Fulltext01.pdf

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

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

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

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

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

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

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

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

CONTENTS B Prestress losses

119

C Initial cable geometry calculation in SAP2000

127

xii

Post-tensioned stress ribbon systems in long-span roofs

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

CHAPTER 2. THEORETICAL BACKGROUND Eurocode 2 divides the shrinkage effects into two parts: drying shrinkage and autogeneous shrinkage [9]. Drying shrinkage has a slower process as the water leaves the concrete after hardening, while autogeneous shrinkage develops during the first days after casting as a result of water consumption for the hydration process [26]. The main factors that affect shrinkage are: relative ambient humidity, concrete class, cross-sectional shape and age of the concrete. Eurocode 2 [9], section 3.1.4 provides guidelines to determine the total shrinkage strain εcs . Losses due to steel relaxation Relaxation is defined as a loss of stress for the same applied strain (constant elongation). Regarding the relaxation of prestressing steel, it is a long-term phenomenon that depends on the magnitude of initial prestress force, the steel properties and the temperature. Eurocode 2 [9] bases the calculation of prestress losses on ρ1000 , which is the relaxation loss (in %) at 1000 hours after tensioning and at a mean temperature of 20◦ C. Steel relaxation reduces the initial prestressing force over time, however, up to 50% of the loss can occur within the first 24 hours. Relaxation losses from temperature are to be considered only for high temperatures, while for normal temperature ranges the effect is rather small and can be disregarded [13]. Manufacturers may define the steel as ordinary strands, low relaxation strands or hot rolled process bars. Different formulas are then used to calculate the relaxation loss (see Eurocode 2, Section 3.2.2(7)[9]). Sum of the long term losses Eurocode 2, Section 5.10.6(2) provides a simplified method to calculate the time dependent losses in the prestress force:

∆Pc+s+r = Ap ·

28

εcs · Ep + 0.8 · ∆σpr + 1+

Ep Ecm

·

Ap Ac

· (1 +

Ac Ic

Ep Ecm

· ϕ(t, t0 ) · σc,Qp

2 ) · [1 + 0.8 · ϕ(t, t )] · zcp 0

(2.4)

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 2. THEORETICAL BACKGROUND where: ∆Pc+s+r εcs ∆σpr ϕ(t, t0 ) σc,Qp Ep Ecm Ap Ac Ic zcp

2.2 2.2.1

is the absolute variation of the prestress force due to the creep, shrinkage and relaxation losses is the absolute value of the concrete shrinkage strain is the absolute value of the variation of stress in the tendons due to the relaxation of the prestressing steel is the creep coefficient is the stress in the concrete adjacent to the tendons, due to self-weight and initial prestress and other quasi-permanent actions where relevant is the prestressing steel modulus of elasticity is the concrete modulus of elasticity is the area of the prestressing tendons is the area of the concrete section is the concrete section second moment of area is the distance between the center of gravity of the concrete section and the tendons

Finite element method Introduction

A short introduction about the Finite Element Method (FEM) is given in this section, based on the book "Concepts and Applications of Finite Element Analysis" [11]. Finite Element Methods (FEM) are used to solve field problems which usually are complex and can’t be solved, or would be very time consuming, by means of analytic equations. Thus, FEM is used to find an approximate solutions to field problems by using systems of algebraic equations. In FEM, a structure is divided into finite, not differential, elements. Such a division, which has a specified arrangement, is named mesh, and all mesh-elements are connected to each other at nodes. The assembly of all elements forms the finite element model. Therefore, the main concept behind the method is that the system is divided in smaller parts, named elements, which are formulated in the same way (generating local stiffness matrices). The elements are then assembled (generating the global stiffness matrix) in the finite element structure which is used to solve the complete equation system, for which the unknowns are solved (displacement field for the nodes). Based on the displacement field, other unknowns such as forces and stresses can be found. Essentially, the method consists of an approximation with elementby-element interpolation of field quantities.

Post-tensioned stress ribbon systems in long-span roofs

29

CHAPTER 2. THEORETICAL BACKGROUND The key of this method is to select the elements with relevant properties, which are usually included in commercial FEM software. The finite elements treated in this thesis are defined in Section 2.2.3.

2.2.2

SAP2000

SAP2000 is a commercial FEM software package from Computer and Structures Inc., which is used for structural analysis and design of general purpose structures [1]. The software offers a graphic interface in both 2D and 3D to generate the structure by introducing its geometry, specifying its material and section properties (from the built-in library or by starting it from scratch), defining the loads and load cases applied and determining the analysis parameters. This constitutes the assembled model which is to be analyzed by the analysis engine. SAP2000 has an integrated analysis engine (SAPfire) which transforms the assembly into a finite element model by meshing it. Different analysis alternatives are possible, such as: static, dynamic, linear, non-linear (material non-linearity or geometric nonlinearity), time dependent (creep, shrinkage and staged construction) and buckling analysis [30]. The analysis types used in this project will be described further in Section 2.2.4. The output of the analysis is displayed in the graphic interface through 3D visualizations of deflections and contours of forces and stresses, or through tables and graphs for specified locations and section cuts. Furthermore, it has a user-friendly interface to visualize the output for each relevant case.

2.2.3

Element types

This section intends to give the reader a general description and an outline of the elements’ types used in this thesis: frame, cable, shell, tendon and links/supports elements.

The frame element A frame element can be described in 2D or 3D. Since this thesis deals with a spatial structure, only 3D frame elements are considered here. This type of element is defined by a line which is delimited by two nodes, and each node allows six degrees of freedom (translations and rotations in the three directions), as illustrated in Figure 2.20. In these elements, the axial dimension (parallel to the longitudinal axis) is predominant when compared with other dimensions in the orthogonal direction [7].

30

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 2. THEORETICAL BACKGROUND

Figure 2.20: Degrees of freedom for a 3D frame element.

A frame element is usually formulated using Euler-Bernoulli theory, Timoshenko beam theory, or sometimes with other general formulations. The main difference between the first two is that, with Euler-Bernoulli theory, the beam section remains plane and normal to the beam axis after deformation (it doesn’t allow shear deformation), thus, it’s more precise for slender beams; as it better represents bending. However, with Timoshenko theory, the beam section is not perpendicular to the beam axis after deformation, introducing shear deformations and, therefore, gives more accurate results for non-slender beams where shear deformations are not negligible [11], at the cost of less accuracy in representing bending. SAP2000 uses a general formulation for frame elements which includes biaxial bending, torsion, axial deformation and biaxial shear deformations. This formulation can be found in [4]. But essentially, since it accounts for shear deformations, it is similar to the Timoshenko theory. This formulation allows the user to assign the frame element for different structural elements, such as: columns, beams, trusses, grillages and cables. Some of those might need additional modifications to obtain more precise results. Property modifiers can be applied to the axial, shear, torsional and bending stiffnesses, besides mass and weight of the section. It allows the user to reduce those properties by a specific ratio [1]. Besides directly specifying property modifiers to an element, the user can also assign them using a "Named Property Set". A Named Property Set includes the six factors mentioned earlier: section stiffnesses, mass and its weight. It’s important to notice that, when Named Property Set is applied to certain elements at a specific stage during a staged construction analysis, it does not alter the structure response up to that stage, but only affects following stages [1]. Other modelling options such as tension/compression limits, end releases, tapered section and joint offsets are also available in SAP2000. The reader is referred to SAP2000 Reference Manual [1] for more detailed descriptions.

Post-tensioned stress ribbon systems in long-span roofs

31

CHAPTER 2. THEORETICAL BACKGROUND

The cable element A cable is defined as a nonlinear element which can only transmit tension forces in the direction perpendicular to its cross-section. It can be seen as a tension-only truss element with nonlinear behaviour, where its self-weight is contributing significantly to the tension forces. Different FE software have diverse methods to deal with the cable element: some of them include a specific formulation, while in many others, the user needs to input specific modifications to truss or beams elements. In the case of SAP2000, cable elements use an elastic catenary formulation to calculate the cable behaviour under its own dead-load and external loads [1]. Its nonlinear behaviour can include both tension-stiffening and large-deflection effects. The element is defined by three translational degrees of freedom on each node, but all rotational degrees of freedom are deactivated. After defining the two main nodes of the element (i and j), SAP2000 provides a shape calculator (also known by: Cable Layout form or Cable Geometry Window, shown in Figure 2.21), the function of which is to find the undeformed shape of the cable by defining different parameters. This provides different starting options for the cable shape, such as specifying the undeformed length, the vertical sag, the horizontal tension component, the axial tension component, etc. By specifying one of these parameters, the software calculates the others based on linear equilibrium. The cable undeformed geometry and definitions are graphically summarized in Figure 2.22. Annex C includes the formulas that SAP2000 uses to calculate those parameters.

Figure 2.21: Cable Layout form in SAP2000, including the starting input parameter options.

32

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 2. THEORETICAL BACKGROUND For slack cables, which are not stable before loading and don’t have a particular position, the program assigns a very small dead load in order to obtain the initial shape. In the case of taut cables, the initial strain or load defines an initial shape.

Figure 2.22: Cable definition in SAP2000. Retrieved from [1]. Unless specific point loads or connecting elements are to be placed between the cable’s ends, it is not necessary to mesh or discretize a cable element, since the program inherently does that when using the cable catenary formulation. Additionally, SAP2000 provides the option of modelling the cable element using frame element formulation. This is recommended when there are highly variable loads or material non-linearity [1]. When frame elements are utilized to model cables, it is important to include large-deflection nonlinear analysis with P-delta, in contrast to cable elements, where those features are automatically activated by the software. Moreover, other property modifiers such as setting the compression limits to zero and reducing the bending stiffness might be needed, which will be discussed further in Section 4.1. Discretizing the element in at least 8 elements is needed to capture the shape variation and simulate cable behaviour [1]. Finally, it’s important to note that, self-weight of cable elements depends on the length; as the cable stretches, the magnitude of dead load per unit weight is reduced, so that the total load remains constant [1].

Shell elements A shell is characterized, according to [11], by a surface that has a small thickness compared with its overall dimensions, and it is defined by its mid-surface and an associated thickness. Shell elements carry load by the combination of membrane stresses and plate bending stresses.

Post-tensioned stress ribbon systems in long-span roofs

33

CHAPTER 2. THEORETICAL BACKGROUND Like for frame elements, there are two important distinctions in shell elements; facet (plane) or curved shells, and Kirchhoff or Mindlin theories. Facet shells don’t include coupling between the membrane and bending effects, while curved shells do. Kirchhoff theory is used for thin shells and neglects shear deformation, while the Mindlin theory is used for thick plates and includes the effect of transverse shear deformation. The combinations of geometry and theory allow many shell formulation in the software, resulting in a wide range of possibilities. Regarding the use of shell elements in SAP2000, taking what is stated in [1] as a reference, a shell element includes the following modeling alternatives. • Membrane: pure membrane behaviour. Only applicable for linear and homogeneous material. • Plate: pure plate behaviour, thick or thin formulation. Applicable for linear and homogeneous material. • Shell: combination of membrane and plate behaviour, thick or thin plate formulation. Applicable for linear and homogeneous material. • Layered: available option for adding multiple layers (defining each layer’s material, behaviour and location), linear or nonlinear behaviour (only for thickplate formulation). Each shell element has its own local coordinate system which is defined by the user’s drawing direction (clockwise or counterclockwise), however it can be changed on a later stage of the modelling by defining a new local axis. It is important to utilize a consistent set of local axis directions to understand the output of the analysis, specially when looking at stresses in the top or bottom faces of the shell. From the different modeling alternatives, the layered shell is the most complex one; where membrane deformations use the strain-projection method and the drilling degrees of freedom are not used. Moreover, the rotations perpendicular to the plane are just loosely tied to prevent instability. Furthermore, layered shells use a Mindlin formulation to simulate the plate behaviour, and thus, it includes transverse shear deformations. Essentially, using layered shells is important when modelling material nonlinearities or when different concrete layers need to be combined in the same element. The relevance of shear deformations starts to be important when the thickness of the shell is exceeding 1/10 to 1/5 of the shell width. Nevertheless, [1] recommends to use thick-plate formulation to all cases, unless the mesh is very distorted; the thick-plate formulation has higher sensitivity to mesh distortion.

34

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 2. THEORETICAL BACKGROUND An interesting feature of shell elements in SAP2000 is the way a uniformly distributed gravity load is transferred between the shells and its adjacent elements. Unless specific measures are taken, the load will be transferred from the field to the nodes which have a support, through membrane and plate action. However, in cases when a single span distribution is wanted, the load should be applied using the option "Uniform to Frame (shell)". This gives the user the possibility to specify the span direction of the shell. Other useful modelling options in SAP2000 regarding shells include: property modifiers, edge constraints, joint offsets and thickness overwrites. The reader is referred to SAP2000 reference manual [1] for further details regarding these features.

Tendon elements The tendon element can be modelled as frame element with specific modifications. This section will only describe how this type of elements is used in SAP2000, based on the description in [1]. Tendons in SAP2000 are objects which are attached to other objects to represent prestressing and post-tensioning by imposing a load. Two main design possibilities are given in SAP2000 when modeling tendons. 1. Tendon as an element: • treated as an independent finite element with its own stiffness, mass and loading; • allows the user to include the losses due to elastic shortening and timedependent effects; • are used when nonlinearities of the tendon’s material or geometry are to be considered; • gives the option of displaying the tendon forces as an output. 2. Tendon as a load: • simulates the prestressing by an external load that applied to the model, no physical objects will be present; • are suitable for linear analysis, when the losses from elastic shortening and time-dependent effects are known in advance, or not considered at all; • do not give tendon forces are not available as an output. Moreover, modelling tendons as elements allows the following design alternatives, for which [36] recommends specific modifications to obtain a realistic results. • Pre-tensioned tendons: friction and anchorage losses are set to zero, as recommended in [36].

