Carbon Fibre Reinforced Polymer (CFRP) Cables for Orthogonally Loaded Cable Structures : Advantages and Feasibility [1 ed.] 9783832594442, 9783832541286

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Yue Liu

Carbon Fibre Reinforced Polymer (CFRP) Cables for Orthogonally Loaded Cable Structures: Advantages and Feasibility

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Carbon Fibre Reinforced Polymer (CFRP) Cables for Orthogonally Loaded Cable Structures: Advantages and Feasibility

vorgelegt von Master of Science – M.Sc. Yue Liu geb. in Chongqing, China Von der Fakultät VI – Planen Bauen Umwelt der Technischen Universität Berlin Institut für Bauingenieurwesen zur Erlangung des akademischen Grades Doktor der Ingenieurwissenschaften Dr.-Ing. genehmigte Dissertation Promotionsauscschuss: Vorsitzender: Prof. Dr.-Ing. Volker Schmid Gutachter: Prof. Dr. sc. techn. Mike Schlaich Gutachterin: Prof. Dr.-Ing. Annette Bögle Gutachter: Dr.-Ing. Arndt Goldack Tag der wissenschaftlichen Aussprache: 26. August 2015  Berlin 2015

Zugl.: Berlin, Technische Universit¨at, Diss., 2015

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.d-nb.de .

c

Copyright Logos Verlag Berlin GmbH 2015 All rights reserved. ISBN 978-3-8325-4128-6

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Contents Preface 

vii 

Vorwort 

ix 

Outline 

xi 

1 

1 

2 

3 

Introduction  1.1 

Characteristic Cable Structures 

1 

1.2 

Structural Forms of Cable Structures 

5 

1.3 

Cable Structures with CFRP Cables 

7 

Concept of Orthogonally Loaded Cable Structures  2.1 

Design Principles of Cable Structures 

21 

2.2 

Pre-tension Theory of Cables 

23 

2.3 

Stiffness Matrices of Cable Elements 

27 

2.3.1  Finite element discretisation of cable structures 

27 

2.3.2  Formulation of stiffness matrices for three dimensional cable elements 

28 

2.3.2.1 

Virtual work theory 

29 

2.3.2.2 

Relationship between displacement and strain 

31 

2.3.2.3 

Stiffness matrices in the local coordinate system 

32 

2.3.2.4 

Transformation of coordinate system 

37 

2.4 

Composition of Structural Stiffness of Cable Structures 

40 

2.5 

Definition of Orthogonally Loaded Cable Structures 

47 

Carbon Fibre Reinforced Polymer (CFRP) Cables  3.1 

Introduction of CFRPs 

49  49 

3.1.1  Composition of CFRPs 

49 

3.1.2  Fabrication of CFRPs 

55 

3.2 

4 

21 

3.1.2.1 

Contact moulding 

56 

3.1.2.2 

Resin transfer moulding (RTM) 

57 

3.1.2.3 

Pultrusion 

58 

Introduction of CFRP Cables 

59 

3.2.1  Structural forms of CFRP cables 

59 

3.2.2  Mechanical properties of CFRP cables 

60 

3.2.3  Unit prices of CFRP cables 

62 

Case Study A: Using CFRP Cables in a Cable Net Facade 

65 

4.1 

Introduction of Cable Net Facade 

65 

i 

 

5 

4.2 

Design Parameters of Investigated Cable Net Facade 

67 

4.3 

Principles of Comparison and Design 

69 

4.4 

Comparison Results 

71 

4.4.1  Comparison result of structural stiffness 

72 

4.4.2  Comparison result of amount of cable used 

76 

4.4.3  Comparison of economic efficiency 

80 

Case Study B: Using CFRP Cables in a spoked wheel Cable Roof 

83 

5.1 

Introduction of spoked wheel Cable Roof 

83 

5.2 

Design Parameters of Investigated Spoked Wheel Cable Roof 

86 

5.3 

Principles of Comparison and Design 

89 

5.4 

Comparison Results 

91 

5.4.1  Comparison result of structural stiffness 

92 

5.4.2  Comparison result of amount of cable used 

96 

5.4.3  Comparison of economic efficiency  5.5 

101 

Influence of  on Advantages of CFRP Orthogonally Loaded Cable Structures104 

5.5.1  Influence of varying  on the advantage in respect of structural stiffness 

104 

5.5.2  Influence of varying  on the advantage in respect of amount of cable used 106  6 

5.5.3  Influence of varying  on the advantage in respect of cable cost 

108 

Anchoring CFRP Cables in Orthogonally Loaded Cable Structures 

111 

6.1 

Existing CFRP Cable Anchorages 

111 

6.1.1  Clamp anchorages 

111 

6.1.2  Bond anchorages 

115 

6.1.3  Pin-loaded anchorages 

118 

6.2 

Design Principles of CFRP Cable Anchorages 

119 

6.3 

New Design A: CFRP Cable Winding-Clamp Anchorage 

125 

6.3.1  Conceptual design 

125 

6.3.2  Actual design 

130 

6.3.3  Finite element analysis 

133 

6.3.3.1 

Finite element model 

133 

6.3.3.2 

Material properties and constitutive relations 

135 

6.3.3.3 

Finite element analysis results 

142 

6.3.4  Experimental verification 

6.4 

146 

6.3.4.1 

Experimental set-up 

146 

6.3.4.2 

Experimental result 

147 

New Design B: CFRP Cable Thimble-Clamp Anchorage 

ii 

150 

  6.4.1  Conceptual design 

150 

6.4.2  Actual design 

153 

6.4.3  Finite element analysis 

155 

6.4.3.1 

Finite element model 

155 

6.4.3.2 

Material properties and constitutive relations 

157 

6.4.3.3 

Finite element analysis results 

157 

6.4.4  Experimental verification 

7 

162 

6.4.4.1 

Experimental set-up 

162 

6.4.4.2 

Experimental result 

163 

Prototype of CFRP Orthogonally Loaded Cable Structure 

167 

7.1 

Conceptual Design 

167 

7.2 

Finite Element Analysis and Experimental Verification 

169 

7.3 

Manufacture of Components and Construction of Actual Structure 

174 

8 

A Novel Design: CFRP Continuous Band Winding System 

179 

9 

Conclusions and Recommendations 

187 

9.1 

Conclusions 

187 

9.2 

Recommendations 

189 

List of Symbols 

191 

List of Tables 

195 

List of Figures 

197 

References 

207 

Appendix A – Numerical Analysis Procedure of Cable Structures 

215 

A.1 

Introduction to Theory 

215 

A.2 

Case Studies 

218 

A.2.1  Case A: initial control angle  = 89° 

219 

A.2.2  Case B: initial control angle  = 85° 

221 

A.2.3  Case C: initial control angle  = 81° 

224 

Appendix B – FORTRAN Source Code: ABAQUS UMAT Subroutine   

iii 

227 

 

To my doctoral supervisor: Prof. Dr. sc. techn. Mike Schlaich. To other experts who helped me a lot: Prof. Dr.-Ing. Volker Schmid, Prof. Dr.-Ing. Annette Bögle, Dr.-Ing. Arndt Goldack and Dipl. -Ing. Bernd Zwingmann. To my parents: Gang Liu and Ling Luo.

v 

 

Preface Reviewing the history of cable structures, it can be found that the development of cable materials can significantly promote the development of structures. Carbon Fibre Reinforced Polymer (CFRP) is an advanced composite material with advantages of high strength, lightweight, no corrosion and high fatigue resistance, which makes it suitable to be made into cables and replace steel cables in a broad range of applications. The application of CFRP cables is now driving the progress of cable structures. The ideal structures for such cables are orthogonally loaded cable structures, which can be defined as cable structures with a majority of cables orthogonally loaded or approximately orthogonally loaded by external loads. In orthogonally loaded cable structures, such as many cable roofs and cable facades, the structural stiffness is mainly comprised of the geometric stiffness which is controlled by the pre-tension force of cables, instead of the elastic stiffness controlled by the Young’s modulus of cables. This means, for such cable structures, increasing the tensile strength of cables, which can bring about the increase of pre-tension force, is a more efficient way to either raise the structural stiffness if the amount of cable used remains unchanged or reduce the amount of cable used if the structural stiffness is maintained, compared to increasing the Young’s modulus of cables. This phenomenon also implies that using CFRP cables, whose tensile strength is considerably greater than that of steel cables, in the orthogonally loaded cable structures will improve the economic efficiency of structures, although the Young’s modulus of CFRP cables is usually smaller than that of steel cables and their unit price is also much higher. To illustrate the above argument, after introducing the history of cable structures and their structural behaviours, the forms of elastic stiffness and geometric stiffness of cables were strictly derived in this book, and then cable structures were classified into orthogonally loaded cables structures and other cable structures according to the proportions of elastic stiffness and geometric stiffness in total structural stiffness, which are determined by the control angle, i.e. the angle between the cable axis and external load. Afterwards, two typical orthogonally loaded cable structures, i.e. the cable net facade and the spoked wheel cable roof, were selected and investigated in two case studies

vii 

  successively. The mechanical properties and economic efficiency of these two structures with CFRP cables of different Young’s moduli and different tensile strengths were compared with those of the corresponding steel cable structures. Because these two orthogonally loaded cable structures have different control angles, the influence of this angle on the advantages of using CFRP cables in orthogonally loaded cable structures was subsequently studied through comparing these two CFRP cable structures with each other. In addition to the advantages, the feasibility of using CFRP cables in orthogonally loaded cable structures is also investigated in this book. In order to solve the challenge of anchoring CFRP cables in such cable structures, new design principles for CFRP cable anchorages were proposed based on analysing the existing anchorages of CFRP cables. According to these principles, two novel CFRP cable anchorages, i.e. the winding-clamp anchorage and the thimble-clamp anchorage, were designed and proposed herein. FEM simulations and experiment verifications of these anchorages were also presented. Then, a prototype CFRP spoked wheel cable roof, which was designed and built by the author and colleagues in Technical University of Berlin, was introduced to show the feasibility of the construction of CFRP orthogonally loaded cable structures based on the present technology. Furthermore, a novel form of using CFRP cables, i.e. the continuous band winding system, which is suitable for CFRP orthogonally loaded cable structures, was also proposed in this book. The CFRP continuous band winding systems do not need any anchorage and thus offering a complete solution for the problem of anchoring CFRP cables. The research results show that CFRP cables can effectively improve the mechanical and economical performances of orthogonally loaded cable structures and this improvement for orthogonally loaded cable structures with the control angle closer to 90° is greater than that in such structures with this angle farther from 90°; the proposed two new designs of anchorages are able to anchor corresponding CFRP cables with 100% anchoring efficiency. The successful construction of the prototype structure shows that design and building of CFRP orthogonally loaded cable structures are feasible based on the present technology and moreover the continuous band winding system is a possible form of using CFRPs in orthogonally loaded cable structures very efficiently.

viii 

 

Vorwort Wenn man auf die Geschichte der Seiltragwerke zurückblickt, wird deutlich, dass neue Seilmaterialien die Entwicklung von Seiltragwerken signifikant gefördert haben. Der kohlenstofffaserverstärkte Kunststoff (CFK) ist ein fortschrittlicher Verbundwerkstoff mit den Vorteilen hohe Festigkeit, geringes Gewicht, keine Korrosion und hohe Dauerfestigkeit. Mit diesen Vorteilen ist CFK für Zugelemente geeignet, die in einem breiten Spektrum von Anwendungen Stahlseile ersetzen können. Die Anwendung von CFK-Zugelementen trägt zur Weiterentwicklung von Seiltragwerken bei. Die mechanisch idealen Tragwerke für solche Zugelemente sind die orthogonal belasteten Seiltragwerke. Bei diesen sind die Seile hauptsächlich orthogonal bzw. annähernd orthogonal zur Seilachse durch äußere Lasten beansprucht.  In den orthogonal belasteten Seiltragwerken, wie bestimmte Seildächer und Seilfassaden, entsteht die Tragwerkssteifigkeit hauptsächlich nicht aus der elastischen Steifigkeit der Seile, sondern aus der geometrischen Steifigkeit. Diese wird durch die Vorspannkraft der Seile bestimmt, jedoch nicht durch deren Elastizitätsmodul. Das bedeutet, dass für ein solches Seiltragwerk, eine Erhöhung der Tragwerkssteifigkeit mit steigender Zugfestigkeit der Seile möglich ist. Im Mittel können dadurch die Seilquerschnitte verringert werden, ohne dass die Verformung ansteigt. Dieser Effekt erklärt, dass die Verwendung von CFK-Zugelementen, deren Zugfestigkeit größer ist als die von Stahlseilen, in orthogonal belasteten Seiltragwerken zu wirtschaftlicheren Tragwerken führen kann. Diese Einsparungen entstehen obwohl der Elastizitätsmodul der CFK-Zugelemente geringer und deren Materialpreis im Vergleich zu Stahlseilen höher ist. Diese Arbeit beginnt mit einer Einführung zur Geschichte der Seiltragwerke und deren Tragverhalten. Um die oben genannte These begründen zu können, werden daraufhin die elastische und die geometrische Steifigkeit des Einzelseiles hergeleitet. Danach werden die Seiltragwerke entsprechend dem Verhältnis von elastischer zu geometrischer Steifigkeit an der Gesamtsteifigkeit klassifiziert. Der maßgebende Parameter zur Unterscheidung zwischen orthogonal belasteten und anderen Seiltragwerken ist der Winkel zwischen der Seilachse und der externen Last (Lastwinkel).

ix 

  Zur Begründung der These werden zwei typische orthogonal belastete Seiltragwerke, eine Seilnetzfassade und ein Ringseildach, ausgewählt und nacheinander in zwei Fallstudien untersucht. Dazu werden CFK-Zugelemente mit verschiedenen Elastizitätsmodulen und Zugfestigkeiten sowie Stahlseile ausgewählt und die Tragwerke damit numerisch untersucht. Die mechanischen Eigenschaften und die Wirtschaftlichkeit der untersuchten Tragwerke werden miteinander verglichen. Da diese zwei Seiltragwerke unterschiedliche Lastwinkel haben, ist es möglich dessen Einfluss auf die Verwendung von CFKZugelementen anhand der Ergebnisse zu beschreiben. Zusätzlich zu den theoretischen mechanischen und wirtschaftlichen Vorteilen, wird die technische

Verwendbarkeit

von

CFK-Zugelementen

in

orthogonal

belasteten

Seiltragwerken untersucht. Um die Herausforderung der Verankerung von CFKZugelementen zu lösen, werden bereits existierende Verankerungen analysiert und neue Verankerungskonzepte vorgeschlagen. Anhand dieser Konzepte, werden zwei neuartige Verankerungen für CFK-Zugelemente, die Wickel-Klemm-Verankerung und die KauscheKlemm-Verankerung, entwickelt. Die numerischen und experimentellen Untersuchungen dieser Verankerungen werden vorgestellt. Dann wird der Prototyp eines CFK-Ringseildachs präsentiert, das vom Autor und seinen Kollegen an der Technischen Universität Berlin gebaut wurde, um die technische Machbarkeit von orthogonal belasteten Seiltragwerken mit CFK-Zugelementen auf der Grundlage des heutigen Stands der Technik zu zeigen. Außerdem wird die neue Strukturform vorgeschlagen, die insbesondere für orthogonal belastete Seiltragwerke geeignet ist. Die kontinuierliche Bandwickelstruktur benötigt keine Verankerung und bietet somit eine innovative Lösung für die Verankerung von CFK-Zugelementen. Die Forschungsergebnisse dieser Arbeit zeigen, dass CFK-Zugelemente die mechanischen und wirtschaftlichen Eigenschaften von orthogonal belasteten Seiltragwerken effiktiv verbessern können. Diese Verbesserung ist umso größer, je näher der Lastwinkel an 90° liegt. Die beiden vorgeschlagenen neuen Verankerungen können CFK-Zuglemente mit 100% Effizienz verankern. Der erfolgreiche Aufbau des Prototyps eines orthogonal belasteten Seiltragwerks zeigt, dass Entwurf und Bau dieser Tragwerke auf der Grundlage der gegenwärtigen Technologie möglich sind. Außerdem wird gezeigt, dass die kontinuierliche Bandwickelstruktur eine mögliche und effiziente Verwendung von CFK in orthogonal belasteten Seiltragwerken ist.

x 

 

Outline This book is organised into nine chapters. In Chapter 1, the development history and structural forms of cable structures are introduced. State of the art, i.e. existing researches about CFRP cable structures and existing real CFRP cable structures, is presented. In Chapter 2, the concept of orthogonally loaded cable structures is explained. First, the design principles of cable structures and the theory of cable pre-tension are introduced. Second, the stiffness matrices of cable element are derived strictly; especially, the form and characteristics of geometric stiffness matrix are described. Then, taking a simple double-curved cable net as the research object, the influence of the control angle (i.e. the angle between the cable axis and external load) on the structural stiffness of cable structures is investigated; based on this angle, the cable structures are classified into two categories, i.e. orthogonally loaded cable structures and other cable structures. The great potential of using CFRP cables in orthogonally loaded cable structures is illustrated theoretically. In Chapter 3, the advanced composite tension members, i.e. the CFRP cables, are introduced. First, the cable material, i.e. the CFRP, and its composition and fabrication are presented. Then, the characteristics of CFRP cables including their forms, mechanical properties and unit prices are illustrated. Chapter 4 and Chapter 5 are devoted to the advantages of using CFRP cables in orthogonally loaded cable structures. In Chapter 4, a cable net facade, which is a typical orthogonally loaded cable structure, is selected and investigated in Case Study A. The mechanical properties and economic efficiency of cable net facades using CFRP cables with different tensile strengths and different Young’s moduli are compared with those of a corresponding steel cable net facade.

xi 

  In Chapter 5, another typical orthogonally loaded cable structure, i.e. a spoked wheel cable roof, is selected and investigated in Case Study B. The mechanical properties and economic efficiency of spoked wheel cable roofs with CFRP cables of different tensile strengths and different Young’s moduli are compared with those of a corresponding steel spoked wheel cable roof. Furthermore, the influence of different control angles on the advantages of CFRP orthogonally loaded cable structures is also analysed through comparing the CFRP cable net facades with the CFRP spoked wheel cable roofs. Chapter 6, Chapter 7 and Chapter 8 are devoted to the feasibility of using CFRP cables in orthogonally loaded cable structures. In Chapter 6, existing anchorages for CFRP cables are classified and analysed; design principles of CFRP cable anchorages are proposed based on this analysis. In accordance with such principles, two novel designs of CFRP cable anchorages, i.e. the winding-clamp anchorage and the thimble-clamp anchorage, are proposed; finite element analyses of these two anchorages are carried out and corresponding verification experiments are performed. In Chapter 7, a prototype CFRP spoked wheel cable roof built by the author and colleagues in Technical University of Berlin is introduced, including the design details and the finite element analysis. In Chapter 8, a novel design of CFRP cable system, i.e. the CFRP continuous band winding system, is also presented to show a possible and efficient way of applying CFRP cables in orthogonally loaded cables structures. Chapter 9 is devoted to the conclusion. In this chapter, the main findings of this research are summarised and the recommendations for further investigations are presented. Novel thoughts and methods were developed and adopted in the research field of this book. Innovation points of this research mainly include: (a)

the term of orthogonally loaded cable structures was proposed;

(b)

the advantages of using CFRP cables in orthogonally loaded cable structures were demonstrated;

xii 

  (c)

the new design principles and two new designs of CFRP cable anchorages for CFRP orthogonally loaded cable structures were proposed;

(d)

a prototype orthogonally loaded CFRP cable structure built by the author and colleagues was presented;

(e)

a novel form of using CFRP cables in orthogonally loaded cable structures, i.e. the CFRP continuous band winding system, was proposed

Through the research of this book, the advantages and feasibility of using CFRP cables in orthogonally loaded cable structures are demonstrated. It is hoped that the application of CFRP cables in this field will thus be heeded and developed in the future.

xiii 

 

1

Introduction

1.1 Characteristic Cable Structures A cable structure can be defined as a structure in which a cable or a system of cables is used as the primary load bearing structural element [Krishna, 1978] The history of human built cable structures can be traced back to the Stone Age. At that time, people utilised vines and creepers as cables to build hammocks [Sima and Watson, 1961] and suspension bridges [Kawada, 2010], which is shown in Figure 1.1. In addition to cable structures built on land, cables have also been used in sailboats on the sea for thousands of years [Carter, 2006], which is shown in Figure 1.2.

Figure 1.1: Suspension bridge with vines and creepers

Figure 1.2: Cable system in a sailboat

1 

 

The materials for cables have developed from natural materials in ancient times to wrought iron to high-strength steel today. The availability of high-strength steel cables allows the construction of long-span cable structures [Buchholdt, 1999], such as modern suspension bridges (see Figure 1.3) and modern cable-stayed bridges (see Figure 1.4).

Figure 1.3: Modern suspension bridge: Golden Gate Bridge, San Francisco, USA, 1937 (a) photo (photo credit: Motion Drives & Controls Ltd) (b) schematic diagram of structural form As can be seen from Figure 1.3, the span of the cable supported bridge has already achieved more than one thousand meters back in 1930s.

2 

 

Figure 1.4: Modern cable-stayed bridge: Ting Kau Bridge, Hong Kong, China, 1997 (a) photo (photo credit: HK Arun) (b) schematic diagram of structural form In addition to cable supported bridges, other application fields of cables include roofs and facades. The first use of high-strength steel cables in a cable roof was for the North Carolina State Fair Arena at Raleigh, USA in 1953 [Berger, 2008], which is shown in Figure 1.5. The cables of this roof were pre-tensioned and anchored at two rigid intersecting concrete arches to form a saddle-shaped (also known as double-curved) cable net. The diameter of this cable net is approximately 90 m and the size of the cable grid is approximately 5 m × 5 m. Since the completion of this arena, many steel cable roofs and facades have been built all over the world.

3 

 

Figure 1.5: North Carolina State Fair Arena at Raleigh, Raleigh, USA, 1953 (a) photo (photo credit: State Archives of North Carolina) (b) schematic diagram of structural form A milestone in the development of cable structures is the Munich Olympic Stadium (see Figure 1.6), which was completed in 1972. Different from the Fair Arena at Raleigh which applies the saddle-shaped cable net with rigid border, the cable net of the Munich Olympic Stadium is tent-shaped and adopts the soft border, which is also made of cables. This stadium shows that the structural ideal of “leicht und weit” can be perfectly achieved using cables [Schlaich et al., 2003].

Figure 1.6: Munich Olympic Stadium, Munich, Germany, 1972 (a) photo (photo credit: Arad Mojtahedi) (b) schematic diagram of structural form One of the most important expert in the field of light-weight and cable structures is Frei Otto. In his classic book, “Das hängende Dach: Gestalt und Struktur” [Otto, 1954], Frei Otto introduced many innovative designs of cable structures, which provided a lot of enlightenment and reference for subsequent designers.

4 

 

1.2 Structural Forms of Cable Structures The ability to achieve long spans with little self-weight and high bearing capacity, as well as to express elegance and beauty, makes cable structures popular. After decades of development, the structural forms of modern cable structures are many and varied; they have been used as cable bridges, cable roofs, cable facades, cable towers, cable masts, cable antennae and etc. Nine typical cable structures are shown in Figure 1.7.

1: Suspension bridge, Highway Bridge Khor al Batah, Oman (photo credit: STRABAG); 2: Cable-stayed bridge, Evripos Bridge, Greece (photo credit: schlaich bergermann und partner); 3: Cable truss roof, Annex Lutherhaus Roof, Germany (photo credit: Monika Nikolic); 4: Cable suspension roof, Glass Canopy of Station Plaza Heilbronn, Germany (photo credit: schlaich bergermann und partner); 5: Cable net facade, Facade Airport Málaga, Spain (photo credit: Roschmann); 6: Saddle-shaped cable net, Canopy in Autostadt Wolfsburg, Germany (photo credit: schlaich bergermann und partner); 7: Cable net tower, Dry Cooling Tower in Schmehausen near Hamm, Germany (photo credit: schlaich bergermann und partner); 8: Arch tower cable supported roof, Moses Mabhida Stadium, South Africa (photo credit: schlaich bergermann und partner); 9: Spoked wheel cable roof, Bay Arena Leverkusen, Germany (photo credit: Bayer 04 Leverkusen Fußball GmbH). Figure 1.7: Typical cable structures

5 

  In general, cable structures can be divided into two categories: (a)

in-plane cable structures;

(b)

spatial cable structures.

The in-plane cable structures are defined as the cable structures in which the loads are in the plane of the cable system, such as the beam string and the single-plane cable-stayed bridge. The spatial cable structures are defined as the cable structures in which the loads are not in the plane of the cable system, such as the cable net facade and the spoked wheel cable roof. Actually, any spatial cable structure can be formed by translating or rotating a corresponding in-plane cable structure. Based on this, the spatial cable structures can be further divided into two categories: (i)

spatial cable structures from translation;

(ii)

spatial cable structures from rotation.

The above classification of cable structures can be illustrated by Figure 1.8. It should be noted that this classification follows the way of classifying structures proposed by Mike Schlaich [Schlaich, 2006].

Figure 1.8: Classification of cable structures

6 

 

1.3 Cable Structures with CFRP Cables The history of building is a history of building materials. The emergence of a new material can usually promote the development of structures. For example, in the field of cable structures, the application of high-strength steel cables has not only considerably increased the span of suspension bridges but also made many designs become reality, such as modern cable-stayed bridges, spoked wheel cable roofs and cable net facades. Carbon Fibre Reinforced Polymer (CFRP) is a new high performance composite material. Many mechanical properties of CFRP, such as the tensile strength, the self-weight and the fatigue performance, are superior to those of high-strength steel. This makes CFRP possess great potential to be made into cables, which can substitute for steel cables in cable structures. One of the experts who first started the research of replacing steel cables with CFRP cables in cable bridges is Urs Meier [Meier, 2012]. In 1987, he demonstrated that the super long CFRP cable supported bridges are superior to the corresponding steel cable supported bridges from the static perspective and proposed building a CFRP cable-stayed bridge with a main span of 8400 m crossing the Strait of Gibraltar [Meier, 1987]. The first use of CFRP cables in a real cable structure dates back to 1996 [Karbhari, 1998]. From then to now, there have been nine real CFRP cable structures over the world, even though all of them were built more or less experimentally. The existing CFRP cable structures are listed and introduced as follows. (1)

Tsukuba FRP Bridge [Karbhari, 1998]

The Tsukuba FRP Bridge, to the author’s knowledge, is the first CFRP cable structure in the world. Moreover, it is also a full FRP structure. This structure, located in Ibaraki, Japan, was designed by Public Works Research Institute in Tsukuba and completed in March 1996. The photo and sketch of Tsukuba FRP Bridge are shown in Figure 1.9.

7 

 

Figure 1.9: Tsukuba FRP Bridge (a) photo (photo credit: Iwao Sasaki) (b) sketch The Tsukuba Bridge is a pedestrian cable-stayed bridge with three spans (see Figure 1.9). The main span between pylons is 11.0 m, while two side spans are 4.5 m. In this bridge, the pylons are made of GFRP, the deck is GFRP profile strengthened by CFRP lamellas and all the 24 stay cables are made of CFRP. Specifically, two types of CFRP cables were used in this bridge, which are indented Leadline rods from Mitsubishi Chemical Company and CFCC 7-wire tendons from Tokyo Rope Manufacturing Co., Ltd. A tailor-made anchorage composed of CFRP pipe and expansive mortar was designed to anchor these CFRP stay cables. The diagrams of this anchorage are shown in Figure 1.10.

Figure 1.10: Anchorage system in Tsukuba FRP Bridge (a) full view (b) transparent view The CFRP cable shown in the above figure is the indented leadline rod. The anchorage for the CFCC 7-wire tendon is exactly the same. In this anchorage system, a CFRP pipe is used as the anchorage socket and an expansive mortar is adopted to generate enough bond force to anchor the cable. (2)

Stork Bridge [Meier et al., 1996]

The Stork Bridge is the first highway cable-stayed bridge with CFRP cables in the world. This bridge is located in Winterthur, Switzerland, and was designed by OMG and Partner Architects AG and Höltschi & Schurter AG. After one and a half years of construction, this bridge was completed and opened to traffic on October 27, 1996. The photo and sketch of Stork Bridge are shown in Figure 1.11.

8 

 

Figure 1.11: Stork Bridge (a) photo (photo credit: EMPA) (b) sketch The Stork Bridge is a single pylon cable stayed bridge with double cable planes (see Figure 1.11). It has 24 stay cables. Two of them are CFRP cables, while others are normal steel cables. These two CFRP cables are 35 m long parallel wires bundles. Each bundle consists of 241 CFRP wires of Φ5 mm. The load bearing capacity of each cable is 12 MN. In order to carry such a large force, a special anchorage called Gradient Anchorage System was developed by EMPA. The diagrams of this anchorage are shown in Figure 1.12.

Figure 1.12: Gradient Anchorage system (a) full view (b) transparent view The above anchorage is composed of a conical steel socket and a mortar called Load Transfer Media (LTM), which was manufactured by mixing Al2O3 ceramic granules of approximately Φ2 mm (light yellow particles in Figure 1.12 (b)) into the epoxy resin (the dark cone in the Figure 1.12 (b)). From the mouth of the steel socket to its end, the distribution density of granules in the resin increases gradually, so as to achieve an anchorage mortar with gradient elastic modulus. Compared with normal conical mortar anchorages, the stress peak of the CFRP cable is considerably reduced in the Gradient Anchorage System due to the mortar with gradually increased E-modulus from the socket mouth to the socket end. As a consequence, the anchorage efficiency can be highly improved.

9 

  (3)

Herning CFRP Bridge [COWI, 1999]

The Herning CFRP Bridge is a pedestrian overpass across a railway switchyard in the vicinity of Herning, Denmark. This bridge was designed by COWI and completed in June 1999. The photo and sketch of Herning Footbridge are shown in Figure 1.13.

Figure 1.13: Herning CFRP Bridge (a) photo (photo credit: COWI) (b) sketch The Herning Footbridge is a single pylon cable stayed bridge with double cable planes (see Figure 1.13). It has 16 stay cables in total and all of them are CFRP cables. These CFRP cables are CFCC 37-wire strands produced by Tokyo Rope Manufacturing Co., Ltd., with a diameter of 40 mm. The load bearing capacity of each cable is 1070 kN and all cables were supplied from factory in fixed lengths with resin filling type anchors. The diagrams of this anchorage are shown in Figure 1.14.

