in long-span roofs. xix.

(20) Nomenclature c. Concrete. d. Design value. k. Characteristic value. p. Prestressing steel. r. Relaxation in the steel. s. Shrinkage in the concrete. T. Temperature. Symbols A. Area. Cc. Exposure coefficient for snow loads. Ct. Thermal coefficient for snow loads. E. Modulus of elasticity. e. Eccentricity. εuk. Characteristic strain of prestressing steel at maximum load. f. Cable sag. fp0,1k Characteristic 0,1% proof-stress of prestressing steel fpk. Characteristic tensile strength of prestressing steel. fyd. Yield strength of steel. G. Permanent point load. g. Permanent uniformly distributed load. I. Moment of inertia. k. Wobble coefficient or stiffness properties. L. Length. M. Moment force. N. Normal force. P. Prestress force. Q. Variable point load. q. Variable uniformly distributed load. xx. Post-tensioned stress ribbon systems in long-span roofs.

(21) NOMENCLATURE R. Support reaction. r. Prestress contact force. s. Snow load. sk. Characteristic value for snow loads. T. Tension force in a cable. t. Thickness. U. Deflection. zcp. distance between the center of gravity of the concrete section and the tendons. Post-tensioned stress ribbon systems in long-span roofs. xxi.

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(23) Chapter 1 Introduction 1.1. Background. A roof is classified as a long-span roof when it exceeds 12 m in length without intermediate supports [23], however, in this thesis, the term "long-spans" will be used for roofs of much longer dimensions. Long-span roofs are usually built to satisfy aesthetics requirements and high functionality levels for the structures. Hence, usually it requires complex design and creates many challenges during the construction process. Different structural systems are used nowadays for the design of this type of roofs, like: truss systems, arches and vaults, domes, cable structures and shell structures. Extensive research presented in [21] suggests to combine more than one of the aforementioned systems to achieve an ideal performance and optimized behaviour for the long-span roofs. For instance, the possibility of combining membrane actions with pre-stretched (tensioned) cables has been studied in [22], this combination is commonly known as stress ribbon system.. 1.2. Problem statement - case study. A new transportation hub is to be built in Västerås, Sweden. Given its location as a commuter link between Stockholm and the Mälaren area, it’s expected to welcome a high number of passengers every day. The building aims to meet the travelers’ demands and link the areas on either side of the tracks under one thin floating roof, shown in Figure 1.1, without utilizing intermediate supports.. Post-tensioned stress ribbon systems in long-span roofs. 1.

(24) CHAPTER 1. INTRODUCTION. Figure 1.1: The entire station united under one floating roof. Retrieved from [38].. As shown in Figure 1.2, the roof structure consists of three main drapes, with the longest span exceeding 200 m, supported on four corner structures. Moreover, the architectural design proposes a very slender roof; a demand that only few structural systems are able to satisfy.. Figure 1.2: Rendering of the travel center. Retrieved from [38].. 1.3. Objective. Aside from loading conditions, long-span cable suspended roofs can be comparable to bridges, thus, this thesis examines the possibility of applying a specific bridge structural system, the stress ribbon system, to the given case study and evaluates its advantages, or disadvantages, over a conventional cable suspended roof design. The two compared systems are schematically shown in Figure 1.3. To reach a conclusion about the applicability of this system for long-span roofs, great attention is paid for the following points throughout this thesis: • Realizing an accurate FE model, that best describes the reality, by: – Investigating the element choice that best captures the cable behaviour.. 2. Post-tensioned stress ribbon systems in long-span roofs.

(25) CHAPTER 1. INTRODUCTION – Modelling options that describe, accurately, the load transfer between adjacent precast concrete panels. – Implementing post-tension tendons in the model, including the effect of short and long term force losses. • Investigating the structural behaviour under temperature loads and support movements.. (a) Conventional cable suspended roof. (b) Stress ribbon system Bearing cables Crossbeams Post-tension tendons Concrete panels. Figure 1.3: Simplification of of the two compared systems.. 1.4. Method. In a short summary, the method is based on investigating the two compared structural systems by, firstly, studying buildings where these systems are already utilized. Secondly, using SAP2000, two simulations representing the two proposals are run, and certain variables are chosen to compare the two models, such as: axial forces and stresses for the cables, stresses in the concrete panels and overall deformation of the structure. The model that behaves best at these selected outputs is then recommended by the authors. To guarantee a better understanding of the software used, and to achieve accurate models, a number of investigations are carried out first in simplified models. Afterwards, the two final models are created and compared.. 1.5. Limitations. As mentioned previously, the roof structure consists of three drapes, however, the scope of this thesis is limited to only the longest one. Assuming that this drape is the most challenging one, based on its geometrical features, the other two drapes are considered to have a less troublesome behaviour.. Post-tensioned stress ribbon systems in long-span roofs. 3.