Post-tensioned stress ribbon systems in long-span roofs

35

CHAPTER 2. THEORETICAL BACKGROUND • Post-tensioned tendons, bonded: behaviour obtained by default. Staged construction is needed to model a more accurate stiffness before the tendon is post-tensioned. • External post-tensioned tendons: model the tendon outside the material domain and connect with deviator blocks. Staged construction, explained in the following Section, is needed to model a more accurate stiffness before the tendon is post-tensioned. • Post-tensioned tendons, unbonded: modelled as external tendons. In order to draw the tendon with a specific geometry, SAP2000 provides different options, such as: straight, parabolic or circular tendon, where point coordinates can be specified to define the shape. A tendon is connected automatically to any element it passes through, using the concept of a "bounding box" that considers the thickness. It uses the discretization points of the tendon and the nodes of the housing element to bound the two objects together. The tendon cross-section is always circular and has axial, shear, bending and torsional properties, resulting in six degrees of freedom (as a frame). However, the axial behaviour is the dominant and the other properties are mainly used by the software to provide stability. Prestress loading can be applied to the tendons, as shown in Figure 2.23, where the jack location, force magnitude and different type of losses are specified.

Figure 2.23: Tendon load form in SAP2000.

36

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 2. THEORETICAL BACKGROUND

Staged construction This type of analysis is also known as sequential construction, segmental construction, or incremental construction. It allows the user to define a certain order of stages in which one, or all, of the following options can be applied to each step: • a duration, in days: necessary to include time-dependent effects. Can be set to zero when time-dependent effects are to be ignored; • addition or removal of any number of structural objects; • application of specific load patterns to a specified structural objects; • variation of section properties of frames, shells, tendons, and link/support objects; • application of property-modifications to frame elements or shell elements. To facilitate the process of applying these options, it’s useful to divide the complete model into smaller groups, based on the modifications needed at later point in the analysis. SAP2000, automatically, creates a group called "ALL" which includes the whole structure. If one stage includes more than one of the aforementioned options, they will be applied according to the following order: 1. the objects to be added, if any, are handled; 2. the objects to be removed, if any, are deleted; 3. section properties are changed, if any; 4. assigned loads are applied gradually. However, load applied through displacement control is not allowed.

Link/support elements According to [1], properties for link elements and support elements are defined similarly. Both elements are composed of six "springs" representing the six degrees of freedom (two bending, torsion, axial & two shear), see Figure 2.24. However, the main difference between the two elements lies in their connectivity options: • A link element connecting two joints, I and J; this option allows for two joints to share the same position in space creating a zero-length element. • A support element connecting a single joint, J, to the ground.

Post-tensioned stress ribbon systems in long-span roofs

37

CHAPTER 2. THEORETICAL BACKGROUND The link/support element can be used to simulate many nonlinear behaviours that including, but not limited to: • gap/compression only and hook/tension only; • damping effects; • multi-linear elasticity; • multi-linear plasticity; • hysteretic (rubber) isolators.

Figure 2.24: Three of the six independent spring hinges in a link/support element. Retrieved from [1]. The orientation of the elements’ local coordinates depends on how they are modelled. It’s important to realize that the direction of the local axes highly affect the forcedeformation properties and might alter the expected outputs. Possible outputs using these elements include the deformation across the element, and the internal forces at the joints of the elements. For each degree of freedom, a linear and nonlinear effective-stiffness can be defined. During a non-linear analysis, the nonlinear force-displacement relations are applied at all degrees of freedom for which a nonlinear behaviour is specified. For the remaining degrees of freedom, the linear effective-stiffness is used even if a nonlinear analysis is run. SAP2000 Reference Manual [1] recommends a value of this effective-stiffness to be 102 -104 times the higher stiffness of the connected elements. However, it’s not recommended to use unreasonably large numbers, which may lead to numerical difficulties.

38

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 2. THEORETICAL BACKGROUND Gap property Gap element is a compression-only spring, used to link two elements that can transfer compression to one another but not tension. Mainly used to model adjacent structures and their effects on each other during earthquakes, they can also be used to describe prefabricated panels lying next to each other. SAP2000 provides an option to control when the spring is activated; the defined stiffness only comes into effect when the deformation is less than the defined separation (physical gap between connected elements). Using gap elements allows the user to specify independent gap properties for each one of the six deformation degrees of freedom; in other words, the opening or closing of a gap for one degree of freedom doesn’t affect the response or behaviour of other degrees of freedom.

Figure 2.25: Different types of link/support elements in SAP2000. Retrieved from [1].

The nonlinear force-displacement relation is given by the following equation:  k · (d + open) if (d + open) = 0 f= (2.5) 0 otherwise

2.2.4

Analysis types

SAP2000 uses "Load Cases" to determine how a specific load is applied, and the kind of structural response requested for that loading type. The various types of analysis can be categorized in two main groups, namely: linear and nonlinear analysis. The results using the former type can be superposed; because they will be calculated using the same stiffness, but that’s not applicable for the latter. Therefore, when running a nonlinear analysis, the loads acting on the structure must, either, be applied all in the same loading step, or, preferably, be chained together and applied one after another to represent a more realistic load application and obtain more accurate results that capture the continuously varying stiffness of the structure.

Post-tensioned stress ribbon systems in long-span roofs

39

CHAPTER 2. THEORETICAL BACKGROUND When applying the loads in sequence, a load case can be either "dependent": which is a load case that depends upon another case taking place first. Or "prerequisite": which is the load case that must occur before a certain "dependent" case. Both Tables 2.5 and 2.6 illustrate some principal differences between the two analysis categories. Table 2.5: Types of analysis in SAP2000. Linear analysis

Nonlinear analysis

Static analysis Static analysis Modal analysis Time-history analysis Moving load analysis Buckling analysis Steady state analysis Response-spectrum analysis

Table 2.6: Differences between linear and nonlinear analyses. Linear analysis

Nonlinear analysis

Structural properties

Constant throughout the analysis

May vary with time, deformation and loading

Initial conditions

Analysis starts with zero stresses

Results of previous nonlinear analysis if existing

Structural response

Proportional to the applied loads

Results are not proportional to the applied loads (Non-zero initial conditions and varying properties)

Superposition Applicable Not applicable The term "Structural properties" here refers to: Stiffness, damping, etc. The term "Results" here refers to: Displacements, stresses, forces, etc.

Nonlinear static analysis Being the most relevant for this thesis, an extended explanation is given for this type of analysis. The practical applications of this type of analysis include: • P-delta and large-displacements analysis; • staged construction analysis, which allows all time-dependent effects; • tension-only bracing; • cable structures; • static pushover test;

40

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 2. THEORETICAL BACKGROUND • any analysis that requires geometric or material nonlinear behaviour; Load application control Using a nonlinear analysis, the load application can be either load-controlled or displacement-controlled. The former is used when the final magnitude of the applied load is known, and the structure is expected to withstand it, while the latter is more suitable when the final deformation is known, but not the corresponding load, or when instabilities are expected in the response. Displacement-controlled loading is typically used for static pushover analysis and snap-through buckling analysis, in other words, when softening behaviour for the load-displacement curve is expected. Nonlinear solution control The assigned load is applied gradually, using as small steps as needed to ensure the convergence to equilibrium. At each step, the nonlinear equations are solved iteratively. The iteration process is continued till the solution converges. If a certain, specified, number of iteration per step is reached without converging, the step is further divided into sub-steps and the iteration process starts again. The step converges when the difference between the internal forces, calculated at the nodes, and the applied external load is less than a certain convergence tolerance set by the user. For large-displacements analysis, such as for cable structures, small values for the convergence tolerance are recommended. Two solution procedures are used in SAP2000. Constant-stiffness iteration is tried first, but, Newton-Raphson iteration is used if convergence fails. If both fail, the step is further divided and the process is repeated. Newton-Raphson iterations are more effective when it comes to cables structures and large deformations, however, each constant-stiffness iteration is faster; the overall efficiency highly depends on the properties of the problem. Figure 2.26 presents graphically the difference in the iteration process between Newton-Raphson (a) and constant stiffness (b), for a softening case.

Figure 2.26: Nonlinear solution controls shown for a one-dimensioned case.

Post-tensioned stress ribbon systems in long-span roofs

41

CHAPTER 2. THEORETICAL BACKGROUND To increase the efficiency of the solution, SAP2000 has a feature called "Line search option". It’s an iteration option that allows the user to scale the solution increment, in a trial-and-error fashion, to find the smallest unbalance. Although this option increases the computational time for each iteration, it often results in better convergence behaviour with less iterations. According to [1], this option is particularly effective with stiffening systems, such as cable structures and when modelling closing gaps.

42

Post-tensioned stress ribbon systems in long-span roofs

Chapter 3 Case Study: Västerås Travel Center 3.1

General description

Västerås is a city in the province of Västmanland, in central Sweden. It has a population of around 150.000 inhabitants, which has increased exponentially on the last 30 years as a result of becoming a commuting center to Stockholm or other municipalities around Lake Mälaren region. Considering the development of the city, a new travel center is needed to accommodate the current population demands. The new travel center will consist of the rebuilding of the old central station and the area around it, to ensure higher capacity and easier accessibility. Even though the project involves a big area around the old station, the case study for this thesis is only dealing with the roof structure of the main station building. The design of the roof is a joint venture between the Danish architect company BIG and the Swedish company Tyréns. Figure 3.1 shows the roof (rendered with a metallic cladding) and the area around it. The new travel center allows the infrastructure to be a welcoming point for commuters and visitors. Moreover, through the emblematic architecture of roof structure, it is intended to become a landmark for the city of Västerås. Figure 3.2 gives an idea of how one of the impressive roof cantilevers will be seen from one of the access ramps. From the architectural point of view, the roof aims to be "a continuous thin sheet gently lifted at its four corners, wrapping the city’s vehicular infrastructure in multiple layers of public program and urban spaces" [38]. The roof consists of three drapes connected through two big corner cantilevers, forming a "U" shape. In some areas the building is closed by glass facades at its edges, but in some other parts it is just open and serves as a rain shelter for people waiting next to the train tracks.

Post-tensioned stress ribbon systems in long-span roofs

43

CHAPTER 3. CASE STUDY: VÄSTERÅS TRAVEL CENTER

Figure 3.1: Rendering of the development area. Retrieved from [37].

Figure 3.2: Rendering of the roof. Retrieved from [37].

This project is currently under preliminary study to investigate the possibility of building such an enormous and challenging project. This thesis will handle the design of the main drape by investigating the applicability of various structural systems and study their differences and the advantages and disadvantages of each. To have a clear picture of the roof structure, it is divided into the following parts, whose locations are sketched in Figure 3.3:

44

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 3. CASE STUDY: VÄSTERÅS TRAVEL CENTER • Drape structures: a cable suspended structure which supports a precast concrete cladding, resulting in a continuous roof. Characterized by a funicular shape which is found by its self-weight. The drapes are supported only at the ends, lacking of intermediate span supports. 1. Main drape: 215 m 2. East drape: 93 m 3. West drape: 12 5m • Shell corner cantilevers: two concrete folded plates, supported only on the corner support structures. Approximately 50 m of free cantilever. • Shell ends: concrete folded plates at the end of the east and west drapes. • Support structures: located under the shell corner cantilevers and the shell ends. Made of a steel frame with the shape as shown in Figure 3.4.

Figure 3.3: Sketch of the roof by BIG Architects.

Post-tensioned stress ribbon systems in long-span roofs

45

MEGASTRUCTURE -V1

CHAPTER 3. CASE STUDY: VÄSTERÅS TRAVEL CENTER

Figure 3.4: Architectural concept for the support structures by BIG Architects. BIG

VTC VÄSTERÅS TRAVEL CENTER

Figure 3.5 shows a 3D rendering of the roof, which doesn’t include the concrete panels that will cover all the three drapes.

Figure 3.5: 3D rendering of the roof superstructure by Tyréns.

In relation to the roof’s geometry, the corner structures have different heights, which means that the drapes start and end at different elevations. Additionally, a constant curvature is used along the east and west drapes, however, they are both tilted towards the center of the building, in other words, the cables at one end of each drape are anchored at different elevations. The complete roof structure forms a highly-complex system with many challenges for the designers, specially to fulfill the architectural demands. This thesis won’t go through all the parts of the roof structure mentioned before, but will instead focus only on the structural design of the main drape.

46

Post-tensioned stress ribbon systems in long-span roofs

2018.03.22

CHAPTER 3. CASE STUDY: VÄSTERÅS TRAVEL CENTER Looking only at the shape and dimensions, the main drape presents many structural challenges. With a span longer than 200 m, and a width variating between 7 and 20 m, it has to be stiff enough to withstand its self-weight and transmit the external loads to the supports with a limited allowable deflection and an acceptable dynamic behaviour. For the sake of simplicity during construction stages, the three drapes that form the roof structure are designed so they are able to be in equilibrium individually, even though they are assembled together. For instance, the main drape should be able to stand by itself without any help from the shell cantilever corners, even though they both share the shell corner structure as a support. This means that it is possible to study the structural behaviour of the main drape separately.

3.2

Main drape

The main drape is planned to consist of three bearing cables which are pulled together by crossbeams in order to define the architectural shape on the plan view. It can be seen as a self-balancing cable system which achieves continuity by a concrete cladding that distributes the self-weight plus any external loads to the rest of the system. The three bearing cables are anchored on the steel frame forming the support structures. Figure 3.6 shows a rendering of the main drape where the three main cables and the crossbeams can be seen, but the concrete panels are not included.

Figure 3.6: 3D rendering of the main drape, without the concrete panels. Given the geometry of the main drape, it is noted that it resembles more a bridge than a common roof structure, and therefore bridge structural systems are considered. Figure 3.7 shows the principal dimensions of the main drape studied in this thesis and the elevation of the anchorage points.

Post-tensioned stress ribbon systems in long-span roofs

47

CHAPTER 3. CASE STUDY: VÄSTERÅS TRAVEL CENTER

+11,1m

215m

+11,1m

5,4m

20m

Figure 3.7: 3D rendering of the main drape, including its main dimensions.

In a preliminary stage of a project, it is essential to study and examine different options that will lastly provide an efficient and effective structural design. With the background given by the literature review and the input from the engineering project team in Tyréns, two structural alternatives are examined for the main drape: a cable suspended roof with precast concrete panels and a stress ribbon system.

3.2.1

Cable suspended roof with precast concrete panels

This is a fairly simple structural system where the concrete panels act as a cladding with the function of distributing the external loads to the crossbeams. Therefore, the system consists of 3 main cables supporting 33 crossbeams and the concrete panels are simply supported along the crossbeams. The number of crossbeams is chosen to have a reasonable span length of the concrete panels. With this system, the concrete panels act like plates in bending and distribute the load to the adjacent crossbeams, with a longitudinal spanning direction, without adding any stiffness through membrane action since they are not continuous. However, it contributes significantly when it comes to dynamic analysis due to its mass. A simplification of this system is presented in Figure 3.11 (a).