Figure 1.14: resin filling type anchorage system (a) full view (b) transparent view The above anchorage system is mainly composed of a cylindrical steel socket and a special resin filling. The anchorage length is approximately 13.5 times that of the cable diameter. The anchor nut is set at the rear of the steel socket to connect the anchorage to other parts of the bridge.

10 

  (4)

Laroin CFRP Footbridge [Geffroy, 2002]

The Laroin CFRP Footbridge crosses the Gave du Pau River and is located in Laroin, France. This bridge was designed by Freyssinet International and completed in 2002. The photo and sketch of Laroin CFRP Footbridge are shown in Figure 1.15.

Figure 1.15: Laroin CFRP Footbridge (a) photo (photo credit: Freyssinet) (b) sketch The Laroin CFRP Footbridge is a single span cable stayed bridge with twin towers and double cable planes (see Figure 1.15). The main span of 110 m long and 2.5 m wide is supported by eight pairs of CFRP stay cables. The back stays, which are standard steel cables, were anchored to the concrete anchor blocks in the ground. These 16 CFRP cables, from 20 m to 45 m long, are modular and each module is composed of a CFRP parallel 7wire bundle. According to the different load conditions, the four pairs of cables near the pylons contain two modules (i.e. two 7-wire bundles), while the other four pairs near the middle span contain three modules (i.e. three 7-wire bundles). The CFRP wires, which were produced by SOFICAR, have a diameter of 6 mm and a load bearing capacity of 70 kN. In order to anchor these CFRP cables, a patented anchorage system called Modular Clamp Anchorage was developed by Freyssinet. The diagrams of this anchorage are illustrated as follows.

Figure 1.16: Modular Clamp Anchorage system (one module) (a) full view (b) sectional view

11 

  Figure 1.16 shows one module of the Modular Clamp Anchorage system. In this figure, one module of CFRP cable, i.e. one CFRP 7-wire bundle, is gripped as a group by the wedge type anchorage. In order to prevent the transversal damage of CFRP from directly clamping, every wire is protected by an aluminium sheath in the anchorage. Each anchorage shown above is a module and each end of CFRP cables contains two or three these modular anchorages. There are two main advantages of applying the modular cable and the modular anchorage. Firstly, a bundle of CFRP wires are anchored integrally in a module, instead of respectively anchoring. This can make the anchorage compact and reduce the size. Secondly, unlike conventional CFRP cable anchorages, which are unique, this type of anchorage is assembled by individual modules, which are standard. Because every module has already been maturely researched, the anchorages for different cable sizes do not need to be investigated any more but simply assembled by several modules. This can facilitate the anchorage design and reduce the cost. (5)

Jiangsu University CFRP Footbridge [Lue and Mei, 2007]

The Jiangsu University CFRP Footbridge is a cable-stayed bridge located in Zhenjiang, China. It was designed by Southeast University, Jiangsu University and Peking TXD Technology Company. After one year’s construction, this bridge was accomplished in the end of May 2005. The photo and sketch of this bridge are shown in Figure 1.17.

Figure 1.17: Jiangsu University CFRP Footbridge (a) photo (photo credit: Kuihua Mei) (b) sketch The Jiangsu University CFRP Footbridge is a single pylon cable-stayed bridge with double cable planes (see Figure 1.17). All the 16 stay cables are CFRP parallel-bar cables. The

12 

  adopted CFRP bars are Φ8 mm Leadline indented bars produced by Mitsubishi Chemical Company. Based on the different load-bearing requirements, three kinds of cables with different number of bars were applied (see Figure 1.17). The load bearing capacities of these cables are 720 kN (6 × 8 mm), 1320 kN (11 × 8 mm) and 1920 kN (16 × 8 mm), respectively. The CFRP cable anchorage was specially designed, which is called Straight Tube and Inner Cone Anchorage. The diagrams of this anchorage are shown in Figure 1.18.

Figure 1.18: Straight Tube and Inner Cone Anchorage system (a) full view (b) transparent view This anchorage consists of a steel socket and the mortar (epoxy resin or expansive cement) inside. The shape of the socket is particularly designed. The front part near the mouth is a cylinder while the rear part is a cone. With this shape of anchorage socket, the stress peak in CFRP cable will start near the joint of cylinder and cone and go on constantly to the socket mouth. In this way, the magnitude of stress peak will be greatly reduced. Therefore, the anchorage efficiency can be significantly improved, compared to normal cone-shaped anchorages. (6)

Penobscot Narrows Bridge [Rohleder Jr et al., 2008]

The Penobscot Narrows Bridge is the first cable-stayed bridge in America, which adopts CFRP strands in stay cables. This bridge, owned by Maine Department of Transportation, is located in Penobscot, Maine, USA. The designer of this bridge was FIGG Engineering Group and its constructor was Cianbro/Reed & Reed, LLC. After three years’ construction, this bridge was completed and opened to the traffic on December 30, 2006. After half year of completion, six steel strands in different stay cables were removed and replaced with six CFRP strands. The Penobscot Narrows Bridge’s photo and sketch are shown below.

13 

 

Figure 1.19: Penobscot Narrows Bridge (a) photo (photo credit: MOT) (b) sketch The Penobscot Narrows Bridge is a twin masts cable-stayed bridge with single cable plane (see Figure 1.19). Its span is 146 m + 354 m + 146 m and the mast height is 136 m. It has 40 stay cables, which are all parallel 7-wire strands cables. These cables were not anchored at the mast but though cradles in the mast and anchored at the bridge deck. From the total of 40 cables, 3 cables with different lengths (approximately 86 m, 198 m and 300 m, respectively) through the western pylon were selected to install CFRP stands. In every selected cable, two steel strands were uninstalled and two new CFRP strands were mounted at the same place. These six CFRP strands are all CFCC 1 × 7 strands, produced by Tokyo Rope Manufacturing Co., Ltd. Based on the existing CFCC strand bond anchorage, the design team of Penobscot Narrows Bridge cooperated with Lawrence Technological University developed a new kind of anchorage for this case. This anchorage can be named as “Highly Expansive Material (HEM) Filling Anchorage System”, which can be illustrated in Figure 1.20.

Figure 1.20: Highly Expansive Material (HEM) Filling Anchorage System (a) full view (b) transparent view This anchorage consists of a long threaded socket with a hollow cylindrical cavity and the mortar inside. Outside the socket, there is often an anchor nut for fixing this system to the structure. The inner diameter of this hollow socket is slightly bigger than the diameter of the strand. The space between the strand and the socket wall is filled by a cement-based 14 

  mortar, which is called Highly Expansive Material (HEM). Through expansion during curing, the HEM will produce enough radial pressure (approximately 75.8 MPa in this case) after hardening, so as to hold the CFRP strand. (7)

EMPA Bowstring Arch Footbridge [Meier et al., 2009]

The EMPA Bowstring Arch Footbridge is a pedestrian bridge over a small pond at the Swiss Federal Laboratories for Materials Science and Technology (EMPA). It is located in Dübendorf near Zürich, Switzerland. This bridge was designed by researchers in EMPA and was installed in spring 2007. The photo and sketch of EMPA Bowstring Arch Footbridge are shown in Figure 1.21.

Figure 1.21: EMPA Bowstring Arch Footbridge (a) photo (photo credit: Urs Meier) (b) sketch The EMPA Bowstring Arch Footbridge is a bowstring arch bridge with a span of 12 m and a width of 3 m (see Figure 1.21). The bridge deck is mainly made of Swiss grown Norway spruce (Picea abies), which was laterally strengthened by CFRP strips. In the longitudinal direction, six CFRP non-laminated strip-loop cables were arranged under the deck and evenly distributed over its width to serve as bowstrings. Each cable is 30 mm wide and its nominal cross section is 60 mm2. The CFRP cable and pin-loaded anchorage are shown in Figure 1.22.

15 

 

Figure 1.22 (a) CFRP non-laminated strip-loop cable and (b) pin-loaded anchorage (photo credit: Urs Meier) The CFRP cable used is a non-laminated pin-loaded strip-loop (see Figure 1.22). Such cable, supplied by Carbo-Link GmbH, Switzerland, is manufactured by winding a very thin (approximately 0.1 mm thick) CFRP strip around two round pins (the pins are made of GFRP in this case) continuously. After winding, only the end of the outermost layer is bonded to the next outermost layer through fusion to form a closed loop (thermoplastics matrix is adopted to produce the CFRP strip). Compared with the laminated strip-loop, the shear and radial stresses at the anchorage zone (especially the layer contacting the pin) of the non-laminated strip-loop will distribute more uniformly, because different layers of the non-laminated one can mutually slide and thus achieving more uniform strain distribution than the laminated one. This will lead to relatively small stress peak in the CFRP nonlaminated pin-loaded strip-loop cables and help increase their ultimate bearing capacity. (8)

TU-Berlin CFRP Stress-Ribbon Footbridge [Schlaich and Bleicher, 2007]

The TU-Berlin CFRP Stress-Ribbon Footbridge is the first CFRP stress-ribbon bridge in the world. It is located in Berlin, Germany, and was designed by the research team of Professor Mike Schlaich in Technical University of Berlin. This bridge was completed in May 2007. After the completion, an active vibration control system was installed in the bridge. The photo and sketch of this bridge are shown in Figure 1.23.

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Figure 1.23: TU-Berlin CFRP Stress-Ribbon Footbridge (a) photo (photo credit: Achim Bleicher) (b) sketch The TU-Berlin CFRP Stress-Ribbon Footbridge (see Figure 1.23) is a single span stressribbon bridge with the sag-to-span ratio of 1/60. Six 50 mm wide CFRP non-laminated pin-loaded strip-loop cables are used as load bearing components in this bridge, which were also provided by Carbo-Link GmbH, Switzerland. In the transverse direction, each cable has 2 × 5 layers and the total thickness is approximately 1.1 mm. The load bearing capacity of single cable is 105 kN. The manufacturing process of these cables is similar to that of cables used in the EMPA Bowstring Arch Footbridge, while the difference is that the steel pins are used in this case. The diagrams of the cable anchorage are illustrated in Figure 1.24.

Figure 1.24: Pin-loaded anchorage system (a) assembled view (b) exploded view The above anchorage system mainly consists of two round pins, a triangular steel box and two bolts to connect to the ground. Among the two pins, the anchor pin, with the diameter of 80 mm, is for anchoring; while the steering pin, with the diameter of 100 mm, is for changing the force direction to facilitate pre-tensioning and anchoring.

17 

  (9)

Cuenca Stress-Ribbon Footbridge [Clemente Ortega and Rodado López, 2011]

The Cuenca Stress-Ribbon Footbridge, across the Jucar River, is located in Cuenca, Spain. It is owned by Cuenca City and it was designed and constructed by ACCIONA, S.A. This bridge was completed and opened to the public in 2011. The photo and sketch of Cuenca Stress-Ribbon Footbridge are shown in Figure 1.25.

Figure 1.25: Cuenca Stress-Ribbon Footbridge (a) photo (photo credit: Juan Rodado Lopez) (b) sketch The Cuenca Stress-Ribbon Footbridge (see Figure 1.25) is a stress ribbon bridge with three spans (72 m × 3). The bridge width is 3 m and the total length is 216 m, which makes it become the longest stressed ribbon bridge in Spain and the eighth in the world. This bridge has two concrete piers, whose heights are 21.6 m and 16.98 m, respectively. The bridge decks are steel reinforced concrete slabs with the dimension 3 m × 3.5 m, and the stress ribbon supporting them consists of 16 CFRP cables. These CFRP cables were divided into 4 groups in transverse direction and fixed under the concrete slabs. Longitudinally, the stress ribbon was divided into 5 sections, and each section is a CFRP cable with a length of 43.7 m and a diameter of 41 mm. These CFRP cables were connected by an “8” shape pin connection with each other. This segmentation measure was to facilitate manufacture and transportation. Outside of the cables are aramid braided sleeves for protecting them. These CFRP cables used in the Cuenca Stress-Ribbon Footbridge were specially manufactured by ACCIONA for this project. Different from CFRP cables used in EMPA Bowstring Arch Footbridge and TU-Berlin CFRP Stress-Ribbon Footbridge, these cables are CFRP laminated strip-loops cables with stainless-steel ring terminations. Indeed, the CFRP non-laminated and laminated pin-loaded strip-loop cables have both merits and demerits. In general, the former has higher ultimate bearing capacity, while the latter has greater stiffness and smaller creep. As a consequence, it is better to say that they have different uses rather than to judge which is better. 18 

 

In order to anchor and connect the CFRP cables in the Cuenca Stress-Ribbon Footbridge, pin-loaded anchorage and connection were designed by engineers of ACCIONA, which are illustrated in Figure 1.26.

Figure 1.26: (a) Anchorage and (b) connection for CFRP cables Figure 1.26 (a) presents the anchorage structure of CFRP cables, which consists of steel fork, steel bar and anchor pin. Figure 1.26 (b) shows the connection structure for CFRP cables, which is formed by “8” shape steel plate and anchor pins. The steel bar of the anchorage is pre-tensioned and anchored at the abutment back wall by spherical nut and anchor plate. As can be seen from the above introduction, the existing CFRP cable structures are all cable bridges. Moreover, majority of them are cable-stayed bridges. Up to now, there is not yet any CFRP cable roof or facade in the world. Furthermore, the existing studies about CFRP cable structures are also mainly confined to the bridge field, especially the CFRP cable-stayed bridge. Meier reviewed the utilisation of CFRP cables in cable bridges before the early 1990s [Meier, 1992]. Zhang and Gu theoretically studied the application of CFRP cables in long span cable-stayed bridges [Zhang and Gu, 1995]. Meier, U. and Meier, H. proposed various possibilities of using CFRPs as cable supports for the bridge engineering [Meier and Meier, 1996]. Noisternig researched the theory of anchoring CFRP cables in cable-stayed bridges and proposed a CFRP cable anchorage [Noisternig, 2000]. Keller reviewed the development of CFRP

19 

  cable bridges in the 1990s [Keller, 2001]. Kao did a thorough study on the static behaviour of long-span cable-stayed bridges using CFRP cables [Kao et al., 2006]. Zang provided a review on the application and research of CFRP cables for cable-stayed bridges up to the mid-000s [Zang et al., 2007]. Wu and Wang investigated a thousand-meter scale cablestayed bridge with CFRP cables [Wu and Wang, 2008]. Xiong proposed the design theory of CFRP and steel stay cables [Xiong, 2009]. Wang and Wu proposed an integrated highperformance thousand-metre scale cable-stayed bridge with hybrid FRP cables including CFRP material [Wang and Wu, 2010]. Xiong studied the mechanical behaviour of super long span cable-stayed bridges with CFRP components [Xiong et al., 2011]. Li investigated the rational structure system for super large-span suspension bridge with CFRP main cables [Li et al., 2011]. Zhang studied the parametric resonance of CFRP cables in long-span cable-stayed bridges [Zhang et al., 2011]. Xie researched the static and dynamic characteristics of a long-span CFRP cable-stayed bridge, especially the combined vibration of CFRP cables and full bridge [Xie et al., 2014]. As can be seen from above literature review, although the number of researches on CFRP cable structures cannot be compared with that of researches on CFRP reinforcements in concrete or strengthening structures with CFRP components, there have been many studies on CFRP cable structures completed by worldwide researchers. However, researches of using CFRP cables in cable roofs and facades, which are two important cable structures, are very few until now. This research gap is just the research focus of this book.

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2

Concept of Orthogonally Loaded Cable Structures

This chapter is devoted to the theoretical backgrounds of this research, especially the definition of orthogonally loaded cable structures. In Section 2.1, the design principles of cable structures are introduced. In Section 2.2, the theory of pre-tension of cables is presented. In Section 2.3, the structural stiffness and its two components, i.e. the elastic stiffness and the geometric stiffness, of cable elements are strictly derived; moreover, the characteristics of elastic stiffness and geometric stiffness are illustrated. In Section 2.4, the influence of the angle between the cable axis and external load on the composition of structural stiffness of cable structures is investigated through taking a simple doublecurved cable net as an example. In Section 2.5, based on the composition of structural stiffness, cable structures are classified into two types, i.e. orthogonally loaded cable structures and other cable structures; the influence of this classification on the design of cable structures is illustrated.

2.1 Design Principles of Cable Structures At present, the method of Limit States Design provided by the Eurocode is used to design the cable structures in the European Union [Eurocode 0, 2002]. The Limit States Design (abbreviated to LSD), also known as the Load and Resistance Factor Design (abbreviated to LRFD), refers to a design method in which the limit states are conditions of a structure beyond those it no longer fulfils the relevant design criteria. Using the method of Limit States Design, the cable structures should be designed to not only satisfy the serviceability requirements but also withstand safely all loads acting on them throughout their lives. [Eurocode 0, 2002] According to the above concept, the limit states mainly include two types, i.e. Serviceability Limit State (abbreviated to SLS) and Ultimate Limit State (abbreviated to ULS), from the static perspective. The Serviceability Limit State defines a limit condition for cable structures under unfactored design loads beyond which their specified requirements for use like limitation of deflection can no longer be met. The Ultimate Limit State defines a limit condition for cable structures under factored design loads beyond which they will collapse due to loss of strength.

21 

  It should be noted that the Serviceability Limit State and Ultimate Limit State are not physical states of cable structures but rather two computational check states [Kotsovos and Pavlovic, 1995]. In addition to them, there are two important physical states in the design of cable structures, i.e. Zero State (abbreviated to ZS) and Initial State (abbreviated to IS). The Zero State, also known as zero-stress state, is a situation of cable structures before the erection of structures and the pre-tension of cables, in which there is no stress or strain in any structural member. The cable lengths of Zero State are just the lofting lengths for cables. The Initial State, also known as pre-tension state, is a situation of cable structures after pretensioning the cables, in which the pre-tension force is balanced by the support reactions as well as the structural self-weight. The geometry of cable structures in the Initial State is determined by the desired architectural shape. Taking a simple cable structure, i.e. a two-link cable, as an example, the design states of cable structures can be listed sequentially in Figure 2.1.

Figure 2.1: Design states of cable structures As shown in Figure 2.1, in the Zero State, there is no force in the cables; in the Initial State, the cables are pre-tensioned by jacks and the pre-tension forces in cables are balanced by

22 

  the support reactions. After the Initial State, that is, the cable pre-tension is completed, the cable structure gets into the working state, in which two computational states, i.e. the Serviceability Limit State and the Ultimate Limit State, should be checked in the design. Usually, in order to ensure cable structures simultaneously reach their service limits in the Serviceability Limit State and their strength limits in the Ultimate Limit State (i.e. the optimal design situation of cable structures), the cross-sectional areas and pre-tension forces of cables should be determined through repeated iterations in the design.

2.2 Pre-tension Theory of Cables Cables in cable structures are always considered to be useful in tension only. They have negligible compressive or flexural stiffness and can transmit or sustain external loads only via tension. If the cable force becomes too small or even zero, the cables will go slack, and this can cause a significant reduction in system stiffness, which may lead to large deformations and could damage the structural members supported or suspended by the cables [Buchholdt, 1999]. Therefore, the tension force level of cable structures must be kept sufficiently high so that the cables do not go slack under any combination of external loads. This can be achieved by pre-tensioning the cables with jacks and/or by introducing gravity loads. The pre-tension forces F in cables have to be balanced by the internal forces of other structural members and/or other cables and/or by the force of gravity. Depending on the form of balancing pre-tension forces, cable structures can be classified into four types, as shown in Table 2.1 [Schlaich et al., 2015].

23 

 

Table 2.1: Classification of cable structures according to the type of pre-tension Type

Sketch

Example Two-link cable

① Pretensioning against supports only

Flat cable net

Description* Pre-tension forces are balanced by support reactions. Increasing pretension causes increased reactions, which require stronger supports.

Cable truss

② Pretensioning against other cables

③ Pretensioning against beam

④ Pretensioning against weight

Pre-tension forces are balanced by support reactions and forces of other cables. Increasing pre-tension causes Double-curved cable net increased reactions and other cable forces, which require stronger supports and greater cross sections of other cables. Beam string Pre-tension forces are balanced by beam's bending moment and/or gravity force. Increasing Cable-stayed bridge pre-tension causes an increased bending moment, which requires a stiffer and/or stronger beam and/or extra pre-camber. Suspension roof Pre-tension forces are balanced by support reactions and gravity force. Increasing pre-tension Stress-ribbon bridge causes increased reactions and gravity force, which requires stronger supports and extra weight.

* Assumption: geometry in Initial State shall not change when pre-tension is increased. As shown in Table 2.1, for any type of cable structure, increasing the pre-tension force while maintaining the specified geometry in the Initial State can be achieved through

24 

  increasing the size of cross sections of other structural members and/or introducing extra weight and/or pre-camber. The above principle can be illustrated as follows. First, take a two-link cable as an example, which is just the type 1: pre-tensioning against supports only (see Table 2.1).

Figure 2.2: Increasing pre-tension while maintaining geometry simultaneously (type 1) As shown in Figure 2.2, in the Zero State, the initial lengths of cables in either the left or the right cable structure is set shorter than the installation lengths, so as to apply the pretension. Moreover, in order to increase the pre-tension forces, the cables of the right cable structure are made even shorter than those of the left cable structure. Then in the Initial State, both cable structures are pre-tensioned by pulling the cables with jacks from their initial lengths to installation lengths. In this state, the pretension forces are balanced with the support reactions, and the geometries of both cable structures are the same though the pre-tension force in the right cable structure is greater than that in the left one. Take a three-cable structure as another example, which is just the type 2: pre-tensioning against other cables (see Table 2.1).

25 

 

Figure 2.3: Increasing pre-tension while maintaining geometry simultaneously (type 2) As shown in Figure 2.3, in order to apply the pre-tension force, two upper cables in either the left or the right cable structure is set shorter than the installation lengths; however, in order to increase the pre-tension, the cables of the right cable structure are made even shorter than those of the left one. Then in the Initial State, both cable structures are pre-tensioned by pulling the cables with jacks. The pretension forces are balanced by not only the support reactions but also the force of lower cable. Because the EA of the lower cable in the right cable structure is greater than that of the lower cable in the left cable structure, the deformations in both cable structures are the same (i.e. S1 = S2), though the pre-tension force in the right cable structure is greater than that in the left one. This demonstrates that the geometry of cable structures in the Initial State can be maintained when the pre-tension force is increased. Moreover, because theoretically the EA of cables can be arbitrarily increased, the pretension force can also be increased to any level. In a similar way, it can also be demonstrated that increasing pre-tension and maintaining geometry simultaneously is possible for type 3 and type 4 cable structures in Table 2.1. Maintaining the geometry in the Initial State is very important, because this makes cable structures keep the required architectural shapes. Table 2.1 and above explanation indicate that every cable structure can be theoretically pre-tensioned to any level without changing the specified geometry in the Initial State; the pre-tension force is a design parameter that can be freely set for all cable structures.

26 

  However, increasing the pre-tension force may result in additional costs of cables, foundations, anchorages and other structural members. Therefore, in practical design, a reasonable pre-tension force level in cable structures must be determined by comprehensively considering all factors.

2.3 Stiffness Matrices of Cable Elements 2.3.1 Finite element discretisation of cable structures Except for the structures with a single cable which may be solved by conventional analytical methods [Palkowski, 2003], most of cable structures with multi-cables can only be solved by numerical methods like the Finite Element Method (FEM). The finite element analysis of cable structures is based on the element stiffness matrix (i.e. the tangential stiffness matrix) of cable elements. The first step of finite element analysis for cable structures is the finite element discretisation. The discretisation process for the cable system of cable structures is illustrated in Figure 2.4, taking a double-curved cable net as an example.

Figure 2.4: Finite element discretisation for the cable system of cable structures (a) whole structure (b) a cable segment modelled with a single cable element (c) a cable segment modelled with several cable elements For the double curved cable net shown in Figure 2.4 (a), between any two natural cable nodes j and j+1, at least one cable element should be created (see Figure 2.4 (b)). This cable element can be a straight truss element hinged at both ends, which can only transmit or sustain external loads via tension. When the cable force is zero or the cable is under compression, this cable element will be deactivated. The external forces will only act at the

27 

  both end nodes of the cable element and the internal force of the cable element will only coincide with the cable axis. If the cable sag or the external loads acting at the middle of the cable should be considered, the cable element shown in Figure 2.4 (b) can be further discretised to several smaller linked cable elements, as shown in Figure 2.4 (c).

2.3.2 Formulation of stiffness matrices for three dimensional cable elements After the finite element discretisation of cable structures, the element stiffness matrix of cable elements should also be determined. The correct element stiffness matrix for cable elements is very important, because it is directly related to the correctness and accuracy of final results. Moreover, the expressions of the stiffness matrices (especially the geometric stiffness matrix) are still not clear and different literature may have different expressions. In this section, the derivation process of the element stiffness matrix for the three dimensional cable element shown in Figure 2.5 is presented. The derivation is strictly based on the virtual work theory, the Green strain and the updated Lagrangian formulation.

Figure 2.5: Cable element in the local coordinate system (xyz) and global coordinate system (XYZ) In Figure 2.5, the cable element has a linear-elastic constitutive relation and is located in the three-dimensional (3-D) space, where xyz (lower case) represents the local coordinate

28 

  system while XYZ (upper case) represents the global coordinate system. The cable element will be first investigated in the local coordinate system and then the coordinate transformation from xyz to XYZ will be performed.

2.3.2.1 Virtual work theory Assume the location and form of the cable element at the equilibrium state of time t is shown in Figure 2.5 and this cable element achieves a new unknown equilibrium at time t+ t, the virtual work equation at the time t+ t can be established as:

 Wintt t   Wextt t

(2.1)

where  Wintt t denotes the virtual work of the internal forces at t+ t and  Wextt t denotes that of the external forces at this time. Assume the length and the cross-sectional area of this cable element at the time t are L and A respectively, Equation (2.1) can be expanded as [De Borst et al., 2012]:

  ε  σ t t T

t t

V

dV    Δs   gdVb    Δs  tdA T

T

Vb

(2.2)

A

where  ε t t denotes the virtual strain tensor and σ t t denotes the stress tensor at t+ t;

  Δs 

T

Vb

 gdVb is the virtual work done by the body force and

  Δs 

T

A

tdA is the virtual

work done by the surface force. For the cable element, because the only non-vanishing strain or stress component is the axial component and the external forces are the concentrated forces only acting at the two end nodes, Equation (2.2) can be simplified to: t t A  xt t xt t dx   Δs  fext L

T

(2.3)

0

where  x and  x denotes the strain and stress in the x (i.e. axial) direction; s denotes the nodal displacement increment vector from t to t+ t and Δs  u1 , v1 , w1 , u2 , v2 , w2  , and T

the virtual displacement increment is  Δs   u1 ,  v1 ,  w1 ,  u2 ,  v2 ,  w2  , where u, v, w T

represent the displacements in x, y, z directions and the subscripts 1 and 2 denote the

29 

  corresponding values of the node 1 and node 2 of the cable element; fext denotes the nodal

force vector and fext    f1x , f1 y , f1z , f 2 x , f 2 y , f 2 z  . T

The cable stress at t+ t can be decomposed as:

 xt t   xt   x

(2.4)

where  xt is the stress at t and  x is the stress increment between t and t+ t. With this decomposition, Equation (2.4) can be written as: t t A  xt t  xt   x  dx   Δs  fext L

T

0

Let the increment

(2.5)

t small enough so that relation between the stress increment  x and

the strain increment  x is linear, that is:  x  Etan  x

(2.6)

where Etan is the tangential stiffness modulus, which is equal to the Young’s modulus for a linear-elastic material, i.e. Etan = E [De Borst et al., 2012]. Therefore,  x  E  x

(2.7)

Substitute Equation (2.7) into Equation (2.5), one can obtain: t t EA  xt t  x dx  A  xt t xt dx   Δs  fext L

L

0

0

T

(2.8)

Furthermore, the cable strain at t+ t can be decomposed as:

 xt t   xt   x

(2.9)

where  xt is the strain at t and  x is the strain increment between t and t+ t. Then,

 xt t   xt   x

30 

(2.10)

 

Because  xt is a known constant,  xt  0 . Equation (2.10) can be written as:

 xt t   x

(2.11)

Substitute Equation (2.11) into Equation (2.8), one can obtain: t t EA  x  x dx  A  x  xt dx   Δs  fext L

L

0

0

T

(2.12)

2.3.2.2 Relationship between displacement and strain According to the geometric relationship, the cable element length at t+ t can be written as: Lt t 

 L  u2  u1    v2  v1    w2  w1  2

2

2

(2.13)

Because the displacement of the cable element is relatively great, the linear strain (i.e. (u1u2)/L) is not suitable in this case. Therefore, the Green strain, which is a typical non-linear strain defined in Equation (2.14), should be used [Green and Adkins, 1970]:  x 

L2t t  L2 2 L2

(2.14)

Substituting Equation (2.13) into Equation (2.14), one can obtain: 1  u  u1   v2  v1   w2  w1   x  1  2      2  L   L   L  2

2

2

  

(2.15)

Simplifying Equation (2.15), one can obtain: 2 2 2 u2  u1 1  u2  u1   v2  v1   w2  w1    x          L 2  L   L   L  

(2.16)

It can be seen from Equation (2.16), the Green strain is composed of the linear strain plus a non-linear term.