(26) CHAPTER 1. INTRODUCTION Only the superstructure of the drape is studied, overlooking the architectural cladding, installations, cables’ anchoring systems, and the foundation. The wind-induced vibrations are another important factor to consider in cable suspended structures, nevertheless, it is not a part of this investigation. In view of the present geometrical complexity of the roof and its surrounding, it is complicated to estimate the wind load accurately, thus, the authors suggest that a wind tunnel test is a must for this project. The deformed geometry specified by the architects can only be realized by an iteration process that requires long computational time, hence, the final geometry is not considered in this work. However, the iteration process itself is outlined and briefly described in Section 7.2. Plain concrete is modelled to simulate precast concrete panels, neglecting the effect of the reinforcement. Moreover, linear properties for the concrete behaviour is used in SAP2000, neglecting possible crack openings and material deterioration. Finally, the studied and compared output is limited to deflections, axial stresses on the cables and longitudinal stresses in the top and bottom face of the concrete. Other results such as concrete stresses on the transverse direction and internal forces in the crossbeams are neglected from the recorded results.. 1.6. Outline of report. Chapter 1 : presents the introduction, aim of the thesis and limitations of the work. Chapter 2 : contains the background theoretical knowledge required to complete the thesis objectives. Chapter 3 : describes the case study, Västerås Travel Center: its geometry, loading types, materials, and the two structural systems investigated. Chapter 4 : includes different investigations of various modelling options in SAP2000, which determines how to proceed for the final, most accurate, models. Chapter 5 : describes the steps followed in SAP2000 to reach the final models, and how to extract the relevant results. Chapter 6 : contains the results obtained from the models and a comparison between the structural systems. Chapter 7 : states the outline of practical considerations, related to the case study, without including numerical investigations. Chapter 8 : summarizes the conclusions obtained throughout the thesis.. 4. Post-tensioned stress ribbon systems in long-span roofs.

(27) Chapter 2 Theoretical Background 2.1 2.1.1. Cable structures History review. Long span structures Long span structures have a long history with outstanding technology developments. Thanks to the advancements in materials sciences, what was considered by the Romans as a long span in the Pantheon (AD 118-128), spanning 43,2 m in the shape of a concrete dome [28], is today surpassed by cable roofs, which easily span up to 200-250 m [16]. Furthermore, in case of bridges, the world record is set to 1991 m by the Akashi Kaiky¯o Bridge in Japan [17]. In the early history of long spans, the structures built were usually cathedrals domes made of stone, working under full compression. This was followed by truss (tension and compression) and beam (bending) systems which were used to create columnfree spaces for public use or bridges, for which, the development of cast iron during the 18th century [28] was essential; due to its high tensile strength. Subsequently, cast iron was developed into steel, which allowed for even larger spans by using also beam or truss elements. Another breakthrough in long spans was the introduction of prestressed concrete, which has been developing since 1928 until present times [15]. Prestressing was used during the forties and sixties mainly for bridges, but nowadays, its use is spread to all kind of long span structures, shells, marine structures, etc. The increase in loading capacity by using prestressed concrete makes it very suitable and economic for long spans, resulting in free lengths of 30 m for precast segments and up to 100 m for in-situ concrete [15].. Post-tensioned stress ribbon systems in long-span roofs. 5.

(28) CHAPTER 2. THEORETICAL BACKGROUND. Cable structures Heading towards recent history, cable systems were adopted, which introduced the use of tension-only structures for long spans. The evolution of cable structures can be divided in two branches: bridge structures and roof structures. Bridge structures using cables have a longer history, being based on simple liana and rope suspension bridges, they have been improved and transformed to the modern cable-stayed and suspension bridges that are built nowadays. The use of cable supported structures in contemporary roof structures has a relatively short history compared to its presence in bridges’ structures. The first notable cable supported roof goes back to the 1950s by the erection of the North Carolina State Fair Arena at Raleigh, USA [6]. Among first efforts made, the Swedish engineer D. Jawerth tried to use a bridge-type structures to roof long span buildings [16] as depicted in Figure 2.1, and his concept was used to design the ice hockey arena Hovet in Stockholm in 1955, shown in Figure 2.2.. Figure 2.1: Sketch of the bridge-type structure developed by D.Jawerth. Retrieved from [25].. Figure 2.2: Section of the Hovet arena in Stockholm. Retrieved from [32]. Another example of a cable roof structure is the Scandinavium Arena in Göteborg, see Figure 2.3, which is made a cable net, anchored in a reinforced concrete ring, creating the shape of a hyperbolic paraboloid.. 6. Post-tensioned stress ribbon systems in long-span roofs.

(29) CHAPTER 2. THEORETICAL BACKGROUND. Figure 2.3: Scandinavium Arena. Retrieved from [2].. As it will be later justified in Section 2.1.4, an additional system is required to increase the stiffness of a cable-only structures; so that it behaves better under additional loads or in vibrations. Many different systems are used to achieve this, such as additional stay cables, arches, trusses or prestress bands [21], however, the most simple method is to simply increase the mass of the system by adding heavy materials, like concrete. This method has been used in many roof structures in recent years, as will be shown in Section 2.1.2. Another advantage of including concrete, is the possibility of adding prestress tendons, which also increases the stiffness. During the next two decades, the sixties and early seventies, it was thought by many engineers that cable-roof structures will be highly demanded in the near future; for their aesthetic and economical potentials. Therefore, an immense amount of research and investigations were conducted to have a closer look on the behaviour of this type of structures. However, the expected bright future for these structures isn’t realized yet, and the total number of similar structures is, fairly, a modest one. A few possible reasons, mentioned in [6], can be summarized as follows. • The demand for long clear spans has been lower than anticipated. • Complicated design process abates engineers’ interest in using them more frequently. • Their usage in earthquake zones hasn’t been sufficiently investigated, compared to other structural types. • The price of tension anchors, when needed, is relatively high and discourages the owners. • The high precision required in manufacture is less attractive.. Post-tensioned stress ribbon systems in long-span roofs. 7.