3.2.2

Stress ribbon system

This structural system follows the principles used for stress ribbon bridges; having external bearing tendons which will support precast concrete panels placed on top of the cables. Those concrete panels will be prestressed by post-tension tendons embedded on the panels, and anchored to the shell’s ends as shown in Figure 3.8. In this case, the crossbeams have exclusively the function of keeping the curved shape of the main cables on the plan view, but they don’t carry any load from the panels.

48

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 3. CASE STUDY: VÄSTERÅS TRAVEL CENTER

Figure 3.8: Simplification of how the tendons (in red) are only anchored in the concrete shell.

In order to see the effect of the post-tensioning and its contribution to the overall behaviour, this systems will be examined in two situations: with and without posttensioning tendons. A simplification of this system is presented in Figure 3.11 (b) and (c). The tendon’s layout shown in Figure 3.11 (c) is chosen to have uniform compression in the concrete after post-tensioning.

3.3 3.3.1

Materials and components Bearing cables

The bearing cables are made of full-locked coil ropes. This type of cables guarantees high bearing capacity since the outer layer, which is made of Z-shaped wires, interlocks the inner wires [31], as shown in Figure 3.9.

Figure 3.9: Full-locked coil rope. Retrieved from [31].

As described in the brochure from the manufacturer Fatzer [31], the material and section properties are as follows: Table 3.1: Cable material and section properties. Elastic modulus (GP a)

160

Weight (kN/m3 )

76,97

Nominal diameter (mm)

200 (3x115 mm bundle)

Design stress (M P a)

866

Characteristic breaking load (M N )

3x13,40 = 40,2

Design load (M N )

3x8,99 = 27

Post-tensioned stress ribbon systems in long-span roofs

49

CHAPTER 3. CASE STUDY: VÄSTERÅS TRAVEL CENTER According to the architectural proposal, the roof is designed as a very slender structure which, in turn, limits the size of cables to be used. However, the cables are expected to resist considerable forces, which requires large diameters; defying the architectural design. To solve this issue, the bundle of cables shown in Figure 3.10 is used, increasing the effective area while keeping the total thickness within an acceptable range.

Figure 3.10: Arrangement of the bearing cables in 3x3 bundle.

3.3.2

Concrete

Due to the overall dimensions of the structure and the high forces that it has to withstand, high strength concrete is needed. The concrete quality is C60/75 with properties, according to Eurocode 2 [9], shown in Table 3.2. Table 3.2: Concrete properties.

3.3.3

Concrete class

C60/75

fck (MPa)

60

fctm (MPa)

4,4

Ecm (GPa)

39

Post-tension tendons

The tendons’ specifications are summarized in Table 3.3. They are chosen according to the brochure from VSL manufacturer [39]:

50

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 3. CASE STUDY: VÄSTERÅS TRAVEL CENTER Table 3.3: Tendon properties. Nominal strand diameter (mm)

15,7

Nominal strand cross section, An (mm2 )

150

Characteristic proof strength, f01k (MPa)

1640

Characteristic tensile strength, fpk (MPa)

1860

Young’s modulus, E (GPa)

195

Unit Plastic duct external diameter (mm) Weight (kN/m3 ) Minimum recommended concrete thickness (mm)

6 strands 63 76,97 200

A critical factor when selecting the tendon’s size is, in this case, the minimum recommended concrete thickness, which is referred here as the total thickness of the concrete shell. Specially for the anchoring zones, the concrete must be thick enough to accommodate the local spiral reinforcement placed around the anchoring systems. To minimize the friction losses, plastic ducts, which have the lowest friction coefficient according to [39], are used to accommodate the tendons running through the concrete panels.

3.3.4

Crossbeams

The crossbeams are standard sections, HEA500, with steel quality S355. As previously mentioned, in the cable suspended system, the crossbeams are the support for the concrete panels, however, for the stress ribbon, its only function is to keep the curved plan shape of the bearing cables.

3.4

Loading

The roof structure must primarily be designed for the following loads: dead load, snow load and wind load. However, in the stage of preliminary design, only the following loads will be considered. • Dead load: only from the superstructure (cables, concrete panels, crossbeams and prestressing tendons). Magnitude depending on each material density. • Snow load: with a value of s = 1, 28 kN/m2 , calculated as follows [8]:

Post-tensioned stress ribbon systems in long-span roofs

51

CHAPTER 3. CASE STUDY: VÄSTERÅS TRAVEL CENTER

s = µi · Ce · Ct · sk = 0, 8 · 0, 8 · 2 = 1, 28 kN/m2

(3.1)

where: µi is the snow load shape coefficient, here taken as 0, 8 [8] Ce is the exposure coefficient, taken as 0, 8 for a windswept topography according to Table 5.1 in [8] Ct is the thermal coefficient, taken as 1 according to Section 5.2(8) in [8] sk is the characteristic value of snow load on the ground, taken as 2 kN/m2 , according to Annex C in [8] and the Swedish National Annex [5] Wind effects to be considered in such structures are natural undamped frequencies, and galloping instabilities. Moreover, precise evaluation of stiffness, mass and damping is required for accurate wind load analysis. The aerodynamics of the crosssectional shape of the cladding covering the drape can also highly affect the wind analysis. However, due to the associated complexity, wind loads will not be considered at this stage. With the roof geometry, the Eurocode specifications are not enough, and a wind tunnel test is necessary to determine the wind loads acting upon the roof. This is considered out of the scope of this thesis.

52

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 3. CASE STUDY: VÄSTERÅS TRAVEL CENTER

Figure 3.11: The three different structural systems considered.

Post-tensioned stress ribbon systems in long-span roofs

53

Chapter 4 Investigation of modelling options in SAP2000 Different verification examples related to distinct modelling capabilities in SAP2000 are included in this section. The intention is to give the reader an outline of how different elements and modelling options can be implemented in the software and what are the consequences of each alternative. Very simple models have been used, which reduce the complexity, focusing on the target of the verification. For instance, the middle cable is ignored and the number of crossbeams is reduced.

4.1

Cable vs. frame elements

The element that governs the behaviour of this structure is a cable. As previously mentioned in Section 2.2.3, cable elements can be treated as a special form of frame elements, for which, geometrical non-linearities must be included. Although cable elements are available in SAP2000, the authors suggest using modified frame elements instead for the following reasons. • The formulation of a cable element, or the assumptions that are made by the software, are not well defined by the software manual [1], and includes some assumptions, which might be correct and relevant, but are unknown. • Given the size of the designed structure, the expected cable cross-sections are large enough to create some bending stiffness, which is ignored when using cable elements. • SAP2000 provides more modelling tools for frame elements than for cable elements. • The output alternatives are more complete using frame elements, such as bending moment diagrams along the elements.

Post-tensioned stress ribbon systems in long-span roofs

55

CHAPTER 4. INVESTIGATION OF MODELLING OPTIONS IN SAP2000

4.1.1

Cable vs. frame elements applied to a 2D case

The first investigation of the suitability of using frame elements instead of cable elements is carried out as follows: a simple supported cable is modeled, as shown in Figure 4.1, using both cable and frame elements. Both results are then compared with an analytical solution given by [16].

Figure 4.1: Initial geometry and loads of the cable and frame model. The curved frame generator in SAP2000 is used to model the structure by frame elements with an initial sag on the undeformed shape (see Figure 4.1), which allows to draw a parabolic shape given three points and divide the frame into small segments (80 segments in this case, to have a smoothly curved shape). The more segments used, the closer the model will represent the cable behaviour, but increasing computational time should be expected. Moreover, setting the compression limit to zero, which is not affecting the bending stiffness [1], and reducing the bending stiffness of the frame element are needed to obtain a cable behaviour. The reduction of bending stiffness is highly dependent on the size of the cable; for small sections, the bending stiffness of the frame is already small, and no further modifications are essential. The load is applied in two steps; first, the dead load of the structure p, and then an additional uniformly distributed load q. The analytical solution provides equations to calculate the axial force in the cable given a certain deformation. Therefore, the final deformation obtained from the model is used to calculate the corresponding axial force. Detailed calculations are included in Appendix A. Table 4.1: Results from the investigation. Deflection m

Axial force kN

Relative difference in axial force

Frame elements

11,033

5017

0,76%

Cable elements

11,034

5019

0,80%

Analytical solution

11,054

4979

-

As presented in Table 4.1, using frame elements instead of cable elements is justified given the negligible relative difference between results.

56

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 4. INVESTIGATION OF MODELLING OPTIONS IN SAP2000 This quick investigation concludes the possibility of using frame elements to model cables, however, given the complexity of the case study, a closer representation to its geometry is used to investigate the difference between the two elements.

4.1.2

Cable vs. frame applied to a simplified version of the case study (3D)

Since the main drape is rather complex and the response highly depends on the chosen geometry, the previous check for a single simply supported cable wasn’t considered enough. Therefore, a simplification of the main drape of the case study has been modelled with both cable and frame elements and the results have been compared. The simplification of the drape consists of two main cables (with a span length of 215 m) which are connected by 17 crossbeams simply supported (only fixed for the three translation degrees of freedom) on the cables in the transverse direction. The main cables are simply supported with pin conditions at their ends. Both models started from the same initial geometry, as shown in Figure 4.2, where the main cables are straight, with no initial deformation in the vertical plane nor in the transverse plane. Afterwards, specific loads, explained later in this section, were applied in order to find the architectural geometry shown in Figure 4.3.

Figure 4.2: Undeformed (initial) shape of the structure.

Figure 4.3: Deformed (after loading) shape of the structure.

Post-tensioned stress ribbon systems in long-span roofs

57

CHAPTER 4. INVESTIGATION OF MODELLING OPTIONS IN SAP2000 The sections used for both the cables and the frames are 0,1 m diameter circular sections, with the material properties presented in Table 4.2. Table 4.2: Material properties.

Weight per unit volume

kN/m3

73,85

Modulus of elasticity

GPa

160

Minimum yield stress

MPa

1689,91

Model with cable elements This model was using cable elements for both the main cables and the crossbeams. The main cables were drawn between the two supports and modified using the "cable geometry window", where the element was broken into 80 segments, with a relative length of 1. The 17 inner crossbeams were drawn equally spaced along the span, applying different relative lengths to the elements (or another way to apply a negative strain load), so they shorten to make the structure find its deformed shape in the plan view, as shown in Figure 4.3.The gravity load is responsible for the deformation in the vertical plane. The magnitude of the strain load applied, due to relative shortening, is always a negative value, and is shown in Figure 4.4.

Figure 4.4: Relative strain load applied to the crossbeams.

Regarding loading, the self-weight of the cables were automatically included by the software, and a uniformly distributed load equivalent to the panel’s weight was applied to the crossbeams.

Model with frame elements This model was using frame elements for both the main cables and the crossbeams. The main cables were drawn using curved frame elements (a feature that breaks the curve into a string of straight elements) with a parabolic shape given three points: the two supports and a midpoint located 0,05 m below the supports. This small initial sag is necessary when using frame elements to reach convergence.

58

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 4. INVESTIGATION OF MODELLING OPTIONS IN SAP2000 Furthermore, the same modifications applied for frame elements in the previous 2D investigation were utilized: compression limit set to zero and a reductiom of the bending stiffness. The frame elements lack the option of applying a relative length. Alternatively, the "shortening" of the crossbeams was obtained by applying a negative strain load. The values of the strain loads were different for each beam to reach the target shape, with magnitudes the same as for the cable model (shown in Figure 4.4). The external vertical loading was the same as for the model using cable elements.

Comparison of the results Table 4.3: Comparison between cable and frame elements. Max. deflection Uz m

Reactions at one support Rx Ry Rz kN kN kN

Cable force Axial kN

Cable elements

-13,6

17258

2757

4476

17958

Frame elements

-13,8

17253

2238

4536

17911

-1,72%

0,02%

18,81%

-1,33%

0,26%

Relative difference

Table 4.3 shows a good similarity between the two models for all the outputs except the reaction of the transverse direction (Ry). This is explained by the different ways used to apply the strain load. Small variations on the applied strain load changes the deformed shape in the transversal direction, which affects the final reactions (Ry). Since a geometrical nonlinear analysis was used, small variations to the deformed shape of the cable structure greatly affect the force required to ensure equilibrium, leading to the observed difference in the reaction force. However, since the drape supports are out of the scope of this thesis, the difference found in the transversal reaction can be disregarded. Therefore, it is concluded that frame elements provide realistic results, and they will be used in the following models.

4.2

Effect of modelling the concrete panels vs. applying an equivalent uniformly distributed load

This section intends to investigate the most suitable technique for modelling simply supported panels in SAP2000.

Post-tensioned stress ribbon systems in long-span roofs

59

CHAPTER 4. INVESTIGATION OF MODELLING OPTIONS IN SAP2000 The importance of such an inspection lies on the many alternatives available in SAP2000 to model an area object (shell, plate or membrane area object) and its load. Different combinations of element type and loading methods will yield in various behaviours for the panels.Therefore, an investigation of which combination best represent a simply supported panel is of a paramount importance for the purpose of thesis. First, a model without the panels was studied, to truly captures the targeted behaviour. This model was obtained by omitting the panels and applying their weight as a uniform line load directly where they would be simply supported: at the crossbeams, as depicted in Figure 4.7 (a). Secondly, the final model was drawn with the panels, following the proposed method which will be explained later. The results from both models were then compared.

Common features of the two models Both models consisted of two simply supported cables with a span of 215 m connected by 17 crossbeams that are simply supported on the cables, as shown in Figure 4.5. Using the conclusion reached in the previous section, frame elements were used to model the cables. Moreover, sections and material used for both models were the same. Finally, nonlinear analyses were run, with geometric non-linearity effects included.

Figure 4.5: Plan view of the models.

Model without panels (load applied as a uniform line load) Instead of physically modeling the panels, a uniform line load was applied directly on the crossbeams with a magnitude corresponding to the equivalent panel’s dead load, as shown in Figure 4.7 (a). The value of the line load, having the same concrete thickness along the span, is different on each crossbeam, depending on the area of the panels, and therefore, it was described as higher at the ends and gradually reduced towards the center of the span.