31 

  Furthermore, let  

u2  u1 L

2 2 2 1  u2  u1   v2  v1   w2  w1            2  L   L   L  

(2.17)

(2.18)

where  is the linear term and  is the quadratic term of the Green strain. 2

 u u  It should be noted that the term  2 1  in Equation (2.18) is ignored in some literature  L 

[Palkowski, 2003] [Palkowski, 2013], which can slightly increase the calculation speed but will decrease the calculation accuracy and change the characteristic of geometric stiffness. Then, Equation (16) can be written as:  x    

(2.19)

The virtual strain increment can be expressed as:

 x    

(2.20)

where

 

 u2   u1 L

(2.21)

 u2  u1    u2   u1   v2  v1    v2   v1   w2  w1    w2   w1        (2.22) L L L  L    L    L  

  

2.3.2.3 Stiffness matrices in the local coordinate system Substituting Equations (2.19) and (2.20) into Equation (2.12), one can obtain:

32 

 

EA   dx  EA   dx  EA   dx  EA   dx  L

L

0

L

0

L

0

0

t t A   xt dx  A   xt dx   Δs  fext L

L

0

0

T

(2.23)

It can be found that, in Equation (2.23), EA   dx and EA   dx are L

L

0

0

quadratic terms and EA   dx is cubic term, which do not contribute to the L

0

tangential stiffness and hence can be ignored [De Borst et al., 2012]. Therefore, Equation (2.23) can be simplified as: t t EA   dx  A   xt dx  A   xt dx   Δs  fext L

L

L

0

0

0

T

(2.24)

t Furthermore, the internal force vector of the cable element at t is f int and

fintt   f1tx , 0, 0, f 2tx , 0, 0 , because the internal forces are only along the x direction, i.e. the T

T

1  1  cable axis. Establish a column vector a and a   , 0, 0, , 0, 0  , then one can obtain: L  L  L

fintt  ALa xt  A a xt dx

(2.25)

0

Moreover,

 

 u2   u1 L

 a T  Δs    Δs  a T

(2.26)

Therefore,

A   xt dx  A  Δs  a xt dx   Δs  fintt L

L

0

0

T

T

(2.27)

Substituting Equation (2.27) into Equation (2.24) and transposing, one can obtain: t t EA   dx  A   xt dx   Δs  fext   Δs  fintt L

L

0

0

T

T

(2.28)

It should be noted that, in the equilibrium state t, the internal forces are equal to the t external forces, i.e. fintt  f ext . Therefore, Equation (2.28) can be written as:

33 

 

t t t EA   dx  A   xt dx   Δs   fext  fext  L

L

0

0

T

(2.29)

Because  

  

u2  u1  a T Δs L

(2.30)

T   u   u1   u2  u1  T  2     Δs  aa Δs L   L 

(2.31)

Substituting Equation (2.30) into the first term in the left side of Equation (2.29), one can obtain:

EA   dx  EA  Δs  aaT Δsdx  EAL  Δs  aaT Δs L

L

0

0

T

T

(2.32)

Because  1  L     0   0   1 aaT     1   L  L   0     0 

0 0

1 L

 1  L2   0   0 0 0     1   L2   0  0

1 L2 0 0 1 L2 0 0

0 0  0 0 0 0 0 0 0 0 0 0

 0 0  0 0 0 0  0 0   0 0 0 0 

(2.33)

Substituting Equation (2.33) into Equation (2.32), one can obtain:

EA   dx   Δs  k e Δs L

0

where

34 

T

(2.34)

   EA  L   0  0 ke    EA  L   0  0

EA  0 0 1 L  0 0 0 0  0 0 0  EA  0   EA L  1 0 0  0 L   0 0 0 0  0 0 0

0 0  0 0 0 0 0 0 0 0 0 0

0 0 1 0 0  0 0 0 0 0  0 0 0 0 0  0 0 1 0 0 0 0 0 0 0  0 0 0 0 0

(2.35)

The above ke (lower case) is the elastic stiffness matrix of the cable element in the local coordinate system. As can be seen from the above derivation process, ke stems from the first order term of Green strain and is only in the direction of cable axis. T

T

1 1  1 1   Establish two column vectors b and c, and b  0,  , 0, 0, , 0  , c  0, 0,  , 0, 0,  , L L  L L  

then one can obtain: T   v2   v1   v2  v1  T      Δs  bb Δs L   L 

(2.36)

T   w2   w1   w2  w1  T      Δs  cc Δs L   L 

(2.37)

Consider Equations (2.22), (2.31), (2.36) and (2.37) together, one can obtain:

   Δs  aaT Δs   Δs  bbT Δs   Δs  ccT Δs T

T

T

(2.38)

Substituting Equation (2.38) into the second term in the left side of Equation (2.29), one can obtain: T T T A   xt dx  A  Δs  aaT Δs   Δs  bbT Δs   Δs  ccT Δs  xt dx  0 0  T  A xt L  Δs   aaT  bbT  ccT  Δs    L

L

(2.39)

Because

35 

   0   1    L  0   1 bbT    0  L  0   1     L   0 

0 0

 0   0     1   L   1 T cc    0 0  L  0   0     1   L 

1 L

0 0

0 0  1 0 L2  0  0 0   0  0  1 0  2 L  0 0 0 0   0 1   L  0 0   0

0 0 0 0  0 0 0 0 0 0 0 0

0

0

0 0

0

0 1 L2 0

0 0

0 0 0

0 0 0 0

0 1 0  2 L

0 0 0 0

0  0  0  0  0  0 

(2.40)

0  0  1  2 L  0  0   1  L2 

(2.41)

0 1 L2 0 0

1 L2 0

Substituting Equations (2.32), (2.40) and (2.41) into Equation (2.39), one can obtain:

A   xt dx   Δs  k g Δs L

T

0

(2.42)

where

 A xt   L   0    0 kg   t   A x  L   0    0 

0

0



A xt L

0

A xt L

0

0

A xt  L

0

A xt L

0

0

0

0

A xt L

0

A xt L

0

0

A xt L

A xt L

0

0



0



    0   A xt   t L   A x  L 0    0  t  A x  L  0

 1 0 0 1 0 0   0 1 0 0 1 0     0 0 1 0 0 1    1 0 0 1 0 0   0 1 0 0 1 0     0 0 1 0 0 1 

(2.43)

36 

  The above kg (lower case) is the geometric stiffness matrix of the cable element in the local coordinate system. As can be seen from the above derivation process, kg stems from the second order terms of Green strain and is not only along but also perpendicular to the cable axis. Substituting Equations (2.34) and (2.42) into Equation (2.29), one can obtain:

 Δs 

T

t t t k e Δs   Δs  k g Δs   Δs   fext  fext  T

T

(2.44)

Since Equation (2.44) must hold for any virtual displacement increment,  Δs  in both T

sides can be eliminated together and one can obtain:

k

e

t t t  k g  Δs  fext  f ext

(2.45)

In Equation (2.45),  k e  k g  Δs is the internal force increment from t to t+ t, while t t t is the external force increment from t to t+ t. Equation (2.45) indicates that the f ext  f ext

cable element achieves a new equilibrium at the time t+ t. t t t Let f ext  f ext  Δf , then Equation (2.45) can also be written as:

k

e

 k g  Δs  Δf

(2.46)

Equation (2.46) shows the relationship between the displacement increment Δs and the external force increment Δf . Through solving Equation (2.46), the unknown displacement or force at any step can be obtained step by step.

2.3.2.4 Transformation of coordinate system Let cos(  , ) ( = x, y, z;  = X, Y, Z) represent the cosine between the coordinate axes of the local coordinate system and the coordinate axes of the global coordinate system. Therefore, the six local displacement components of Δs can be expressed by the six global T displacement components ΔS ( ΔS  U1 , V1 , W1 , U 2 , V2 , W2  ) as:

37 

 

u1  U1 cos( x, X )  V1 cos( x, Y )  W1 cos( x, Z )

(2.47)

v1  U1 cos( y, X )  V1 cos( y, Y )  W1 cos( y, Z )

(2.48)

w1  U1 cos( z, X )  V1 cos( z, Y )  W1 cos( z , Z )

(2.49)

u2  U 2 cos( x, X )  V2 cos( x, Y )  W2 cos( x, Z )

(2.50)

v2  U 2 cos( y, X )  V2 cos( y, Y )  W2 cos( y, Z )

(2.51)

w2  U 2 cos( z, X )  V2 cos( z, Y )  W2 cos( z , Z )

(2.52)

Introducing a coordinate transformation matrix as:  cos   cos   cos  λ     

x, X y, X z, X

  

cos  x, Y

cos  y, Y cos  z , Y

  

cos  x, Z

cos  y, Z cos  z , Z

0

0

0

0

0

0

0

0

0

  

0

0

0

0

0

0

cos  x, X

cos  y, X cos  z , X

  

cos  x, Y

cos  y, Y cos  z , Y

  0   0  (2.53)  cos  x, Z   cos  y, Z    cos  z , Z   0

  

The Equations from (47) to (52) can be expressed as the form of matrix equation: Δs  λΔS

(2.54)

The transformation relationship between the local nodal force vector f ext and the global nodal force vector Fext is the same as that between Δs and ΔS , that is: t t t t t t t t t f ext  f ext  λFext  λFext  λ  Fext  Fext 

(2.55)

Therefore, in the global coordinate system, Equation (45) can be expressed as:

k

e

t t t  k g  λΔS  λ  Fext  Fext 

Left multiplying λ 1 at the both sides of Equation (56), one can obtain:

38 

(2.56)

 

t t t λ -1  k e  k g  λΔS  λ 1λ  Fext  Fext 

(2.57)

λ 1 λ  I

(2.58)

Because

and the coordinate transformation matrix λ is a orthogonal matrix, which means its transpose is equal to its inverse, that is λ 1  λ T

(2.59)

Equation (2.56) can be simplified to

λ

T

t t t k e λ  λ T k g λ  ΔS  Fext  Fext

(2.60)

Furthermore, Equation (2.60) can also be written as:

 K E  K G  ΔS  ΔF

(2.61)

where ΔF is the external force increment in the global coordinate system, K E is the element elastic stiffness matrix in the global coordinate system and K G is the element geometric stiffness matrix in the global coordinate system. Let

K E  KG  K

(2.62)

where K is the structural stiffness matrix of cable element, also known as the tangential stiffness matrix at the time t. Therefore, Equation (2.61) can be written as:

KΔS  ΔF

(2.63)

Furthermore, let

39 

 

cos( x, X )  l

(2.64)

cos( x, Y )  m

(2.65)

cos( x, Z )  n

(2.66)

The l, m, n are three direction cosines of the cable element in the global coordinate system. Through matrix operation, one can obtain:

 l2 l 2 lm ln  lm ln   m2 mn lm m 2 mn   lm mn n2 ln mn n 2  EA  ln K E  λT k eλ   2  L  l lm ln l2 lm ln   lm m 2 mn lm m2 mn    2 ln mn n 2   ln mn n

(2.67)

 1 0 0 1 0 0   0 1 0 0 1 0    t   0 0 1 0 0  1 A  x K G  λT k g λ    L  1 0 0 1 0 0   0 1 0 0 1 0     0 0 1 0 0 1 

(2.68)

and

Through comparing Equation (2.35) with Equation (2.67) and comparing Equation (2.43) with Equation (2.68), it can be seen that, after the coordinate transformation, KE ≠ ke while

KG ≡ kg. This indicates that the elastic stiffness will vary with the variation of the coordinate system, while the geometric stiffness will remain unchanged when the coordinate system is changed. This phenomenon demonstrates that the geometric stiffness is independent from the orientation of cable element and is a rotation invariant.

2.4 Composition of Structural Stiffness of Cable Structures As argued in Section 2.3, the structural stiffness K of cable element is composed of two 40 

  parts, i.e. the elastic stiffness KE and the geometric stiffness KG. Furthermore, because cable structures are constituted by cable elements, the structural stiffness of cable structures is also composed of the elastic stiffness and the geometric stiffness. These two stiffnesses have very different properties. The elastic stiffness derives from the Young’s modulus E and the cross-sectional area A of the cable and exists along the cable axis; the geometric stiffness derives from the pre-tension force F of the cable and exists not only along but also perpendicular to the cable axis. Their magnitudes and directions at the middle node of a two-link cable can be shown in Figure 2.6 [Zhang, 2009]. It should be noted that KE and KG are now scalars in this figure, because they represent the stiffness values in only one degree of freedom.

Figure 2.6: Magnitudes and directions of KE and KG at the middle node of a two-link cable The proportions of elastic stiffness and geometric stiffness in total structural stiffness can considerably influence the structural behaviours of cable structures and thus determining their designs. This point can be demonstrated by investigating a simply cable structure shown in Figure 2.7, which is a double-curved cable net with only four cables [Noesgen, 1974]. Without considering self-weight and cable sag, every cable of the investigated cable net can be set to one cable element, whose corresponding stiffness matrices are determined in Section 2.3.

Figure 2.7: Double-curved cable net with four cable elements

41 

 

As can be seen from Figure 2.7, the cable net is composed of four cable elements and five nodes. In this cable net, all cables have the same length L and the same cross-sectional area

A; the external load P acts at the node 1 in the global Z direction; these parameters are all constant and are set to L = 1 m, A = 100 mm2, P = 10 kN. The Young’s modulus of all cables is expressed as E; the angles between all cable axes and P are the same and expressed as ; all cables are identically pre-tensioned and the pre-tension force is expressed as F. According to the derivation in Section 2.3, the structural stiffness of this cable net can be written as:

K  K E  KG

(2.69)

In Equation (2.69), K, KE and KG are not matrices any more but three scalars, because they represent the stiffness values in only one degree of freedom, i.e. the direction of X axis. Moreover, KE and KG can be expressed as: KE  4

EA 2 EA n 4 cos 2 θ L L

(2.70)

A xt F 4 L L

(2.71)

KG  4

where the n is the direction cosine of four cable elements, which is equal to cos θ ; A xt is the cable force at the time t, which is expressed as F. In particular, if the time is 0, i.e. if the cable structure is in the Initial State, F is the pre-tension force. As can be seen from Equations 2.70 and 2.71, the parameters which can influence KE and

KG are the angle , the Young’s modulus E and the pre-tension force F. First, E and F are set to constants: E = 160 GPa and F = 100 kN, while the angle  is varied from 90° to 0°. In this case, KE is a function of  (see Equation 2.70), while KG is a constant (see Equation 2.71). Their values and relative sizes with varying  are compared in Figure 2.8 and Figure 2.9, respectively.

42 

 

Figure 2.8: Comparing values of KE and KG with the variation of  Figure 2.8 shows the variation of values of KE and KG for  varying from 90° to 60° (in order to highlight the straight line of KG, especially the intersection point of KE and KG, only the variation trends of KE and KG in the range of  from 90° to 60° are drawn in this figure). In this figure, KE and KG are stiffness values, which can be defined as the force divided by the displacement, and are measured in kilonewtons per metre (kN/m). As shown in this figure, the absolute size of elastic stiffness KE increases from 0 quickly with the decrease of , while the geometric stiffness KG remains constant. They are equal in size approximately at  = 85°. When  decreases to 60°, KE becomes much greater than KG.

Figure 2.9: Comparing relative sizes of KE and KG with the variation of 

43 

  Figure 2.9 shows the variation of relative sizes of KE and KG, which are expressed as a percentage (%) of the total structural stiffness K, with the angle varies from 90° to 0°. As shown in the figure, the proportion of geometric stiffness KG in the total stiffness K declines rapidly from 100% with the decrease of , while the proportion of elastic stiffness

KE rises rapidly from 0% as  decreases. If  is smaller than 60°, KG can be ignored, and K will be almost entirely composed of KE. As can be seen from Figure 2.8 and Figure 2.9, when the angle  is near 90° (i.e., the cable structure is orthogonally loaded by the external load), KG is greater than KE; as  decreases from 85°, KE becomes much greater than KG, primarily because EA is orders of magnitude greater than the pre-tension force F. If a smaller Young’s modulus or a higher pre-tension force were set for the cable structure, the lines of KE and KG in these two figures would intersect at a smaller angle . The arguments above indicate that the structural stiffness of orthogonally loaded cable structures is primarily composed of the geometric stiffness, which is generated by the pretension force in the cables, while the structural stiffness of cable structures which are not orthogonally loaded is primarily composed of the elastic stiffness, which is generated by the Young’s modulus of the cables. Second, as can be seen from Equations (2.69), (2.70) and (2.71), if the angle  and the cable cross-sectional area A are fixed, i.e. the initial geometry and the amount of cable used of this cable net are determined, there are two ways to increase the structural stiffness, that is, increasing the Young’s modulus E or increasing the pre-tension force F. In what follows, the angle  in Figure 2.7 is set to 89°, 85° and 81°, respectively. For each case of , the Young’s modulus E of cables increases from 160 GPa to 313 GPa with a constant ratio of 5/4 (i.e., 200/160 = 250/200 = 313/250 = 5/4), and their pre-tension force

F increases from 100 kN to 195 kN with the same ratio of 5/4 (i.e., 125/100 = 156/125 = 195/156 = 5/4). When the angle  is 89°, the structural stiffness K with different E and different F is shown below.

44 

 

Figure 2.10: Structural stiffness K with different E and different F when  = 89° As seen from Figure 2.10, when the  is 89°, increasing F of cables can significantly raise K of cable structure. However, increasing E of cables can hardly increase K. If the angle  is 85°, the structural stiffness K with different E and different F is shown below.

Figure 2.11: Structural stiffness K with different E and different F when  = 85° As seen from Figure 2.11, when the  is 85°, increasing F or increasing E have almost the same effect to raise the structural stiffness K. If the angle  is 81°, K with different E and different F is shown below.

45 

 

Figure 2.12: Structural stiffness K with different E and different F when  = 81° As seen from Figure 2.12, when the  equals 81°, increasing E of cables can significantly raise K of cable structure. However, increasing F of cables plays a relatively small role in increasing the structural stiffness. In summary, if the angle  is near 90° (i.e., the cable structure is orthogonally loaded), increasing the pre-tension force of cables is more efficient than increasing the Young’s modulus of cables to increase the structural stiffness of cable structures; if the angle  is far from 90° (i.e., the cable structure is not orthogonally loaded), increasing the Young’s modulus is more helpful than increasing the pre-tension force to increase the structural stiffness. Furthermore, since the cross-sectional area of cables is kept the same, increasing the pretension force means choosing cables with higher tensile strength. This indicates that in orthogonally loaded cable structures, increasing the tensile strength of the cables to increase the pre-tension force level is more helpful to reduce the deformation of structures than increasing the Young’s modulus of the cables. It can also be seen from the above analysis that the angle  between the external load and the cable axis, which can be called the control angle, is an important parameter for cable structures, and different angles give the structure different mechanical properties.

46 

  In addition, the analysis procedure, especially the solution process, of cable structures is introduced in Appendix A, taking the above double-curved cable nets with three cases of control angles (i.e.  = 89°,  = 85° and  = 81°) as examples. In this Appendix, changes of cable length and cable force as well as KE, KG and K during the loading are also presented.

2.5 Definition of Orthogonally Loaded Cable Structures The orthogonally loaded cable structure is, therefore, defined as a cable structure with a majority of cables orthogonally loaded or approximately orthogonally loaded by external loads; there is no clear distinction, however. In this context, the cable structures in the left third of Figure 2.13 are regarded as “orthogonally loaded” because their control angles are 90° or near 90°.

Figure 2.13: Classification of cable structures according to the control angle  Classifying cable structures like Figure 2.13 is significant for the design of these types of structures. Orthogonally loaded cable structures (cable structures in the left one-third of Figure 2.13), such as stress-ribbon bridge, cable net facade and spoked wheel cable roof, have at least this common aspect: the tensile strength u of cables used in these structures determines not only the ultimate load bearing capacity but also the structural stiffness of the cable structure. This means that if the cross-sectional area A of cables is kept constant,

47 

  increasing u of cables allows increasing the ultimate load bearing capacity and the structural stiffness of cable structure at the same time. Therefore, in the design of orthogonally loaded cable structures, choosing cables with a higher u and the same or even lower Young’s modulus E (e.g., substituting steel cables with CFRP cables which are well known for the high tensile strength) can lead to reducing the cross-sectional area A of cables and hence reducing the amount of cable used without decreasing both the ultimate load bearing capacity and the structural stiffness. On the contrary, cables structures in the right two-thirds of Figure 2.13, such as cablestayed bridge, suspension bridge and basket handle arch bridge, are not orthogonally loaded. In such cable structures, u of cables can only determine the ultimate load bearing capacity of cable structures but hardly influence their structural stiffness, which is determined by E of cables. If the A of cables remains unchanged, increasing their u can only increase the ultimate load bearing capacity of cable structures. This means, in the design of such cable structures, it is impossible to reduce the cross-sectional areas of cables and save the amount of cable used through using cables with higher u and the same or lower E, because this will result in excessive structural deformation under external load, even though the structural ultimate load bearing capacity is maintained.

48 

 

3

Carbon Fibre Reinforced Polymer (CFRP) Cables

3.1 Introduction of CFRPs This section is devoted to the cable material used, i.e. CFRPs, short for Carbon Fibre Reinforced Polymers (or Carbon Fibre Reinforced Plastics). The composition and fabrication of CFRPs are introduced in this section, respectively.

3.1.1 Composition of CFRPs The Carbon Fibre Reinforced Polymer (CFRP) is an advanced composite material with outstanding properties. As its name suggests, it is composed of carbon fibres as the reinforcement embedded in a polymer resin as the matrix [Bhargava, 2004]. The properties of these two components are presented in the following text. Carbon fibres refer to fibres which contain at least 90 wt.% and up to 100 wt.% carbon. They can be produced from polymeric precursor materials, such as polyacrylonitrile (PAN), cellulose, pitch and polyvinylchloride. These precursors can be converted into carbon fibres through a series of treatment operations of heating and tensioning. [Donnet, 1998] From a macro perspective, carbon fibres are very thin filaments (about 5 m – 10 m in diameter), which are just visible to the human eye. The size of a carbon fibre is compared with that of a human hair in Figure 3.1.

49 

 

Figure 3.1: Carbon fibre compared with human hair (photo credit: Anton) From a micro perspective, the structure of carbon fibres can be amorphous, partly crystalline or crystalline [Chung, 1994]. A schematic diagram of the typical micro structure of a partly crystalline carbon fibre is shown in Figure 3.2 [Morgan, 2005].

Figure 3.2: Schematic diagram of the micro structure of a carbon fibre [Morgan, 2005]

50 

  Microscopically, the carbon fibre consists of many carbon atom layers (see Figure 3.2). Some of these layers arrange relatively neatly and form a layered structure similar to the graphite crystal, while other layers arrange randomly and are in a non-crystalline state. The crystallisation degree of carbon fibres is usually controlled by the temperature of heating treatment. The higher the temperature is, the higher the crystallisation degree will be, which can be shown in the following figure [Morgan, 2005].

Figure 3.3: Crystallisation degree of carbon fibres under different temperatures [Morgan, 2005] As can be seen from Figure 3.3, when the heating temperature is relatively low, i.e. 1000 °C, carbon fibres are basically in the amorphous state; with the rise of the temperature, the layered structures begin to appear and the crystallisation degree increases; when the temperature reaches 2000 °C, all of the amorphous structures are converted into the crystalline structures and the carbon fibres in this state are also called the graphite fibres. The crystalline structure of carbon fibres is illustrated in Figure 3.4 [Chung, 1994].

51 

 

Figure 3.4: Schematic diagram of the crystalline structure of carbon fibres [Chung, 1994] As shown in the above figure, the crystalline structure of carbon fibres is very similar to that of graphite. In layer planes, carbon atoms are hexagonally arranged with strong covalent bonds; while the layers are connected together by weak Van de Waals forces. This structure makes carbon fibres possess a strong orthotropic property, that is, the strength and modulus in the direction of fibre axis are orders of magnitude greater than those in the direction of perpendicular to the fibre axis. The crystallisation degree of carbon fibres can considerably influence their mechanical properties. Usually, with the increase of the crystallisation degree (i.e. with the increase of the temperature of heating treatment), the modulus of carbon fibres will increase while their strength will decrease, and vice versa. The mechanical properties (in the direction of fibre axis) of three types of typical carbon fibres [Morgan, 2005] are listed in Table 3.1, compared with two types of commonly used steel materials [Eurocode 3, 2005] [Eurocode 3, 2006].

52 

 

Table 3.1: Mechanical properties of carbon fibres compared with typical steel materials Density Material type

 (kg/m3)

Carbon fibre

Tensile Breaking Young's length strength u modulus E (GPa) u/(g) (km) (GPa)

Standard

1760

3.53

230

205

High strength

1820

7.06

294

396

High modulus

1870

3.45

441

188

S355

7850

0.50

210

6

Wire*

7850

1.77

210

23

Steel * Round wire for full-locked coil rope, made of cold drawn non-alloy carbon steel. As shown in Table 3.1, the tensile strengths of all carbon fibres are higher than those of steel materials, while their densities are much lower. The breaking length, which is defined as the maximum length of a hanging bar that could suspend its own weight, is a good parameter to show the high strength and lightweight characteristics of certain materials. It can be calculated by u/(g), where g is the standard gravity constant of 9.8 m/s2. As can be seen from Table 3.1, the breaking lengths of carbon fibres are one order of magnitude larger than those of steel materials. Although the mechanical properties of CFRPs are determined primarily by the fibre type, fibre volume and fibre direction, the polymer resin as the matrix is also indispensable. The functions of polymer resins in CFRPs can be described as follows [Michaeli et al., 1989]: (a)

keeping the chosen orientation of the carbon fibres according to the design;

(b)

transferring the load into the carbon fibres;

(c)

providing lateral support to prevent buckling under compressive loading;

(d)

protecting the fibres from the chemicals in the environment.

The polymer resins for CFRPs are of two main types, namely, thermoplastic resin and thermosetting resin. This classification of polymers is based on their thermal (thermomechanical) response [Ebewele, 2000]. Although the thermoplastic resin and thermosetting resin sound similar, they have different molecular structures and thus having very different properties.

53 

 

Thermoplastic resins, also known as thermosoftening plastics, are polymers linked by intermolecular interactions or Van der Waals forces, forming a linear or branched molecular structure. This linear or branched molecular structure can only provide a relatively small restriction for the motion of molecular chains, which makes the thermoplastics remeltable and tractable upon the application of heat and pressure after curing. [Winistoefer, 1999] Thermosetting resins, also known as thermosets, are polymers joined together by chemical bonds, forming a highly cross-linked molecular structure. This cross-linked structure can greatly restrict the motion of molecular chains, which makes the thermosets unmeltable and intractable upon the application of heat after curing. [Winistoefer, 1999] The molecular structures of thermoplastic and thermosetting resins are illustrated in Figure 3.5 [Ebewele, 2000].

Figure 3.5: Molecular structures of thermoplastic and thermosetting resins [Ebewele, 2000] The mechanical properties of some commonly used polymer resins are listed in Table 3.2 [Chung, 1994] [Morgan, 2005].

54 

 

Table 3.2: Mechanical properties of commonly used polymer resins

Name

Polyethersulfone

1370

0.084

2.4

Polyetherether ketone

1310

0.070

3.8

Polyetherimide

1270

0.105

3.0

Orthophthalic polyester

1350

0.070

3.2

Vinylester

1250

0.075

3.3

Epoxy

1250

0.115

3.0

Type

Thermoplastic

Thermosetting

Tensile Young's strength u modulus E (GPa) (GPa)

Density  (kg/m3)

Comparing Table 3.1 and Table 3.2, it can be seen that the densities of polymer resins are slightly smaller than those of carbon fibres, while the strengths and moduli of polymer resins are orders of magnitude smaller than those of carbon fibres. The differences of strength and modulus between carbon fibres and polymer resins make CFRP have a strong orthotropic characteristic. In the fibre direction, CFRP mainly exhibits the mechanical properties of fibres, i.e. relatively high strength and high modulus. However, in the direction perpendicular to the fibre axis, CFRP mainly exhibits the mechanical properties of resins, i.e. relatively low strength and low modulus. Moreover, the orthotropy of carbon fibres themselves makes the orthotropy of CFRP even more evident.

3.1.2 Fabrication of CFRPs The fabrication of CFRPs is the process of embedding carbon fibres into corresponding polymer resins. There are now many methods available to fabricate CFRPs. The selection of methods mainly depends on the type of resin used, the amount of CFRP products fabricated, the requirement of product surface and etc. In this section, three commonly used methods for the fabrication of CFRPs are introduced, respectively.

55 

  3.1.2.1 Contact moulding Contact moulding is basically a hand lay-up method, which is shown in Figure 3.6. It is the oldest method for fabricating CFRPs but also the most widely used around the world. In contact moulding, the applied fibre reinforcement is layered and placed on a specially designed contact mould and then well wetted out layer by layer with the chosen polymer resin system. During the wetting, a consolidation roller is used by the fabrication operator to roll and apply pressure to remove the entrapped air. After the wetting, the mould and the semi-finished CFRP product are then enclosed by a vacuum bag and the air in the bag is removed to cure the CFRP. Sufficient time must elapse before removing the CFRP from the mould. Usually, a gel coat based on epoxy or unsaturated polyester resin is used to provide a smooth surface for the CFRP and a release coat is used to keep the CFRP from sticking to the mould. [Morgan, 2005]

Figure 3.6: Schematic diagram of contact moulding (photo credit: Laurensvan Lieshout) The contact moulding method has a lot of advantages, such as simplicity, cheapness and etc. However, its disadvantages are also prominent. First, the quality of the CFRP product is mainly dependent on the skill of the fabrication operator and the standardized production is difficult. Second, the CFRP product can have only one finished surface, which is in contact with the mould. Third, because the applied pressure is relatively low and the CFRP is typically cured at room temperature, the fibre volume fraction is limited to the natural packing density.

56 

  3.1.2.2 Resin transfer moulding (RTM) Resin transfer moulding, abbreviated to RTM, is an automated, closed-mould and vacuumassisted process, which is shown in Figure 3.7. In RTM, the resin matrix is transferred through a port, or series of ports, under moderate pressure (0.35 MPa - 0.70 MPa) into a closed and clamped mould in which the fibre reinforcement has already been positioned. During the transferring, a vacuum is generated to assist the matrix flow and reduce the formation of voids. After the transferring, the curing is started during which the mould is heated. Gel coats may be used to provide high-quality surfaces for the CFRP product. [Morgan, 2005]

Figure 3.7: Schematic diagram of resin transfer moulding (RTM) (photo credit: Net composites) RTM is a good method to achieve high fibre volume fractions for complex CFRP shapes. Its main advantages can be described as follows: (i)

ability to use automated preforms and achieve repeatable high quality;

(ii)

the CFRP product has two finished surfaces;

(iii)

speed of production is relatively quick and cost of production is competitive;

(iv)

environmentally friendly due to the use of closed moulds.

Its disadvantages include: the CFRP product is usually limited to a small size; during RTM, unimpregnated areas may occur in CFRP, which usually results in very expensive scrap

57 

  parts. Moreover, it should be noted that the low-viscosity polymer resin needs to be applied in RTM.