(30) CHAPTER 2. THEORETICAL BACKGROUND In the past, the use of cable roofs was linked mainly to the need of wide column-free areas. Consequently, its presence was primarily limited to buildings such as concert halls, theatres, stadiums, swimming pools, hangars, warehouses and few other types of buildings. However, recent investigations and studies suggest that cable roofs can be reasonable alternatives for short span structures [6]. With their high architectural value and good economic potential, compared to the rising cost of steel nowadays, the use of cables is becoming an attractive alternative. The developments in material technology besides the presence of FEM software as a design tool has made a substantial positive impact on long span structures. The latter cause is clearly seen in the introduction of parametric design: which is an innovative design tool specially useful for handling the design of cable structures, whose final geometry depends on many parameters.. 2.1.2. Examples of existing structures. In this section, relevant examples of long span structures are summarized to give the reader an overview of those projects that have similar structural design and behaviour to the building analyzed in this thesis as a case study.. Stress Ribbon Bridges Stress ribbon bridges are tension structures where the superstructure becomes also the walking deck; because of its low-sag catenary shape. This is accomplished through supporting the prestressed concrete slab by slightly slack suspension cables anchored at the abutments, resulting in a rather simple structure which involves very high loads on the cables. The book "Stress Ribbon and Cable-supported Pedestrian Bridges", written by J.Strasky [21] has been used as the main reference for this section unless otherwise is stated. This section only summarizes the history of stress ribbon structures, more technical understanding of this structural typology is given in Section 2.1.5. The origin of this structural typology may be found on primitive bamboo or liana ropes, where the bridge deck hangs directly on the ropes, finding its shape from the applied loads. Improving, this primitive system, by stronger tension materials (steel cables) and utilizing the ability of prestresed concrete to increase the stiffness of the structure, gives rise to what is called a stress ribbon structure. As mentioned earlier, the deck follows a catenary shape that has a very small sag compared to its span, usually the sag to span ratio f /L varies from 1/33 to 1/48 [14]. Due to the slightly steep slope of the catenary, these bridges are generally designed for pedestrian use only.. 8. Post-tensioned stress ribbon systems in long-span roofs.

(31) CHAPTER 2. THEORETICAL BACKGROUND This bridge typology is not very common, and very few experts work on this area; being J.Strasky the engineer who developed most of the design procedures used nowadays [18]. The first bridge of this type was built in 1965 in Switzerland, with a span of 48 m. Most of the bridges built after that are found in central Europe (Germany, former Czechoslovakia and Switzerland), United States, Japan and United Kingdom [10]. To go further into the structural characteristics for stress ribbon bridges, the two main structural elements involved will be, briefly, introduced: • Concrete deck: in-situ or precast panels. The deck transfers the external loads to the bearing cables, besides, it increases stiffness and stability. • Cables, for a few different functions: – Erection cables: temporary cables which are removed after construction – Bearing tendons: main cables which support the complete system and transfer the load to the abutments. – Prestressing tendons: these induce compression forces in the concrete, increasing the overall stiffness. They can be present as: ∗ embedded in the concrete: bonded or un-bonded. ∗ external prestressing: under or on the sides of the deck. – Combination: the same cable is used for bearing and prestressing (external prestressing). To provide more specific characteristics of a stress ribbon bridge, the Sacramento River Trail Bridge, located in the city of Redding (USA), is taken as an example. With a single span of 127 m, it was built as a stress ribbon bridge to avoid having piers in the river and preserve the local environment. The bridge structure utilizes precast elements which are post-tensioned by four prestressing tendons, and are suspended on four bearing tendons. To support the pull-out forces on the bearing cables, they are anchored on the rock just below the abutments. The bridge is shown in Figure 2.4. The construction sequence for the superstructure of Sacramento River Trail Bridge was as follows: first, the bearing tendons were positioned. Afterwards, precast segments were suspended on the bearing tendons and shifted along the cables to their final location. Subsequently, the prestressing tendons were installed and the joints were cast, which introduced additional stiffness to the system.. Post-tensioned stress ribbon systems in long-span roofs. 9.

(32) CHAPTER 2. THEORETICAL BACKGROUND. Figure 2.4: Sacramento River Trail Bridge. Retrieved from [34]. Table 2.1: Main facts about the Sacramento River Trail Bridge. Use. Pedestrian bridge. Structural system. Stress ribbon, precast concrete shell. Span. 127 m. Width. 4m. Thickness. 38 cm. Bearing cables. 4 tendons (28× 1,27 cm strands). Dulles International Airport Built in 1962, Dulles International Airport is considered as the first roof to be built using a stress ribbon system [21]. It is located in Washington DC, and it serves as a passenger terminal for the city international airport. It was designed by the architect Eero Saarinen and Whitney Engineers, with the main idea of creating a "large open room under sweeping roof flanked by colonnades" [27]. To achieve this, 16 columns were placed at each side of the big open space, with one side higher than the other, and the roof spanned between them with a catenary shape, as shown in Figure 2.5.. 10. Post-tensioned stress ribbon systems in long-span roofs.