60

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 4. INVESTIGATION OF MODELLING OPTIONS IN SAP2000

Model with panels Panels were modeled using shell elements spanning between crossbeams, as shown in Figure 4.7 (b) . Although SAP2000 includes the dead load of the panels automatically, the load is transferred to the frame as point loads as shown in Figure 4.6 (a), disregarding any internal forces in the shell or the frame. To transfer the dead load as a one-span load, as shown in Figure 4.6 (b), one must use SAP2000 tool "Uniform to frame (shell)" and set the panels’ weight to zero.

Figure 4.6: Different options to apply area loads in SAP2000.

Moreover, to simulate the realistic behaviour of uniformly simply supported concrete panels, where the axial stiffness is negligible compared to the overall structural stiffness, the membrane stiffness of the shell was eliminated by modelling the shells using thick plate formulation, and applying end moment releases. This simplification ignored the compressive axial stiffness, which could be accounted for using gap elements if more accurate results were needed, as will be the case in the final model used in Section 5.3.

Post-tensioned stress ribbon systems in long-span roofs

61

CHAPTER 4. INVESTIGATION OF MODELLING OPTIONS IN SAP2000

Results and conclusion Table 4.4: Comparison between the panel’s model and UDL model. Deflection U3 m

Axial force T kN

Reactions at each support Rx Ry Rz kN kN kN

Model with panels

-7,53

6777

6587

851

1468

Model with UDL

-7,58

6802

6618

856

1485

Relative difference

-0,58%

-0,37%

-0,47%

-0,58%

-1,18%

As seen in Table 4.4, the similarities captured on the vertical reaction assures that the values used for UDL were correctly estimated. Moreover, the percentage difference between all results are insignificant. This concludes that the proposed method for modelling simply supported panels is accurate enough for the present purposes.

62

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 4. INVESTIGATION OF MODELLING OPTIONS IN SAP2000

Figure 4.7: The two options to apply the load.

Post-tensioned stress ribbon systems in long-span roofs

63

CHAPTER 4. INVESTIGATION OF MODELLING OPTIONS IN SAP2000

4.3

Investigation of support movements

The three drapes, catenary, structures are supported by four corner structures, as shown in Figure 4.8, and horizontal movements in the support corner structures, where the cables are anchored, is expected to take place due to the high pull-out forces exerted by the cables. This movement will have an effect on the drape structures.

Figure 4.8: 3D rendering of the support structures and expected movements. A steel moment-frame system is suggested for the corner structures, as shown in Figure 4.9, which will function as an anchoring system for the drape’s cables and a support system for the folded plate roof. Obviously, beams with the right size, to accommodate the anchoring devices, and sufficient amount of stiffeners, to withstand the pull-out forces, have to be employed for the frame. The authors suggest that the cables can be anchored directly to the moment-frame system as schematically shown in Figure 4.9. However, this suggestion needs further investigation in later stages of the project. The steel frame is made of closed box sections, made of steel grade S355, with dimensions ranging between 800x800x40 mm and 1200x1200x60 mm. All the connections are moment resistant. The load from the cables follows a path through the steel frame, which finally ends as vertical and horizontal forces on the supports. How the supports work in reality is out of the scope of this work, however, due to extreme magnitude of the forces, big foundations or piles are expected. The following subsections aim to examine the expected movement of the corner structures, and the effect this may have on the cable forces and final deformations.

64

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 4. INVESTIGATION OF MODELLING OPTIONS IN SAP2000

Figure 4.9: 3D and plan view of the support structure frame.

4.3.1

Movement of the anchoring point throughout the construction stages

The anchoring points are expected to sway in the direction of the pull forces. Figure 4.10 illustrates the proposed construction sequence and the corresponding movement of an anchoring point. First, the corner structure is built (a). Then, the main cables are anchored in the corner structure and partially loaded with the precast concrete panels, thus, a horizontal movement towards the main drape is expected (b). Afterwards, the concrete folded plate is cast which will cause a horizontal movement in the opposite direction due to the cantilever (c). Finally, the main drape is fully loaded, causing another movement of the anchoring point (d). Alternative construction sequence were proposed and tested numerically. The results suggested to cast the the folded plates before anchoring the cables. However, the recorded horizontal movement towards the cantilevering parts was considered too large. Therefore, a balancing force, to reduce the final movement towards the cantilevering side, was introduced via the partially loaded cables. The horizontal movement of the anchoring point was evaluated numerically using a simplified model only containing the corner structure and the folded plates. A point load was applied at the anchoring point to simulate the pull-out forces exerted by the cables. The final location of the anchoring point, after each stage, is shown in Table 4.5. The "Final position" used in Table 4.5 describes the distance between the final spot of the anchoring point and its original position in the corner structure, before installing the cables or casting the folded plates. This distance can either be negative or positive, with a negative sign indicating a sway towards the main drape.

Post-tensioned stress ribbon systems in long-span roofs

65

CHAPTER 4. INVESTIGATION OF MODELLING OPTIONS IN SAP2000

δ

δ

δ

Figure 4.10: Proposed construction sequence. Table 4.5: Displacement at the cable anchorage point for the different construction stages.

66

Applied loads

Final position mm

Stage 1

-

-

Stage 2

Half of the pull-out forces from the cables

-25,36

Stage 3

Previous load + weight of the folded plate

-13,21

Stage 4

Previous load + rest of the pull-out forces from the cables

-26,87

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 4. INVESTIGATION OF MODELLING OPTIONS IN SAP2000

4.3.2

Quantification of the effect on the main drape

Once the displacement of the anchoring point was known, the next step was to determine how that movement affects the cable forces and displacements of the main drape. For this, the stiffness of the corner structurewas first determined. The stiffness of the support structure was calculated by applying different loads and reporting the corresponding displacement of the anchoring point. As shown in Figure 4.11, the load-displacement relation is linear, indicating a constant stiffness that can be found as follows: k=

F = 738552, 44 kN/m δ

(4.1)

where: F is the load applied on the anchoring point δ is the output displacement from F on the same point As a simplification, the load from the main drape was applied in one single point, instead of three, representing the total pull-out force from the three cables. Moreover, the stiffness was calculated by applying the load in the direction of the main drape. However, the corner structure also supports another drape in the perpendicular direction, whose effect was disregarded, but should be considered to be more precise. 35000 30000 Force (kN)

25000 20000 15000 10000 5000 0 13

18

23

28

33

38

Displacement (mm)

Figure 4.11: Force-displacement graph for the corner support.

With the stiffness known, the same model from Section 4.1.2 was used, but in this case, the supports were modeled with springs, as shown in Figure 4.12, using the stiffness value previously given in Equation 4.1. For both end-support conditions, simulations were run with the model subjected only to its self-weight. Then, a brief comparison between the two models was obtained with results displayed in Table 4.6.

Post-tensioned stress ribbon systems in long-span roofs

67

CHAPTER 4. INVESTIGATION OF MODELLING OPTIONS IN SAP2000

Figure 4.12: Spring support conditions.

Table 4.6: Comparison between pinned and spring support conditions. Model

Max. vertical displacement mm

Axial force in the cable kN

Pinned supports

-7,532

6777

Spring supports

-7,528

6666

Relative difference

-0,06%

-1,64%

It is observed that the movement of the supports results in an insignificant change of 1.64 % in the cable forces, therefore, it can be disregarded in further stages of this work. A final analysis, should, however, consider this further as a detailed description of the individual cable forces can be somewhat affected by the support movement.

4.4

Temperature load effects

Given the length of the cables, the expected climate exposure, and the strict geometrical requirements by the architect, the authors realized the significance of investigating whether or not the temperature change will have a considerable effect in fulfilling the final design requirements. Change in temperature can be categorized, according to its distribution over the cross-section, into linear, nonlinear and average temperature change [33]. The latter refers to the annual variation in temperature, exerting the highest temperature gradients on the structure. Consequently, this variation was used to estimate the temperature gradients applied in the following investigations. Different materials tend to change their initial volume as the surrounding temperature changes, which is also known as thermal strain [1]. These strains are calculated as follows: εT = α · ∆T

68

(4.2)

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 4. INVESTIGATION OF MODELLING OPTIONS IN SAP2000 where: εT α ∆T

is the strain due to temperature change is the coefficient of thermal expansion, which is a material property is the change in temperature

4.4.1

Temperature effect on a simplified model

Before applying the temperature effects in the final model, thermal loading was first applied on the two simplified models described previously in Sections 4.1.2 and 4.3.2, both models can be seen in Figure 4.13. This preliminary investigation aimed to examine the effect of restraining the thermal movements; the same thermal loading was applied to two similar models except for their restraining conditions. Both positive and negative temperature gradients, beside a coefficient of thermal expansion of 12 · 10−6 [31], were used to estimate the thermal loading, which was later applied only to the two bearing cables.

Figure 4.13: Two simplified models to apply the temperature load.

SAP2000 applies the temperature load as a strain on the elements, where the user only specifies the temperature difference, since the coefficient of thermal expansion is already specified as a material property. The temperature load was applied to the two bearing cables in the model. The temperature difference is taken as ±50◦ C, considering that the temperature in Västerås can reach −25◦ C in winter and rise up to +25◦ C in summer. Checking both, positive and negative gradients is essential, since the cable behaves differently at each case, as shown in Figure 4.14. The thermal loading was applied in a separate load case subsequent to the dead load case.

Post-tensioned stress ribbon systems in long-span roofs

69

CHAPTER 4. INVESTIGATION OF MODELLING OPTIONS IN SAP2000

Figure 4.14: Deflections under different loads: dead load (blue) and temperature load (orange).

Tables 4.7 and 4.8 present the results obtained from the two models. As expected, the model with higher degree of restraint is more influenced by the temperature changes. This observation will be further discussed later in this thesis. Also, the results are displayed in the simple sketch shown in Figure 4.14. Table 4.7: Support movements and deflection due to temperature load in the bearing cables, simplified models.

∆T ◦

Spring support Pinned support

C -50 50 -50 50

Dead load mm 8,73

Support movement Deflection at midspan Dead + Temp. Relative Dead Temperature Relative loads difference load load difference mm m m 8,9 2% -7,36 -2% -7,53 8,62 -1% -7,65 2% -7,29 -3% n/a -7,48 -7,81 4%

Table 4.8: Axial load in the cables due to temperature load in the bearing cables, simplified models.

∆T ◦

Spring support Pinned support

70

C -50 50 -50 50

Axial force in the cable Dead + Temp. Relative loads difference kN 6790 2% 6666 6584 -1% 6839 2% 6703 6472 -3%

Dead load kN

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 4. INVESTIGATION OF MODELLING OPTIONS IN SAP2000

4.4.2

Temperature effect on the stress ribbon model

Since the positive temperature gradient applied to pinned supported model showed the greatest effects in the previous check, these conditions were applied to the final stress ribbon model, whith specifications in Section 5.3.1. This section aims only to give the reader an idea of how the thermal load applied to the bearing cables can affect the final output. A thermal strain was simulated with a temperature gradient of +50◦ C applied only to the three bearing cables, in a load case that followed a dead load case. Table 4.9 shows the effect of the temperature on the deflection and the axial force of the cable. A significant effect is noted in the cable deflection, increasing by 0,5 meters relative to the dead load deflection. The increasing deflection requires a corresponding increase in the cable’s sag, leading to 6% reduction in the cable force. Table 4.9: Deflection and axial forces in the bearing cables from the temperature load. Deflection at midspan ∆T

Axial force in the cable

C

Dead load m

Dead + Temp. loads m

Relative difference %

Dead load kN

-50

-6,4

-6,9

8%

14309

◦

4.5

Dead + Temp. Relative loads difference kN 13381

-6%

Effect of post-tension on the bearing cable forces

Initially, the application of post-tensioning was meant to control the concrete tensile stresses and to contribute to the overall structural stiffness. However, [21] suggests that applying a post-tensioning in a stress ribbon system can highly reduce the tension force taken by the bearing cables. This section aims to investigate this force reduction, and to establish an empirical relationship between the force reduction with the amount of the applied post-tension force, which can later be used to quantify a suitable post-tensioning force that can limit the cable axial force. The study was carried out using the same model developed for the final design, which is later explained in Section 5.3.2. However, it’s important to notice that the post-tensioning tendons were anchored in the concrete panels and not in the supporting corner structures, where the bearing cables were anchored.

Post-tensioned stress ribbon systems in long-span roofs

71

CHAPTER 4. INVESTIGATION OF MODELLING OPTIONS IN SAP2000 A simple description of the study can be stated as follows. First, the model was run subjected to self-weight only, and the axial force in the middle cable was recorded. Starting from this dead load case, post-tensioning was applied to the concrete panel, and the same results were recorded. Varying the magnitude of the post-tensioning force, the process was repeated seven times, and the obtained results are displayed in Figure 4.15. Bearing in mind the adopted 200 mm thickness of the concrete panels, the final point used, an unrealistically high post-tensioning force of 2000 kN, just to examine whether this effect continues with increasing the post-tension force or it abates eventually. As expected, Figure 4.15 describes the nonlinear relation between the two variables: the cable force reduces as the prestressing force increases. 50000

Cable force (kN)

40000 30000 20000 10000 0 0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

Prestress force applied per tendon (kN)

Figure 4.15: Relation between the bearing cable force and the prestress force.

To conclude, as the panels are prestressed, their stiffness increase, thus, the panels attract more of the external loads, reducing the total force taken by the cables, and therefore, its axial force.

72

Post-tensioned stress ribbon systems in long-span roofs

Chapter 5 FE models 5.1

Structural concepts

For this thesis, two different structural systems have been considered. First, the concrete panels are simply supported on crossbeams, thus, they contribute very little to the stiffness of the structure. The second models a stress ribbon concept where the concrete panels can be post-tensioned which will highly increase the stiffness. Both systems were modeled using SAP2000, starting with the same undeformed geometry and having the same material properties in order to be comparable. The chosen geometry is inspired by the main drape of the Västerås project.