3.1.2.3 Pultrusion Pultrusion is also an automated process, which is used for producing continuous lengths of CFRP shapes with constant cross-sectional area (see Figure 3.8). As shown in Figure 3.8, the process of pultrusion can be divided into six operations. The first operation is fibre reinforcement handling, that is, the requisite number of carbon fibre tows are positioned in a suitable creel to prepare for the entry into the resin bath. The second operation is matrix impregnation, that is, the carbon fibres are pulled out from the creel and then well saturated in the resin bath. The third operation is pre-die forming. In this operation, the preform plates gently shape the material and remove all but about 10 % of the excess resin prior to entering into the heated die. The fourth operation is curing and shaping in heated die. In this operation, the resin-impregnated carbon fibres are pulled through a heated stationary die where the resin undergoes polymerization and then a rigid CFRP profile is formed that corresponds to the shape of the die. The fifth operation is providing traction by pulling unit. The pultruded CFRP product is cooled prior to the pulling unit, which can be a caterpillar puller or a reciprocating puller. Typical pulling line speeds vary in the range 1.5 m/h - 100 m/h, depending on the section being produced. The sixth operation is cutting off by saw. Once the pultruded section has left the die and cooled sufficiently, it is clamped and a flying saw moves along with the clamped section to cut off required lengths. [Morgan, 2005]

Figure 3.8: Schematic diagram of pultrusion (photo credit: Universal pultrusions)

58 

  The pultrusion is an old and widely used method for the fabrication of long carbon fibre reinforced polymers. Its advantages are prominent: (i)

production speed can be very fast and hence production cost can be relatively low;

(ii)

carbon fibre volume fraction can be accurately controlled;

(iii)

handling of fibre reinforcement is easy and its cost is minimised since the reinforcement is just taken from a creel;

(iv)

size of the CFRP product can be very large and the product quality is very good.

The main disadvantage of pultrusion is that the cross section of CFRP product is limited to constant or near constant.

3.2 Introduction of CFRP Cables This section is devoted to the introduction of cable fibre reinforced polymer (CFRP) cables. The structural forms and mechanical properties of such cables are presented, respectively.

3.2.1 Structural forms of CFRP cables As an advanced composite material, the CFRP has many merits, including high strength, lightweight, no corrosion, high fatigue resistance, low relaxation and low thermal expansion. This makes CFRP an excellent material in construction industry. The first practical utilization of CFRPs in construction was in 1991 for strengthening the Ibach Bridge in Lucerne, Switzerland [Meier, 1992]. From then on, more and more CFRP products were used not only in strengthening, repairing, reinforcing, pre-stressing but also as cables. According to their structural forms, the existing CFRP cables can be mainly classified into four types, whose appearances and cross sections are shown in Figure 3.9.

59 

 

Figure 3.9: Schematic diagrams of four types of CFRP cables Figure 3.9 (a) shows the CFRP cable in the form of lamella, which can be fabricated by pultrusion or lamination. Figure 3.9 (b) shows the CFRP cable in the form of rod, which is usually fabricated by pultrusion; such CFRP cable can be made up of a single rod or a rod bundle, and the CFRP rod can be plain round or deformed. Figure 3.9 (c) shows the CFRP cable in the form of strip-loop, which is fabricated by winding a continuous CFRP strip on two pins; then the strip-loop can choose to be laminated or non-laminated. Figure 3.9 (d) shows the CFRP cable in the form of wire rope, which is fabricated by twisting several CFRP wires into a helix; the CFRP wires used are usually produced by pultrusion.

3.2.2

Mechanical properties of CFRP cables

The mechanical properties along the fibre direction of CFRPs, such as tensile strength u and Young’s modulus E, are usually approximately 60% of those of the carbon fibres because the fibre volume fraction is usually 60%. It should also be noted that u and E of CFRP cables are slightly smaller than those of corresponding CFRPs; this is similar to the fact that u and E of steel cables are slightly smaller than those of steel wires. The mechanical properties of three CFRP cables from different producers and a steel fulllocked coil rope are listed in Table 3.3 [Schlaich et al., 2015].

60 

 

Table 3.3: Mechanical properties of CFRP cables compared with steel cable Mitsubishi Leadline

Tokyo Rope CFCC

EMPA CFRP strip-loop

Steel fulllocked coil rope

Description

Parallel CFRP deformed rods

Twisted CFRP round wires

Non-laminated looped CFRP thin strip

Twisted steel round and zprofile wires

Density  (kg/m3)

1600

1500

1500

7850

2.3

2.1

2.0

1.5

147

137

120

160

Cable type Structural form

Tensile strength u (GPa) Young's modulus E (GPa)

The above three CFRP cables are made from standard cable fibres (see Table 3.1) and have already been used in cable structures [Schlaich et al., 2012]. The steel full-locked coil rope is a commonly used cable type in cable roofs and facades. As seen from Table 3.3, the tensile strengths of CFRP cables are higher than that of steel cable while their densities are only approximately 1/5 of steel cable’s density. In addition to high strengths and low weights, CFRP cables have better corrosion and fatigue resistance and lower thermal expansion than steel cables. Because carbon fibres have excellent creep resistance, the stress relaxation of CFRP cables is very small, and there is no need to limit sustained tensile stresses [Morgan, 2005]. Except for the advantages mentioned above, CFRP cables also have some disadvantages; these include lower Young’s moduli than that of steel cables (see Table 3.3), which may have a negative effect on the structural stiffness of certain CFRP cable structures. However, utilising CFRP cables in orthogonally loaded cable structures is a feasible way to utilise the advantages of CFRP cables while bypassing their disadvantages. As discussed in Chapter 2, substituting CFRP cables for steel cables in such structures can fully exploit the

61 

  high tensile strength of CFRP cables and effectively avoid the unfavourable influence of their relatively low Young’s modulus, thus reducing the amount of cable used and improving economic efficiency.

3.2.3 Unit prices of CFRP cables The cost of CFRP cables is of particular interest to the industrial community. Usually, the unit prices of CFRP cables are influenced by two parameters, i.e. the tensile strength and the Young’s modulus. Because the prices of carbon fibres or CFRPs from their producers are the commercial secret to some extent, it is difficult to obtain the accurate unit prices of CFRP cables. Through personal inquiries of the producers and also considering the information from the Internet, the unit prices (prices per unit weight) of the CFRP cables with different tensile strengths u and different Young’s moduli E (the selected values of u and E would be used in following case studies) were estimated and then listed in Table 3.4.

Table 3.4: Unit prices of CFRP cables (unit: €/kg)

u (GPa)

E (GPa)

2

2.7

3.6

4.7

120

55

70

115

235

160

85

115

185

370

210

165

210

325

645

280

345

445

680

1350

As shown in Table 3.4, the unit prices of CFRP cables are much higher than those of steel cables (for example, the unit price of steel full-locked coil rope is only approximately 10 €/kg). The variation tendency of unit prices of CFRP cables can be shown in Figure 3.10. The percentages represent the ratios of unit prices of CFRP cables to that of steel cable (10 €/kg) and the directions of arrows indicate the increasing directions of u and E.

62 

 

Figure 3.10: Variation tendency of unit prices of CFRP cables compared to that of steel cable As seen from the above figure, the unit price rises with the increase of either u or E of CFRP cables, and the increase of E makes the price increase slightly faster than that of u.

63 

 

4

Case Study A: Using CFRP Cables in a Cable Net Facade

According to the theoretical analysis in Chapter 2 and the mechanical properties of CFRP cables introduced in Chapter 3, utilising CFRP cables in orthogonally loaded cable structures is advantageous. In order to prove this assertion, two typical orthogonally loaded cable structures, i.e. a cable net facade and a spoked wheel cable roof, are selected and investigated in case studies in Chapter 4 and Chapter 5, respectively. In this chapter, the mechanical properties and economic efficiency of the cable net facade with CFRP cables are compared with those of corresponding steel cable structure. Furthermore, to address possible designs in the future, not only normal CFRP cables with standard carbon fibres but also CFRP cables with very high tensile strengths and/or Young’s moduli are examined.

4.1 Introduction of Cable Net Facade The cable net facade is an innovative facade structure, which was first proposed by Jörg Schlaich in 1993 for the Kempinski Hotel in Munich [Schlaich, 1995]. Such facade substitutes conventional support structure for highly pre-tensioned flat cable net to offer a transparent and seamless look with all the capability of conventional facade systems. After just twenty years of development, now cable net facades have been used in many buildings around the world due to their advantages in aesthetics and mechanics. Four typical cable net facades are shown in Figure 4.1.

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1: Cable net facade of the Kempinski Hotel in Munich, Germany (photo credit: schlaich bergermann und partner); 2: Cable net facade of the Time Warner Centre in New York, USA (photo credit: Jürgen Schmidt); 3: Cable net facade of the Jia Ming Centre in Peking, China (photo credit: schlaich bergermann und partner); 4: Cable net facade of the Ministry of Foreign Affairs in Berlin, Germany (photo credit: schlaich bergermann und partner); Figure 4.1: Typical cable net facades As seen from the above figure, cable net facades primarily consist of a flat cable net and several glass panels supported by it. In the cable net, pre-tensioned horizontal and vertical cables are connected by cable clamps. The corners of each glass panel are fixed by clamps, which are connected to the cable net.

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4.2 Design Parameters of Investigated Cable Net Facade In this case study, a simply cable net facade was selected for investigation. Furthermore, only the flat cable net was modelled; the glass panels and the connection members were simplified as dead load. The geometry and boundary conditions of the studied steel or CFRP flat cable nets are shown in Figure 4.2:

Figure 4.2: Geometry and boundary conditions of steel or CFRP flat cable nets It was assumed that the cable net facades were located in Berlin, Germany. Based on the location, the wind load = 1.0 × 1.7 × 0.39 × (30 / 10) 0.37 = 1.0 kN/m2 [Eurocode 1, 2005] [Goris and Schneider, 2006], which was assumed to act at the nodes and perpendicularly to the cable net plane. The dead load of the glass panels and connection members, which was set to 0.5 kN/m2, was also assumed to act at the nodes but in the direction of gravity (i.e., along the vertical cables). The flat cable nets were designed with the Limit States Design method provided by the Eurocode using an Ultimate Limit State (ULS) factor of 1.5 for the wind load [Eurocode 0, 2002] and a partial safety factor of 1.65 for the resistance of steel cables [Eurocode 3, 2006] [PFEIFER, 2011]. For comparison purposes, the partial safety factor for the CFRP cables was also set to 1.65.

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  Additionally, according to design experience, the deformation (i.e., deflection) limit of flat cable nets in the Serviceability Limit State (SLS) was set to L/50 = 30/50 = 0.6 m = 600 mm. The steel flat cable net with full-locked coil rope cables, whose form and mechanical properties are shown in Table 3.3, was used as a reference. The mechanical properties of the investigated CFRP cables are listed in Table 4.1.

Table 4.1: Material properties of investigated CFRP cables in Case Study A Mechanical property

Value

Density  (kg/m3)

1500

Tensile strength u (GPa)

2.0, 2.7, 3.6 and 4.7

Elastic modulus E (GPa)

120, 160, 210 and 280

As shown in Table 4.1, the tensile strength of CFRP cables varies from 2.0 GPa to 4.7 GPa with a constant increment ratio of 4/3 (i.e., 2.7/2.0 = 3.6/2.7 = 4.7/3.6 = 4/3), and their Young’s modulus varies from 120 GPa to 280 GPa also with the same ratio of 4/3 (i.e., 160/120 = 210/160 = 280/210 = 4/3). The above values apply to the products available on the market currently and in the future. Furthermore, these values of CFRP cables used in this research can also cover the range of

E and u of CFRP cables applied in existing CFRP cable structures, which can be illustrated in Figure 4.3.

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Figure 4.3: E and u of cables used in this research and some existing CFRP cable structures It should be noted that the investigated CFRP cables can possess any combination of u and E. This means, for every type of comparison, 16 different CFRP flat cable nets were designed and compared to the reference steel flat cable net.

4.3 Principles of Comparison and Design In this study, the following three items were compared: (a)

structural stiffness, expressed by the mid-span deflection in the Service Limit State;

(b)

amount of cable used, expressed by volume as cubic metres (m3);

(c)

economic efficiency, based on comprehensive consideration.

During the comparison, six rules were followed: (i)

the geometry in the Initial State and the boundary condition of the steel or CFRP flat cable nets were kept the same;

(ii)

the external load conditions of the steel and CFRP flat cable nets were identical and their ultimate bearing capacities were kept the same;

(iii)

every cable reached its ultimate tensile strength in the Ultimate Limit State;

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  (iv)

if the structural stiffness was compared, the amounts of cable used in the CFRP flat cable nets were kept the same as that of the steel flat cable net;

(v)

if the amount of cable used was compared, the deflections of the CFRP or steel flat cable nets in the Service Limit State were kept the same and equal to the deformation limit;

(vi)

when the economic efficiency was compared, the cable cost was found to be the primary factor. Other minor factors, such as cable size, support reaction and cable weight, were also considered.

Based on the design parameters and the above rules, all flat cable nets were designed with the same procedure, as shown in Figure. 4.4.

Figure 4.4: Design procedure of steel or CFRP flat cable nets As can be seen from Figure 4.4, the cross-sectional areas and pre-tension forces of cables were determined through repeated iterations to ensure the cable net facades simultaneously reached their deformation limit in the Service Limit State and ultimate strength limit in Ultimate Limit State (i.e., are optimally designed). It should be noted that for comparison reasons, the deflections of the CFRP flat cable nets could be less than or equal to the deformation limit in the Service Limit State, if the structural stiffness was compared.

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It should also be noted that, for flat cable nets in Ultimate Limit State, usually only the cables near mid-span could reach their ultimate tensile strengths. As a design practice, 19 vertical cables had the same cross-sectional area, so did 19 horizontal cables, and the extra cross-sectional areas of cables near boundary were taken as the safety margin. The design and calculation were completed using the general finite element software SOFiSTiK [SOFiSTiK, 2012]. Because the deformations of the flat cable nets were expected to be relatively large, the geometrical non-linearity of them was also considered in the finite element analysis, so as to ensure the accuracy of calculation.

4.4 Comparison Results First, the design results of the reference steel flat cable net are listed in Table 4.2.

Table 4.2: Design results of reference steel flat cable net Item

Value

Mid-span deflection (mm)

600

Pre-tension force (kN)

130

Cross-sectional area of vertical cable (mm2)

241

Cross-sectional area of horizontal cable (mm2)

227

Amount of cable used (m3)

0.27

Cable weight (kg)

2090

Support reaction of vertical cable (kN)

215

Support reaction of horizontal cable (kN)

200

Cable unit price (€/kg)

10

Cable cost (103 €)

20.9

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  It should be noted that, due to the need to bear dead load, the vertical cables were slightly stronger than the horizontal ones and the support reactions of vertical cables were also slightly greater than those of horizontal cables.

4.4.1

Comparison result of structural stiffness

For the comparison of structural stiffness, the result is shown in Figure 4.5. The structural stiffness is expressed as the mid-span deflection of flat cable nets; moreover, a greater deflection value represents a smaller structural stiffness and vice versa. The percentages represent the ratios of mid-span deflection values of CFRP flat cable nets to that of steel flat cable net and the directions of arrows indicate the increasing directions of u and E.

Figure 4.5: Comparison of mid-span deflections between CFRP and steel flat cable nets Figure 4.5 indicates that utilising CFRP cables in flat cable nets can achieve much higher structural stiffness compared to steel flat cable net. Increasing the tensile strength u of the CFRP cables to increase the pre-tension forces will significantly raise the structural stiffness. However, increasing the Young’s modulus E of the cables will not increase but rather slightly decrease the overall structural stiffness. This phenomenon can be attributed to the influence of geometrical non-linearity.

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Because the geometrical non-linearity is considered in the finite element analysis, the geometry and cable force of investigated flat cable nets will be updated in every iteration or step (see Appendix A). In this example, it means that the deflection of flat cable nets will gradually increase (i.e. the angle  will gradually decrease from 90°) and the cable force of flat cable nets will also increase. With the decreasing angle , the elastic stiffness

KE will increase (see Equation 2.70); with the increasing cable force, the geometric stiffness KG will increase (see Equation 2.71). This means that the structural stiffness K which is composed of KE and KG will gradually increase as the solution proceeded. The additional stiffness can be called the second order stiffness [Chen, 1997], which is due to the consideration of geometrical non-linearity and will make the deflection of flat cable nets decrease. Furthermore, according to the method of Limit State Desgin, the structural deformation is limited in Service Limit State, while the cable force is limited in Ultimate Limit State [Eurocode 0, 2005]. This cable force in the Ultimate Limit State is determined not by the pre-tension force but by the external load and the flat cable net deflection in the Ultimate Limit State, i.e. the angle between the external load and the cable axis at this time. Thus, a higher E of the cables will produce greater second order stiffness and thus lead to smaller deflections of the flat cable net in the Ultimate Limit State, although its deflection in the Service Limit State is greater than (see Figure 4.5) or equal to those of flat cable nets with lower E cables if the amount of cable used is compared (see following Section 4.4.2). Therefore, if u of CFRP cables is kept the same, increasing their E will increase the cable force in the Ultimate Limit State and decrease the applied pre-tension force, which eventually results in lower structural stiffness or a larger required cross-sectional area for the CFRP cables, that is, more cable will be used (see following Section 4.4.2). The above theory can also be demonstrated in detail by the following figure.

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Figure 4.6: Influence of geometrical non-linearity on flat cable nets In Figure 4.6, the flat cable net is expressed as the two-link cable, because they have the same calculation theory. The u of CFRP cables for both left and right investigated flat cable nets are kept the same. In the Initial State, the pretension force F1 is applied to the left flat cable net using higher E CFRP cables, while the pretension force F2 is applied to the right flat cable net using lower E CFRP cables. In the Service Limit State, if the structural stiffness (i.e. the deflection of flat cable net) is compared, the cross-sectional areas of CFRP cables for both flat cable nets are kept the same and it can be found that the deflection of flat cable net with higher E cables is greater than that of flat cable net with lower E cables; moreover, if the amount of cable used is compared, the cross-sectional areas of CFRP cables of left or right flat cable nets can be different in this case, while the deflections of both flat cable nets are identical, which are set to 600 mm. In the Ultimate Limit State, the external load P is multiplied with a factor 1.5. It can be found that the deflection of left flat cable net, which applies higher E CFRP cables, is

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  smaller than that of right flat cable net, which applies lower E CFRP cables, no matter the structural stiffness or the amount of cable used is compared. Therefore, in the Ultimate Limit State, the angle 1 will always be greater than the angle

2. Furthermore, because F1' 

1.5 P cos α1 2

(4.1)

F2' 

1.5 P cos α 2 2

(4.2)

it can be easily found that the final cable force greater than the final cable force

´

´

in the left flat cable net will also be

in the right flat cable net. This means, for the flat cable

net, if u of CFRP cables remains unchanged, increasing their E will lead to a greater final cable force in the Ultimate Limit State, which will cause the decline of pretension force applied in the Initial state (see following Figure 4.7 and Figure 4.10) and will result in a smaller structural stiffness if the structural stiffness is compared or a greater cross-sectional area of CFRP cables (i.e. more amount of cable will be used) if the amount of cable used is compared. As can be seen from the above analysis, choosing CFRP cables of relatively high Young’s modulus for flat cable nets is disadvantageous to the structures; if their tensile strengths are kept the same, using CFRP cables of relatively low Young’s modulus in flat cable nets can not only achieve relatively high structural stiffness but also help reduce the amount of cable used. The pre-tension forces of CFRP or steel flat cable nets in this case are compared in Figure 4.7. The percentages represent the ratios of pre-tension force values of CFRP flat cable nets to that of steel flat cable net and the directions of arrows indicate the increasing directions of u and E.

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Figure 4.7: Comparison of pre-tension forces between CFRP and steel flat cable nets The increase of structural stiffness is because of the increase of pre-tension force. As can be seen in Figure 4.7, the pre-tension forces of CFRP cable nets are much greater than that of steel cable net; to provide clarity, the increasing directions of u and E are opposite to those in Figure 4.5. Furthermore, increasing u of CFRP cables can help increase the pre-tension force significantly, while increasing E of the cables will lead to a slight decrease of pre-tension force due to the effect of geometrical non-linearity.

4.4.2 Comparison result of amount of cable used For the comparison of amounts of cable used, result is shown in Figure 4.8. The percentages represent the ratios of amount of cable used of CFRP flat cable nets to that of steel flat cable net and the directions of arrows indicate the increasing directions of u and E.

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Figure 4.8: Comparison of amounts of cable used between CFRP and steel flat cable nets In order to give readers visual impression, the comparison result of the cross-sectional areas of the vertical cables is also shown in Figure 4.9; the cross-sectional areas of the horizontal cables have identical tendency. The percentages represent the ratios of corresponding cross-sectional areas of CFRP flat cable nets to that of steel flat cable net and the directions of arrows indicate the increasing directions of u and E.

Figure 4.9: Comparison of cross-sectional areas between CFRP and steel flat cable nets As can be seen from Figure 4.8 and Figure 4.9, using CFRP cables in the flat cable net can considerably decrease the amount of cable used as well as the required cable crosssectional area. Furthermore, choosing CFRP cables with a higher u can lead to a reduced amount of cable or cable cross-sectional area. However, increasing E of CFRP cables cannot reduce the amount of cable used or the cable cross-sectional area, but rather slightly increases both values due to the influence of geometrical non-linearity. 77 

 

The comparison result of pre-tension forces of vertical cables in this case is shown in Figure 4.10; the pre-tension forces of horizontal cables show a similar trend. The percentages represent the ratios of pre-tension force values of CFRP flat cable nets to that of steel flat cable net and the directions of arrows indicate the increasing directions of u and E. For straightforward comparison, the pre-tension force of steel cable net is drawn as a light red plane in this figure.

Figure 4.10: Comparison of pre-tension forces between CFRP and steel flat cable nets As can be seen from the above figure, the pre-tension forces of most CFRP cable nets are slightly higher than that of the steel cable net. Furthermore, a higher u or a lower E is found to yield higher applied pre-tension forces. Conversely, a decreasing u or increasing E will help decrease the pre-tension force. It should be noted that the pre-tension forces in Figure 4.10 are not the same as those in Figure 4.7 due to different design conditions used in the comparison of structural stiffness and the comparison of amounts of cable used. Compared to the applied pre-tension forces, the final support reactions in the Ultimate Limit State, which are equal to the cable forces in this state, are more significant, because they determine the design of supports and anchorages and thus directly influencing the cost of flat cable nets.

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  The comparison result of support reactions of vertical cables in this case is shown in Figure 4.11; the support reactions of horizontal cables show similar trend. The percentages represent the ratios of support reaction values of CFRP flat cable nets to that of steel flat cable net and the directions of arrows indicate the increasing directions of u and E. For straightforward comparison, the support reaction of steel cable net is drawn as a light red plane in this figure.

Figure 4.11: Comparison of support reactions between CFRP and steel flat cable nets As shown in the above figure, only one support reaction of CFRP flat cable net is slightly greater than that of steel flat cable net. This means, compared to steel cables, applying CFRP cables in flat cable nets can help decrease the support reaction and thus reducing the final costs of structures. Furthermore, comparing Figure 4.10 and Figure 4.11, it can be found that the greater the pre-tension force is, the smaller the support reaction will be. This indicates that the CFRP flat cable nets with higher pre-tension forces will lead to lower support reactions, which will bring about cheaper supports and anchorages and also lower structural costs, than the CFRP flat cable nets with lower pre-tension forces. The comparisons of structural stiffness and the amount of cable used confirmed that the stiffness of flat cable nets is primarily governed by the geometric stiffness KG (i.e., the cable’s tensile strength u). In such a structure with CFRP cables, whose u is much higher than that of the steel cables, pre-tension forces in the cables can be significantly higher, which can raise the structural stiffness if the amount of cable used is maintained; this may

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  also significantly reduce the amount of cable used if the structural stiffness is maintained instead.

4.4.3

Comparison of economic efficiency

For the comparison of economic efficiency, the comparison result of cable costs is most important. First, based on the amounts of cable used (see Figure 4.8) as well as the density of CFRP cables (see Table 4.1), the cable weights of CFRP flat cable nets were obtained and compared with that of steel flat cable net in Figure 4.12. The percentages represent the ratios of cable weight values of CFRP flat cable nets to that of steel flat cable net and the directions of arrows indicate the increasing directions of u and E.

Figure 4.12: Comparison of cable weights between CFRP and steel flat cable nets As shown in Figure 4.12, the cable weights of CFRP flat cable nets are much smaller than that of steel flat cable net not only because using CFRP cables can help reduce the amount of cable used but also because the density of CFRP cables is much smaller than that of steel cables. In addition, the higher u or the lower E is, the smaller the cable weight will be. Then, multiplying the above cable weights with the unit prices of CFRP cables (see Table 3.4), the cable costs of CFRP flat cable nets were obtained and are compared with that of steel flat cable net in Figure 4.13. The percentages represent the ratios of cable cost values of CFRP flat cable nets to that of steel flat cable net and the directions of arrows indicate

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  the increasing directions of u and E. For straightforward comparison, the cable cost of steel cable net is drawn as a light red plane in this figure.

Figure 4.13: Comparison of cable costs between CFRP and steel flat cable nets As shown in Figure 4.13, the tendency of the cable cost is different from that of the cable weight; for clarity, the increasing directions of u and E are opposite to those in Figure 4.12. Though all CFRP flat cable nets were much lighter than the steel flat cable net, the cable costs of most CFRP flat cable nets were greater than that of the steel flat cable net because the CFRP cable prices per unit weight are much higher than that of steel cable (see Table 3.4). Increasing u of CFRP cables will reduce the amount of cable used but raise the cable unit price concurrently; therefore, the cable nets using CFRP cables with ultra-high tensile strength (u = 4.7 GPa) cannot achieve economic efficiency. Increasing E of CFRP cables will increase the amount of cable and the unit cable price as well, giving CFRP flat cable nets very poor economic efficiency. However, there are still three flat cable nets with CFRP cables with lower Young’s modulus (E = 120 GPa) that can achieve lower cable costs compared to the steel flat cable net. The most economic CFRP flat cable net with respect to cable cost is the design with CFRP cables of u = 2.7 GPa and E = 120 GPa.

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  In addition to the cable cost, other minor factors, such as the cable size, the support reaction and the cable weight, were also considered in the comparison of economic efficiency. As shown in Figure 4.9, cable cross-sectional areas of all CFRP flat cable nets are smaller than that of steel flat cable net, that is, all CFRP cables have smaller sizes. This means that the required sizes of the anchorages and connection members for CFRP cables can be smaller than those for steel cables, which will save material and thus benefiting the economic competitiveness of CFRP flat cable nets. As shown in Figure 4.11, support reactions of most CFRP flat cable nets are smaller than that of the steel flat cable net, which will lead to simpler and cheaper anchorages and foundations and make the CFRP flat cable nets cheaper than the steel flat cable net in this regard. As shown in Figure 4.12, cable weights of all CFRP flat cable nets are significantly smaller than that of the steel flat cable net. With such a light cable system, the construction of CFRP flat cable nets can be easier than that of steel one and thus achieve economic efficiency in this regard. In conclusion, three flat cable nets with CFRP cables of E = 120 GPa are more economic than the steel flat cable net with respect to cable cost, which is the primary factor affecting economic efficiency. They are also more economic than the steel flat cable net with respect to the abovementioned minor factors, while their economies are similar to each other in these regards. Among these three CFRP flat cable nets, the author would currently recommend the one with CFRP cables of E = 120 GPa and u = 2.7 GPa as the most economic choice, primarily because it will minimise the cable cost.

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5

Case Study B: Using CFRP Cables in a spoked wheel Cable Roof

In addition to the cable net facade, spoked wheel cable roof is another typical orthogonally loaded cable structure. They have common ground. However, the s of most cables in spoked wheel cable roofs are smaller than 90°, which makes the advantages of using CFRP cables in spoked wheel cable roofs may be different from those of using CFRP cables in cable net facades. In this chapter, the mechanical properties and economic efficiency of a spoked wheel cable roof with CFRP cables are compared with those of a corresponding steel spoked wheel cable roof. To address possible designs in the future, not only normal CFRP cables with standard carbon fibres but also CFRP cables with very high tensile strengths and/or Young’s moduli are examined. Furthermore, the influence of varying control angles on the advantages of CFRP orthogonally loaded cable structures is also discussed.

5.1 Introduction of spoked wheel Cable Roof The spoked wheel cable roof is a highly efficient structural solution that encloses largevolume space with very light and slender cable system. As its name suggests, such cable structure consists of wheel and spoke cables. The design philosophy of spoked wheel cable roofs stems from the tensile spoked wheel [Schlaich et al., 2003], which is shown in Figure 5.1 [Masubuchi, 2013].

Figure 5.1: A typical tensile spoked wheel

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As shown in Figure 5.1, the horizontal wheel, which is in compression, is connected by several inclined spokes to the central hub. It is important to note that the tensile spoked wheel, which was invented less than 200 years ago, is much different from the conventional compressive spoked wheel, which has been used for thousands of years. In conventional spoked wheels, all spokes are compression members. However, in tensile spoked wheel, every spoke is a tension member, which makes the spokes much thinner than the compressive spokes and thus considerably reducing the structural weight und the material used. In approximately 1960s, the idea of tensioned spoked wheel was introduced into the field of cable structures. The American Pavilion completed in 1958 at the Brussels Universal and International Exposition, the New York State Pavilion completed in 1964 and the Oracle Arena completed in 1966 in Oakland are three well-known examples of this early stage. [Schlaich et al., 2003] [Masubuchi, 2013] Since the late 1980s, the spoked wheel principle has been applied for roofs of large stadiums, because the spoked wheel cable roof can efficiently cover a huge area because the annular shape of wheel can fit the external rims of stadiums, which are also usually annular. One of the most well-known examples at this date is the roof of Gottlieb-DaimlerStadium completed in 1993 in Stuttgart. [Schlaich et al., 2003] [Masubuchi, 2013] After half century of development, now spoked wheel cable roofs have already been built all over the world and their structural forms include not only the original one (see Figure 5.1) but also several variants originated from it. Six common structural forms of spoked wheel cable roofs and their corresponding real structures are listed in Figure 5.2 and Figure 5.3, respectively [Schlaich et al., 2014].

Figure 5.2: Six common structural forms of spoked wheel cable roofs

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1: Original tensile spoked wheel form: inner roof of National Stadium in Warsaw, Poland (photo credit: schlaich bergermann und partner); 2: Monolayer cable system with heavy weight: Cape Town Stadium, South Africa (photo credit: PHParsons); 3: Monolayer cable system with undulated compression ring: Jaber Al Ahmad Stadium, Kuwait (photo credit: schlaich bergermann und partner); 4: Form of one compression ring and two tension rings: Bao’an Stadium in Shenzhen, China (photo credit: schlaich bergermann und partner); 5: Form of two compression rings and one tension ring: National Sports Complex Olimpiyskiy in Kiev, Ukraine (photo credit: schlaich bergermann und partner); 6: Hybrid form: AOL Arena in Hamburg, Germany (photo credit: schlaich bergermann und partner); Figure 5.3: Six real spoked wheel cable roofs corresponding to Figure 5.2 In Figure 5.2, spoked wheel cable roofs with the forms from No. 1 to No. 5 are selfbalancing systems, that is, the tension forces of spoke cables are balanced by the compression force in the compression ring and thus producing no bending moment in roof support structures. From the original tensile spoked wheel form, substituting two tension rings and several compression stabs for the central hub, the form of one compression ring and two tension rings can be obtained; changing the central hub to only one tension ring and one compression ring to two compression rings, the form of two compression rings and one tension ring can be obtained. In addition to the multilayer cable systems, monolayer cable systems are also available for spoked wheel cable roofs. In the form of monolayer cable system with heavy weight, the wind suck will be balanced by the additional weight suspended by the cable system; in the

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  form of monolayer cable system with undulated compression ring, the cable system is shaped like a saddle to achieve sufficient stiffness against vertical loads and thus eliminating the heavy weight. However, the hybrid form of spoked wheel cable roof (No. 6) is no longer a self-balancing system. In this spoked wheel cable roof, the tensile forces of spoke cables are not only balanced by the compression force in compression ring but also directly bore by the foundation. As can be seen from Figure 5.3, in addition to the compression ring and the cable system, membrane cladding and other secondary structural members are also applied in spoked wheel cable roofs to ensure the structures have specific functions.