(33) CHAPTER 2. THEORETICAL BACKGROUND. Figure 2.5: Dulles International Airport after the expansion in 1997. Retrieved from [19]. On top of the columns, an edge beam was cast in-situ, with a curved profile, which is used to anchor the cables. The continuity of the roof was achieved by using precast concrete panels made of lightweight concrete [27]. Thereafter, fresh concrete was cast on top of the panels to increase the roof weight, raising its resistance against upwards wind. An important feature of this building is the columns’ inclination to overcome the high tensile pull-force from the cables. Table 2.2: Main facts about the Dulles International Airport. Use. Suspension roof for airport terminal. Structural system. Bearing cables and post-tensioned precast concrete shell. Span. 61 m. Width. 182 m. Concrete thickness 20 cm Cables. n/a. Table 2.2 summarizes the main features of the original building of Dulles International Airport. However, it is important to mention that, the terminal was expanded in 1997, adding about 90 m to each end (perpendicular to the drape), while replicating the original architecture [35].. Post-tensioned stress ribbon systems in long-span roofs. 11.

(34) CHAPTER 2. THEORETICAL BACKGROUND. The Portuguese National Pavilion The Portuguese National Pavilion was built for the Lisbon World Exposition which took place in Lisbon in 1998, and this building was the central masterpiece of the exposition due to its impressive concrete canopy. Designed by the architect Alvaro Siza and the structural engineer Cecil Balmond, the roof consists on a 70 meters concrete canopy spanning between adjacent buildings, as shown in Figure 2.6.. Figure 2.6: Portuguese National Pavilion. Retrieved from [29]. The engineer, Cecil Balmond, explains in his book "Informal" [3] the design process of the structure, including key decisions about the materials used for the canopy, and interesting geometry relations to optimize the flow of forces. The following paragraphs summarize the relevant information from [3]. With the main goal of a light, paper-thin structure, the design began with the idea of a net of cables supporting a metal or fabric cladding. However, weight was needed to overcome upwards wind effects, which led to the idea of a steel truss structure hidden by a cladding, but, using a steel truss would result in a thickness which wouldn’t represent the paper-thin structure, so this option was disregarded. Other designs like having extra supporting cables, similar to a cable-stayed bridge, acting as hangers from the abutments were also investigated, but the ideas were not taken forward. Thus, the best solution was to introduce a heavier material suspended on the cables which would naturally determine the shape of the drape and act as a counterweight for the upwards wind action. The logical choice was therefore to use concrete, creating a shell suspended on the cables. The main concern at that point was how to reduce the thickness of the concrete, so the structure would "look" light, and have a smaller pull-out force. The thickness of the concrete was set to 20 cm, giving a ratio of t/L = 0.0028. It is mentioned in [3] that the thickness could have been thinner according to calculations, but the engineers judgment and intuition raised it to 20 cm.. 12. Post-tensioned stress ribbon systems in long-span roofs.

(35) CHAPTER 2. THEORETICAL BACKGROUND The drape follows a catenary shape, with single curvature along the main span, for which the relations between the sag, span, and abutments height were carefully studied from the architectural point of view to be "pleasant to the eye". It is mentioned in [21], that two types of cables are used: bearing cables, anchored in the abutments, and prestressing tendons anchored at the end of the concrete shell. Taking into account the long span, and to avoid cracking due to temperature effects and shrinkage, the concrete was de-bonded from the cables, thus the suspended cable becomes the main structural element, and the prestressed concrete adds stiffness and weight to the system, but doesn’t carry any load to the abutments by itself. Perhaps one of the most interesting facts about the project described by Balmond in [3] is the construction sequence, which is reproduced in Figure 2.7, showing that the cables are stressed gradually after the concrete is poured, so the final shape is only achieved on the last stage with fully stressed cables.. Figure 2.7: Construction sequence of the Portuguese National Pavilion. Redrawn from [29].. Table 2.3 summarizes the main facts about the Portuguese National Pavilion. Table 2.3: Main facts about the Portuguese National Pavilion. Use. Suspended roof canopy for the Lisbon World Exposition. Structural system. Bearing cables and prestressed in-situ concrete shell. Span. 65 m. Width. 50 m. Thickness. 20 cm. Cables. n/a. Post-tensioned stress ribbon systems in long-span roofs. 13.

(36) CHAPTER 2. THEORETICAL BACKGROUND. Braga Municipal Stadium The Braga Municipal Stadium is an example of another long-span roof in Portugal, it was built for the UEFA European Championship (EURO 2004) in the city of Braga. The design of the stadium was executed by the architect Eduardo Souto de Moura, and the firm AFA Consult was in charge of the engineering work. A technical report by AFA Consult [12] describes the main technical facts of the project, and relevant parts of it are summarized in the following paragraphs. From the early stages of the design, two main challenges were found; geotechnical uncertainties, which is not discussed further in this thesis but more information can be found in [12], and the feasibility of the long-span roof. Again, as in the Portuguese National Pavilion, the main concept of "lightness" ruled the design and determined a cable suspension roof with concrete cladding as the natural solution. As observed in Figure 2.8, where the finished stadium is shown, the concrete roof doesn’t cover the complete span, due to natural light requirements, and, it is divided into two parts with free standing cables in the middle. This introduced an additional challenge regarding dynamic behaviour under wind load.. Figure 2.8: Braga Municipal Stadium. Retrieved from [12]. The suspension cables are full locked coil cables which were grouped in pairs and anchored in large beams on top of the uprights. Accounting for the high horizontal forces from the cables, the uprights were dimensioned accordingly. As seen in Figure 2.9, one of the uprights (the one in the right part of the Figure) has the advantage of rock presence at the same level, and therefore, the cables are anchored to the rock there. The concrete covering part of the span is made of precast panels which are connected to the cables only in the longitudinal direction. By allowing relative movements in the transverse direction, problems coming from loads such as shrinkage or thermal action are reduced. The connection between the panels was done by linking bolts and pouring in-situ concrete on the joints so the slab becomes continuous.. 14. Post-tensioned stress ribbon systems in long-span roofs.