5.1.1

Common aspects for all models

Geometry All models consist of three main cables with a span of 215 m and an initial sag of -0,05 m at midspan.The cables are connected with 33 crossbeams, evenly distributed along the cable length, in order to keep the architectural shape in the plan view, as presented in Figure 5.1. To account for the expected pull-out forces from the cables, the chosen support conditions must be able to resist forces in all three directions but free to rotate, and therefore the three cables are pinned at both ends. Given the geometry, the models consist of 3 main cables, 33 crossbeams, and 32 concrete panels.

Post-tensioned stress ribbon systems in long-span roofs

73

CHAPTER 5. FE MODELS

Figure 5.1: Plan view and geometry of the final models.

Sections and material properties. There are three materials and sections used in all models, as described in Tables 5.1, 5.2 and 5.3. Table 5.1: Steel material properties. Effective elastic modulus GPa

Yield strength MPa

Weight kN/m3

Steel (cables)

160

1300

76,97

Steel (tendons)

195

1640

76,97

Steel (crossbeams)

210

355

78,5

Table 5.2: Concrete material properties. Elastic Characteristic modulus compressive strength GPa MPa Concrete C60/75

39

Characteristic tensile strength MPa

Poisson ratio

4,4

0,2

60

Weight

kN/m3 24,99

Table 5.3: Section properties.

Bearing cables Post-tension tendons Crossbeams Panels

74

Section

Material

0,2 m diameter

Steel (cables)

Unit 6-7 VSL

Steel (tendons)

HE500A

Steel

200 mm thickness

Concrete C60/75

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 5. FE MODELS

Loading steps Since cable systems are highly non-linear structures, superposition is not possible and the load cases used have significant effects on the obtained results. Therefore, to simulate the real behaviour of the structure, the following loads are considered. 1. Dead load: cables, crossbeams and panels. 2. Prestress: (only for the post-tensioned stress ribbon system) + previous load. 3. Live load: snow (1,28 kN/m2 ) + previous load. Each step combination starts using the ending solution from the previous one to make the prediction for the equilibrium iterations in the new step.

Analysis type When modelling cables as structural elements, high deflections are expected and therefore geometrical nonlinear analyses including P-delta and large displacements effects are needed. The parameters of the performed nonlinear analyses are described in Table 5.4. Moreover, the maximum number of Constant-Stiffness and Newton-Raphson iterations per step are set to 10 and 40, respectively. Rather coarse equilibrium rolerance was used, but is not believed to affect the results significantly. Table 5.4: Convergence parameters. Convergence criterion

Convergence norm

Tolerance

Newton-Raphson & Constant-Stiffness

Force control

1 · 10−2

5.1.2

Initial model

This model was used as a template for the two structural systems explained in Sections 5.2 and 5.3. The cables were modelled as series of straight frame elements that follow a parabolic arc (discretized in a series of small straight elements). To obtain a more accurate shape, the curve was broken into 80 equal segments, seen as meshing nodes. The exact geometry used was obtained using the curved frame geometry generator in SAP2000 shown in Figure 5.2.

Post-tensioned stress ribbon systems in long-span roofs

75

CHAPTER 5. FE MODELS

Figure 5.2: Example of the curved frame geometry generator in SAP2000.

As previously proven in Section 4.1, a cable behaviour is obtained using frame elements by applying the following modifications. • The compression capacity is eliminated by activating the SAP2000 feature: compression limit equals zero. • The bending stiffness is reduced to one percent of the geometrically calculated stiffness by using the SAP2000 feature: setting the property modifier for moment of inertia about both axis equals 0,01. • An initial sag of -0,05 m is necessary for convergence. The crossbeams were modeled as straight frame elements simply supported by the outer cables, but were allowed to slide horizontally in the connection with the mid cable, as sketched in Figure 5.3. This was obtained by using the following steps: • dividing crossbeams at its intersection with the main cable, creating two nodes at each intersection; • applying "Equal" constraints, which link the vertical (z) and transverse (y) transitional degrees of freedom, between the two nodes on each intersection.

76

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 5. FE MODELS

Figure 5.3: Connection between the crossbeams and the cables.

To ensure a simply supported connection between the crossbeams and the cables, one end was released for the two moments and torsion, while the other end was released only for the two moments, as recommended in [1]. This generated the template model, which has only cables and crossbeams, but no external load or concrete panels applied yet.

5.2

Cable suspended roof, with precast concrete panels

As a repetition, Figure 5.4 shows the main structural elements of the cable suspended roof model and the spanning direction of the concrete panels.

Post-tensioned stress ribbon systems in long-span roofs

77

CHAPTER 5. FE MODELS

Figure 5.4: Main structural elements of the cable suspended roof model.

Starting from the template model, two shell elements between crossbeams are modelled, using the end nodes and the midpoint node of the crossbeam. The layout of the shell elements is shown in Figure 5.5.

Figure 5.5: Example of how the shell elements are divided in between crossbeams for the cable suspended roof.

To make sure that the shells are not working continuously, i.e. each panel is simply supported in two edges on the crossbeams, thick plate formulation was used, and moments were released at the edges of each shell. To control the span direction, the loading option "Uniform to Frame (Shell)" in SAP2000 was used, where the dead load, as well as any other external load, was applied manually (as a gravity load on the panel’s area), and the automatic calculation of the dead load was turned off. This feature of SAP2000 was previously explained in Section 4.2.

78

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 5. FE MODELS

5.3

Stress ribbon roof

As previously mentioned in this report, the stress ribbon system can be used without or with post-tensioning system, thus, in order to evaluate the prestressing effects, both cases studied. Figure 5.6 and 5.7 show the main characteristics of both models, respectively.

Figure 5.6: Main structural elements of the stress ribbon without post-tension.

Figure 5.7: Main structural elements of the stress ribbon with post-tension.

Post-tensioned stress ribbon systems in long-span roofs

79

CHAPTER 5. FE MODELS

5.3.1

Model without post-tension

Figure 5.8: Example of how the shell are connected to the adjacent elements for the stress ribbon system.

Prefabricated concrete panels are expected to be used, the connection between adjacent panels should be able to transmit a compression forces when they touch, but no tension forces is transmitted if they are separated for any reason. To obtain this, gap elements which simulate compression only behaviour were modelled. Figure 5.8 shows the connection between the shell elements and the cables for the stress ribbon system. Although the concrete panels are not separated in reality, a physical space of 12 cm was introduced in the model to draw the gap element between two nodes. However, the structural behaviour was expected to be similar to reality by setting the gap opening equals zero, as shown in Figure 5.9. The aforementioned figure shows the properties given to the gap property in SAP2000; the nonlinear stiffness value and the opening for U 1, which is in the longitudinal direction of the drape.

80

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 5. FE MODELS

Figure 5.9: Properties of the gap link used in SAP2000.

To obtain a constant physical space between each two consecutive panels, an additional node in the cables was created, located 6cm from the crossbeam. SAP2000 has different options to do this, the one used is described in Figure 5.10.

Figure 5.10: Sketch of how the required nodes for the gap distance were achieved in SAP2000.

Post-tensioned stress ribbon systems in long-span roofs

81

CHAPTER 5. FE MODELS Since the panels are not rectangular, the tool "draw poly area" was used to model them using the newly created nodes, following a counter-clockwise direction in order to be consistent with the definition of the shell top face in SAP2000, as explained in [1]. Thick shells with linear behaviour were used. Figure 5.11 shows the model after drawing the shell elements in SAP2000.

Figure 5.11: The model after drawing the concrete panels. Detail of the physical space between the panels.

Then, to simulate the real structure, the shells were offset 25 cm above in relation to the cable nodes. To apply a gap element along the interfaces between the concrete panels, internal nodes were needed. These were obtained by dividing each panel to 10 by 5 smaller elements. Subsequently, the gap elements were used to link each two adjacent nodes as shown in Figure 5.12, creating 11 gap links.

Figure 5.12: Gap links (in red) between the panels divided in 10x10 elements.

According to [1], the stiffness of a gap element is sufficiently estimated by multiplying the axial stiffness of the connected elements by any value between 102 to 104 . The gap stiffness was here calculated as: kgap =

82

EA · 103 = 50 · 106 kN/m L

(5.1)

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 5. FE MODELS where: E is the concrete modulus of elasticity A is the cross-sectional area of the largest panel L is the length of the panel Finally, the panels were connected to the cables using equal constraints which linked the vertical (z) and transverse (y) transitional degrees of freedom, as shown in Figure 5.13. The panel connection with the six pinned end supports was defined by equal constraints fixed for all transitional degrees of freedom.

Figure 5.13: Sketch of the equal constraints (in red) applied to nodes between the shells and the cables (in blue).

Loading Given the cross-sections and densities of each structural element, SAP2000 calculated and applied the corresponding dead-load automatically. The live load was applied as a uniform downwards pressure on the shell elements.

5.3.2

Model with post-tension

In order to, accurately, introduce the post-tension effects to the aforementioned model in Section 5.3.1, a staged construction analysis was performed. The suggested tendon layout is shown in Figure 3.11 (c). The spacing between the tendons was chosen as 0,5 m and the length varied along the span, since the changing width of the panels won’t allow the usage of the same number of tendons on each panel; the wider the panel, the more tendons are needed to achieve a uniform compressive stresses. Also, the number of tendons at any cross section along the span depends on the available panel width at that location. Finally, due to the small thickness of the panels, the tendons were placed concentrically at the panel midplane.

Post-tensioned stress ribbon systems in long-span roofs

83

CHAPTER 5. FE MODELS The tendons could be drawn using the "straight line" tool, where the user uses the Graphical User Interface (GUI) to, accurately, locate each tendon by clicking on its starting and ending point. Drawing 62 tendons this way is an onerous process, not to mention the difficulties the user would face in specifying the exact location of each tendon. Luckily, SAP2000 provides a more convenient modelling tool named "Interactive Database Editing", which is usually used for models where high level of accuracy and repetition is expected. Interactive Database Editing allows the user to export all, requested, modelling information to an excel sheet, where it can be easily manipulated, and then retrieve all modified information and apply it to the model. Obviously, the latter modelling method was adopted by the authors to locate the tendons in their final positions. Eventually, the tendons were assigned to one group, to facilitate the application of stage construction analysis, as will be explained below. Beside the post-tension force, other parameters used to define the tendon elements are shown in Table 5.5. Table 5.5: Tendon parameters. Jack

From both ends

Shape

straight line

Discretization (m)

0,5

To, meticulously, simulate realistic post-tenioning effects, the tendons must only be introduced after the roof’s self-weight is applied. The only way to do this, in SAP2000, is to run a staged construction analysis. However, before defining the construction stage load-case, the following steps were followed by the authors. • The complete model was drawn, including the tendons. • The whole model, including the tendons, was assigned to a single group, named "ALL". • Another group, containing only the tendons, was created and named "Tendons". • A Named Property Set, named "Ghost", was created and defined as described in Table 5.6. • Another Named Property Set, named "Normal", was created with all modification factors set to unity (default properties). • A separate load pattern, named "Snow load", was used to define the snow load on the concrete panels.

84

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 5. FE MODELS Finally, the construction stage analysis was defined as follows. • Stage 1: Group "All" was added to the model. • Stage 2: Named Property set "Ghost" was applied to group "Tendons". • Stage 3: Self-weight load was applied to group "All". • Stage 4: Named Property Set "Normal" was applied to group "Tendons". • Stage 5: Prestressing force was applied to group "Tendons". • Stage 6: Snow load was applied to applied to group "All". The description of Named Property Sets in SAP2000 was introduced to the reader in Section 2.2.3. Although the tendons were modelled at the first stage, their contribution to the overall structural stiffness was disabled until stage 4; using the applied property modifier set "Ghost", which helps simulating a non-existing tendon situation. Table 5.6: Tendon section modification. Ghost section modifier Axial stiffness

10−6

Shear stiffness

10−6

Torsional constant

1

Bending stiffness

1

Weight, mass

10−6

The bending and torsional stiffness modifier in the "Ghost" property set were kept as unity to ensure numerical stability, as recommended by [1], which states that the bending and torsional stiffness have little importance on the tendon’s structural behavior, but they are essential for numerical stability. On the other hand, the value 10−6 , used to modify the other properties, was chosen randomly by the authors to simulate a non-existing stiffness, since SAP2000 does not allow the user to set any stiffness to zero. Moreover, a modification to the frame properties was done in order to find convergence. Using the current diameter of 0,2 m, SAP2000 wasn’t able to find a solution due to numerical errors. This issue was solved by using a bigger diameter, of 0,5 m. However, cables’ weight and material properties had to be modified, accordingly, to reach the same total self-weight and axial stiffness as would be obtained by using the 0,2 m diameter. The modified elastic modulus and weight ratio are calculated as follows:

Post-tensioned stress ribbon systems in long-span roofs

85

CHAPTER 5. FE MODELS

Ereal · A0,2 = Emodified · A0,5 ⇒ Emodified = 26, 4 GPa

weight modifier =

A0,2 = 0, 16 A0,5

(5.2)

(5.3)

where: A0,2 A0,5 Emodified Ereal

is the area for a cable with 0,2 m diameter is the area for a cable with 0,5 m diameter is the modified elastic modulus is the elastic modulus of the steel cable used for design as shown in Table 5.1 weight modifier is the input ratio between the areas to have the correct final weight of the cables

Finally, to include the effect of prestressing losses, the previously mentioned steps were repeated twice; changing the applied prestress force each time. The various applied forces are shown in Table 5.7. This describes the effect of the prestress in three critical stages: when the maximum force was initially applied, when the load was quickly reduced by short-term losses, and when the load was reduced over a longer period of time due to long-term losses. The calculation of the force reduction due to losses is defined in Appendix B. Table 5.7: Prestressing force applied.

5.4

1

Initial force

1547 kN

2

After short-term losses

1250 kN

3

After long-term losses

1194 kN

Results reporting

The relevant output was extracted from the models in SAP2000 using a built-in tool that extracts the requested data and exports it directly to spreadsheets, where they were displayed in forms of graphs and summary tables. For all three bearing cables, axial stresses were taken at the neutral axis of the cross-section. Other stress locations were disregarded since the bending stresses are considered insignificant in the cables. The global deflection of the system was extracted from the cable’s deflection. Transverse deflections were negligible, thus, only longitudinal deflections along the span length were recorded.

86

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 5. FE MODELS SAP2000 assesses the stresses by 2 × 2 Gauss integration points in each element and extrapolates those values to the joints. Moreover, the stresses are manually averaged on the shared nodes to obtain continuous stress distribution. Figure 5.14 schematically shows this process.