5.2 Design Parameters of Investigated Spoked Wheel Cable Roof In this case study, a simply spoked wheel cable roof in the form of one compression ring and two tension rings was selected for investigation. Furthermore, only the compression ring and the cable system were modelled; the membrane cladding and the other secondary structural members were simplified as dead load. The geometry and boundary conditions of the studied steel or CFRP spoked wheel cable roofs are shown in Figure 5.4:

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Figure 5.4: Geometry and boundary conditions of investigated spoked wheel cable roofs As can be seen in Figure 5.4, the investigated spoked wheel cable roof is a symmetrical structure in geometry. The diameter of the outer compression ring is 250 m, while the diameters of two inner tension rings are 100 m. The heights of all the compression stabs are 15 m. Cross sections of compression ring and compression stab, which are not the focus of this research, are illustrated as follows:

Figure 5.5: Cross-sectional view of compression ring (unit: mm)

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Figure 5.6: Cross-sectional view of compression stab (unit: mm) In the compression ring, a steel web was set at every anchoring point of spoke cable. The compression ring and compression stabs were designed conservatively, so as to prevent these structures from any failure or instability for all the studied steel or CFRP spoked wheel cable roofs. It was assumed that the spoked wheel cable roofs were located in Berlin, Germany. Based on the location, the snow load = 0.8 × 1.0 × 1.0 × 0.85 = 0.68 kN/m2 [Eurocode 1, 2003] [Goris and Schneider, 2006], which was assumed to act at the upper end nodes of compression stabs (also the cable nodes) and in the direction of gravity; the wind load was set to 0.6 kN/m2 [Eurocode 1, 2005] [Goris and Schneider, 2006] and assumed to act at the same nodes as the snow load but in the reverse direction. The dead load of the membrane and its supports was set to 0.1 kN/m2; the dead load of the catwalk and equipment along the upper inner ring was set to 3 kN/m; the dead load of every cable node was set to 10 kN; the dead loads of cables and compression stabs were automatically calculated by the software. All the above dead loads were assumed to act at the upper end nodes of compression stabs. The spoked wheel cable roofs were designed with the Limit States Design method provided by the Eurocode using an Ultimate Limit State (ULS) factor of 1.5 for both the snow load and the wind load, as well as 1.35 for the dead loads if they were combined with the snow load [Eurocode 0, 2002].

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A partial safety factor of 1.65 was set for the resistance of steel cables [Eurocode 3, 2006] [PFEIFER, 2011]. For comparison purposes, the partial safety factor for the CFRP cables was also set to 1.65. Additionally, according to design experience, the deformation (i.e., the deflection) limit of spoked wheel cable roofs under snow load or wind load in the Serviceability Limit State (SLS) was set to 2000 mm. The steel spoked wheel cable roof with full-locked coil rope cables, whose form and mechanical properties are shown in Table 3.3, was used as a reference. The mechanical properties of investigated CFRP cables, which are the same as those of CFRP cables used in Case Study A, can also be seen in Table 5.1.

Table 5.1: Material properties of investigated CFRP cables in Case Study B Mechanical property

Value

Density  (kg/m3)

1500

Tensile strength u (GPa)

2.0, 2.7, 3.6 and 4.7

Elastic modulus E (GPa)

120, 160, 210 and 280

The investigated CFRP cables can possess any combination of u and E. This means, for every type of comparison, 16 different CFRP spoked wheel cable roofs were designed and compared to the reference steel spoked wheel cable roof.

5.3 Principles of Comparison and Design In this study, the following three items were compared: (a)

structural stiffness, expressed by the mid-span deflection in the Service Limit State;

(b)

amount of cable used, expressed by volume as cubic metres (m3);

(c)

economic efficiency, based on comprehensive consideration.

During the comparison, six rules were followed: 89 

 

(i)

the geometry in the Initial State and the boundary condition of the steel or CFRP spoked wheel cable roofs were kept the same;

(ii)

the external load conditions of the steel and CFRP spoked wheel cable roofs were identical and their ultimate bearing capacities were kept the same;

(iii)

every cable reached its ultimate tensile strength in the Ultimate Limit State;

(iv)

if the structural stiffness was compared, the amounts of cable used in the CFRP spoked wheel cable roofs were kept the same as that of the steel spoked wheel cable roof;

(v)

if the amount of cable used was compared, the deflections of the CFRP or steel spoked wheel cable roofs in the Service Limit State were kept the same and equal to the deformation limit;

(vi)

when the economic efficiency was compared, the cable cost was found to be the primary factor. Other minor factors, such as cable size, compression forces in compression ring and compression stabs and cable weight, were also considered.

Based on the design parameters and the above rules, all spoked wheel cable roofs were designed with the same procedure, as shown in Figure. 5.7.

Figure 5.7: Design procedure of steel or CFRP spoked wheel cable roofs

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  As can be seen from Figure 5.7, the cross-sectional areas and pre-tension forces of cables should be determined through repeated iterations to ensure the spoked wheel cable roofs reached not only the deformation limit in the Service Limit State but also the ultimate strength limit in Ultimate Limit State at the same time. Furthermore, the specified geometry in the Initial State should also be maintained simultaneously, that is, the dead loads of spoked wheel cable roofs were balanced by the pretension forces and hence the deflection of spoked wheel cable roofs in the Initial State was zero. It should be noted that for comparison reasons, the deflections of the CFRP spoked wheel cable roofs could be less than or equal to the deformation limit in the Service Limit State, if the structural stiffness was compared. The design and calculation were completed using the general Finite Element software SOFiSTiK [SOFiSTiK, 2012]. Because the deformations of the spoked wheel cable roofs were expected to be relatively large, the geometrical non-linearity of them was also considered in the finite element analysis, so as to ensure the accuracy of calculation.

5.4 Comparison Results First, the design results of the reference steel spoked wheel cable roof were obtained, which are listed in Table 5.2.

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Table 5.2: Design results of reference steel spoked wheel cable roof Item

Value

Mid-span deflection due to snow load (mm)

2000

Mid-span deflection due to wind suction (mm)

2000

Pre-tension force of lower spoke cables (kN)

1640

Pre-tension force of upper spoke cables (kN)

1185

Cross-sectional area of lower spoke cables (mm2)

4964

Cross-sectional area of upper spoke cables (mm2)

2165

Cross-sectional area of lower inner ring (mm2)

31416

Cross-sectional area of upper inner ring (mm2)

13685

Amount of cable used (m3)

35.66

Cable weight (kg)

279931

Compression force in compression ring (kN)

28890

Compression force in compression stabs (kN)

630

Cable unit price (€/kg)

10

Cable cost (103 €)

2799.4

It should be noted that, due to the different load conditions, the cross-sectional area of lower spoke cables was not the same as that of upper spoke cables, so did their pre-tension forces.

5.4.1 Comparison result of structural stiffness The structural stiffness can be expressed as the mid-span deflection due to the snow load combined with the dead loads (abbreviated to “due to snow”) or due to the wind load combined with the dead loads (abbreviated to “due to wind”) of spoked wheel cable roofs;

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  moreover, in order to compare with the result in Case Study A, the change tendency of average deflections of these two directions is also presented. A greater deflection value represents a smaller structural stiffness and vice versa. The comparison results of structural stiffness are shown in Figure 5.8, Figure 5.9 and Figure 5.10, respectively. The percentages represent the ratios of corresponding mid-span deflection values of CFRP spoked wheel cable roofs to that of steel spoked wheel cable roof; the directions of arrows indicate the increasing directions of u and E.

Figure 5.8: Comparison of mid-span deflections due to snow between CFRP and steel spoked wheel cable roofs

Figure 5.9: Comparison of mid-span deflections due to wind between CFRP and steel spoked wheel cable roofs

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Figure 5.10: Comparison of average deflections between CFRP and steel spoked wheel cable roofs As can be seen from the above three figures, utilising CFRP cables in spoked wheel cable roofs can considerably help to raise the structural stiffness compared to the spoked wheel cable roof with steel cables. Furthermore, the change tendencies of deflections due to snow and deflections due to wind as well as average deflections are similar, that is, increasing either the Young’s modulus E or the tensile strength u of the CFRP cables can raise the structural stiffness, but increasing u is a more efficient way to increase the structural stiffness. The pre-tension forces of CFRP lower or upper spoke cables in this case are compared with those of corresponding steel spoke cables in Figure 5.11 and Figure 5.12, respectively. The pre-tension forces of lower inner rings show similar trend as those of lower spoke cables, while the pre-tension forces of upper inner rings show similar trend as those of upper spoke cables. The percentages represent the ratios of pre-tension force values of CFRP cables to that of steel cables and the directions of arrows indicate the increasing directions of u and E. It should be noted that, to provide clarity, the increasing directions of u and E in Figure 5.11 and Figure 5.12 are opposite to those in Figure 5.8 and Figure 5.9.

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Figure 5.11: Comparison of lower spoke cable pre-tension forces between CFRP and steel spoked wheel cable roofs

Figure 5.12: Comparison of upper spoke cable pre-tension forces between CFRP and steel spoked wheel cable roofs The above two figures have similar variation trend, which indicates that the increase of structural stiffness is a product of the increase of pre-tension force. As can be seen from these two figures, the pre-tension forces of CFRP spoke cables are much greater than those of steel spoke cables; moreover, increasing u of CFRP cables can help increase the pre-tension force significantly, while increasing E of CFRP cables will lead to a slight decrease of pre-tension force due to the effect of geometrical non-linearity (see Section 4.4.1).

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  5.4.2

Comparison result of amount of cable used

For the comparison of amounts of cable used, the result is shown in Figure 5.13. The percentages represent the ratios of amount of cable used of CFRP spoked wheel cable roofs to that of steel wheel cable roof and the directions of arrows indicate the increasing directions of u and E.

Figure 5.13: Comparison of amounts of cable used between CFRP and steel flat cable nets In order to give readers visual impression, the comparison results of the cross-sectional areas of the lower and upper spoke cables are also shown in Figure 5.14 and Figure 5.15, respectively. The cross-sectional areas of lower inner rings show similar trend as those of lower spoke cables, while the cross-sectional areas of upper inner rings show similar trend as those of upper spoke cables. The percentages represent the ratios of corresponding cross-sectional areas of CFRP spoked wheel cable roofs to that of steel spoked wheel cable roof and the directions of arrows indicate the increasing directions of u and E.

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Figure 5.14: Comparison of cross-sectional areas of lower spoke cables between CFRP and steel flat cable nets

Figure 5.15: Comparison of cross-sectional areas of upper spoke cables between CFRP and steel flat cable nets As can be seen from Figure 5.13, Figure 5.14 and Figure 5.15, using CFRP cables in the spoked wheel cable roof can considerably decrease the amount of cable used as well as the cross-sectional areas of cables. Furthermore, increasing either u or E of CFRP cables can lead to a reduced amount of cable used or required cable cross-sectional areas. However, applying CFRP cables with higher u is more efficient than using CFRP cables with higher E to reduce the amount of cable used or the cross-sectional areas of cables in the spoked wheel cable roof. The comparison results of pre-tension forces of lower and upper spoke cables in this case are shown in Figure 5.16 and Figure 5.17, respectively; the pre-tension forces of lower

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  inner rings show a similar trend as those of lower spoke cables, while the pre-tension forces of upper inner rings show similar trend as those of upper spoke cables. The percentages represent the ratios of pre-tension force values of CFRP spoked wheel cable roofs to that of steel spoked wheel cable roof and the directions of arrows indicate the increasing directions of u and E. For straightforward comparison, the corresponding pre-tension forces of steel spoked wheel cable roof are drawn as light red planes in these two figures.

Figure 5.16: Comparison of pre-tension forces of lower spoke cables between CFRP and steel spoked wheel cable roofs

Figure 5.17: Comparison of pre-tension forces of upper spoke cables between CFRP and steel spoked wheel cable roofs

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  As can be seen from the above two figures, the pre-tension forces of most CFRP spoked wheel cable roofs are higher than that of the steel spoked wheel cable roof. Furthermore, a higher u or a lower E is found to yield higher applied pre-tension forces. Conversely, a decreasing u or increasing E will help decrease the pre-tension forces. It should be noted that the pre-tension forces in Figure 5.16 and Figure 5.17 are not the same as those in Figure 5.11 and Figure 5.12, because different design conditions were used in the comparison of structural stiffness and the comparison of amounts of cable used. Compared to the applied pre-tension forces, the compression forces of compression ring and compression stabs in the Ultimate Limit State are more significant, because they determine the design of compression ring and compression stabs and thus directly influencing the cost of flat cable nets. The comparison results of compression forces of compression ring and compression stabs in this case are shown in Figure 5.18 and Figure 5.19, respectively. The percentages represent the ratios of compression force values of CFRP spoked wheel cable roofs to that of steel spoked wheel cable roof and the directions of arrows indicate the increasing directions of u and E. For straightforward comparison, the corresponding compression forces of steel spoked wheel cable roof are drawn as light red planes in these two figures.

Figure 5.18: Comparison of compression forces of compression rings between CFRP and steel spoked wheel cable roofs

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Figure 5.19: Comparison of compression forces of compression stabs between CFRP and steel spoked wheel cable roofs As shown in the above two figures, majority of compression forces in both compression rings and compression stabs of CFRP spoked wheel cable roofs are greater than those of steel spoked wheel cable roof. Moreover, Figure 5.18 and Figure 5.19 have similar change tendencies, that is, the compression forces increase with either the increase of u of CFRP cables or the decrease of E of CFRP cables, while they decrease with the decrease of u or the increase of E. Furthermore, comparing Figure 5.18 with Figure 5.16 as well as comparing Figure 5.19 with Figure 5.17, it can be found that the higher the pre-tension forces are, the greater the compression forces will be. This indicates that the CFRP spoked wheel cable roofs with lower pre-tension forces will lead to lower compression forces in compression ring and compression stabs, which will bring about cheaper compression ring and compression stabs and also lower structural costs, than the CFRP spoked wheel cable roofs with higher pre-tension forces. The comparisons of structural stiffness and the amount of cable used confirmed that the stiffness of spoked wheel cable roofs is primarily governed by the geometric stiffness KG (i.e., the cable’s tensile strength u). In a spoked wheel cable roof with CFRP cables, whose u is much higher than that of the steel cables, pre-tension forces in the cables can be significantly higher, which can raise the structural stiffness if the amount of cable used is maintained; this may also significantly reduce the amount of cable used if the structural stiffness is maintained instead.

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5.4.3

Comparison of economic efficiency

For the comparison of economic efficiency, the comparison result of cable costs is most important. First, based on the amounts of cable used (see Figure 5.13) as well as the density of CFRP cables (see Table 5.1), the cable weights of CFRP spoked wheel cable roofs were obtained and compared with that of steel spoked wheel cable roof in Figure 5.20. The percentages represent the ratios of cable weight values of CFRP spoked wheel cable roofs to that of steel one and the directions of arrows indicate the increasing directions of u and E.

Figure 5.20: Comparison of cable weights between CFRP and steel spoked wheel cable roofs As shown in the above figure, the cable weights of CFRP spoked wheel cable roofs are much smaller than that of steel spoked wheel cable roof not only because using CFRP cables can help reduce the amount of cable used but also because the density of CFRP cables is much smaller than that of steel cables. Furthermore, increasing either u or E of CFRP cables can lead to a smaller cable weight. However, increasing u is more efficient than increasing E to reduce the cable weight. Then, multiplying the above cable weights with the unit prices of CFRP cables (see Table 3.4), the cable costs of CFRP spoked wheel cable roofs were obtained and are compared with that of steel spoked wheel cable roof in Figure 5.21.

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  The percentages represent the ratios of cable cost values of CFRP spoked wheel cable roofs to that of steel spoked wheel cable roof and the directions of arrows indicate the increasing directions of u and E. For straightforward comparison, the cable cost of steel spoked wheel cable roof is drawn as a light red plane in this figure.

Figure 5.21: Comparison of cable costs between CFRP and steel spoked wheel cable roofs It can be seen through comparing Figure 5.21 with Figure 5.20, the change tendency of the cable cost is different from that of the cable weight. For clarity, the increasing directions of

u and E in Figure 5.21 are opposite to those in Figure 5.20. Though all CFRP spoked wheel cable roofs were much lighter than the steel spoked wheel cable roof, the cable costs of most CFRP spoked wheel cable roofs were greater than that of the steel spoked wheel cable roof, mainly because the prices per unit weight of CFRP cables are much higher than that of steel cable (see Table 3.4). Increasing either u or E of CFRP cables will reduce the amount of cable used but raise the cable unit price at the same time. Therefore, the spoked wheel cable roofs using CFRP cables with relatively high u or relatively high E cannot achieve economic efficiency. However, compared to increasing E of cables, increasing their u can reduce more amount of cable used while increase less unit price of cables. This makes two spoked wheel cable roofs using CFRP cables with relatively low Young’s modulus (E = 120 GPa) still be able to achieve lower cable costs compared to the steel spoked wheel cable roof. In respect to the cable cost, these two designs, i.e. spoked wheel cable roofs with CFRP cables of E = 120 GPa and u = 2.0 GPa or u = 2.7 GPa, have the same economic characteristics. 102 

 

In addition to the cable cost, other minor factors, such as the cable size, the compression forces in compression ring and compression stabs and the cable weight, were also considered in the comparison of economic efficiency. As shown in Figure 5.14 and Figure 5.15, cable cross-sectional areas of CFRP spoked wheel cables are smaller than that of steel flat cable net, that is, CFRP cables have smaller sizes. This means that the required sizes of the anchorages and connection members for CFRP cables can be smaller than those for steel cables, which will save material and thus benefiting the economic competitiveness of CFRP spoked wheel cable roofs. As shown in Figure 5.18 and Figure 5.19, majority of compression forces in compression rings and compression stabs of CFRP spoked wheel cable roofs, especially some spoked wheel cable roofs using CFRP cables with relatively high tensile strength and low Young’s modulus, are greater than those of steel spoked wheel cable roof. This means that the required sizes of the compression rings and compression stabs in these CFRP spoked wheel cable roofs should be bigger than those in the steel one, which will go against the economic competitiveness of these CFRP spoked wheel cable roofs. As shown in Figure 5.20, cable weights of all CFRP spoked wheel cable roofs are significantly smaller than that of the steel spoked wheel cable roof. With such a light cable system, the construction of CFRP spoked wheel cable roofs can be easier than that of steel one and thus achieve economic efficiency in this regard. In conclusion, two spoked wheel cable roofs with CFRP cables of E = 120 GPa (u = 2.0 GPa or u = 2.7 GPa) are more economic than the steel spoked wheel cable roof with respect to cable cost, which is the primary factor affecting economic efficiency; moreover, these two designs have the same economic characteristics in this regard. They are also more economic than the steel spoked wheel cable roofs with respect to the abovementioned two minor factors, i.e. the cable size and the cable weight, though they are less economic than their steel counterpart in terms of the compression forces in compression ring and compression stabs. Furthermore, comparing these two designs with each other, applying CFRP cables of E = 120 GPa and u = 2.7 GPa can achieve smaller cable size and lower cable weight (two minor factors) than applying CFRP cables of E = 120 GPa and u = 2.0 GPa, while applying CFRP cables of E = 120 GPa and u = 2.0 GPa can only achieve

103 

  smaller compression forces in compression ring and compression stabs (one minor factor) than applying CFRP cables of E = 120 GPa and u = 2.7 GPa. Therefore, among these two CFRP spoked wheel cable roofs, the author would currently recommend the one with CFRP cables of E = 120 GPa and u = 2.7 GPa as the most economic choice.

5.5 Influence of  on Advantages of CFRP Orthogonally Loaded Cable Structures In Chapter 4 and Chapter 5, two typical orthogonally loaded cable structures, i.e. the cable net facade and the spoked wheel cable roof, with CFRP cables were investigated in case studies. The advantages of using CFRP cables in orthogonally loaded cable structures are demonstrated by comparing them with corresponding steel counterparts. However, as an important coefficient, the angle  (i.e. the control angle) is able to influence the advantages of using CFRP cables in orthogonally loaded cable structures. In this section, this influence is investigated though comparing these two CFRP orthogonally loaded cable structures with each other. First, the exact control angles of the investigated orthogonally loaded cable structures should be determined. Because all cables are perpendicular to the external load, the angle  of cable net facade is unquestionably 90°. For the spoked wheel cable roof, the angle  of spoke cables is 84.29°, while the  of two inner tension rings is 90°; through calculating from cable volume weighted average (i.e. 84. 29° × 21.49 m3 / 35.66 m3 + 90° × 14.17 m3 / 35.66 m3), the control angle of spoked wheel cable roof is determined as 86.56°. As can be seen, the control angle of spoked wheel cable roof is 3.44° smaller than that of cable net facade. The influence of this difference on the advantages of CFRP orthogonally loaded cable structures is illustrated in following three respects, i.e. the structural stiffness, the amount of cable used and the cable cost.

5.5.1 Influence of varying  on the advantage in respect of structural stiffness Through comparing Figure 4.5 with Figure 5.10, it can be found that the change tendency of structural stiffness of CFRP cable net facades is not the same as that of CFRP spoked

104 

  wheel cable roofs. Specifically, for CFRP cable net facades, increasing the tensile strength

u of CFRP cables can significantly help reducing the deflection of cable net facades, i.e. raising the structural stiffness, while increasing the Young’s modulus E of CFRP cables cannot increase the structural stiffness but slightly decrease it; for CFRP spoked wheel cable roofs, increasing either u or E is able to raise the structural stiffness of spoked wheel cable roofs, but increasing u is nevertheless a more efficient way to raise the stiffness. Furthermore, the increment degrees of structural stiffness brought by applying CFRP cables in the cable net facade and in the spoked wheel cable roof were also different. Since the structural stiffness can be expressed as the external load divided by the deflection, the increment degree of structural stiffness was set to “(1 / deflection of CFRP cable structure  1 / deflection of steel cable structure) / (1 / deflection of steel cable structure) × 100 %”. The increment degrees of structural stiffness of CFRP cable net facades are compared with those of CFRP spoked wheel cable roofs in Figure 5.22. The light red surface represents the corresponding stiffness increment degrees of investigated 16 CFRP cable net facades compared to the reference steel cable net facade, while the green surface represents the corresponding values of investigated 16 CFRP spoked wheel cable roofs compared to the reference steel cable structure; the directions of arrows indicate the increasing directions of

u and E.

Figure 5.22: Comparison of increment degrees of structural stiffness between CFRP cable net facades and CFRP spoked wheel cable roofs

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  As can be seen from Figure 5.22, most of CFRP cable net facades have higher stiffness increment degrees than corresponding CFRP spoked wheel cable roofs, except for the cable net facade with CFRP cables of E = 280 GPa and u = 2.0 GPa, which is the combination of greatest Young’s modulus and smallest tensile strength of investigated CFRP cables. Furthermore, this trend, i.e. the stiffness increment degrees of CFRP cable net facades are higher than those of CFRP spoked wheel cable roofs, will expand with the increase of u of CFRP cables and the decrease of E of CFRP cables; while it will shrink with the increase of E and the decrease of u. The aforementioned phenomenon indicate that, for the aspect of structural stiffness, using CFRP cables with relatively high tensile strength in the orthogonally loaded cable structures with relatively great control angle  like the cable net facade can achieve greater advantage than using such CFRP cables in the orthogonally loaded cable structures with relatively small  like the spoked wheel cable roof; however, the smaller the  is, the greater the help of increasing the Young’s modulus of CFRP cable for the increment of structural stiffness will be, which means that using CFRP cables with relatively high Young’s modulus is also an efficient way to raise the structural stiffness in some orthogonally loaded cable structures with relatively small .

5.5.2 Influence of varying  on the advantage in respect of amount of cable used Through comparing Figure 4.9 with Figure 5.13, it can be found that the change tendencies of amounts of cable used for CFRP cable net facades and CFRP spoked wheel cable roofs are not the same. As shown in Figure 4.9, for CFRP cable net facades, increasing the tensile strength u of CFRP cables can lead to the significant reduction of the amount of cable used, while increasing the Young’s modulus E of CFRP cables cannot save any amount of cable used but slightly increase it. However, for CFRP spoked wheel cable roofs (see Figure 5.13), increasing either u or E is able to reduce the amount of cable used, while increasing u is found to be a more efficient way to reduce it.

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  Furthermore, the reduction degrees of amount of cable used resulting from using CFRP cables in the cable net facade and in the spoked wheel cable roof were also different. This reduction degree can be calculated by “(amount of cable used of steel cable structure  amount of cable used of CFRP cable structure) / amount of cable used of steel cable structure × 100 %”. The reduction degrees of amount of cable used of CFRP cable net facades are compared with those of CFRP spoked wheel cable roofs in Figure 5.23. In this figure, the light red surface represents the corresponding reduction degrees of amount of cable used for investigated 16 CFRP cable net facades compared to the reference steel cable net facade, while the green surface represents the corresponding values of investigated 16 CFRP spoked wheel cable roofs compared to the reference steel spoked wheel cable roof; the directions of arrows indicate the increasing directions of u and E.

Figure 5.23: Comparison of reduction degrees of amount of cable used between CFRP cable net facades and CFRP spoked wheel cable roofs As can be seen from Figure 5.23, most of CFRP cable net facades have higher reduction degrees of amount of cable used than corresponding CFRP spoked wheel cable roofs, except for the cable net facade with CFRP cables of E = 280 GPa and u = 2.0 GPa, which is the combination of greatest Young’s modulus and smallest tensile strength of investigated CFRP cables.

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  Furthermore, this trend, i.e. the reduction degrees of amount of cable used of CFRP cable net facades are higher than those of CFRP spoked wheel cable roofs, will expand with the increase of u of CFRP cables and the decrease of E of CFRP cables; while it will shrink with the increase of E and the decrease of u. The above phenomenon indicate that, for the respect of amount of cable used, using CFRP cables with relatively high tensile strength in the orthogonally loaded cable structures with relatively great control angle  like the cable net facade can achieve greater advantage than using such CFRP cables in the orthogonally loaded cable structures with relatively small  like the spoked wheel cable roof; however, with the decrease of the , the positive influence of increasing the Young’s modulus of CFRP cable on the reduction of amount of cable used will increase, which means that using CFRP cables with relatively high Young’s modulus is also an efficient way to reduce the amount of cable used in some orthogonally loaded cable structures which have relatively small .

5.5.3 Influence of varying  on the advantage in respect of cable cost The cable cost is an important factor to influence the economic efficiency of cable structures. Comparing the cable costs of CFRP cable net facades and CFRP spoked wheel cable roofs can also show the economies of corresponding CFRP orthogonally loaded cable structures. As can be seen from Figure 4.13 with Figure 5.21, for the cable cost, the investigated CFRP cable net facades and CFRP spoked wheel cable roofs have similar but not identical change tendencies. To be specific, applying CFRP cable with either exorbitant tensile strength u or exorbitant Young’s modulus E in either cable net facades or spoked wheel cable roofs cannot reduce the cable cost but raise it; for both orthogonally loaded cable structures, utilising CFRP cables with relatively low Young’s modulus (E = 120 GPa) was possible to achieve lower cable cost than the reference steel cable structures. However, the trends of cable costs of CFRP cable net facades and CFRP spoked wheel cable roofs were not exactly the same. For example, for the cable net facade, using CFRP

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  cables of u = 2.7 GPa and any E would lead to lower cable cost than using CFRP cables of u = 2.0 GPa and the same E; while for the spoked wheel cable roof, applying CFRP cables of u = 2.0 GPa and any E could bring about lower cable cost than using CFRP cables of u = 2.7 GPa and the same E. Furthermore, the reduction degrees of cable cost through using CFRP cables in the cable net facade and in the spoked wheel cable roof were also different. The reduction degree of cable cost can be defined as “(cable cost of steel cable structure  cable cost of CFRP cable structure) / cable cost of steel cable structure × 100 %”, which is able to show the economic efficiency of corresponding CFRP cable structures. Moreover, the plus reduction degree indicates that using CFRP cables in corresponding cable structures can save the cable cost, while the minus reduction degree means that using CFRP cables cannot reduce the cable cost but raise it. The reduction degrees of cable cost of CFRP cable net facades are compared with those of CFRP spoked wheel cable roofs in Figure 5.24. In this figure, the light red surface represents the corresponding reduction degrees of investigated 16 CFRP cable net facades, while the green surface represents the corresponding values of investigated 16 CFRP spoked wheel cable roofs; the directions of arrows indicate the increasing directions of u and E.

Figure 5.24: Comparison of reduction degrees of cable cost between CFRP cable net facades and CFRP spoked wheel cable roofs

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  As can be seen from Figure 5.24, most of CFRP cable net facades have higher reduction degrees of cable cost than corresponding CFRP spoked wheel cable roofs, except for the cable net facade with CFRP cables of E = 280 GPa and u = 2.0 GPa, which is the combination of greatest Young’s modulus and smallest tensile strength of investigated CFRP cables. In addition, there are three CFRP cable net facades (i.e. E = 120 GPa and u = 2.0 GPa, E = 120 GPa and u = 2.7 GPa and E = 120 GPa and u = 3.6 GPa) which can achieve plus reduction degrees (i.e. 28%, 34% and 19%, respectively); while there are only two CFRP spoked wheel cable roofs (i.e. E = 120 GPa and u = 2.0 GPa and E = 120 GPa and u = 2.7 GPa) which can achieve plus reduction degrees (i.e. 7% and 7%, respectively). This indicates that not only the number of economically efficient CFRP cable net facades compared to the reference steel cable structure is more than that of economically efficient CFRP spoked wheel cable roofs but also the improvement degrees of such CFRP cable net facades are greater than those of corresponding CFRP spoked wheel cable roofs. Furthermore, this trend, i.e. the reduction degrees of cable cost of CFRP cable net facades are higher than those of CFRP spoked wheel cable roofs, will expand with the increase of

u of CFRP cables. However, with the increase of E of CFRP cables, this trend will expand when u of CFRP cables is equal to 3.6 GPa or 4.7 GPa, while it will shrink when u is equal to 2.0 GPa or 2.7 GPa. From the comparison results of cable costs between CFRP cable net facades and CFRP spoked wheel cable roofs, it can be found that using CFRP cables in the orthogonally loaded cable structures with relatively great control angle  like the cable net facade can usually achieve lower cable cost and hence better economic efficiency than using CFRP cables in such structures with relatively small  like the spoked wheel cable roof, especially when the tensile strength of CFRP cables is relatively high.