(37) CHAPTER 2. THEORETICAL BACKGROUND. Figure 2.9: Section diagram of Braga Municipal Stadium. Retrieved from [12]. The utilization of precast concrete panels was a crucial choice for the construction process. It allowed to "slide" the panels over the cables, in a similar way as usually done for stress ribbon bridges [21]. After the panels are placed on its location, the longitudinal joints were filled with concrete to make a continuous slab. Table 2.4 summarizes the main facts concerning the Braga Municipal stadium. Table 2.4: Main facts about the Braga Municipal Stadium.. 2.1.3. Use. Suspension roof for a football stadium. Structural system. Bearing cables and precast concrete shell. Span. 200 m. Width. n/a. Thickness. 24 cm. Cables. 34 pairs of cables, φ 86 mm. Categorization of cable roofs. Cable roofs are either self-balancing or non-self-balancing. According to [6], a selfbalancing structure is defined as: "a building in which the structure supporting the cables has a geometry which permits the forces in the cables to be balanced internally", whereas the non-self-balancing building is defined as: "a building which the geometry of the building supporting the roof structure is unable to resist the cable forces without the aid of ground anchors". Categorizing the roof as a selfbalancing or a non-self-balancing structure is highly dependent on the way used to support the roof cladding. Four main ways of support can be distinguished, as shown in Figure 2.10. Since simply suspended cables is relevant to the case study object of this thesis, a brief description will be given here. For further details about the later three types, the reader is referred to [6].. Post-tensioned stress ribbon systems in long-span roofs. 15.

(38) CHAPTER 2. THEORETICAL BACKGROUND. Figure 2.10: Different typologies of cable roofs. Retrieved from [6]. If the cladding of the roof is rectangular or trapezoidal in plan, it can be supported by two, or more, simply suspended cables hanging in vertical plane, as presented in Figure 2.10 (a). While for circular roofs in plan, the cables are suspended radially and attached at the perimeter of the roof to a compression ring and at the centre to a tension ring.. In some cases, a combination of the two geometrical systems may be used. To increase the stability of cable suspended structures and limit the movements under various types of loading, the cladding must either be very heavy or act as a shell, [6] suggests that concrete is the most suitable roofing material for simply suspended cables roofs. Both in-situ concrete, at which plywood clamped below the cables can be used as formwork, and prefabricated panels are used. Pretensioning of the cables can be beneficial to stiffen the structure during construction and to prevent concrete cracks. Moreover, cracks are prevented in prefabricated concrete panels by applying an overload on the roof before grouting the gaps between adjacent panels, then removing the extra load when the grout has set [6].. 2.1.4. General structural characteristics. In general, cable structures are fundamentally nonlinear in their response to loading for two main reasons according to [6]: to be able to load a cable structure, a pretension is almost always necessary for the cables to reach certain stiffness and stability, thus, cable structures are structural mechanisms rather than true structures. Moreover, the steel type used for cables, high-tensile steel, can withstand strains approximately six times those sustained by ordinary steel. If all the cables. 16. Post-tensioned stress ribbon systems in long-span roofs.

(39) CHAPTER 2. THEORETICAL BACKGROUND are in tension, cable structures exhibit an increase in stiffness with increasing deformations. Consequently, one should be aware that using linear analysis for cable structures will result in largely overestimated forces and displacements [6], which could, to some extent, make linear analyses more or less useless. As mentioned in [6], the stiffness of cables structures is dependent upon: • the curvature of the cables; • the cross-sectional areas of the cables; • the level of pretension; • the stiffness of the support structure; • cladding material (in case of cable roofs). Cladding elements are considered as a main source for dynamic stability of the structure. However, unless the cladding is made as a concrete shell, its contribution to the overall stiffness of the structure is negligible [6]. The cable’s initial strain is an important factor in the cables’ structural behaviour. It this context, it is understood as the strain in the cable due to its initial length and its own weight; depending how long the cable is, the sag varies and thus the strain increases or decreases. It should not be confused with the term "pretension", which is referred to the tension applied to tendons in prestressed concrete structures. If cables go slack under a certain loading combination, the structure will show a softening behaviour and large deformations will occur [6], which may damage the cladding as mentioned earlier. Furthermore, a sufficient level of initial strain will ensure a good level of dynamic stability. Using cables allows the design of long spans thanks to their low weight and high tensile capacity. As a result of low bending stiffness, the deformation follows a funicular curve, which means that the deformation highly depends on the load pattern applied. This characteristic feature of cables allows for infinite creation of architectural compositions, nonetheless, it could sometimes be a problem for the structural design. Let’s imagine a simply supported cable, with undeformed shape as shown in Figure 2.11 (a). The cable is first loaded under a point load F 1, which results in the deformed shape (b). If that cable is afterwards loaded with an additional point load F 2, its deformed shape will turn into something similar to (c). If this cable net is to be used for instance as a roof in a building, such change in deformation can’t be allowed. Figure 2.12 is taken as an example of the increase of the deflection under an uniform variable load Q for a cable structure with dead load G. As the relation Q/G increases, the deflection from the variable load increases gradually, until a certain point when the curve starts to flatten. If the uniform variable load is taken as a. Post-tensioned stress ribbon systems in long-span roofs. 17.