Figure 5.14: Extrapolation and averaging of stresses in the shells.

Post-tensioned stress ribbon systems in long-span roofs

87

Chapter 6 Results The two main structural elements involved are the cables and the concrete panels, therefore, the only relevant output results of interest to the authors were: cable stresses and deformation and stresses in the concrete shells. Locations where results were extracted are depicted in Figure 6.1, where Path SC is the line along which the stresses in the shell were recorded. Results displayed in this chapter are extracted from the three models previously described in Sections 5.2 and 5.3. The extracted results are displayed for all load cases, however, it’s important to note the cumulative load cases. In other words, a snow load case always includes dead load effect as well.

Figure 6.1: Location of the output results.

6.1

Cable suspended roof

Figure 6.2 shows the axial stresses along the three bearing cables for various load cases. As expected, due to symmetry, cables A and C have similar stresses, whereas cable B displays slightly higher stress; because of its location in the middle of the section, which forces it to cover more loading area. The figure also shows the increase in stresses at each cable due to snow load.

Post-tensioned stress ribbon systems in long-span roofs

89

Stress (MPa)

CHAPTER 6. RESULTS

540 520 500 480 460 440 420 400 0

50

100

150

200

Distance (m) Cable A, dead load Cable A, snow load

Cable B, dead load Cable B, snow load

Cable C, dead load Cable C, snow load

Figure 6.2: Axial stresses in the bearing cables, for the cable suspended roof model.

Figure 6.3 shows the deflection of the three bearing cables along the span. All three cables deflect by the same magnitude under dead load, which is further increased by 0,4 m due to snow load. This increase can not be accepted for the final design, thus, possible solutions for this issue will be discussed later in this thesis.

Deflection (m)

0 -1 0

50

100

150

200

-2 -3 -4 -5 -6 -7 -8 Cable A, dead load Cable A, snow load

Distance (m) Cable B, dead load Cable B, snow load

Cable C, dead load Cable C, snow load

Figure 6.3: Deflection in the bearing cables, for the cable suspended roof model. Note: similar deflections, some colours don’t show

Figure 6.4 shows the variation in longitudinal stresses, at top and bottom faces, along the concrete panels for half of the span. The repeated pattern verifies the adopted modelling procedure of the simply supported panels. The highest values of stresses are observed around the center of each panel: compressive stress at the top face and tensile stress at the bottom. Unfortunately, tensile stresses recorded at panels’ mid-field exceed the concrete tensile capacity, which means that the concrete cracks and the force would be taken by the mild reinforcement. However, these stresses drop remarkably when approaching panels’ ends.

90

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 6. RESULTS

8 Stress S11 (MPa)

6 4 2 0 -2 0

20

40

60

80

100

-4 -6 Top face, dead load

Distance (m) Bottom face, dead load

Top face, snow load

Bottom face, snow load

Tension capacity

Figure 6.4: Stresses in the concrete along path SC, for the cable suspended roof model.

The maximum value of each investigated output quantity is displayed in Table 6.1, where they are compared to the maximum allowable values and the desired architectural shape. Regarding the target architectural shape, an iteration process is necessary to reach the right deflection value. An outline of the iteration process, which is, however, not implemented in this thesis, is given in Section 7.2. Table 6.1: Final results for the cable suspended roof model. Bearing cables Axial stress MPa

Deflection m

Max.

Max.

Max.

Min.

Results

511

-6,96

6,9

-6,5

Target

≤ 866

≈ -9

≤ 4,4

≥ -60

YES

n/a

NO

YES

Target check

6.2 6.2.1

Concrete panels Shell stress MPa

Stress ribbon Without post-tension

Figure 6.5 shows the axial stresses along the three bearing cables. Unlike the previous model, a small difference in stresses, about 5 MPa, between cables A and C is noted under snow load case, however, such a small difference is not enough to indicate any serious problems with the model.

Post-tensioned stress ribbon systems in long-span roofs

91

CHAPTER 6. RESULTS

Stress (MPa)

530 520 510 500 490 480 470 460 450 440 430 0

50

100

150

200

Distance (m) Cable A, dead load Cable A, snow load

Cable B, dead load Cable B, snow load

Cable C, dead load Cable C, snow load

Figure 6.5: Axial stresses in the bearing cables, for the stress ribbon model without post-tension.

Figure 6.6 shows the deflections of the three bearing cables for dead and snow load cases, which indicates similar result as the one obtained for the suspended roof model.

Deflection (m)

0 -1 0

50

100

150

200

-2 -3 -4 -5 -6 -7 -8

Distance (m) Cable A, dead load Cable A, snow load

Cable B, dead load Cable B, snow load

Cable C, dead load Cable C, snow load

Figure 6.6: Deflection in the bearing cables, for the stress ribbon model without post-tension. Note: similar deflections, some colours don’t show

Regarding stresses in the longitudinal direction of the concrete panels, which they are supported in points instead of line supports, Figure 6.7 shows a similar pattern to the one given by the suspended roof model. In this case, however, the tension stresses due to dead load only are below the concrete capacity. 6 Stress S11 (MPa)

4 2 0 -2

0

20

40

60

80

100

-4 -6 Top face, dead load

Top face, snow load

Distance (m) Bottom face, dead load

Bottom face, snow load

Tension capacity

Figure 6.7: Stresses in the concrete along path SC, for the stress ribbon model without post-tension.

92

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 6. RESULTS Finally, Table 6.2 summarizes the maximum values obtained for each variable studied. Table 6.2: Final results for the stress ribbon model without post-tension. Bearing cables Axial stress MPa

Deflection m

Max.

Max.

Max.

Min.

Results

518

-6,96

5,2

-5,2

Target

≤ 866

≈ -9

≤ 4,4

≥ -60

YES

n/a

NO

YES

Target check

6.2.2

Concrete panels Shell stress MPa

With post-tension

To be able to interpret the obtained results correctly, it is important to fully understand what each load case actually represents. The utilized construction stage analysis contained the following load stages. 1. Dead load: after the full self-weight is applied. 2. Full prestress: dead load + maximum prestress load. 3. Short term losses: dead load + prestress load after short term losses. 4. Long term losses: dead load + prestress load after long term losses. 5. Snow load: dead load + prestress load after long term losses + snow load. The results obtained from stage 1, the dead load case, are the same as those reported for the same load case of the stress ribbon model without post-tension. Sections 6.1 and 6.2.1 have proved that the output for the three bearing cables follows a trend: cables A and C have the same results and cable B provides slightly higher values. Therefore, in this Section, only results for cable B are reported in order to simplify the graphs. Figure 6.8 shows the axial stresses along cable B for the different load cases. The maximum stress occurs for the dead load case. With the full post-tension load applied, the stresses reach lower values, the lowest for any cosidered load case, but start to rise again as the prestressing losses increase. Finally, subjected to snow loads, the cables’ stresses continue to develop, nonetheless, the final stress is less than the initial stress recorded under dead load case.

Post-tensioned stress ribbon systems in long-span roofs

93

CHAPTER 6. RESULTS

460

Stress (MPa)

450 440 430 420 410 400 0

50 (1) Dead load

100 Distance (m)

(2) Full prestress

(3) Short term losses

150 (4) Long term losses

200 (5) Snow load

Figure 6.8: Axial stresses along cable B, for the stress ribbon model.

Figure 6.9 shows the deflection along the span for cable B. Just like the cable stresses, dead load results in maximum deflection, which is reduced as the prestressing takes place and increases again under snow load. Table 6.4 shows how the maximum deflection at midspan increases or decreases as the loading process develops. It is observed that the long term prestress losses have an almost negligible effect on the deflections. 0 -1

0

50

100

150

200

Deflection (m)

-2 -3 -4 -5 -6 -7 -8 (1) Dead load

(2) Full prestress

Distance (m) (4) Long term losses

(3) Short term losses

(5) Snow load

Figure 6.9: Deflection along cable B, for the stress ribbon model.

94

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 6. RESULTS Table 6.3: Variation of the deflection during the different load cases, for the stress ribbon model. Deflection Increase referred m to previous load case (1) Dead load

-6,39

-

(2) Full prestress

-6,08

-5%

(3) Short term losses

-6,14

1%

(4) Long term losses

-6,15

0%

(5) Snow load

-6,27

2%

+% means increased deflection -% means decreased deflection

The shell stresses, due to all load cases, are shown in Figure 6.10. Again, the same pattern is noticed: dead load provides the maximum stresses, then the stresses decrease for the full prestressed load. Afterwards, a slight increase is noted as the losses take place. Finally, the final increase takes place under snow loading. 7

Stresses (MPa)

2 -3 -8 -13 -18

0

20 (1) Dead load, top face (1) Dead load, bottom face (4) Long term losses, top face (4) Long term losses, bottom face

40

Distance (m) 60 (2) Full prestress, top face (2) Full prestress, bottom face (5) Snow load, top face (5) Snow load, bottom face

80

100

(3) Short term losses, top face (3) Short term losses, bottom face Tension capacity

Figure 6.10: Stresses in the concrete along path SC, for the stress ribbon model.

It is interesting to see the differences between the top and bottom faces for different load cases. For dead load, and after accounting for all post-tensioning losses, the top face stresses follow a horizontal trend line; indicating negligible differences in stress values along the span. On the other hand, the slightly inclined trend line, describing the changes in the bottom face stresses, denotes a gradual rise in the tensile stresses when approaching mid-span, as depicted in Figure 6.11. However, when subjected to snow loads, both faces exhibit a gradual reduction in their compressive stresses towards mid-span, as illustrated by the inclined trend lines shown in Figure 6.12.

Post-tensioned stress ribbon systems in long-span roofs

95

CHAPTER 6. RESULTS 7

Stresses (MPa)

2 -3 -8 -13 -18 0

20

(4) Long term losses, top face

40 60 80 Distance (m) (4) Long term losses, bottom face Bottom face stress trendline

100

Top face stresses trendline

Figure 6.11: Stresses trendline during post-tension.

7

Stresses (MPa)

2 -3 -8 -13 -18

0

20 (5) Snow load, top face

40

60

Distance (m)

(5) Snow load, bottom face

80

100

Bottom face stress trendline

Top face stress trendline

Figure 6.12: Stresses trendline for snow load case.

Finally, Table 6.4 includes the summary of the maximum values recorded for this model. The stages prior to long term prestress losses are not considered; since their effect is only present for a limited time. For this model, all the targets are fulfilled; the tension stresses are below the concrete capacity. Table 6.4: Final results for the post-tensioned stress ribbon model. Bearing cables Axial stress MPa

Deflection m

Max.

Max.

Max.

Min.

Results

420

-6,15

1,63

-13,2

Target

≤ 866

≈ -9

≤ 4,4

≥ -60

YES

n/a

YES

YES

Target check

96

Concrete panels Shell stress MPa

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 6. RESULTS

6.3

Model comparison

This section presents a simple comparison between the three models, using the same three output variables studied in previous sections. However, only two load cases are used in this comparison, namely: dead load, plus long term losses where applicable, and snow load. Figure 6.13 compares the axial stresses in cable B for the three models. The maximum values are summarized in Table 6.5, where the percentage difference, relative to the cable suspended roof model, is calculated. It is notable that the cable suspended model and the stress ribbon model without post-tensioning show insignificant differences between each other (1%), however, the post-tensioned stress ribbon model reveals a stress reduction up to -9% and -16% for dead load and snow load, respectively. 540

Axial stress (MPa)

520 500 480 460 440 420 400 0

50

100

Cable suspended, dead load Stress ribbon, without postension, dead load Stress ribbon, dead load

150

Distance (m)

200

Cable suspended, snow load Stress ribbon, without postension, snow load Stress ribbon, snow load

Figure 6.13: Comparison of axial stresses along cable B for the different models.

Table 6.5: Comparison of maximum axial stress along cable B for the different models. Axial stresses in cable B Dead load MPa

Difference

Snow load MPa

% difference

Cable suspended

460

523

Stress ribbon, without post-tension

456

-1%

518

-1%

Stress ribbon, with post-tension

420

-9%

438

-16%

Post-tensioned stress ribbon systems in long-span roofs

97

CHAPTER 6. RESULTS Regarding deflections, Figure 6.14 compares the three models graphically, while Table 6.6 compares the deflections at mid-span. The percentage increase represents the increase in deflections due to snow load, for each system. For serviceability requirements, the increase in deflection due to snow load should be as small possible, which is the case for the post-tensioned stress ribbon system. 0 -1

0

50

100

150

200

Deflection (m)

-2 -3 -4 -5 -6 -7 -8 Cable suspended, dead load Stress ribbon, without postension, dead load Stress ribbon, dead load

Distance (m)

Cable suspended, snow load Stress ribbon, without postension, snow load Stress ribbon, snow load

Figure 6.14: Comparison of deflections along cable B for the different models.

Table 6.6: Comparison of deflections at midspan for the different models. Maximum deflection Dead load

Snow load

Percentage increase

m

Deflection increase from snow load m

m Cable suspended

-6,54

-6,96

0,42

6%

Stress ribbon, without post-tension

-6,39

-6,80

0,41

6%

Stress ribbon with post-tension

-6,15

-6,27

0,12

2%

A comparison of the stresses is shown in Figure 6.15 and Table 6.7 for the shell’s top face and in Figure 6.16 and Table 6.8 for the shell’s bottom face. In the top face, with compression stresses, the post-tensioning applied in the stress ribbon model shows a considerable effect, increasing the maximum compressive stresses in the concrete by 215% under dead load and 169% under snow load, compared to the cable suspended model. However, this effect is reduced as approaching mid-span, especially for the snow load case. The stress ribbon system without post-tensioning also shows higher compressive stresses, but the maximum increase obtained is limited to 15%.

98

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 6. RESULTS

Stresses (MPa)

Similar trends are observed in the bottom face, but in this case, the maximum stress reduction, obtained by the post-tensioned model, is limited to 71% under dead load. 2 0 -2 -4 -6 -8 -10 -12 -14

0

20

40

60

Distance (m)

80

100

Cable suspended, dead load

Cable suspended, snow load

Stress ribbon, without postension, dead load

Stress ribbon, without postension, snow load

Stress ribbon, dead load

Stress ribbon, snow load

Figure 6.15: Comparison of stresses in the shell top face along path SC.