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6

Anchoring CFRP Cables in Orthogonally Loaded Cable Structures

As explained in Chapter 3, the CFRP is a typical orthotropic material, that is, its mechanical properties, such as the strength and the modulus, perpendicular to the fibre direction is considerably lower than those in the fibre direction. For CFRP cables, the fibre direction is along the cable axis, in which the cables show relatively high tensile strength and relatively high Young’s modulus, while perpendicular to the cable axis, the cables mainly show the mechanical properties of polymer resins, which are low-strength, lowhardness and brittleness. Consequently, the anchorage designs for CFRP cables cannot directly copy those for steel cables, in which the steel is an isotropic material, but need to be innovatively and carefully treated. Anchoring CFRP cables is a main challenge to achieve the feasibility of CFRP orthogonally loaded cable structures. In this chapter, existing CFRP cable anchorages are classified and introduced; design principles of reliable CFRP cable anchorages are established, based on the analysis of existing anchorages; then two new anchorage designs are proposed in accordance with these design principles.

6.1 Existing CFRP Cable Anchorages Over the last few decades, several anchorage systems for use with CFRP cables have been proposed in the world. Based on the different structural forms, the existing CFRP cable anchorages can be mainly classified into three categories, that is, clamp anchorages, bond anchorages and pin-loaded anchorages. These three types of CFRP cable anchorages are introduced in this section, respectively.

6.1.1 Clamp anchorages The clamp anchorage, as its name suggests, is a type of anchorages relying on the clamp function, or more specifically, the friction force and/or the mechanical interlocking force, to anchor the CFRP cables. As a consequence, a compressive force perpendicular to the CFRP cable axis has to be applied [Schmidt et al., 2012]. There are many ways to obtain enough compressive force, such as clamping with plates and fastening bolts or using

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  wedges, and different ways have led to different anchorage systems. Several representative clamp anchorages for CFRP cables are listed and introduced as follows. (1)

Clamp anchorage with gradiently fastened bolts [Horvatits et al., 2004]

Figure 6.1: Clamp anchorage with gradient pre-stressed bolts (a) assembled view (b) exploded view As can be seen from Figure 6.1, in the above anchorage, the CFRP rod is clamped by two metal plates, which are fastened by two rows of metal clamping bolts. Especially, the fastening forces of the bolts will gradually increase from the anchorage mouth to the anchorage end. The hole in the rear of anchorage is to facilitate connection this anchorage to other parts of the structure. (2)

LAP strain anchorage [Dehn et al., 2005]

Figure 6.2: LAP strain anchorage (a) assembled view (b) exploded view As shown in the above figure, the LAP strain anchorage is a clamp anchorage, whose stiffness increases from the anchorage mouth to the anchorage end. This gradiently increased anchorage stiffness is achieved by gradually increasing the cross-sectional areas 112 

  of two steel plates. The trapezoid structure in the rear of anchorage is to facilitate the pretension. (3)

SIKA Leoba-Carbodur 2 (SLC II) [Dehn et al., 2005]

Figure 6.3: SIKA Leoba-Carbodur 2 (a) assembled view (b) exploded view As shown in Figure 6.3, the SIKA Leoba-Carbodur 2 anchorage consists of two parts, i.e. a temporary part (red) for pre-tensioning and a permanent part (white) for anchoring. The threaded metal rods at the sides of the anchorage are designed to facilitate the pretension. (4)

SIKA Stress Head 220 [Fischli et al., 2007]

Figure 6.4: SIKA Stress Head 220 (a) assembled view (b) exploded view As can be seen from the above figure, the SIKA Stress Head 220 anchorage is a full plastic anchorage. In this anchorage, the CFRP lamella is clamped by 2 plastic wedges in a conic CFRP cylinder. (5)

Composite wedge anchorage [Burtscher, 2008]

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Figure 6.5: Composite wedge anchorage (a) assembled view (b) exploded view As shown in Figure 6.5, the wedges of this anchorage are consisted of steel and epoxy. The epoxy part is in contact with the CFRP lamella, and especially, its thickness decreases from the anchorage mouth to the anchorage end. Moreover, the surface of the epoxy part is indented rather than smooth, so as to increase the coefficient of friction. (6)

CFCC strand wedge anchorage [Schmidt et al., 2012]

Figure 6.6: CFCC strand wedge anchorage (a) assembled view (b) exploded view As can be seen from above figure, the CFCC strand wedge anchorage is similar to the common wedge anchorage for the steel strand. However, an aluminium sleeve is added to protect the CFRP strand from local crushing. (7)

Thermoplastic clamp anchorage [Decker, 2008]

Figure 6.7: Thermoplastic clamp anchorage (a) full view (b) sectional view

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The thermoplastic clamp anchorage can also be called the thermoplastic clamp joint. It should be noted that only the CFRP cables using thermoplastic matrix are suitable for this anchorage. As shown in Figure 6.7, this anchorage is composed of a metal socket, which is usually made of aluminium. After the CFRP rod is placed in the socket, the temperature at the anchorage will be increased above the glass transition temperature of the matrix. Then the metal socket is compressed to be wavy, so as to generate enough mechanical interlocking force to anchor the CFRP rod.

6.1.2

Bond anchorages

The bond anchorages rely on the bond force to anchor the CFRP cables. Usually, the bond materials used are the resin-based mortar (e.g. epoxy resin) or the cement-based mortar (e.g. high-expansion cement). The bond anchorages can be applied to anchor almost all types of CFRP cables, including lamellas, rods and strands. In order to increase the bond force and thus raising the anchorage efficiency, the anchored CFRP cables can be engraved beforehand. Several typical CFRP cable bond anchorages are listed and introduced as follows. (1)

Inverse conical bond anchorage [Kollegger, 2001]

Figure 6.8: Inverse conical bond anchorage (a) full view (b) sectional view As can be seen from Figure 6.8, in the inverse conical bond anchorage, the inner cavity of steel socket is inverse conical, instead of cylindrical. Furthermore, the shape of mortar inside is also inverse conical, which will help achieve uniformly distributed radial compression force and shear force on the anchored CFRP cable. Consequently, the anchorage efficiency of this anchorage will be improved, compared with the bond anchorage using cylindrical cavity.

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  (2)

Inverse conical bond anchorage with indentation [Kollegger, 2001]

Figure 6.9: Inverse conical bond anchorage with indentation (a) full view (b) sectional view As shown in Figure 6.9, the above anchorage also applies the inverse conical inner cavity and mortar. However, at the mouth of the anchorage, the mortar has an indentation, which will reduce the stress concentration of CFRP cable near the mouth of the anchorage and thus increasing the anchorage efficiency. (3)

Segmented conical anchorage [Kollegger, 2001]

Figure 6.10: Inverse segmented cone anchorage (a) full view (b) sectional view As can be seen from the above figure, the inner cavity of this anchorage is shaped uniquely, which is like a segmented cone; moreover, the size of the cone gradually increases from the anchorage end to the anchorage mouth. This special design is to make the radial compression force as well as the shear force on the CFRP cable distribute as uniformly as possible, so as to raise the anchorage efficiency. (4)

Highly expansive mortar anchorage [Karbhari, 1998]

Figure 6.11: Highly expansive mortar anchorage (a) full view (b) sectional view

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As shown in Figure 6.11, the above anchorage utilises the expansion of the mortar to generate enough radial compression force, so as to anchor the CFRP cable. Furthermore, because the highly expansive mortar (e.g. high-expansion cement) is used, the creep of the anchorage will be significantly reduced. (5)

Gradiently reduced cable end anchorage [Mönig and Preis, 1979]

Figure 6.12: Gradiently reduced cable end anchorage (a) full view (b) sectional view As can be seen from Figure 6.12, this anchorage applies normal conical inner cavity. However, the anchored CFRP rod has gradiently reduced end, that is, the cross-sectional area of the CFRP rod decreases gradiently from the anchorage mouth to the anchorage end. With this measure, the inner fibres of the CFRP rod can also be directly bonded by the mortar, the same as the outer fibres. Hence the anchorage efficiency will be improved. (6)

Forked cable end anchorage [Schwegler, 2005]

Figure 6.13: Forked cable end anchorage (a) full view (b) sectional view As its name suggests, in the forked cable end anchorage (see Figure 6.13), the tail of anchored CFRP cable is forked. This special design has two functions: upsetting the cable end to make its size greater than that of anchorage mouth; increasing the bond surface. (7)

Gradient Anchorage System [Meier and Farshad, 1996]

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Figure 6.14: Gradient Anchorage System (a) full view (b) sectional view As shown in Figure 6.14, the Gradient Anchorage System applies normal conical inner cavity. However, the stiffness of mortar inside increases from the anchorage mouth to the anchorage end, which is achieved by increasing the density of Al2O3 ceramic granules in the epoxy resin from the anchorage mouth to the anchorage end. With this type of mortar, the stress peak of the CFRP cable near the anchorage mouth can be greatly reduced and the anchorage efficiency will be improved. This anchorage has been successively used in the Stork Bridge to anchor the CFRP stay cables [Meier et al., 1996].

6.1.3

Pin-loaded anchorages

The “pin-loaded” is a novel concept for CFRP cable anchorages, which was proposed by Professor Urs Meier of EMPA. With the pin-loaded anchorages, the CFRP strip-loop cable will be anchored by the self-balance of tension force in the cable. According to the different forms of the CFRP strip-loops, these anchorages can be classified into two types, i.e. the CFRP non-laminated strip-loops pin-loaded anchorage and the CFRP laminated strip-loop pin-loaded anchorage. These two types of pin-loaded anchorages are listed and introduced as follows. (1)

Non-laminated CFRP strip-loop pin-loaded anchorage [Winistoefer, 1999]

Figure 6.15: Non-laminated CFRP strip-loop pin-loaded anchorage (a) full view (b) schematic structural view 118 

 

For the non-laminated CFRP strip-loop pin-loaded anchorage (see Figure 6.15), a very thin CFRP strip is continuously wound on two round pins to form the loops and then the striploops are never bonded together or only the beginning and the end of the strip are bonded. In this anchorage, different CFRP layers can slide between each other during the loading, which will help achieve uniform stress distribution in the CFRP cable and thus improving the anchorage efficiency. (2)

Laminated CFRP strip-loop pin-loaded anchorage [Carbo-Link, 2012]

Figure 6.16: Laminated CFRP strip-loop pin-loaded anchorage (a) full view (b) schematic structural view As shown in Figure 6.16, for the Laminated CFRP strip-loop pin-loaded anchorage, a CFRP prepreg strip is usually used to form the loops and then all the loops are laminated together. The lamination will raise the stiffness of the CFRP cable. Furthermore, the CFRP prepreg loops can be pre-tensioned before hardening, so as to decrease the stress concentration in use and thus increasing the load bearing capacity.

6.2 Design Principles of CFRP Cable Anchorages As noted earlier, the key problem facing the application of CFRP cables in orthogonally loaded cable structures is how to anchor them. The advantages of CFRP cables mentioned previously can only be valid if their mechanical properties (i.e. strength and modulus) are fully exploited. However, the CFRP is a typical orthotropic material, which makes the strength and modulus of CFRP cables perpendicular to the cable axis are orders of magnitude smaller than those of CFRP cables in the cable axis. This orthotropic property makes the issue of successful CFRP cable anchorage a difficult one [Schmidt et al., 2012].

119 

  As early as 1954, Rubinsky and Rubinsky have proposed a clamp anchorage system for FRP tension members [Rubinsky and Rubinsky, 1954]. However, corresponding researches and further developments came to a standstill after a few years, mainly because the efficiencies of most designed anchorages were relatively low [Schmidt et al., 2012]. Early CFRP cable clamp anchorages were usually simple imitations of clamp anchorages for steel cables. For example, a typical early plate type clamp anchorage for CFRP cables can be illustrated in Figure 6.17.

Figure 6.17: Early plate type clamp anchorage (a) diagrammatic sketch (b) shear stress distribution diagram For the above clamp anchorage, the fastening forces of the bolts are constant (see Figure 6.17 (a)). This will result in a shear stress concentration at the anchored CFRP cable near the anchorage mouth (see Figure 6.17 (b)), which will cause premature failure of CFRP cable and reduce the anchorage efficiency. After repeated tests and studies, a method was found to solve this stress concentration problem, which can be shown in Figure 6.18 (also see Section 6.1.1 No. 1 clamp anchorage) [Horvatits et al., 2004].

Figure 6.18: Improved plate type clamp anchorage (a) diagrammatic sketch (b) shear stress distribution diagram

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As can be seen from Figure 6.18 (a), the bolt fastening forces in the improve plate type clamp anchorage decrease from the anchorage end to the anchorage mouth, which can effectively reduce the shear stress concentration of CFRP cable near the anchorage mouth (see Figure 6.18 (b)) and thus raising the anchorage efficiency. Another example is a typical early wedge type clamp anchorage for CFRP cables, which is illustrated in Figure 6.19.

Figure 6.19: Early wedge type clamp anchorage (a) diagrammatic sketch (b) shear stress distribution diagram As can be seen from Figure 6.19 (a), the early wedge type clamp anchorage for CFRP cables is very similar to that for steel cables. Both anchor body and wedges are made of steel. As a consequence, a high stress peak (lateral stress as well as shear stress) occurs at the anchored CFRP cable near the anchorage mouth, which will cause the failure of CFRP cable greatly ahead of its tensile strength. In view of this, an improved anchorage system with composite wedges is proposed, which can be illustrated in Figure 6.20 (also see Section 6.1.1 No. 5 clamp anchorage) [Burtscher, 2008].

Figure 6.20: Improved wedge type clamp anchorage (a) diagrammatic sketch (b) shear stress distribution diagram 121 

 

In the above improve wedge type clamp anchorage, the wedge is made of steel and plastic, instead of only steel, and the plastic part is in contact with the anchored CFRP cable (see Figure 6.20 (a)). Moreover, the cross-sectional area of plastic part increases from the anchorage end to the anchorage mouth. These composite wedges will make the anchorage stiffness decrease from the anchorage end to the anchorage mouth, thereby eliminating the stress concentration of CFRP cable near the anchorage mouth (see Figure 6.20 (b)). Corresponding numerical test and experiment have demonstrated that this design can successfully prevent the premature failure of anchored CFRP cable and improve the anchorage efficiency. For early CFRP cable bond anchorages, the shape of inner cavity filled with bond material is usually conical, as shown in Figure 6.21.

Figure 6.21: Early conical bond anchorage (a) diagrammatic sketch (b) shear stress distribution diagram As can be seen from the above figure, in the early conical bond anchorage, the stiffness, i.e. the elastic modulus, of the filled mortar over the anchorage length is constant, which will result in a high stress peak (lateral stress as well as shear stress) at the anchored CFRP cable near the anchorage mouth. This unfavourable stress concentration will result in pullout or tensile failure far below the tensile strength of CFRP cable [Meier, 2012]. In order to solve this problem, an improved design with gradient stiffness mortar is proposed, which can be illustrated in Figure 6.22 (also see Section 6.1.2 No. 7 bond anchorage) [Meier, 2012].

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Figure 6.22: Improved conical bond anchorage (a) diagrammatic sketch (b) shear stress distribution diagram As shown in Figure 6.22 (a), the mortar used in the improved conical bond anchorage is a material with gradient stiffness, whose elastic modulus decreases from the anchorage end to the anchorage mouth. With this mortar, the stress peak of CFRP cable near the anchorage mouth can be avoided (see Figure 6.22 (b)) and the anchorage efficiency will consequently be raised. This improved conical bond anchorage with gradient mortar has been successfully used in the Stork Bridge of Winterthur, Switzerland. [Meier, 2012] The pin-loaded anchorage is a relatively new anchorage system for CFRP cables, which is different from the clamp or bond anchorage, as shown in Figure 6.23.

Figure 6.23: Pin-loaded anchorage compared with clamp or bond anchorage

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As can be seen from the above figure, the main part of pin-loaded anchorage is only a pin or a thimble. CFRP cables are anchored through winding on the pin. With such “pin-loaded” design concept, the anchorage structure can be greatly simplified and the length of anchorage can be considerably shortened. Moreover, when the anchored CFRP cable is subjected to tension, the pressure will be generated between the CFRP cable and the pin, which will generate an extra friction force and help to anchor the CFRP cable. After reviewing the existing CFRP cable anchorages and analysing their design experiences, several design principles for CFRP cable anchorages can be evolved and summarised as follows. First, the steel cable anchorages can hardly be applied to anchor CFRP cables without any targeted improvement. If such anchorage systems are directly used to anchor CFRP cables, severe stress concentration will occur at the cables near the anchorage mouth, which will cause the premature failure of CFRP cables and thus making the anchorage efficiency far below 100%. Second, an effective way to considerably decrease the above mentioned stress concentration and prevent the premature failure of CFRP cables is anchoring them with gradient intensity decreasing from the anchorage end to the anchorage mouth. Such gradient anchorage intensity means that the CFRP cables will be anchored more intensively at the anchorage end than at the anchorage mouth. In this way, the CFRP cables can elongate relatively uniformly in the anchorages under tensile loads and the stress distributions of CFRP cables along the anchorage length will also be relatively uniform. In practice, the gradient anchorage intensity can be achieved by gradually decreasing the clamping forces or reducing the stiffness of wedges, bond materials or anchorages themselves from the anchorage end to the anchorage mouth. Third, the “pin-loaded” concept can benefit the design of CFRP cable anchorages, which helps reduce the length of anchorage and makes the structure of anchorage also very simple. Especially, pin-load anchorages are particularly well suited for anchoring CFRP cables in orthogonally loaded cable structures, such as cable roofs and cable facades, where the structural spaces are usually too limited to place the clamp anchorages or the bond anchorages. Moreover, beams and columns with round cross sections in such cable

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  structures can provide natural pins for the pin-loaded CFRP cable anchorages and thus leading to even smaller anchorage sizes and simpler anchorage structures.

6.3 New Design A: CFRP Cable Winding-Clamp Anchorage Based on the design principles for CFRP cable anchorages established in the above section, two novel designs, i.e. the winding-clamp anchorage and the thimble-clamp anchorage, were proposed by the author. These two anchorages are especially suitable for anchoring CFRP cables in orthogonally loaded cable structures. In this section, the CFRP cable winding-clamp anchorage is introduced. This anchorage is applicable to anchoring the lamella-type CFRP cables (see Figure 3.9). The conceptual design, the actual design, the Finite Element Analysis and the experimental verification about this anchorage are presented, respectively.

6.3.1

Conceptual design

Consider a swimming pool shown in Figure 6.24, which needs a cable-membrane roof supported by a frame structure. As illustrated in Chapter 2, this roof will be a typical orthogonally loaded cable structure (see Figure 6.25). Therefore, it is suitable for the application of CFRP cables.

Figure 6.24: Swimming pool which needs a cable-membrane roof

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Figure 6.25: Diagrammatic sketch of cable system of swimming pool However, the existing clamp or bond anchorages are hardly suitable for the CFRP cables in this case, mainly because they are too long and too cumbersome and thus destroying structural aesthetic effect (see Figure 6.26).

Figure 6.26: Cable structure using ordinary clamp or bond anchorages In view of this, pin-loaded anchorages are good choices for this cable structure and the longitudinal girders with round cross sections can act as natural pins. However, although the two types of pin-loaded anchorages presented in Section 6.1 are already able to achieve relatively high anchorage efficiency (more than 90%) [Meier and Farshad, 1996] [CarboLink, 2012], their capacities still cannot reach 100% of ultimate bearing capacities of anchored CFRP cables [Schlaich et al., 2012].

126 

  In this book, a novel design, which combines the pin-loaded anchorage and the clamp anchorage, was proposed. Based on its structural form, it can be called “winding-clamp anchorage”, which is illustrated as follows.

Figure 6.27: Winding-clamp anchorage with CFRP lamella (a) assembled view (b) exploded view As can be seen from Figure 6.27 (a), the winding-clamp anchorage is used to anchor CFRP lamellas. The suitable CFRP lamella should be a relatively thin band, because it will be wound around a circular ring and its tensile strength will reduce with the increase of the lamella thickness if the radius of the ring is fixed [Winistoefer, 1999]. Moreover, this anchorage is not limited to anchor only one CFRP lamella; if necessary, several CFRP lamellas can be stacked up to form a multi-layer CFRP cable and anchored by this anchorage at the same time. As can be seen from Figure 6.27 (b), the winding-clamp anchorage is composed of a metal ring, two bent metal plates and several pairs of clamping bolts. The metal used can be steel or aluminium. The anchored CFRP lamella is wound almost a circle around the metal ring and then clamped by these two bent metal plates with clamping bolts. It should be noted that the fastening forces of the bolts are designed to gradually increase from the anchorage mouth to the anchorage end, which is shown in Figure 6.28.

127 

 

Figure 6.28: Winding-clamp anchorage (a) diagrammatic sketch (b) shear stress distribution diagram As shown in Figure 6.28, with the gradient clamping force, the shear stress of the CFRP lamella will distribute relatively uniformly along the anchorage length instead of concentrating at the anchorage mouth. As illustrated in Section 6.2, the anchorage efficiency can be effectively increased by this measure. Furthermore, when the CFRP lamella is subjected to tension, the pressure and the corresponding extra friction force will be generated between the CFRP lamella and the metal ring, which can also help to anchor the CFRP lamella. Effect drawings of installing winding-clamp anchorages and corresponding CFRP cables in the structure are shown below.

Figure 6.29: Cable structure with winding-clamp anchorage (overall view)

128 

 

Figure 6.30: Cable structure with winding-clamp anchorage (local view) As can be seen from Figure 6.29 and Figure 6.30, CFRP cables are anchored relatively concisely by the winding-clamp anchorages. Furthermore, the longitudinal girders act as natural pins; the size and the length of anchorages are considerably reduced. As a consequence, the aesthetic effect of the cable structure is maintained to the utmost. Assembling the membrane, the final rendering of swimming pool cable roof with CFRP cables and winding-clamp anchorages is shown in Figure 6.31.

Figure 6.31: Final rendering of swimming pool cable roof with CFRP cables and windingclamp anchorages

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  It should be noted that the winding-clamp anchorage is not only suitable for the above cable roof but also applicable to other CFRP orthogonally loaded cable structures, such as cable facades and stress-ribbon bridges.

6.3.2

Actual design

According to the conceptual design result as well as the existing conditions of materials and machines in the laboratory of Technical University of Berlin (for the subsequent verification test), the investigated winding-clamp anchorage was designed, which is presented in this section. The design result is shown below.

Figure 6.32: Investigated winding-clamp anchorage (a) assembled view (b) exploded view As can be seen from Figure 6.32, the investigated winding-clamp anchorage is basically the same as the conceptual design result shown in Figure 6.27. To facilitate the connection with the test machine, the aluminium ring was replaced by an aluminium cylinder with two holes. Furthermore, because the bent aluminium plates are relatively thin (due to the limitation of material), six steel cuboids were added to transfer the clamping forces of the bolts more evenly to the CFRP lamella. The geometries of the components in the designed winding-clamp anchorage are illustrated as follows. It should be noted that these geometries were mainly determined by the geometries of available materials in the laboratory.

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Figure 6.33: Geometry of aluminium cylinder

Figure 6.34: Geometry of bent aluminium plate

Figure 6.35: Geometry of steel cuboid It should be noted that the surfaces of the aluminium cylinder and the aluminium plates, which are in contact with the CFRP lamella, were sanded to increase the friction, as shown in Figure 6.36.

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Figure 6.36: Photo of aluminium cylinder and plate In addition, the steel bolts used were M 6 screws with a length of 20 mm. The geometry of the CFRP lamella anchored by the investigated anchorage is shown below.

Figure 6.37: Geometry of CFRP lamella anchored by winding-clamp anchorage This CFRP lamella was produced by SGL Group. Its ultimate tensile bearing capacity in the fibre direction is 4 kN (data from the producer). Moreover, in order to increase the friction, the CFRP lamella has special uneven surface, as shown in Figure 6.38.

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Figure 6.38: Photo of CFRP lamella

6.3.3

Finite element analysis

In order to test and verify the aforementioned design result, the finite element analysis was performed and is presented in this section.

6.3.3.1 Finite element model In terms of the dimensionality, finite element models can be mainly classified into two types, i.e. 2-dimensional (2-D) models and 3-dimensional (3-D) models. The 2-D models are simpler and require less computing resource, but the 3-D models are more accurate and closer to the actual situation. Therefore, the 3-D finite element model was adopted in this research. The actual design result (see Section 6.3.2) has to be simplified, so as to facilitate the establishment of finite element model and the subsequent finite element calculation. First, the steel bolts were simplified to the uniform pressure on the steel cuboids. Then, the two holes in the middle of the aluminium cylinder were simplified to one hole with a diameter of 50 mm. Because of the structural symmetry, only half of the anchorage in the thickness direction was modelled. Moreover, in order to reduce the amount of unnecessary calculation, the anchorage thickness was further decreased. The simplified geometric model including boundary conditions is shown in Figure 6.39.

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Figure 6.39: Geometric model including boundary conditions In the above model, the aluminium cylinder was fixed and one end of the CFRP lamella was under tension loading. The pressures acted at the six steel cuboids were set to 5 MPa, 10 MPa, 15 MPa, 20 MPa, 25 MPa and 30 MPa, respectively, from the anchorage mouth to the anchorage end. These pressures were mainly limited by the corresponding strengths of the CFRP lamella and were determined through tentative calculation. In addition, because the surfaces of the aluminium cylinder and the bent aluminium plates were sanded (see Figure 6.36) and the CFRP lamella had a special uneven surface (see Figure 6.38), the friction coefficient between the aluminium components and the CFRP lamella was set to 0.3, greater than 0.2, i.e. the friction coefficient between smooth aluminium and smooth CFRP [Schön, 2004]. Based on the above geometric model, the finite element model was established in the general FEM software ABAQUS, as shown in Figure 6.40.

Figure 6.40: Finite element model of investigated winding-clamp anchorage

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  In the above finite element model, a commonly used three-dimensional solid element C3D8, which is an 8-node linear brick element [Dassault Systèmes, 2013], was adopted for the CFRP lamella and two bent aluminium plates; another commonly used threedimensional solid element C3D8R, which is an 8-node linear brick element with reduced integration [Dassault Systèmes, 2013], was adopted for the other parts of this anchorage.  Furthermore, since the CFRP lamella is the research focus, the mesh at the lamella was refined. The final element sizes were determined through tentative calculations, so as to maintain a balance between the accuracy and the computation speed.

6.3.3.2 Material properties and constitutive relations The two metal materials used in the model, i.e. aluminium and steel, are typical isotropic materials, whose mechanical properties are listed in Table 6.1. The fourth strength criterion, i.e. the von Mises stress criterion, which is suitable for ductile materials, was adopted to predict the failure of these metal materials.

Table 6.1: Mechanical properties of metal material used Material

Elastic modulus

Poisson's ratio

Strength

Aluminium

70 GPa

0.33

120 MPa

Steel

210 GPa

0.3

235 MPa

However, the CFRP is a typical orthotropic material, whose material properties in the fibre direction are considerably different from those perpendicular to the fibre direction. The mechanical properties of the CFRP used in the model are listed in Table 6.2.

Table 6.2: Mechanical properties of CFRP used Engineering constant

Value

Strength

Value

100 GPa

1500 MPa

=

6 GPa

800 MPa

=

3 GPa

=

70 MPa

2.8 GPa

=

140 MPa

0.3

=

90 MPa

=

0.33

80 MPa

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In the above table, the subscript 1 represents the fibre direction, while the subscript 2 and 3 represent two directions perpendicular to the fibre direction; the superscript t represents tension, while the superscript c represents compression. As can be seen from Table 6.2, the elastic modulus and the strength in the fibre direction are significantly greater than those in two directions perpendicular to the fibre direction; moreover, the tensile strength in the fibre direction is greater than the compressive strength in the same direction, while the tensile strength perpendicular to the fibre direction is smaller than the compressive strength in the same direction. This phenomenon indicates that the CFRP has strong orthotropic characteristics. For the orthotropic material, the constitutive model expressed in terms of stress-strain relationship can be written as [Reddy, 1997]:

σ  Cε

(6.1)

where  is the stress vector and can be written as:

σ  11 , 22 , 33 ,12 ,13 , 23

T

(6.2)

 is the strain vector and can be written as:

ε  11 ,  22 ,  33 , 12 , 13 ,  23

T

(6.3)

and C is the stiffness matrix of orthotropic material, which can be written as [Reddy, 1997]:

 C11 C12 C  21 C22 C C32 C   31 0  0  0 0  0  0

C13

0

0

C23 C33

0 0

0 0

0

C44

0

0 0

0 0

C55 0

0  0  0   0  0   C66 

(6.4)

The relationship between the engineering constants (see Table 1) and the terms in the stiffness matrix of the orthotropic material can be expressed as [Reddy, 1997]:

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C11 

1  v23v32 E2 E3

(6.5)

C22 

1  v13v31 E1E3

(6.6)

C33 

1  v12v21 E1E2 

(6.7)

C12 

v21  v31v23 E2 E3

(6.8)

C21 

v12  v13v32 E1E3

(6.9)

C13 

v31  v21v32 E2 E3

(6.10)

C31 

v13  v12v23 E1E2 

(6.11)

C23 

v32  v31v12 E1E3

(6.12)

C32 

v23  v13v21 E1E2 

(6.13)

C44  2G12

(6.14)

C55  2G13

(6.15)

C66  2G23

(6.16)

where,



1  v12v21  v13v31  v23v32  2v12v13v23 E1E2 E3

(6.17)

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Furthermore, because

v23 v32 v12 v21 v13 v31    , and E2 E3 E1 E2 E1 E3

(6.18)

E2  E3 , v12  v13 and G12  G13

(6.19)

C12  C21 , C13  C31 , C23  C32 and C44  C55

(6.20)

.

and

one can obtain

Through solving Equation (6.1), unknown stress or strain will be calculated, which is a solution process in the FEM software ABAQUS. In order to predict the failure of the CFRP lamella, a strength criterion has to be adopted. Up to now, thousands of strength criteria for orthotropic materials, especially for FRP composites, have been developed by researchers and engineers worldwide [Hinton and Kaddour, 2013]. In the early stages of this research topic, Whitney and Nuismer [Whitney and Nuismer, 1974] established the point stress criterion (PSC) and average stress criterion (ASC) to predict the strength of notched FRP laminate. Karlak [Karlak, 1977] proposed the modified PSC based on research from Whitney and Nuismer. Pipes [Pipes et al., 1979] discussed the influence of stress concentrations on the strength of notched FRP laminate. Based on a large number of failure experiments of FRP single ply, many strength criteria for unidirectional orthotropic plates were proposed to predict the strengths of the FRP laminates, such as the Hankinson criterion, the Strussi criterion, the Norris–Fischer criterion, the Kopnov criterion etc. [Echaabi et al., 1996]. Among them, some strength criteria were presented in the form of quadratic polynomials, such as the Hill–Tsai criterion, the Hoffman criterion and the Tsai–Wu criterion. These are still commonly used today, mainly because of their simple forms and relatively good accuracy [Chamis, 2007].