(40) CHAPTER 2. THEORETICAL BACKGROUND. (a) Undeformed state.. (b) First loading stage.. (c) Second loading stage.. Figure 2.11: Effect of different loading patterns in a simply supported cable. constant value (certain snow load value given by the Eurocode, for instance), an increase in dead load results in a lower Q/G ratio and therefore a lower deflection for the variable load. Therefore, the initial dead load of the structure is critical for the behaviour in later stages when variable loads are applied; in most cases, the higher the self-weight of the structure is from the initial stage, the lower is the additional deflection under external applied loads in a second stage.. Figure 2.12: Relation between the ratio of applied loads and the deflection. Redrawn from [24].. Finally, the main load-carrying members in cable roof structures are only subjected to tensile forces, which generally are simpler to erect and not labour intensive. With the rising costs of labour and material, the cost effectiveness of cable roof structures will improve compared to other forms of structural systems.. 18. Post-tensioned stress ribbon systems in long-span roofs.

(41) CHAPTER 2. THEORETICAL BACKGROUND. 2.1.5. Stress ribbon structures. In this type of structures, slightly-sagging tensioned cables constitute the main loadbearing members. The deck, which can be made of steel, concrete or timber, has a very thin thickness compared to the span, and is mainly for distributing the load and ensuring the continuity of the structure [21]. Although this structural system is primarily used for bridges, its concept can be applied to other structures such as roofs. In older designs, the stiffness of these structures is mainly inherited from the cables’ axial stiffness and the bending stiffness of the deck. However, in recent designs, additional axial stiffness from the deck is obtained by means of prestressing. One characteristic feature of this type of structures is its inherent large local slope, which limits the usage to pedestrian bridges rather than highway bridges, except for very rare cases. Another characteristic feature, in which the engineering and aesthetic value of this type of structures lies, is the fact that the structure itself is the suspended walkway. It carries its own weight without any piers, masts or any supporting structural elements [21]. Moreover, because of the minimal amount of material needed and the fact that its construction is independent of the surrounding terrain, it’s considered environmentaly friendly. Also, compared to other forms of long span bridges, it does not require any bearings or expansion joints, which highly reduces the long-term maintenance. Although stress ribbon bridges may seem very flexible, they are able to withstand large concentrated forces and even extremely large forces caused by flooding in rivers if properly designed. According to [21], stress ribbon bridges can behave very well dynamically under disturbances caused by walking pedestrians or vandalism actions. Normally, stress ribbon structures are mainly used as pedestrian bridges, therefore, two main features are required [21]: first, since the deck follows the shape of the cables, the sag should be limited to ensure an acceptable slope. Secondly, sufficient stiffness is needed in order to guarantee a comfortable walking and dynamic stability for the bridge users. These two requirements can be achieved by increasing the stiffness of the structure, which can be realized by many ways, such as: introduction an initial strain to the cables or by using a prestressed concrete band [21]. Alternatively, different cables’ alignment can be used to ensure higher stiffness as can be seen in Figure 2.13. For further details about ways to stiffen the stress ribbon structure, the reader is advised to visit [21]. Focusing on achieving an increase in the stiffness by using a prestressed concrete band, Figure 2.14 shows the differences in the additional deflection w (a) from a load p applied in half of the span for different structural systems (b). The deflection w is higher for light materials, such as timber boards, and lower for heavier materials, such as concrete. Moreover, including a fully prestressed band reduces the relative deflection even more, due to the added membrane stiffness [21]. Therefore, stress ribbon bands are considered to be beneficial to increase the overall stiffness of the system and reduce deflections from external loads.. Post-tensioned stress ribbon systems in long-span roofs. 19.

(42) CHAPTER 2. THEORETICAL BACKGROUND. Figure 2.13: Possible alternatives to increase the stiffness. Retrieved from [21].. Figure 2.14: Effect of different materials, thicknesses and prestressing to the overall stiffness. Retrieved from [21].. 20. Post-tensioned stress ribbon systems in long-span roofs.