Table 6.7: Comparison of minimum stresses in the top face for the different models. Stresses - top face

Stress (MPa)

Dead load MPa

% difference

Snow load MPa

Difference

Cable suspended

-3,97

-4,50

Stress ribbon, without post-tension

-4,34

9%

-5,16

15%

Stress ribbon

-12,51

215%

-12,09

169%

8 6 4 2 0 -2 -4 -6 -8 -10

0

20 Cable suspended, dead load Stress ribbon, without postension, snow load

40 Distance (m) 60 Cable suspended, snow load Stress ribbon, dead load

80 100 Stress ribbon, without postension, dead load Stress ribbon, snow load

Tension capacity

Figure 6.16: Comparison of stresses in the shell bottom face along path SC.

Post-tensioned stress ribbon systems in long-span roofs

99

CHAPTER 6. RESULTS Table 6.8: Comparison of minimum stresses in the bottom face for the different models. Stresses - bottom face Dead load MPa

100

Difference

Snow load MPa

Difference

Cable suspended

6,12

6,92

Stress ribbon, without post-tension

4,39

-28%

5,24

-24%

Stress ribbon

1,76

-71%

3,46

-50%

Post-tensioned stress ribbon systems in long-span roofs

Chapter 7 Practical considerations 7.1

Construction methods

As described previously, the roof structure is composed of different parts, namely: corner structures, folded plates and drapes. Their individual behaviour and their interactions are highly affected by the adopted construction sequence. Although the main drape analysis is performed for a final usage state, this section intends to give an outline of the construction sequence. Figure 7.1 depicts one alternative for a construction sequence suggested by the authors, which can be described as follows. Stage 1: the steel frames for the support structures are built on its respective locations. Stage 2: the bearing cables for the three drapes are placed and attached to the support structures. The cables are only loaded by their self-weight at this stage. Stage 3: the crossbeams are placed on the bearing cables in all three drapes. This will introduce some load into the bearing cables, which can be assumed as uniformly distributed along the span length. Stage 4: the folded plates are cast on top of the four support structures. Stage 5: the concrete panels are placed on the drapes. This is done by first placing the panels at midspan, and then continuing towards the ends (not shown on the figure). If the behaviour in different construction stages were to be analyzed, it would be important to determine all the loads acting on each stage and the relative movements between the drapes and the supports. A simple investigation of how the support structure movements would affect the behaviour of the main drape was included in Section 4.3.

Post-tensioned stress ribbon systems in long-span roofs

101

CHAPTER 7. PRACTICAL CONSIDERATIONS

Figure 7.1: Construction sequence. Figure 7.1 is merely a graphical representation of the construction sequence, the rendering is missing the top horizontal bracing of the steel frame, which is an essential part to transmit the horizontal forces to the foundation.

102

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 7. PRACTICAL CONSIDERATIONS

7.2

Design process

Cable elements are characterized by their flexibility, in other words, their final shape is highly dependent on the applied loads. Therefore, their design must be an iterative process. Regarding the roof structure for Västerås Travel Center, an iterative approach must be followed to reach a design that satisfies the architectural requirements while meeting the structural design regulations simultaneously. Although the implementation of this iteration process is out the scope of this thesis, Figure 7.2 illustrates the authors’ proposal for the design steps to be carried out. Di = Dfinal Li = Lfinal

Check Yes

Check

Yes

No

Increase Di

Di

or

Input

Increase Li

Li

No

Decrease Di Check

or

Yes

Increase Li

No

Increase Di or

Decrease Li

Figure 7.2: Flowchart for the iteration process.

In the figure, the following notation is used: Di Li δi δr σi fyd Dfinal Lfinal

is the initial diameter for the bearing cables is the initial length for the bearing cables is the resulting deflected shape of the drape is the required architectural shape of the drape is the resulting maximum stress in the cables is the yield strength of the cables symbolizes the final chosen diameters for the cables symbolizes the final lengths for the cabless

Post-tensioned stress ribbon systems in long-span roofs

103

CHAPTER 7. PRACTICAL CONSIDERATIONS The iteration process starts by choosing an initial diameters and unstressed lengths for the cables. Good starting values can be approximated by simple analytic calculations, as suggested by [16]. For each of the cables, this essentially leads to the equations: s P =

(

q · Lspan 2 q · Lspan 2 ) +( ) 8·f 2 s 4·P Di = π · fyd

Li = Lspan + where: P q Lspan

8 · f2 3 · Lspan

(7.1)

(7.2) (7.3)

is the tension force in the cable is the load in the cable, including its self-weight is the span length of the cable

With these initial values for the cable diameter and length, the analysis is run and the final deformation and stress is obtained. Testing the different checkpoints included in the flowchart will lead to one of the following conclusions. • Di = Dfinal and Li = Lfinal : the input Di and Li fulfilled the architectural and structural requirements and could be used for the final design. • Increase Di or increase Li : can be a solution when the maximum stress in the cable is over the limit. The stress may be reduced by increasing the cables cross section (D) or increasing the cable sag (L) which, in turn, reduces the pull-out forces. However, it is important to consider the change in self-weight and the additional stresses, attracted to the stiffer sections, in the second run. • Decrease Di or increase Li : the obtained deformation is less than the architectural requirements. Reducing the overall stiffness, by decreasing the diameter, or increasing the cables’ sag, by increasing its unloaded length, will increase the deflections. • Increase Di or decrease Li : the obtained deformation is more than the architectural requirements. Increasing the overall stiffness, by increasing the diameter, or decreasing the cables’ sag, by reducing its length, will reduce the deflections.

104

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 7. PRACTICAL CONSIDERATIONS The proposed iteration procedure contains many simplifications, for instance, concrete panels were not checked during the process. Nevertheless, it gives the reader a general idea of the work flow. Moreover, the cables are not necessarily equal in length and diameters, which may induce a more complex response optimization. Parametric design software can be of great help in these situations; they allow an easy manipulation of the input factors, and can be connected with FEM software for a faster iteration process.

Post-tensioned stress ribbon systems in long-span roofs

105

Chapter 8 Discussion and conclusions 8.1

Modelling and analysis

A series of investigations were performed in order to obtain realistic models, these investigations tested possible element choices, loading types and how these were applied. Moreover, the drape support condition, and its effect, were examined. Frame elements proved to give the same behaviour as cable elements, when specific modifications were applied. Since it’s easier to manipulate a frame element than a cable element, frame elements were here chosen to simulate the cable behaviour. Two distinct area elements were utilized for different purposes. A thick plate formulation, which lacks the membrane stiffness, was used to simulate a simply supported panel for the conventional cable suspended model. In contrast, the stress ribbon roof utilizes the membrane stiffness of the panels, therefore, thick shell formulation was used. Gap elements, also known as compression only links, were utilized to account for the contact pressure taking place between adjacent concrete panels in the deformed stress ribbon model. To allow for a relative movement between the bearing cables, sliding on the longitudinal direction was permitted, for the middle cable, using specific constraints between the cable and the crossbeams. Consequently, an easier construction process is possible, and no relative force re-distribution between cables will happen unintentionally. Bearing in mind the lack of back-stayed cables at each end of the drape, the effect of the supports’ relative movements were analyzed. The simple investigation carried out showed the insignificant effect these movements have in the drape structural design and geometrical restrictions. Therefore, pinned supports were considered close enough to reality.

Post-tensioned stress ribbon systems in long-span roofs

107

CHAPTER 8. DISCUSSION AND CONCLUSIONS The structural elements that govern the behaviour of the case studied here are the bearing cables. Therefore, a sensitivity analysis was only performed for them to determine the optimal discretization size that provides, sufficiently, accurate results with acceptable computational time.

8.2

Results evaluation

The results in Chapter 6 indicate the overall superiority of the post-tensioned stress ribbon model over the other models. Reaching the target architectural shape is out of the scope of this work, thus, more attention is given to examine the increased deflection under the snow load case. Sufficient stiffness in needed to prevent substantial deformations under snow load case. Unfortunately, both studied models result in significant deformation increase under snow load. Nonetheless, the post-tensioned stress ribbon model showed a better behavior, limiting the increase to 12 cm, compared to 42 cm recorded in the conventional cable suspended model. This deformation could be better controlled by increasing the dead to live load ratio, as was previously explained in Figure 2.12. Up to this point, both models showed acceptable performance. As soon as the shell stresses were examined, however, the suspended roof model performed poorly; the obtained concrete tensile stresses were 57% over the design capacity, thus, a big amount of reinforcement would be needed. During construction, the stress ribbon design will lack of prestressing action for a period of time, hence, the results of the stress ribbon model without post-tension are of interest as well. This model performs slightly better than the cable suspended model, yet, the stresses overpass their limit under snow load case. This issue can be solved by choosing a suitable time for placing the panels, ensuring the application of post-tension before it snows. In the post-tensioned stress ribbon model, the stresses comfortably fulfill the design requirements near the ends, however, they increase towards mid-span, getting closer to the limiting value. This phenomenon may be explained by two main reasons: first, although the prestress losses were set to zero, SAP2000 always includes the elastic deformation losses when the tendons are modelled, which is why the prestress effect abates as mid-span is approached. Secondly, the tensile stresses increase near mid-span due to the bending effect, which has more influence closer to mid-span due to the shape of the global deformation, which has slightly steep slopes near the ends and flat part closer to mid-span. Figure 8.1 shows a simple explanation of this behaviour.

108

Post-tensioned stress ribbon systems in long-span roofs

CHAPTER 8. DISCUSSION AND CONCLUSIONS

Figure 8.1: Local bending effect due to the deformed shape.

The aforementioned matter could be practically solved by using pre-tensioned concrete panels, which can locally limit the tensile stresses. However, using pre-tensioned panels would not replace the importance of the post-tensioning system which is responsible for increasing the overall stiffness and reducing the cable stresses. Temperature effects have proved to give unacceptable results regarding deflections. Positive temperature gradient results in the highest changes in deformations, increasing the dead load deflection by half a meter; as the cable elongates, or relaxes, under increasing temperature. To minimize the temperature effects, the following can be considered: employing insulation materials, installing cooling devices, or simply have a light colour in the cladding that attracts less heat.

8.3

Conclusion

Prestressed, or not prestressed, stress ribbon systems can be applied to cable suspended structures, offering many advantages over a conventional cable suspended system. Due to the reduced design forces, more efficient usage of the materials is achieved, besides, enhanced overall stiffness. With present assumptions and models, a prestressed stress ribbon system showed the best performance within the three tested models, achieving the lowest cable forces and shell stresses. Furthermore, its benefits surpass the drape to the corner structures, being designed for less vertical and pull-out forces. Consequently, this system is highly recommended by the authors to be used for the roof structure of Västerås Travel Center.

8.4

Recommendations for future research

Generally, stress ribbon systems are utilized for pedestrian bridges, which are characterized by their small width, which was the case for the studied roof. However, whether or not using this system for more conventional roof geometries would be as beneficial as it was for the studied case is yet to be investigated.

Post-tensioned stress ribbon systems in long-span roofs

109

CHAPTER 8. DISCUSSION AND CONCLUSIONS The dynamic characteristics of the roof, such as the natural frequencies and mode shapes, should be examined to assure a satisfactory behaviour against wind and traffic vibrations. As previously mentioned, the targeted architectural shape wasn’t perfectly matched in this design, but the outline of an iteration process, describing how to achieve it, was suggested. How to practically implement this process need to be further study. A parametric modelling software could be used in cooperation with SAP2000 for an efficient and fast manipulation of the relevant variables.

110

Post-tensioned stress ribbon systems in long-span roofs

Bibliography R R [1] CSI America. CSI Analysis Reference Manual For SAP2000 , ETABS , R and CSiBridge . R Berkeley, 15th edition, July 2016. xiv, xiv, xiv, 30, SAFE 31, 32, 33, 34, 35, 37, 38, 39, 42, 55, 56, 68, 77, 82, 85

[2] V.F. Arcaro. Scandinavium Arena. Institute for Membrane and Shell Technologies. http://www.arcaro.org/tension/album/scandinavium.htm. Accessed: 2018-04-09. xiii, 7 [3] C. Balmond, J. Smith, and C. Brensing. Informal. Architecture Series. Prestel, 2002. 12, 13 [4] K.J. Bathe. Numerical methods in finite element analysis. Civil engineering and engineering mechanics series. Prentice Hall, Englewood Cliffs, N.J, 1976. 31 [5] Boverket. Boverkets konstruktionsregler, EKS 10, Karlskrona, 2016. Boverket. 52 [6] H.A. Buchholdt. Introduction to Cable Roof Structures. Cambridge University Press, Cambridge, 1985. xiii, 6, 7, 8, 15, 16, 17 [7] E. Carrera, G. Giunta, and M. Petrolo. The Euler Bernoulli and Timoshenko Theories, pages 9–22. John Wiley & Sons, Ltd, 2011. 30 [8] CEN. EN 1991-1-3 Eurocode 1 - Actions on structures - Part 1-3: General actions - Snow loads, Brussels, 2003. CEN. 51, 52 [9] CEN. EN 1992-1-1 Eurocode 2 - Design of concrete structures - Part 1-1: General rules and rules for buildings, Brussels, 2004. CEN. xiv, xiv, 25, 26, 27, 28, 50 [10] D. Cobo Del Arco, A. Aparicio, and A. Mari. Preliminary design of prestressed concrete stress ribbon bridge. Journal of Bridge Engineering, 6(4):234–242, August 2001. 9 [11] R.D. Cook. Concepts and Applications of Finite element analysis. Wiley, New York, 4th edition, 2002. 29, 31, 33 [12] R. Furtado, C. Quinaz, and R. Bastos. The Engineering of Braga Municipal Stadium. Engineering Report. AFA Associates. xiii, xiii, 14, 15