138 

 

The strength criteria mentioned above can only generally predict the failure of the FRP materials, rather than considering the failure modes and the difference between tensile failure and compressive failure. Zvi Hashin proposed a stress criterion called the Hashin criterion, which can distinguish the fibre failure mode and the matrix failure mode as well as the tensile failure condition and the compressive failure condition [Hashin, 1980]. In this way, the prediction accuracy of the strength criterion will generally be improved and the obtained failure will also have clear physical meaning. Because of the above advantages, the Hashin criterion was adopted in this research to predict the failure of the CFRP lamella. The three dimensional (3-D) Hashin criterion can be expressed as [Hashin, 1980]: (1)

Fibre tensile failure for 11  0 2

2

2

  11    12   13   1 no failure    t     S S S  11   12   13   1 failure (2)

(6.21)

Fibre compressive failure for 11  0 2

 11   1 no failure  c    S11   1 failure (3)

Matrix tensile failure for  22   33  0 2

2

2

  22   33   12   13   232   22 33  1 no failure         t S232  1 failure  S22   S12   S13  (4)

(6.22)

(6.23)

Matrix compressive failure for  22   33  0

2      2   2   2  2      22   33   S22c   1 no failure 22 33 13 23 22 33 12         1            c S232   2S23   S12   S13   1 failure  S22   2S23 

(6.24)

139 

 

The meanings of symbols, subscripts and superscripts in the above four strength inequalities are the same as those in Table 6.2. If the term on the left side of any of the four inequalities is greater than or equal to one, the corresponding failure will occur. To quantify the degree of failure, the failure index k, defined as the ratio between the applied stress and the ultimate strength was introduced. For the 3-D Hashin criterion, the terms on the left side of the strength inequalities can be written in the tensor form [Tsai, 1992]:

Gij i j  Gi i

(6.25)

Letting every  reach its ultimate value when:

Gij iultimate ultimate  Gi iultimate  1 j

Substituting

i k

(6.26)

ultimate for  i results in:

2

 Gij i j   k1    Gi i   1k   1 Solving the quadratic equation, where a  Gij i j , b  Gi i and x 

(6.27)

1 : k

ax 2  bx  1  0

(6.28)

b  b 2  4a x  2a

(6.29)

Getting the positive quadratic root:



Therefore

k

140 

2a b  b 2  4a



2Gij i j Gi i 

 Gi i 

2

 4Gij i j

(6.30)

 

According to Equation (6.30), four failure indices for the four strength inequalities of 3-D Hashin criterion can be written as follows. (1)

Fibre tension failure index: 2

2

      k   11t    12    13   S11   S12   S13 

2

t f

(2)

Fibre compression failure index:

  k   11c   S11 

2

(6.32)

c f

(3)

Matrix tension failure index: 2

2

2

  22   33    12    13   232   22 33 k         t S232  S22   S12   S13  t m

(4)

(6.31)

(6.33)

Matrix compression failure index:

kmc 

    2   2   2  2     33 13 23 22 33 12  2  22       2 S23  2S23   S12   S13   2   22   33    S22c       1   c S S 2  22     23  

where 2

      S c 2        2    2    2  2     22 33 22 33 13 23 22 33 22 12       1   4          c 2 2 2 S S S S S S      23  22 23 23   12   13   

(6.34)

141 

  In the above four equations, t and c, i.e. the subscripts of the failure indices, indicate tension and compression, respectively; f and m, i.e. the superscripts of the failure indices, indicate fibre and matrix, respectively. If four failure indices are smaller than one, it means no failure occurs; if a failure index is greater than one, it means corresponding failure occurs, and the greater the failure index is, the greater the degree of failure will be. The aforementioned constitutive model of the CFRP and 3-D Hashin criterion were implemented using the ABAQUS subroutine UMAT with the FORTRAN language (see Appendix B). Four internal variables of ABAQUS, i.e. SDV9, SDV10, SDV11 and SDV 12, represented the four failure indices, i.e.

,

,

and

, respectively. The non-

linear iteration calculation would be terminated when any failure occurred at the CFRP lamella or the anchorage itself.

6.3.3.3 Finite element analysis results First, the final stress conditions (at the ultimate tensile load = 4.170 kN) of the anchorage itself, i.e. the aluminium cylinder and the bent aluminium plates as well as the steel cuboids, are presented as follows.

Figure 6.41: Contour of von Mises stress of aluminium cylinder (unit: MPa) As can be seen from Figure 6.41, the maximum von Mises stress of the aluminium cylinder is considerably smaller than the strength of the aluminium, which indicates that there was no failure at the aluminium cylinder during the tension of the CFRP lamella.

142 

 

Figure 6.42: Contour of von Mises stress of bent aluminium plates (unit: MPa) As can be seen from Figure 6.42, the maximum von Mises stress of the two bent aluminium plates is also considerably smaller than the strength of the aluminium, which means that there was no failure at the bent aluminium plates during the tension of the CFRP lamella.

Figure 6.43: Contour of von Mises stress of steel cuboids (unit: MPa) As can be seen from Figure 6.43, the maximum von Mises stress of the six steel cuboids is significantly smaller than the strength of the steel, which indicates that there was no failure at the steel cuboids during the tension of the CFRP lamella. Then, the distributions of the four failure indices (at the ultimate tensile load = 4.170 kN) of the CFRP lamella are presented as follows.

143 

 

Figure 6.44: Contour of fibre tension failure (failure index

Figure 6.45: Contour of fibre compression failure (failure index

Figure 6.46: Contour of matrix tension failure (failure index

144 

= SDV9)

= SDV10)

= SDV11)

 

Figure 6.47: Contour of matrix compression failure (failure index

= SDV12)

As can be seen from the above four figures, only the fibre tension failure index is slightly greater than one, which indicates that the failure mode of the CFRP lamella is the fibre tension failure. Furthermore, as shown in Figure 6.44, the fibre tension failure is only in the free section of the CFRP lamella, instead of the anchorage section. The load-displacement curve during the numerical test is presented as follows.

Figure 6.48: Load-displacement curve during numerical test As can be seen from Figure 6.48, the load-displacement curve (blue one) from start to failure is an inclined straight line, which shows the linear elasticity of the CFRP lamella. Furthermore, the highest point of the load-displacement curve is slightly over the

145 

 

schematic line of the theoretical ultimate load, i.e. the tensile bearing capacity of the CFRP lamella provided by the producer, which is 4 kN. This indicates that the CFRP lamella can be fully used and the anchorage efficiency is 100 %.

6.3.4 Experimental verification To verify the results of the aforementioned numerical test, the corresponding experiment was carried out by the author, which is presented in this section.

6.3.4.1 Experimental set-up The experiment was conducted in a laboratory at the Technical University of Berlin. Two specimens of winding-clamp anchorages with CFRP lamellas were tested. A universal testing machine (Instron 8502) was used to pull the specimens. For each specimen, the CFRP lamella was clamped at the one end by the test machine, while at the other end it was anchored by the winding-clamp anchorage, which was connected to the test machine with two bolts. The experimental set-up is shown in Figure 6.49.

146 

 

Figure 6.49: Experimental set-up The loading velocity was set to 1 mm/s. During the tension, load cells and displacement sensors recorded the tension force and the deformation. The loading did not terminate until visible failure occurred. The results of the experiment are present in following section.

6.3.4.2 Experimental result First, the specimen at failure is shown in the following figure (the situations of two specimens were very similar when the failure occurred).

147 

 

Figure 6.50: Final failure situation of specimen As can be seen from Figure 6.50, the failure exists at the free section of the CFRP lamella, instead of its anchorage section or the metal components of the winding-clamp anchorage. Furthermore, the failure happened very suddenly, which shows that it was a brittle failure, instead of a ductile failure. Then, the load-displacement curves of these two specimens are presented as follows.

148 

 

Figure 6.51: Load-displacement curves of two specimens As can be seen from Figure 6.51, the highest points of the load-displacement curves of two specimens are slightly over or very close to the schematic line of the theoretical ultimate load, i.e. the tensile bearing capacity of the CFRP lamella provided by the producer, which is 4 kN. This demonstrates that the CFRP lamella can be fully used and the anchorage efficiency can reach 100 %. Furthermore, comparing Figure 6.51 with Figure 6.48, it can be found that the loaddisplacement curve from the numerical test (i.e. the finite element analysis) is very similar to those from the real tests, which proves the correctness of the results from the finite element analysis. The numerical and experimental ultimate tensile load results are compared in Table 6.3. Table 6.3: Comparison of numerical and experimental ultimate tensile loads FEM result

Test 1 result

Test 2 result

Average test result

Error

4.170 kN

4.297 kN

3.727 kN

4.012 kN

3.8 %

As can be seen from Table 6.4, the ultimate tensile load result of finite element analysis coincides well with those of experiment (the error between the numerical result and the average experimental result is merely 3.8 %), which proves the correctness of the finite element analysis again.

149 

 

In summary, both the finite element analysis and the experiment show that the CFRP lamella anchored by the winding-clamp anchorage is able to achieve 100 % of its tensile bearing capacity; the final failure is a brittle tensile failure occurring at the free section of the CFRP lamella, while its anchorage section and the anchorage itself remain undamaged during the loading. The results of the finite element analysis and the verified experiment demonstrate that the winding-clamp anchorage proposed in this research is a reliable design to anchor CFRP cables; moreover, its anchorage efficiency can reach 100 %.

6.4 New Design B: CFRP Cable Thimble-Clamp Anchorage In this section, the CFRP cable thimble-clamp anchorage is introduced. This anchorage is suitable for the lamella-type CFRP cables (see Figure 3.9) too. The conceptual design, the actual design, the finite element analysis and the experimental verification about this anchorage are presented, respectively.

6.4.1

Conceptual design

A stress-ribbon bridge shown in Figure 6.52, which is similar to the TU-Berlin CFRP Stress-Ribbon Footbridge [Schlaich and Bleicher, 2007], was used here to illustrate the conceptual design. As illustrated in Chapter 2, this bridge is a typical orthogonally loaded cable structure (see Figure 6.53). Therefore, it is suitable for the application of CFRP cables.

Figure 6.52: Studied stress-ribbon bridge

150 

 

Figure 6.53: Diagrammatic sketch of cable system of stress-ribbon bridge However, the existing clamp or bond anchorages are hardly suitable for the CFRP cables in this case, mainly because they are too long and too cumbersome, as shown in Figure 6.54.

Figure 6.54: Cable bridge using ordinary clamp or bond anchorages In view of this, pin-loaded anchorages are good choices for this cable bridge. However, the two types of pin-loaded anchorages presented in Section 6.1 cannot achieve 100% anchorage efficiency [Schlaich et al., 2012]. In this book, a novel design, which combines the pin-loaded anchorage and the clamp anchorage, is proposed. Based on its structural form, it can be called “thimble-clamp anchorage”, which is illustrated as follows.

Figure 6.55: Thimble-clamp anchorage with CFRP lamella (a) assembled view (b) exploded view

151 

  As can be seen from Figure 6.55 (a), the thimble-clamp anchorage is used to anchor CFRP lamellas. The suitable CFRP lamella should be a relatively thin band, because it will be wound around the thimble and its tensile strength will reduce with the increase of the lamella thickness if the radius of the thimble is fixed [Winistoefer, 1999]. It should be noted that this anchorage is not limited to anchor only one CFRP lamella; if need be, several CFRP lamellas can be stacked up to form a multi-layer CFRP cable and anchored by this anchorage at the same time. As can be seen from Figure 6.55 (b), the winding-clamp anchorage is composed of a thimble, two clamping plates and several pairs of bolts and nuts. The material of thimble can be aluminium or steel, while the material of clamping plates should ideally have high hardness like steel. The anchored CFRP lamella is wound more than a circle around the thimble and then clamped by the clamping plates at the anchorage mouth. The fastening forces of the bolts are set to constant. Furthermore, in the clamping area, an agglutinant can be used to glue the CFRP lamella together and the clamping plates with the CFRP lamella. Effect drawings of installing thimble-clamp anchorages and corresponding CFRP cables in the stress-ribbon bridge are shown below.

Figure 6.56: Cable structure with thimble-clamp anchorage (overall view)

152 

 

Figure 6.57: Cable structure with thimble-clamp anchorage (local view) As can be seen from the above two figures, CFRP cables are anchored very concisely by the thimble-clamp anchorages. Furthermore, the size and the length of anchorages are considerably reduced. As a consequence, the aesthetic effect of the cable structure is maintained to the utmost. It should be noted that the thimble-clamp anchorage is not only suitable for the above stress-ribbon bridge but also applicable to other CFRP orthogonally loaded cable structures, such as cable roofs and cable facades.

6.4.2

Actual design

According to the conceptual design result as well as the existing conditions of materials and machines in the laboratory of Technical University of Berlin (for the subsequent verification test), the investigated thimble-clamp anchorage was designed and is presented in this section. The design result is shown below.

153 

 

Figure 6.58: Investigated thimble-clamp anchorage (a) assembled view (b) exploded view As can be seen from Figure 6.58, the investigated thimble-clamp anchorage is basically the same as the conceptual design result shown in Figure 6.55. Due to the limitation of the material, two relatively big steel clamping plates were replaced by six relatively small steel clamping plates. In addition, the epoxy adhesive was used to glue the CFRP lamella together and the steel clamping plates with the CFRP lamella. The geometries of the components of the designed thimble-clamp anchorage including the CFRP lamella are illustrated as follows. It should be noted that these geometries were mainly determined by the geometries of available materials in the laboratory.

Figure 6.59: Geometry of aluminium thimble

154 

 

Figure 6.60: Geometry of steel plate

Figure 6.61: Geometry of CFRP lamella The CFRP lamella is the same lamella used in Section 6.3 (see Figure 6.38), which was supplied by SGL Group. Its ultimate tensile bearing capacity in the fibre direction is 4 kN. In addition, the steel bolts used were M 6 screws with a length of 20 mm and the steel nuts used were M 6 too.

6.4.3 Finite element analysis In order to test and verify the aforementioned design result, the finite element analysis (i.e. the numerical test) of the thimble-clamp anchorage was performed and is presented in this section.

6.4.3.1 Finite element model The same as in Section 6.3, the 3-D finite element model was also adopted here to simulate the thimble-clamp anchorage. The actual design result (see Section 6.4.2) had to be simplified, so as to facilitate the establishment of finite element model and the subsequent numerical calculation. First, the

155 

  steel bolts and the steel nuts were simplified to the uniform pressure on the steel clamping plates. Furthermore, because of the structural symmetry, only half of the anchorage in the thickness direction was modelled. In addition, to reduce the amount of unnecessary calculation, the anchorage thickness was further decreased. The simplified geometric model including boundary conditions is shown in Figure 6.62.

Figure 6.62: Geometric model including boundary conditions In the above model, the aluminium cylinder was fixed and one end of the CFRP lamella was under tension loading. The pressures acted at the three pairs of steel plates were set to 10 MPa and these steel plates were tied to the CFRP lamella. Because the CFRP lamella has a special uneven surface (see Figure 6.38), the friction coefficient between the aluminium components and the CFRP lamella was set to 0.25, greater than 0.2, i.e. the friction coefficient between smooth aluminium and smooth CFRP [Schön, 2004]. In addition, based on the actual situation of the specimens, the thickness of the epoxy adhesive layers was set to 0.5 mm. According to the above geometric model, the finite element model was established in the general FEM software ABAQUS, which is shown in Figure 6.63.

Figure 6.63: Finite element model of investigated thimble-clamp anchorage In the above finite element model, a commonly used three-dimensional solid element C3D8, which is an 8-node linear brick element [Dassault Systèmes, 2013], was adopted for 156 

 

the CFRP lamella; another commonly used three-dimensional solid element C3D8R, which is an 8-node linear brick element with reduced integration [Dassault Systèmes, 2013], was adopted for the other parts of this anchorage.  Since the CFRP lamella was the research focus, the mesh at the lamella was refined. The final element sizes were determined through tentative calculations, so as to maintain a balance between the accuracy and the computation speed.

6.4.3.2 Material properties and constitutive relations The mechanical properties of aluminium, steel and CFRP have already been described in Section 6.3.2.2 (see Table 6.1 and Table 6.2). The mechanical properties of the epoxy adhesive used are presented as follows. Table 6.4: Mechanical properties of epoxy adhesive used Material

Elastic modulus

Poisson's ratio

Strength

Epoxy adhesive

70 GPa

0.35

100 MPa

The constitutive relations of the CFRP have already been presented in Section 6.3.2.2 and the same strength criterion, i.e. the 3-D Hashin criterion (see Section 6.3.2.2), was still used to predict the failure of the CFRP lamella anchored by the thimble-clamp anchorage. In addition, the fourth strength criterion, i.e. the von Mises stress criterion, which is commonly used for ductile materials, was used to predict the failure of the aluminium thimble and the steel plates; the first strength criterion, i.e. the maximum principal stress criterion, which is suitable for fragile materials, was used to predict the failure of the epoxy adhesive layer. The results of the numerical test are present in following section.

6.4.3.3 Finite element analysis results First, the final stress conditions of the anchorage itself (at the ultimate tensile load = 4.038 kN), i.e. the aluminium thimble and the steel plates as well as the epoxy adhesive layers, are presented as follows. 157 

 

Figure 6.64: Contour of von Mises stress of aluminium thimble (unit: MPa) As can be seen from Figure 6.64, the maximum von Mises stress of the aluminium thimble is considerably smaller than the strength of the aluminium, which indicates that there was no failure at the aluminium thimble during the tension of the CFRP lamella.

Figure 6.65: Contour of von Mises stress of steel plates (unit: MPa) As can be seen from Figure 6.65, the maximum von Mises stress of the six steel plates is smaller than the strength of the steel (the initial pressure on the steel plates was 10 MPa, see Section 6.4.3.1), which means that there was no failure at the steel plates during the tension of the CFRP lamella.

158 

 

Figure 6.66: Contour of maximum principal stress of epoxy adhesive layer gluing CFRP lamella (unit: MPa) As can be seen from Figure 6.66, the greatest maximum principal stress of the epoxy adhesive layer is smaller than the strength of the epoxy adhesive, which means that there was no failure at the epoxy adhesive layer during the tension of the CFRP lamella.

Figure 6.67: Contour of maximum principal stress of epoxy adhesive layers gluing steel plates to CFRP lamella (unit: MPa) As can be seen from Figure 6.67, the greatest maximum principal stress of the epoxy adhesive layers is smaller than the strength of the epoxy adhesive, which means that there was no failure at the epoxy adhesive layers during the tension of the CFRP lamella. Then, the distributions of the four failure indices (at the ultimate tensile load = 4.038 kN) of the CFRP lamella are presented as follows.

159 

 

Figure 6.68: Contour of fibre tension failure (failure index

Figure 6.69: Contour of fibre compression failure (failure index

Figure 6.70: Contour of matrix tension failure (failure index

160 

= SDV9)

= SDV10)

= SDV11)

 

Figure 6.71: Contour of matrix compression failure (failure index

= SDV12)

As can be seen from the above four figures, only the fibre tension failure index is slightly greater than one, which indicates that the failure mode of the CFRP lamella is the fibre tension failure. Furthermore, as shown in Figure 6.68, the fibre tension failure is only in the free section of the CFRP lamella, instead of the anchorage section. The load-displacement curve during the numerical test is presented in Figure 6.72.

Figure 6.72: Load-displacement curve during numerical test

161 

 

The load-displacement curve (blue one) from start to failure is an inclined straight line (see Figure 6.72), which shows the linear elasticity of the CFRP lamella. Furthermore, the highest point of the load-displacement curve is slightly over the schematic line of the theoretical ultimate load, i.e. the tensile bearing capacity of the CFRP lamella provided by the producer, which is 4 kN. This indicates that the CFRP lamella can be fully used and the anchorage efficiency is 100 %.

6.4.4 Experimental verification To verify the results of the numerical test, the corresponding experiment was carried out by the author, which is presented in this section.

6.4.4.1 Experimental set-up The experiment was conducted in a laboratory at the Technical University of Berlin. Two specimens of thimble-clamp anchorages with CFRP lamellas were tested. A universal testing machine (Instron 8502) was used to pull the specimens. For each specimen, the CFRP lamella was clamped at the one end by the test machine, while at the other end it was anchored by the thimble-clamp anchorage, which was connected to the test machine with a bolt. The experimental set-up is shown in Figure 6.62.

162 

 

Figure 6.73: Experimental set-up The loading velocity was set to 1 mm/s. During the tension, load cells and displacement sensors recorded the tension force and the deformation. The loading did not terminate until visible failure occurred. The results of the experiment are present in following section.

6.4.4.2 Experimental result First, the specimen at failure is shown in the following figure (the situations of two specimens were very similar when the failure occurred).

163 

 

Figure 6.74: Final failure situation of specimen As can be seen from Figure 6.74, the failure exists at the free section of the CFRP lamella, instead of its anchorage section or the metal components of the thimble-clamp anchorage. Furthermore, the failure happened very suddenly, which shows that it was a brittle failure, instead of a ductile failure. Then, the load-displacement curves of these two specimens are presented as follows.

Figure 6.75: Load-displacement curves of two specimens

164 

 

As can be seen from Figure 6.75, the highest points of the load-displacement curves of two specimens are slightly over or very close to the schematic line of the theoretical ultimate load, i.e. the tensile bearing capacity of the CFRP lamella provided by the producer, which is 4 kN. This demonstrates that the CFRP lamella can be fully used and the anchorage efficiency can reach 100 %. Furthermore, comparing Figure 6.75 with Figure 6.72, it can be found that the loaddisplacement curve from the numerical test (i.e. the finite element analysis) is very similar to those from the real tests, which proves the correctness of the results from the finite element analysis. The numerical and experimental ultimate tensile load results are compared in Table 6.5. Table 6.5: Comparison of numerical and experimental ultimate tensile loads FEM result

Test 1 result

Test 2 result

Average test result

Error

4.038 kN

4.167 kN

3.981 kN

4.074 kN

0.9 %

As can be seen from Table 6.5, the ultimate tensile load result of finite element analysis coincides well with those of experiment (the error between the numerical result and the average experimental result is merely 0.9 %), which proves the correctness of the finite element analysis again. In summary, both the finite element analysis and the experiment show that the CFRP lamella anchored by the thimble-clamp anchorage is able to achieve 100 % of its tensile bearing capacity; the final failure is a brittle tensile failure occurring at the free section of the CFRP lamella, while its anchorage section and the anchorage itself remain undamaged during the loading. The results of the finite element analysis and the verified experiment demonstrate that the thimble-clamp anchorage proposed in this research is a reliable design to anchor CFRP cables; moreover, its anchorage efficiency can reach 100 %.

165 

 

7

Prototype of CFRP Orthogonally Loaded Cable Structure

To investigate the feasibility of applying CFRP cables to orthogonally loaded cable structures based on the present technology, a prototype of CFRP orthogonally loaded cable structure was designed and built by the author and colleagues in 2013 at Technical University of Berlin, which is briefly introduced in this chapter.

7.1 Conceptual Design The spoked wheel cable roof, which is a typical orthogonally loaded cable structure, was selected as the structural form of the prototype. First, based on available materials and funds, a prototype CFRP spoked wheel cable roof was preliminarily designed, as shown in the following figure.

Figure 7.1: Hand drawing of preliminary design of prototype CFRP spoked wheel cable roof As can be seen from Figure 7.1, the prototype CFRP spoked wheel cable roof has one compression ring (diameter is approximately 4 m) and one tension ring (diameter is approximately 1 m). The spoke cables are loop-shaped CFRP cables and the tension ring is a closed octagon CFRP loop. The compression ring and all nodes are made of aluminium. The rendering of the prototype CFRP spoked wheel cable roof with aluminium pillars is shown below.

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Figure 7.2: Rendering of prototype CFRP spoked wheel cable roof As can be seen from the above figure, this prototype is a very concise and aesthetic structure. The design details of this prototype are shown in the following figures.

Figure 7.3: Diagrammatic sketch of node at outer compression ring

Figure 7.4: Diagrammatic sketch of node at inner tension ring

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  As can be seen from Figure 7.3 and Figure 7.4, the node at outer compression ring for the CFRP spoke cable is cylinder-shaped, while the node at inner tension ring to connect the CFRP spoke cable with the CFRP tension ring are composed of two half-cylinders, which are linked together by four bolts. Through fastening the bolts and thus shortening the distance between these two half-cylinders, the CFRP cable system of the prototype will be pre-tensioned to a sufficient level.

7.2 Finite Element Analysis and Experimental Verification In order to verify the feasibility of aforementioned conceptual design result, especially the load bearing capacity of CFRP spoke cables, corresponding finite element analysis and experiment were performed and are presented in this section. Due to the limitation of experimental equipment, only a pin-loaded loop-shaped CFRP cable, which is shorter than the CFRP spoke cable used, was analysed and tested. The photo of the specimen is shown in Figure 7.5.

Figure 7.5: Loop-shaped CFRP cable specimen This loop-shaped CFRP cable is a thin CFRP laminated strip-loop (see Figure 7.5). Its geometry is illustrated as follows.

Figure 7.6: Geometry of CFRP cable specimen

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It should be noted that the material used and the cross section of this specimen are the same as those of CFRP spoke cable and CFRP tension ring, while the bending radius of the former is smaller than those of the latter two. As a consequence, the specimen is in a more adverse situation, because the tensile strength of CFRP lamella will reduce with the decrease of the bending radius if the thickness of the lamella is fixed [Winistoefer, 1999]. The material properties of the CFRP cable specimen are illustrated in Table 7.1. Table 7.1: Material properties of CFRP cable specimen Engineering constant

Value

Strength

Value

145000 MPa

1700 MPa

=

8000 MPa

1000 MPa

=

3000 MPa

=

80 MPa

2800 MPa

=

180 MPa

0.29

=

100 MPa

=

0.35

90 MPa

In Table 7.1, the subscripts 1, 2 and 3 represent the fibre direction and two directions perpendicular to the fibre direction, respectively. The material constitutive relation and the material strength criterion (i.e. 3-D Hashin Theory) for CFRP have been already illustrated in Section 6.3. In addition, for two steel pins, E = 210000 MPa and v = 0.3. Because of the structural symmetry, only one fourth of the pin-loaded CFRP cable specimen was modelled. The simplified geometric model including boundary conditions is shown in Figure 7.7.

170 

 

Figure 7.7: Geometric model of pin-loaded CFRP cable specimen The friction coefficient between the CFRP cable and the steel pins was set to 0.2 [Friedrich, 1986]. Based on the above geometric model, the finite element model was established in the general FEM software ABAQUS, as shown in Figure 7.8.

Figure 7.8: Finite element model of pin-loaded CFRP cable specimen In the above finite element model, a commonly used three-dimensional solid element C3D8, which is an 8-node linear brick element [Dassault Systèmes, 2013], was adopted for the CFRP cable; another commonly used three-dimensional solid element C3D8R, which is an 8-node linear brick element with reduced integration [Dassault Systèmes, 2013], was adopted for the steel pins.  Since the CFRP cable was the research focus, the mesh at the lamella was refined.

171 

  The corresponding experiment was conducted in a laboratory at the Technical University of Berlin. Two pin-loaded CFRP cable specimens were tested. The experimental set-up is shown in Figure 7.9.

Figure 7.9: Experimental set-up The finite element analysis and the experiment were to test the failure mode, the failure location and the tensile capability of the CFRP cable specimen. First, the results of failure mode and failure location are shown below, which corresponds to the ultimate tensile load 54.1 kN.

Figure 7.10: Contour of fibre tension failure (failure index

172 

= SDV9)

 

Figure 7.11: Contour of fibre compression failure (failure index

Figure 7.12: Contour of matrix tension failure (failure index

= SDV10)

= SDV11)

Figure 7.13: Contour of matrix compression failure (failure index = SDV12) 173 

 

As can be seen from the above figures, only the fibre tension damage index is slightly greater than one, which indicates that the failure mode of the CFRP cable specimen is the fibre tensile failure. Furthermore, as shown in Figure 7.10, the failure is located in the region that the straight segment of CFRP lamella meets its arc segment, which is consistent with the experimental phenomenon. The ultimate tensile loads obtained by the finite element analysis and the experiment are listed in Table 7.2. Table 7.2: Comparison of numerical and experimental ultimate tensile loads FEM result

Test 1 result

Test 2 result

Average test result

Error

54.1 kN

51.7 kN

55.7 kN

53.7 kN

0.7 %

As can be seen from Table 7.2, the ultimate tensile load result of finite element analysis coincides well with that of experiment (the error between the FEM result and the average experimental result is merely 0.7 %). Furthermore, because the design tensile load of the CFRP spoke cable is 30 kN and the ultimate tensile load withstood by the specimen is over 50 kN, the CFRP spoke cables have enough load bearing capacity and safety margin.

7.3 Manufacture of Components and Construction of Actual Structure All the structural components of the prototype CFRP spoked wheel cable roof were manufactured in the workshop or the laboratory at the Technical University of Berlin. The manufacturing processes of CFRP inner tension ring and CFRP spoke cable are illustrated as follows.

174 

 

Figure 7.14: Manufacture of CFRP inner tension ring

Figure 7.15: Manufacture of CFRP spoke cable As can be seen from Figure 7.14 and Figure 7.15, both CFRP inner tension ring and CFRP spoke cables were produced by winding continuous carbon fibre tows coated with epoxy

175 

  resin on the corresponding moulds. Then, the CFRP components were laminated through using a vacuum technique. The finished prototype CFRP spoke-wheel cable roof is shown in Figure 7.16.