(43) CHAPTER 2. THEORETICAL BACKGROUND Stress ribbon structures usually have bearing and prestressing tendons uniformly distributed in the concrete deck. The prestressing tendons are usually embedded in the concrete deck, however the bearing tendons can also be placed externally. In some cases, the bearing and prestressing tendons are the same structural member. Both the bearing and prestressing tendons are usually considered as ordinary prestressing tendons and are checked according to the appropriate national standards. The concrete deck can be formed of a prestressed continuous band (cast in-situ), post-tensioned precast panels or a combination of both. Considering that the deck is mainly stressed by normal forces on the longitudinal direction, the section can be relatively slender. Its minimum cross-section is calculated to result in zero tension stresses and compression stresses below the concrete compressive strength. The limiting thickness requirement is usually determined by the minimum reinforcement cover. Figure 2.15 shows the conceptual design of the tendons’ arrangement in the DS-L Bridge (Czech Republic). There are different possibilities regarding the construction process for a stress ribbon superstructure, but if precast panels are used, the procedure is generally as follows: 1. The bearing tendons are drawn and anchored to the abutments/support structures; 2. The precast panels are hung from the bearing tendons. They can be erected one by one by a crane and placed on their specific locations directly, or, they can be hung on the sides and "slide" towards midspan. In any case, the sequence starts on the midspan’ panels and ends with the sides’ panels, achieving symmetry during construction; 3. The prestressing tendons are inserted and the deck is post-tensioned; 4. Concrete is poured in the joints between the precast panels.. Figure 2.15: DS-L Bridge conceptual design. Retrieved from [21].. Post-tensioned stress ribbon systems in long-span roofs. 21.

(44) CHAPTER 2. THEORETICAL BACKGROUND By the construction sequence, it is obvious that the stresses in the structure vary at each stage; before post-tensioning the prestressing tendons in the concrete, the structure acts as a cable system and all the dead load is taken by the bearing tendons. However, once the structure is post-tensioned, the dead load is "shared" by the bearing and the prestressing tendons, see Figure 2.15. Therefore, it is crucial to evaluate the structural behaviour during different construction stages , as well as the final service stage.. Figure 2.16: DS-L Bridge under construction. Retrieved from [21]. Figure 2.16 shows the DS-L bridge under construction, in the phase where few of the precast panels are hung from the bearing tendons. The reader is referred to [21] for more information about the construction sequences for non-precast segments or special cases. The paragraphs above are concentrated on the original system for stress ribbon bridges, nonetheless, there are other possibilities that allow to extend the use of structures, such as using stay cables, arch supports, external tendons, etc. The interested reader is referred to [21] for more information about those special arrangements.. 2.1.6. Prestress. Introduction Concrete is a material characterized by having high compressive strength but relatively low tensile strength, therefore, it is usually enhanced by an additional component to achieve a higher tensile resistance or by geometrical shapes reducing tensile stresses. This is done either by steel reinforcement or by compensating for the tensile forces by arching or prestressing [26]. This section is treating the principle of prestressing.. 22. Post-tensioned stress ribbon systems in long-span roofs.

(45) CHAPTER 2. THEORETICAL BACKGROUND Concrete structures have to be designed to fulfill crack requirements, SLS (Serviceability Limit State) deflection limits and attain satisfactory resistance under ULS (Ultimate Limit State). In many cases, all of the previous can be realized by only using reinforcing steel, however, according to [26], factors like the following restrict the use of reinforcing steel. • Increasing the amount of reinforcement doesn’t infinitely increase the loading capacity; high reinforcement ratios can lead to inadmissible brittle failure when the element is in bending. Additionally, it becomes practically difficult to introduce large amounts of reinforcement. • For long span beams, the bending moment increases considerably, and regular reinforcement steel is not enough to keep the tension stresses within the limits. • For long spans, the deflection have to be kept under certain values that are usually not met by using only reinforcement. Using the principle of prestressing can overcome the problems described previously for reinforced concrete. Moreover, according to [20] and [26], it results in many improvements to non-prestressed or mild steel-reinforced concrete, for instance: • less or nonexistent crack formation, which results in better corrosion resistance; • smaller deflections; • achieve more slender sections for the same span length, which leads to smaller dead load; • the possibility for longer spans, using the same depth of structural elements. However, the main drawback of prestessed concrete is the high cost; principally, due to the cost of the anchorage systems, but also processes as tensioning and grouting (when needed) requires intensive labour on-site [26]. The following three main systems are differentiated in prestressing. 1. Pre-tensioning: the tendons are stressed before the concrete is cast. When the concrete is hardened enough, the ends of the tendons are gradually released. Finally, the tendons are anchored by bonding to the concrete. 2. Post-tensioning, bonded tendons: concrete is cast first over the ducts containing the tendons. After hardening, the prestressing is applied by jacks and anchorages at the ends. Finally, a special grout in injected in the ducts to bond the tendons.. Post-tensioned stress ribbon systems in long-span roofs. 23.

(46) CHAPTER 2. THEORETICAL BACKGROUND 3. Post-tensioning, unbonded tendons: the same principle as for post-tensioned bonded systems, however, here the tendons are covered with grease or a bituminous material, which prevents corrosion and allows slipping between the tendon and the ducts. Thus, there is no force transfer between the concrete and the tendon along the duct, and the force is only transferred at the active and passive anchor points. The alignment of the tendons in the concrete cross-section plays an important role in the structural behaviour. If the tendon is placed at the neutral axis of the section, the tendon will only induce compressive axial forces on the concrete. However, if the tendon is placed with an eccentricity, not passing through the section’s neutral axis, it induces axial forces and a moment into the concrete section. All in all, the placement of the tendon in relation to the section’s neutral axis highly determines its effectiveness, and therefore, a tendon layout that counteracts the applied load is usually sought. Figure 2.17 shows examples of how different tendon layouts generate distinct equivalent forces.. ·. ·. φ. φ. α. ·. ·. ·. ·α. ·. Figure 2.17: Tendon layouts and correspondent prestress equivalent forces. Redrawn from [20].. Prestressing steel The steel used for prestressing must have specific characteristics that are not available in regular reinforcement steel. To fully realize the prestressing effect, the shortening of the concrete must be kept relatively small compared to the elongation of the prestressing steel, and that can only be achieved by high quality steel, which has. 24. Post-tensioned stress ribbon systems in long-span roofs.