Post-tensioned stress ribbon systems in long-span roofs

111

BIBLIOGRAPHY [13] N. Hewson. Prestressed Concrete Bridges - Design and Construction. ICE Publishing, 2nd edition, 2012. 26, 27, 28 [14] H. Ki-Jang, L. Nam-Hyoung, K. Man-Gi, and K. Kee-Dong. Efficient assumption of design variables for stress ribbon footbridges. KSCE Journal of Civil Engineering, 20(1):250–260, January 2016. 8 [15] N. Krishna Raju. Prestressed Concrete. Tata McGraw-Hill, New York, 4th edition, 2007. 5 [16] V. Kulbach. Cable Structures, Design and Static Analysis. Estonian Academy Publishers, Tallinn, 2007. 5, 6, 56, 104 [17] A. Pipinato. Chapter 26 - Case study: the Akashi-Kaikyo bridge . In Innovative Bridge Design Handbook, pages 681 – 699. Butterworth-Heinemann, Boston, 2016. 5 [18] J. Radnic, D. Matesan, and D. Buklijas-Kobojevic. Numerical model for analysis of stress-ribbon bridges. Gradevinar, 67(10):959–973, October 2015. 9 [19] J. Ravi. Dulles International Airport. https://commons.wikimedia.org/ wiki/File:Washington_Dulles_International_Airport.jpg. Accessed: 2018-02-10. xiii, 11 [20] J. Silfwerbrand and H. Sundquist. Prestressed concrete structures. Trita-BKN. Rapport, 122. Architecture and the Built Environment, KTH Royal Institute of Technology, Stockholm, 2008. xiv, 23, 24 [21] J. Strasky. Stress Ribbon and Cable-supported Pedestrian Bridges. Thomas Telford Publishing, 2004. xiii, xiii, xiii, xiii, 1, 7, 8, 10, 13, 15, 19, 20, 21, 22, 71 [22] J. Strasky, P. Juchelková, P. Kaláb, and R. Nečas. Roofs of prestressed concrete membranes. Structural Concrete, 18(1):5–18, 2016. 1 [23] C.V. Subramanian and L. Clemmie. Structural Systems for Large Span Sports Complex Structures. International Journal of Recent Trends in Engineering & Research (IJRTER), 03, 01 2017. 1 [24] H. Sundquist. Cable supported structures. Trita-BKN. Rapport ; 106. KTH, Stockholm, 2007. xiii, 18 [25] G. Tibert. Numerical analyses of cable roof structures. Master’s thesis, KTH Royal Institute of Technology, Stockholm, 1999. xiii, 6 [26] J.C. Walraven and C.R. Braam. Prestressed Concrete: CIE3150/CT4160. TU Delft, 2015. 22, 23, 25, 28 [27] R. Whitehead. Formative Experiences: Saarinen’s Shells and the Evolutionary Impact of Construction Challenges. Architecture Conference Proceedings and Presentations, 3:593–601, June 2015. 10, 11

112

Post-tensioned stress ribbon systems in long-span roofs

BIBLIOGRAPHY [28] C. Wilkinson. Supersheds : the Architecture of Long-span, Large-volume Buildings. Butterworth Architecture, 1991. 5 [29] The Portuguese National Pavilion, Balmond Studio. http://www. balmondstudio.com/work/portuguese-national-pavilion.php. Accessed: 2018-02-10. xiii, xiii, 12, 13 [30] CSI America. SAP2000. https://www.csiamerica.com/products/sap2000. Accessed: 2018-02-18. 30 [31] Fatzer structural wire ropes brochure. http: //www.fatzer.com/wp-content/uploads/2018/02/ FATZER-Seilbau-Structural-Ropes-Brochure-metrisch-DE-EN.pdf. Accessed: 2018-03-23. xiv, 49, 69 [32] Hovet. SGA Fastigheter. http://www.sgafastigheter.se/arenorna/hovet/. Accessed: 2018-04-20. xiii, 6 [33] Effects of thermal actions from environment on concrete structures. http://www.kstr.lth.se/english/research/alla-projekt/ effects-of-thermal-actions-from-environment-on-concrete-structures/. Accessed: 2018-05-14. 68 [34] Bridge Automation. Sacramento River Bridge. http://bridgeautomation. com/blog/2014/08/sacramento-river-trail-bridge/. Accessed: 2018-0222. xiii, 10 [35] Washington Dulles International Airport, Main Terminal Expansion. https://www.som.com/projects/washington_dulles_international_ airport__main_terminal_expansion. Accessed: 2018-02-13. 11 [36] The tendon element in SAP2000, CSI Knowledge Base. https://wiki. csiamerica.com/display/kb/Modeling+different+types+of+tendons. Accessed: 2018-03-23. 35 [37] Västerås Kommun. Resecentrum Projekt. https://www.vasteras.se/ kommun-och-politik/vasteras-utvecklas/malarporten/resecentrum. html. Accessed: 2018-02-21. xiv, xiv, 44 [38] BIG Architects. Västerås Travel Center project. https://www.big.dk/ #projects-vtc. Accessed: 2018-03-28. xiii, xiii, 2, 43 [39] VSL Strand Post-tensioning systems. http://www.vsl.com/brochures/ post-tensioning-strand-systems.html. Accessed: 2018-04-17. 50, 51

Post-tensioned stress ribbon systems in long-span roofs

113

Appendix A Analytical calculation for simply supported cable

Post-tensioned stress ribbon systems in long-span roofs

115

APPENDIX A. ANALYTICAL CALCULATION FOR SIMPLY SUPPORTED CABLE

nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn nnnnnnnnnnnn n nnnnnnnnnnnnnnnn nnnnnnnnnnnnnnnnnnnnnnnnnnn n nnnn n nnnnnn


      

                                     

nnnnnnnnnnnnnnnnnnnnnnnnnnnnnn nnnnnnnnnnnnnnnnnnnnnnnnnnnn

nnnnnnnnnn nnnnnnnnnnnnnnnnnnnnn nnnnnnnnnnnnnnnnnnnnnnnnn nnnnnnnnnnnnnn nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn nnnnnnnnnnnnnnn

nnnnn nnnnnnnnnnnnnnnnnnnnnnnnnnnn

  &


        
        !"# $%   ''''''''        

nnnnnnnnnnnnnnnnn nnnnnnnnnnnnnnnn nnnnnnnn


   () !"# $%  +     ,-      nnnnnnnnnnnnnnnnnnnnnnnnnnnn  )   )*  -.  '''''''''''''''''''''''' ( ( , ,
      &    )!"# $%     ) +  -.-.     nnnnnnnnnnnnnnnnnnnnnnnn    * *     () +   ,-,-    /   ()!"# $%     *  .. * (  (  ,   $ % ,  -   !"#  -     0   )  *  )* .  .

116

nnnnnnnnnn nnnn

nnnnnnnnnnnnnn

Post-tensioned stress ribbon systems in long-span roofs

APPENDIX A. ANALYTICAL CALCULATION FOR SIMPLY SUPPORTED CABLE

nnnnn nnnnnnnnnnnnnnnnnnn

1 2

      

  3    ( (  ,3  , )*  -.   )*  -.

nnnnnnnnnnnnnn  nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn

3 ( ,      ) -.  $ 2%  *
4    

      5          (*    7 ,.  6   ,   $ 2%   () *  -.    
4  6 1 +     8   1  6  8  (* 8,.    

nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn

Post-tensioned stress ribbon systems in long-span roofs

117

APPENDIX A. ANALYTICAL CALCULATION FOR SIMPLY SUPPORTED CABLE

118

Post-tensioned stress ribbon systems in long-span roofs

Appendix B Prestress losses

Post-tensioned stress ribbon systems in long-span roofs

119

APPENDIX B. PRESTRESS LOSSES

::::::::::::::::




:
:::::

::::

:

:
:::::
::::

:


::::
::
::::::::


::
:
::::::::
:
::: :
:::::
::

:


:


:


:
:



::
:::
:::
:


:
::

:

:: :


::::
:


:
:::::

:::: :::


:::

:




::
::
::::::::


:
:
:

:


::
:: 
:::



    
 
:

::



: 
::
::


:
:::::
:
:
::


 

aaaaaaaaaaaaa  

:

::

:::


 



:
::
:
:

:

 
:::

:


        

                     !"  #  

 $   % $ &  %




:


::::

:

:
:
:::

:

:
:: :
:: ::::::
::
: :
::
::::

:


:
::
:


::::

:

:
::::: :
:::

:::
::
::
:

:
::
:


:::

:

:
::::: 
::
::::

:::


::

:
:


 


::

::::

::

:

'          !"   '   ( '   $'&  % $')*+&   $'&     !" %



:
::
:

:



::
::
:

:
::::::
::
: :
::
:


:

::

::::

::

: :::::::
::
:


:
:
:::


::
::
::
:


::::

:

:
::


::


:
::
:::
::
::
:
:
:::
:
:::

:::::
:
::::

::

:

:
::
:

::

::::

::

:
:
:::

:::::


:
::: ::

::
::
:

::

::::

::

:
:


120

Post-tensioned stress ribbon systems in long-span roofs

APPENDIX B. PRESTRESS LOSSES

,+ 
:
,- 
:
./ 01 ,+  '  $'& 2 ,-  '  $')*+&!"     !" ,3

aa

::: :
:::::::
:

::

::
::
:



 


aa a a a aa a

Post-tensioned stress ribbon systems in long-span roofs

121

APPENDIX B. PRESTRESS LOSSES

aa

aa a a a aa a

 
::

::
:
::
:


::::

:


45 5 *67

./  %   45

 
:::

:

:

:: :



8 9)  :

    

     8        # 

 = :!" ;<: 

  >>>>>  ! 9  = :!"   ?) @ " ,A 

:
::::::
::
::

::
::
:


::::

:

::
:


:
::


::


:
::
::
:


:
:
::: ::

::
::
:


::::

:

:
::
::
::
:


:
:
:::
::

::
:
::
:
::
:

:
:
::



::
::
:::
::::::::  

:
:::::
   

::
:


::
:


::::

:


:::


::


::





:::
:
:::

::
:


::::

:

::
::





::: :::::: :::::::
::



::
:


::::

:



::

::
:


:::
::
:::::







::
:


::::

:



::

::
:





::::

::


::

:

:: :



:
:::::
::


:
:
::

::
:


::::::::
::


 


::

 


 :
:::::
::
::

:
:
::
:::::

 

::

 :
:::::
::
::

:
:
::

:::::

 

::

E ::
::


::::



::
::
 :::


:::
:::::: E  CD !  !   ? = ? @    CD) "@ " JKNOPQR M SSS TUV \]!  WX YZ[ ^"@   N_  ;FG    N# L ? H <*)   

   B<:+!"  I  

::

B<:+ B<:CD CD) ;FG

H < ;<:  ,A  H <*)    N# ;.:  = I NX` M a bcd 

122


::

:

:: :



Post-tensioned stress ribbon systems in long-span roofs

APPENDIX B. PRESTRESS LOSSES

H . H . H :

$ & =  %!" N_ N#

   %       ;.:  H .     N#  ::

:
: :
:

:: :

H < H .    N#
::::
:

:: :



 
:::

:

:


:

)  ef*f) 

 

:
:::::
  




::
:


::::

:

::
:::

::
:::
::



::
::::
:


:
::
:::::
::

 


::

::::

::

:


:::::::

 

:
:::::
  




./     ! % 5'g  ::: :


::

::
:


:::::::
:

::

:: " '  h+))) E ::

::


:::::::
::::



:::


:
:::::::

::

::



::

:
::
:
5 'g ijkl  $'&  X`mr M JLW N nopq\^ Nr  !  _`m n M opq 45'j 5'g    h+)))  I  ?    %   "@ :::
:



::
::

:


:::::::

::::



aa

a

a a

Post-tensioned stress ribbon systems in long-span roofs

123

APPENDIX B. PRESTRESS LOSSES

aa

a

aaaaa

a a a aa a

  
::::::


::
::::::
::
::::

:


s 

:
:::::
::


:
:

::
::
: :


::
:
:
:::
:

::

::


::
:


:::

:::





::::
:
:



s  45 4kl '  '    

:::
:



::

::::::


::
::::::
::
::::

:


  

::::::
::::
:

 

:
:::::
   

t  u  i   u u ,     4v wxy ./   = I Nn M z| { } M ~!" 4v w y  ,3
!@  ,3 4v*g<:'.€ 4v ?  " 4v w
y 4v w y  ,3   
::
:
:


::::
:

4v*g<:'.€   ,3 4  


4‚   >>>>>>>>> 4‚  '  '   ƒ  4 4.€  4  ƒ   ,3

a aa

124

 

:
:::::
   

: :
::
::
::


::::::
:
:::
:




::::::

 
::::::
::
:::::
::

:
::::
:::::::
::
::


::::::
:
::

:
:::
:



::
:
::::::

 

:
:::::
   

:

:::
::
::
:

::
:


:::
:
:





::


: ::
::




: 
: ::
::




:

:


::::::

:::
::



:




: ::
::




: :::
:



::


: ::
::




:

aa a aa aaa

Post-tensioned stress ribbon systems in long-span roofs

APPENDIX B. PRESTRESS LOSSES

a aa

aa a aa aaa

'  e  5 H :  '   45'j    f*f) *67  ,3 4 *:*j '   '   '  ?    (  !@     e !   f*f)"      ' " 

::: :


::

::
:


::::
::
::
:

::
:


:

::

::
:
::
:


:

:

:: :


::



:::::::

./     !" ,3 .„fkj:A…jfl…::k: ./ = 4v*g<:'.€ 4kl 4.€!"      !" ,3 .„fkjf…f.ll…::k: ./ = 4v*g<:'.€ 4kl 4.€ 4 *:*j!"     !" ,3 .„fkjf…f.ll…::k:   ./



:
:::


::
:


:::::
::
:


::
:::


:::
:


::::
:

Post-tensioned stress ribbon systems in long-span roofs

125

APPENDIX B. PRESTRESS LOSSES

126

Post-tensioned stress ribbon systems in long-span roofs

Appendix C Initial cable geometry calculation in SAP2000

Post-tensioned stress ribbon systems in long-span roofs

127

APPENDIX C. INITIAL CABLE GEOMETRY CALCULATION IN SAP2000

:::::: ttttttttttt t

t tt

ttttttt tttt tt t ttt

:::::: tttttt ttttt tttttttt tttttttt tttttttt

 tttttttt t

tt tttttt t ttt ttttttttttt

128


 

                   !     !            
"#$%
 & '  
 (    
"#$%    
  
     ! %) 
"#$% %) 
"#$% 
    ! *  

%) 
"#$%    ! +   *  & +    . ! , -------

Post-tensioned stress ribbon systems in long-span roofs