(47) CHAPTER 2. THEORETICAL BACKGROUND a much higher strength (in the order of 900 to 2000 MPa) [26] allowing the required prestressing level to be reached. High quality steel is used to produce tendons made of bars, wires or strands (wrapped wires), however, the most common choice for the tendons is 7-wire strands. Prestressing steel is characterized by not having a clear "yield plateau", as shown in Figure 2.18. Therefore, the reference point for yielding is set to fp0,1k , defined as the stress that will cause a permanent deformation of 0,1% after unloading. The characteristic maximum stress in axial tension is denominated fpk [9], where: fpk fp0,1k = 0, 9 · fpk εuk = 3.5%. is dependent on steel grade recommended Eurocode value, if not specified by manufacturer [9]. Figure 2.18: Stress-strain diagram for prestressing steel. Retrieved from [9]. However, for design purposes, the Eurocode EN1992-1-1 [9] proposes using the simplification shown in Figure 2.19, where the design value fpd is used.. Prestress losses The main goal of prestressing is to counteract tensile stresses in the concrete by inducing compressive stresses, and that highly depends on the magnitude of the prestress force applied, which is considerably reduced due to force losses overtime. Eurocode 2 [9] divides the losses into two groups. First, immediate losses, which consists of elastic deformation of concrete, friction and anchorage slip. Secondly, time-dependent losses, which include losses due to creep and shrinkage of the concrete and relaxation of the prestressing steel. Each of those losses are explained in the following paragraphs.. Post-tensioned stress ribbon systems in long-span roofs. 25.

(48) CHAPTER 2. THEORETICAL BACKGROUND. Figure 2.19: Idealized and design stress-strain diagram for prestressing steel. Retrieved from [9]. Losses due to elastic shortening of the concrete When the tendon is stressed, an elastic shortening takes place in the concrete. When multiple tendons are to be stressed, the ones stressed early in the sequence will suffer a stress loss as consecutive tendons are stressed, which causes the concrete to further shorten [13]. During post-tensioning process, the first stressed tendon will suffer the highest loss, while no loss at all will be recorded for the last tendon stressed. However, for simplicity purposes, the design calculations are done for an average loss which applies to all tendons. According to Eurocode 2 [9], Section 5.10.5.1, the losses due to the instantanous deformation of concrete shall be calculated as follows: ∆Pel = Ap · Ep ·. X j · ∆σ(t) Ecm (t). (2.1). where: ∆σ(t) is the variation of stress at the centre of gravity of the tendons applied at time t j is a coefficient equal to (n -1)/2n where n is the number of identical tendons successively prestressed. As an approximation j may be taken as 1/2 Losses due to friction Friction between the tendon and the duct is usually significant when the tendon follows a curved profile. Since the tendon will be in contact with the sides of the duct when stressed, the force is opposed by friction, which leads to a loss of prestressing force [13]. Friction due to curvature of the tendon is considered in design by the. 26. Post-tensioned stress ribbon systems in long-span roofs.

(49) CHAPTER 2. THEORETICAL BACKGROUND friction coefficient (µ), which is usually available from manufacturers. Furthermore, friction losses can also occur at straight tendons due to possible local misalignment of the duct causing a contact between the duct and the strands. This is considered in design by the wobble coefficient (k), which is also given by the manufacturer. The loss in force due to friction can be calculated as follows [9]: ∆Pµ (x) = Pmax · (1 − e−µ·(θ+k·x) ). (2.2). where: Pmax µ θ k. is is is is. the the the the. initial prestress force applied friction coefficient sum of angular displacements over the distance x wobble coefficient. Alternatively, some textbooks [13] use a different representation of Formula 2.2: ∆Pµ (x) = Pmax · (1 − e−µ·θ+k. ∗ ·x. ). (2.3). where the adjusted wobble coefficient k ∗ is the product of the original wobble coefficient k and the friction factor µ. Losses due to anchorage slip This loss represents the reduction in the prestressing force due to the loss in the stretched length of the strands, which is taking place prior to seating of the anchorage gripping device. The change in tendons’ length, which depends on the anchorage systems used, is a specific value given by the manufacturer, but it is usually not more than 7 mm [13]. Losses due to concrete creep Concrete subjected to compressive stresses is prone to shortening due to creep effects. Consequently, concrete strain will change, changing the tendons’ strain and reducing the force. The concrete creep depends on the ambient relative humidity, the dimensions of the concrete element, the concrete composition and the age of the concrete when the element is loaded. Eurocode 2 [9], Section 3.1.4 provides guidelines to determine the creep factor ϕ(t, t0 ). Losses due to concrete shrinkage When the concrete shortens due to shrinkage, it induces a corresponding shortening on the prestressing tendons. As the tendons are less stretched, an equivalent stress reduction will take place.. Post-tensioned stress ribbon systems in long-span roofs. 27.

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