Figure 7.16: Photo of prototype CFRP spoke-wheel cable roof The geometry of this prototype and details of the nodes are illustrated as follows.

176 

 

Figure 7.17: Geometry of prototype CFRP spoke-wheel cable roof

177 

 

Figure 7.18: Details of nodes This prototype demonstrates that orthogonally loaded CFRP cable structures are already feasible today. According to the experiences of the author with this prototype, new cable forms and anchoring strategies different from those for steel cables should be developed for CFRP cables according to their unique characteristics.

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8

A Novel Design: CFRP Continuous Band Winding System

A novel form of using CFRP cables in orthogonally loaded cable structures, i.e. the CFRP continuous band winding system, is conceptually introduced in this chapter. As its name suggest, the cables used in such cable structure are CFRP continuous bands, which will be wound through all the intermediate nodes and only anchored at the both end nodes or which will form closed loops without any anchorages. The greatest advantage of the CFRP continuous band winding system is that the number of anchorages is minimised and thus exploiting to the full favourable conditions of CFRP cables and avoid the unfavourable one. Consider the same swimming pool as in Section 6.3 (see Figure 6.24), which needs a cable roof supported by a frame structure. The CFRP continuous band winding system is very suitable for this case, which can be shown in Figure 8.1 and Figure 8.2.

Figure 8.1: Swimming pool cable roof with CFRP continuous band winding system

179 

 

Figure 8.2: Swimming pool cable roof with CFRP continuous band winding system (including membrane) As can be seen from the above figures, there is only one cable, i.e. a CFRP continuous band, used in the structure, which makes the cable structure very concise and aesthetically pleasing. The CFRP continuous band used and details of the structure are illustrated as follows.

Figure 8.3: CFRP continuous band anchored at both ends

Figure 8.4: Diagrammatic sketch of end node

180 

 

Figure 8.5: Diagrammatic sketch of intermediate node As can be seen from Figure 8.3, the CFRP continuous band is only anchored at both end nodes (see Figure 8.4), while is wound through every intermediate node (see Figure 8.5). A suitable anchorage for the CFRP continuous band is the winding-clamp anchorage, which is shown below.

Figure 8.6: Winding-clamp anchorage for CFRP continuous band (a) assembled view (b) exploded view As can be seen from the Figure 8.6, this anchorage is very similar to the winding-clamp anchorage presented in Section 6.3 (see Figure 6.27). The difference is that the axis of the CFRP lamella in this case is oblique to (instead of orthogonal to) the central axis of the metal ring and the CFRP lamella can be wound several circles around the metal ring. In addition to the above orthogonally loaded cable structure, the CFRP continuous band winding system is also applicable to many other orthogonally loaded cable structures. For example, it can be used in the following spoked wheel cable roof in the form of two compression rings and one tension ring, which are shown in Figure 8.7 and Figure 8.8.

181 

 

Figure 8.7: Spoked wheel cable roof #1 using CFRP continuous band winding system

Figure 8.8: Spoked wheel cable roof #1 using CFRP continuous band winding system (including membrane) As can be seen from the above figures, there are only two cables in the structure, i.e. a CFRP continuous band acting as the spoke cables and a CFRP inner ring, which are shown below.

Figure 8.9: CFRP continuous band and CFRP inner tension ring 182 

 

As can be seen from Figure 8.9, the CFRP continuous band used here has no anchorage. It forms a closed loop and is wound around all nodes. Three types of nodes in this cable structure are shown in the following figures.

Figure 8.10: Node at lower outer compression ring

Figure 8.11: Node at upper outer compression ring

Figure 8.12: Node at inner tension ring

183 

 

Furthermore, the CFRP continuous band system can also be used in the following spoked wheel cable roof in the form of one compression ring and two tension rings, which are shown in Figure 8.13 and Figure 8.14.

Figure 8.13: Spoked wheel cable roof #2 using CFRP continuous band winding system

Figure 8.14: Spoked wheel cable roof #2 using CFRP continuous band winding system (including membrane) As can be seen from the above figures, there are only three cables used in the cable structure, i.e. a CFRP continuous band acting as the spoke cables and two CFRP inner rings, which are shown below.

184 

 

Figure 8.15: CFRP continuous band and CFRP inner tension rings As can be seen from Figure 8.15, the CFRP continuous band used here has no anchorage. It forms a closed loop and is wound around all nodes. Three types of nodes in this cable structure are shown in the following figures.

Figure 8.16: Node at outer compression ring

Figure 8.17: Node at lower inner tension ring 185 

 

Figure 8.18: Node at upper inner tension ring Apart from the aforementioned cases, the CFRP continuous band winding system is well suited for many other orthogonally loaded cable structures. Applying such system, the number of the required anchorages is significantly reduced or eliminated and thus making the best use of the advantages of CFRP cables and bypassing their disadvantages. Furthermore, cable structures will become very concise and aesthetic too, with the CFRP continuous band winding systems. Admittedly, except for the advantages, the CFRP continuous band winding system also has some disadvantages. For example, the failure of a single segment of the CFRP continuous band may result in the failure of the whole system; therefore, the safety factor for the CFRP continuous band might be set higher than that for ordinary cables. In general, the CFRP continuous band winding system is an interesting and promising way of using CFRP cables in orthogonally loaded cable structures. More researches will be needed for its realisation.

186 

 

9

Conclusions and Recommendations

The conclusions of this book and the recommendations for future research are presented in this chapter.

9.1 Conclusions In this book, a new field of application for CFRPs is proposed: using CFRP cables in orthogonally loaded cable structures. The orthogonally loaded cable structures can be defined as cable structures with a majority of cables orthogonally loaded or approximately orthogonally loaded by external loads. In other words, the angles between cable axes of most cables and external loads, i.e. the control angles, are just or close to 90° in such structures. For orthogonally loaded cable structures, structural stiffness is primarily derived from the geometric stiffness, which is governed by the tensile strength of cables. This indicates that CFRP cables, whose tensile strength is higher than that of steel cables, in orthogonally loaded cable structures can improve the mechanical and economic performances of these cable structures. To demonstrate this point, two typical orthogonally loaded cable structures, i.e. a cable net facade and a spoked wheel cable roof, with CFRP cables was studied. The mechanical properties and economic efficiencies of these two structures with CFRP cables having different Young’s moduli and different tensile strengths were compared to those of counterpart structures with steel cables. The influence of varying control angle on the advantages of using CFRP cables in orthogonally loaded cable structures was also investigated through comparing these two CFRP orthogonally loaded cable structures. Furthermore, to solve the challenge of anchoring CFRP cables in orthogonally loaded cable structures and thus making such CFRP cable structures more feasible, new design principles for anchoring CFRP cables were proposed based on analysing the existing anchorages for CFRP cables. According to these principles, two novel CFRP cable anchorages, i.e. the winding-clamp anchorage and the thimble-clamp anchorage were designed and presented in this book. Corresponding finite element analyses and experiments of these new anchorages were performed to verify the feasibility of the designs.

187 

 

Moreover, a prototype CFRP spoked wheel cable roof designed and built by the author and colleagues was introduced to show the feasibility of the construction of CFRP orthogonally loaded cable structures based on the present technology. In addition, a new form of using CFRP cables in orthogonally loaded cable structures, i.e. the continuous band winding system, was also presented in this book. Such cable system can eliminate anchorages and thus offering a complete solution for the problem of anchoring CFRP cables. Key conclusions drawn from this research include: 1. Compared to steel cables, using CFRP cables in orthogonally loaded cable structures can significantly raise the structural stiffness if the amount of cable used is maintained; using CFRP cables can also significantly reduce the amount of cable used if the structural stiffness is maintained instead. Moreover, using CFRP cables in cable net facade can generally increase more structural stiffness or decrease more amount of cable used than using them in spoked wheel cable roof, because the control angle  of the cable net facade is closer to 90° than that of the spoked wheel cable roof. 2. Increasing the tensile strength u of CFRP cables, which increases the pre-tension force, will significantly reduce the structural deformation or the amount of cable used of orthogonally loaded cable structures. However, increasing the Young’s modulus E of the CFRP cables will lead to a decrease in pre-tension forces, which results in slightly increasing the structural deformation or the amount of cable used in the cable net facade or only indistinctively reducing them in the spoked wheel cable roof. 3. Although the unit prices of CFRP cables are much higher than that of steel cables, there are still several CFRP orthogonally loaded cable structures modelled with lower Young’s modulus achieving lower cable costs than the steel cable structures used for comparison and thus having better economy, primarily because the material savings successfully offset the adverse effect of the high unit prices of CFRP cables. Moreover, using CFRP cables in cable net facade can generally

188 

 

achieve better economic efficiency than using them in spoked wheel cable roof, because the control angle  of the cable net facade is closer to 90° than that of the spoked wheel cable roof. 4. CFRP cable anchorages should be designed in accordance with the material characteristics of CFRP instead of directly copying from steel cable anchorages. Anchoring CFRP cables with gradient intensity decreasing from the anchorage end to the anchorage mouth is an effective way to decrease stress concentration and prevent premature failure of CFRP cables, which can be realised by gradually decreasing the clamping forces or reducing the stiffness of wedges, bond materials or anchorages themselves from the anchorage end to the anchorage mouth. Moreover, “pin-loaded” can not only help reduce the length of the anchorage but also simplify the anchorage structure and thus benefiting the design of CFRP cable anchorages, especially for CFRP orthogonally loaded cable structures. 5. Both finite element analysis and experiment demonstrate that two new CFRP cable anchorages proposed in this book, i.e. the winding-clamp anchorage and the thimble clamp anchorage, are reliable designs for anchoring CFRP cables and their anchorage efficiencies are able to reach 100 %. 6. The successful construction of a CFRP orthogonally loaded cable structure prototype shows that the design and building of such cable structures are feasible with present knowledge and technology. 7. Applying CFRP continuous band winding systems, the number of the required anchorages can be significantly reduced or eliminated and thus making the best use of the advantages of CFRP cables and bypassing their disadvantages. Moreover, with the CFRP continuous band winding systems, cable structures will become very concise and aesthetic.

9.2 Recommendations To the author’s knowledge, this is the first book in the world about the systematic investigation of using CFRP cables in orthogonally loaded cable structures. Researches

189 

 

related to this topic are very few up to now. Although the advantages are obvious, substituting CFRP cables for steel cables in orthogonally loaded cable structures may still have a long way to go. Further studies on this issue are really needed in the future. Key points related to this research, which are recommended to be further investigated, are listed as follows. 1. Some simplifications were made to establish the calculation models of investigated orthogonally loaded cable structures in this book. For example, hanger cables of the spoked wheel cable roof were omitted. In the future, it is recommended to select a real spoked wheel cable roof as the research subject and investigate the influence of replacing all steel cables with CFRP cables. 2. Only two types of CFRP orthogonally loaded cable structures were studied in this book. It is recommended that more such structures, e.g. CFRP stress-ribbon bridge and CFRP beam string structure, are investigated in the future. 3. Only the static properties of the two CFRP cable anchorages proposed in this book were studied. In the future, their fatigue properties and creep properties should also be investigated. 4. The size of the prototype CFRP spoked wheel cable roof presented in this book is relatively small. It is recommended to build larger prototype or replace some steel cables with CFRP cables in a real orthogonally loaded cable structure experimentally. 5. Only the conceptual design of the CFRP continuous band winding system was presented in this book. Its static and dynamic properties need to be further investigated.

190 

 

List of Symbols English Letters A

Cross-sectional area

a

The first column vector

b

The second column vector

C

Stiffness matrix of orthotropic material

c

The third column vector

E

Young’s modulus

E1

Young’s modulus in fibre direction

E2

In-plane Young’s modulus perpendicular to fibre direction

E3

Out of plane Young’s modulus perpendicular to fibre direction

Etan

Tangential stiffness modulus

F

Pre-tension force

Fext

External force vector in global coordinate system

F

External

force

increment

vector

in

global

coordinate system Fun

Unbalanced force vector in global coordinate system

fext

External force vector (nodal force vector) in local coordinate system

fint

Internal force vector in local coordinate system

fx

Nodal force in x direction

fy

Nodal force in y direction

fz

Nodal force in z direction f

External force increment vector in local coordinate system

G

Gravity force

g

Standard gravity constant

G12

In-plane shear modulus

G13

Out of plane shear modulus across fibre

G23

Out of plane shear modulus not across fibre 191 

 

Gij

Coefficient of quadratic term of strength inequality

Gi

Coefficient of linear term of strength inequality

I

Area moment of inertia

I

Identity matrix

i

Time of iteration

j

Number of cable node

K

Structural stiffness value in one degree of freedom

KE

Elastic stiffness value in one degree of freedom

KG

Geometric stiffness value in one degree of freedom

K

Structural stiffness matrix in global coordinate system

KE

Elastic stiffness matrix in global coordinate system

KG

Geometric stiffness matrix in global coordinate system

ke

Elastic stiffness in local coordinate system

kg

Geometric stiffness in local coordinate system

k

Failure index Fibre tension failure index Fibre compression failure index Matrix tension failure index Matrix compression failure index

L

Length of cable element

l

Direction cosine 1

m

Direction cosine 2

n

Direction cosine 3

P

External load

p

Pressure

R

Radius

S

Displacement vector in global coordinate system S

Displacement

increment

vector

in

global

coordinate system Displacement vector in local coordinate system

s s

Displacement increment in local coordinate system Tensile strength in fibre direction Compressive strength in fibre direction

192 

 

In-plane tensile strength perpendicular to fibre direction In-plane compressive strength perpendicular to fibre direction Out of plane tensile strength perpendicular to fibre direction Out of plane compressive strength perpendicular to fibre direction In-plane shear strength Out of plane shear strength across fibre Out of plane shear strength not across fibre Time (step)

t t

Time increment

U

Displacement in X direction

u

Displacement in x direction

V

Displacement in Y direction

v

Displacement in y direction

Vb

Arbitrary body

W

Displacement in Z direction

w

Displacement in z direction

Wint

Virtual work of internal forces

Wext

Virtual work of external forces

X

X-axis of global coordinate system

x

X-axis of local coordinate system

Y

Y-axis of global coordinate system

y

Y-axis of local coordinate system

Z

Z-axis of global coordinate system

z

Z-axis of local coordinate system

Greek Letters 

Angle between cable axis and external load after deformation



Strain tensor 

Strain increment tensor

193 

 

x

Strain in x direction



Diameter



Linear term of Green strain



Quadratic term of Green strain



Coordinate transformation matrix

v12

In-plane Poisson's ratio

v13

Out of plane Poisson's ratio 1

v23

Out of plane Poisson's ratio 2



Angle between cable axis and external load, i.e. control angle



Density



Stress tensor 

Stress increment tensor

u

Tensile strength

x

Stress in x direction



Coordinate axes of local coordinate system



Coordinate axes of global coordinate system

Mathematical Operators d□

Finding derivative

□

Finding partial derivative

□T

Matrix transpose

Abbreviations FEM

Finite Element Method

IS

Initial State

PAN

Polyacrylonitrile

RTM

Resin transfer moulding

SLS

Serviceability Limit State

SDV

Internal variable of ABAQUS

ULS

Ultimate Limit State

ZS

Zero State

194 

 

List of Tables

Table 2.1: Classification of cable structures according to the type of pre-tension

24

Table 3.1: Mechanical properties of carbon fibres compared with typical steel materials 53 Table 3.2: Mechanical properties of commonly used polymer resins

55

Table 3.3: Mechanical properties of CFRP cables compared with steel cable

61

Table 3.4: Unit prices of CFRP cables (unit: €/kg)

62

Table 4.1: Material properties of investigated CFRP cables in Case Study A

68

Table 4.2: Design results of reference steel flat cable net

71

Table 5.1: Material properties of investigated CFRP cables in Case Study B

89

Table 5.2: Design results of reference steel spoked wheel cable roof

92

Table 6.1: Mechanical properties of metal material used

135

Table 6.2: Mechanical properties of CFRP used

135

Table 6.3: Comparison of numerical and experimental ultimate tensile loads

148

Table 6.4: Mechanical properties of epoxy adhesive used

156

Table 6.5: Comparison of numerical and experimental ultimate tensile loads

163

Table 7.1: Material properties of CFRP cable specimen

168

Table 7.2: Comparison of numerical and experimental ultimate tensile loads

172

Table A.1: Solution process and results in MS Excel (control angle  = 85°)

208

Table A.2: Comparison of results from Excel, SOFiSTiK and ABAQUS ( = 89°)

209

Table A.3: Solution process and results in MS Excel ( = 85°)

211

Table A.4: Comparison of results from Excel, SOFiSTiK and ABAQUS ( = 85°)

212

Table A.5: Solution process and results in MS Excel ( = 81°)

213

Table A.6: Comparison of results from Excel, SOFiSTiK and ABAQUS ( = 81°)

214

195 

 

List of Figures Figure 1.1: Suspension bridge with vines and creepers

1

Figure 1.2: Cable system in a sailboat

1

Figure 1.3: Modern suspension bridge: Golden Gate Bridge, San Francisco, USA, 1937 (a) photo (photo credit: Motion Drives & Controls Ltd) (b) schematic diagram of structural form

2

Figure 1.4: Modern cable-stayed bridge: Ting Kau Bridge, Hong Kong, China, 1997 (a) photo (photo credit: HK Arun) (b) schematic diagram of structural form

3

Figure 1.5: North Carolina State Fair Arena at Raleigh, Raleigh, USA, 1953 (a) photo (photo credit: State Archives of North Carolina) (b) schematic diagram of structural form 4 Figure 1.6: Munich Olympic Stadium, Munich, Germany, 1972 (a) photo (photo credit: Arad Mojtahedi) (b) schematic diagram of structural form

4

Figure 1.7: Typical cable structures

5

Figure 1.8: Classification of cable structures

6

Figure 1.9: Tsukuba FRP Bridge (a) photo (photo credit: Iwao Sasaki) (b) sketch

8

Figure 1.10: Anchorage system in Tsukuba FRP Bridge (a) full view (b) transparent view 8 Figure 1.11: Stork Bridge (a) photo (photo credit: EMPA) (b) sketch

9

Figure 1.12: Gradient Anchorage system (a) full view (b) transparent view

9

Figure 1.13: Herning CFRP Bridge (a) photo (photo credit: COWI) (b) sketch

10

Figure 1.14: resin filling type anchorage system (a) full view (b) transparent view

10

Figure 1.15: Laroin CFRP Footbridge (a) photo (photo credit: Freyssinet) (b) sketch

11

Figure 1.16: Modular Clamp Anchorage system (one module) (a) full view (b) sectional view

11

Figure 1.17: Jiangsu University CFRP Footbridge (a) photo (photo credit: Kuihua Mei) (b) sketch

12

Figure 1.18: Straight Tube and Inner Cone Anchorage system (a) full view (b) transparent view

13

Figure 1.19: Penobscot Narrows Bridge (a) photo (photo credit: MOT) (b) sketch

14

197 

 

Figure 1.20: Highly Expansive Material (HEM) Filling Anchorage System (a) full view (b) transparent view

14

Figure 1.21: EMPA Bowstring Arch Footbridge (a) photo (photo credit: Urs Meier) (b) sketch

15

Figure 1.22 (a) CFRP non-laminated strip-loop cable and (b) pin-loaded anchorage (photo credit: Urs Meier)

16

Figure 1.23: TU-Berlin CFRP Stress-Ribbon Footbridge (a) photo (photo credit: Achim Bleicher) (b) sketch

17

Figure 1.24: Pin-loaded anchorage system (a) assembled view (b) exploded view

17

Figure 1.25: Cuenca Stress-Ribbon Footbridge (a) photo (photo credit: Juan Rodado Lopez) (b) sketch

18

Figure 1.26: (a) Anchorage and (b) connection for CFRP cables

19

Figure 2.1: Design states of cable structures

22

Figure 2.2: Increasing pre-tension while maintaining geometry simultaneously (type 1) 25 Figure 2.3: Increasing pre-tension while maintaining geometry simultaneously (type 2) 26 Figure 2.4: Finite element discretisation for the cable system of cable structures (a) whole structure (b) a cable segment modelled with a single cable element (c) a cable segment modelled with several cable elements

27

Figure 2.5: Cable element in the local coordinate system (xyz) and global coordinate system (XYZ)

28

Figure 2.6: Magnitudes and directions of KE and KG at the middle node of a two-link cable

41

Figure 2.7: Double-curved cable net with four cable elements

41

Figure 2.8: Comparing values of KE and KG with the variation of 



Figure 2.9: Comparing relative sizes of KE and KG with the variation of 



Figure 2.10: Structural stiffness K with different E and different F when  = 89°

45

Figure 2.11: Structural stiffness K with different E and different F when  = 85°

45

Figure 2.12: Structural stiffness K with different E and different F when  = 81°

46

Figure 2.13: Classification of cable structures according to the control angle 



Figure 3.1: Carbon fibre compared with human hair (photo credit: Anton)

50

198 

 

Figure 3.2: Schematic diagram of the micro structure of a carbon fibre [Morgan, 2005] 50 Figure 3.3: Crystallisation degree of carbon fibres under different temperatures [Morgan, 2005]

51

Figure 3.4: Schematic diagram of the crystalline structure of carbon fibres [Chung, 1994]

53

Figure 3.5: Molecular structures of thermoplastic and thermosetting resins [Ebewele, 2000]

54

Figure 3.6: Schematic diagram of contact moulding (photo credit: Laurensvan Lieshout) 56 Figure 3.7: Schematic diagram of resin transfer moulding (RTM) (photo credit: Net composites)

57

Figure 3.8: Schematic diagram of pultrusion (photo credit: Universal pultrusions)

58

Figure 3.9: Schematic diagrams of four types of CFRP cables

60

Figure 3.10: Variation tendency of unit prices of CFRP cables compared to that of steel cable

63

Figure 4.1: Typical cable net facades

66

Figure 4.2: Geometry and boundary conditions of steel or CFRP flat cable nets

67

Figure 4.3: E and u of cables used in this research and some existing CFRP cable structures

69

Figure 4.4: Design procedure of the steel or CFRP flat cable nets

70

Figure 4.5: Comparison of mid-span deflections between CFRP and steel flat cable nets 72 Figure 4.6: Influence of geometrical non-linearity on flat cable nets

74

Figure 4.7: Comparison of pre-tension forces between CFRP and steel flat cable nets

76

Figure 4.8: Comparison of amounts of cable used between CFRP and steel flat cable nets77 Figure 4.9: Comparison of cross-sectional areas between CFRP and steel flat cable nets 77 Figure 4.10: Comparison of pre-tension forces between CFRP and steel flat cable nets

78

Figure 4.11: Comparison of support reactions between CFRP and steel flat cable nets

79

Figure 4.12: Comparison of cable weights between CFRP and steel flat cable nets

80

Figure 4.13: Comparison of cable costs between CFRP and steel flat cable nets

81

Figure 5.1: A typical tensile spoked wheel

83

Figure 5.2: Six common structural forms of spoked wheel cable roofs

84

199 

 

Figure 5.3: Six real spoked wheel cable roofs corresponding to Figure 5.2

85

Figure 5.4: Geometry and boundary conditions of investigated spoked wheel cable roofs 87 Figure 5.5: Cross-sectional view of compression ring (unit: mm)

87

Figure 5.6: Cross-sectional view of compression stab (unit: mm)

88

Figure 5.7: Design procedure of steel or CFRP spoked wheel cable roofs

90

Figure 5.8: Comparison of mid-span deflections due to snow between CFRP and steel spoked wheel cable roofs

93

Figure 5.9: Comparison of mid-span deflections due to wind between CFRP and steel spoked wheel cable roofs

93

Figure 5.10: Comparison of average deflections between CFRP and steel spoked wheel cable roofs

94

Figure 5.11: Comparison of lower spoke cable pre-tension forces between CFRP and steel spoked wheel cable roofs

95

Figure 5.12: Comparison of upper spoke cable pre-tension forces between CFRP and steel spoked wheel cable roofs

95

Figure 5.13: Comparison of amounts of cable used between CFRP and steel flat cable nets

96

Figure 5.14: Comparison of cross-sectional areas of lower spoke cables between CFRP and steel flat cable nets

97

Figure 5.15: Comparison of cross-sectional areas of upper spoke cables between CFRP and steel flat cable nets

97

Figure 5.16: Comparison of pre-tension forces of lower spoke cables between CFRP and steel spoked wheel cable roofs

98

Figure 5.17: Comparison of pre-tension forces of upper spoke cables between CFRP and steel spoked wheel cable roofs

98

Figure 5.18: Comparison of compression forces of compression rings between CFRP and steel spoked wheel cable roofs

99

Figure 5.19: Comparison of compression forces of compression stabs between CFRP and steel spoked wheel cable roofs

100

Figure 5.20: Comparison of cable weights between CFRP and steel spoked wheel cable roofs

200 

101

 

Figure 5.21: Comparison of cable costs between CFRP and steel spoked wheel cable roofs

102

Figure 5.22: Comparison of increment degrees of structural stiffness between CFRP cable net facades and CFRP spoked wheel cable roofs

105

Figure 5.23: Comparison of reduction degrees of amount of cable used between CFRP cable net facades and CFRP spoked wheel cable roofs

107

Figure 5.24: Comparison of reduction degrees of cable cost between CFRP cable net facades and CFRP spoked wheel cable roofs

109

Figure 6.1: Clamp anchorage with gradient pre-stressed bolts (a) assembled view (b) exploded view

112

Figure 6.2: LAP strain anchorage (a) assembled view (b) exploded view

112

Figure 6.3: SIKA Leoba-Carbodur 2 (a) assembled view (b) exploded view

113

Figure 6.4: SIKA Stress Head 220 (a) assembled view (b) exploded view

113

Figure 6.5: Composite wedge anchorage (a) assembled view (b) exploded view

114

Figure 6.6: CFCC strand wedge anchorage (a) assembled view (b) exploded view

114

Figure 6.7: Thermoplastic clamp anchorage (a) full view (b) sectional view

114

Figure 6.8: Inverse conical bond anchorage (a) full view (b) sectional view

115

Figure 6.9: Inverse conical bond anchorage with indentation (a) full view (b) sectional view

116

Figure 6.10: Inverse segmented cone anchorage (a) full view (b) sectional view

116

Figure 6.11: Highly expansive mortar anchorage (a) full view (b) sectional view

116

Figure 6.12: Gradiently reduced cable end anchorage (a) full view (b) sectional view

117

Figure 6.13: Forked cable end anchorage (a) full view (b) sectional view

117

Figure 6.14: Gradient Anchorage System (a) full view (b) sectional view

118

Figure 6.15: Non-laminated CFRP strip-loop pin-loaded anchorage (a) full view (b) schematic structural view

118

Figure 6.16: Laminated CFRP strip-loop pin-loaded anchorage (a) full view (b) schematic structural view

119

Figure 6.17: Early plate type clamp anchorage (a) diagrammatic sketch (b) shear stress distribution diagram

120

201 

 

Figure 6.18: Improved plate type clamp anchorage (a) diagrammatic sketch (b) shear stress distribution diagram

120

Figure 6.19: Early wedge type clamp anchorage (a) diagrammatic sketch (b) shear stress distribution diagram

121

Figure 6.20: Improved wedge type clamp anchorage (a) diagrammatic sketch (b) shear stress distribution diagram

121

Figure 6.21: Early conical bond anchorage (a) diagrammatic sketch (b) shear stress distribution diagram

122

Figure 6.22: Improved conical bond anchorage (a) diagrammatic sketch (b) shear stress distribution diagram

123

Figure 6.23: Pin-loaded anchorage compared with clamp or bond anchorage

123

Figure 6.24: Swimming pool which needs a cable-membrane roof

125

Figure 6.25: Diagrammatic sketch of cable system of swimming pool

126

Figure 6.26: Cable structure using ordinary clamp or bond anchorages

126

Figure 6.27: Winding-clamp anchorage with CFRP lamella (a) assembled view (b) exploded view

127

Figure 6.28: Winding-clamp anchorage (a) diagrammatic sketch (b) shear stress distribution diagram

128

Figure 6.29: Cable structure with winding-clamp anchorage (overall view)

128

Figure 6.30: Cable structure with winding-clamp anchorage (local view)

129

Figure 6.31: Final rendering of swimming pool cable roof with CFRP cables and windingclamp anchorages

129

Figure 6.32: Investigated winding-clamp anchorage (a) assembled view (b) exploded view

130

Figure 6.33: Geometry of aluminium cylinder

131

Figure 6.34: Geometry of bent aluminium plate

131

Figure 6.35: Geometry of steel cuboid

131

Figure 6.36: Photo of aluminium cylinder and plate

132

Figure 6.37: Geometry of CFRP lamella anchored by winding-clamp anchorage

132

Figure 6.38: Photo of CFRP lamella

133

202 

 

Figure 6.39: Geometric model including boundary conditions

134

Figure 6.40: Finite element model of investigated winding-clamp anchorage

134

Figure 6.41: Contour of von Mises stress of aluminium cylinder (unit: MPa)

142

Figure 6.42: Contour of von Mises stress of bent aluminium plates (unit: MPa)

143

Figure 6.43: Contour of von Mises stress of steel cuboids (unit: MPa)

143

Figure 6.44: Contour of fibre tension failure (failure index

= SDV9) = SDV10)

Figure 6.45: Contour of fibre compression failure (failure index Figure 6.46: Contour of matrix tension failure (failure index

= SDV11)

Figure 6.47: Contour of matrix compression failure (failure index

= SDV12)

144 144 144 145

Figure 6.48: Load-displacement curve during numerical test

145

Figure 6.49: Experimental set-up

147

Figure 6.50: Final failure situation of specimen

148

Figure 6.51: Load-displacement curves of two specimens

149

Figure 6.52: Studied stress-ribbon bridge

150

Figure 6.53: Diagrammatic sketch of cable system of stress-ribbon bridge

151

Figure 6.54: Cable bridge using ordinary clamp or bond anchorages

151

Figure 6.55: Thimble-clamp anchorage with CFRP lamella (a) assembled view (b) exploded view

151

Figure 6.56: Cable structure with thimble-clamp anchorage (overall view)

152

Figure 6.57: Cable structure with thimble-clamp anchorage (local view)

153

Figure 6.58: Investigated thimble-clamp anchorage (a) assembled view (b) exploded view

154

Figure 6.59: Geometry of aluminium thimble

154

Figure 6.60: Geometry of steel plate

155

Figure 6.61: Geometry of CFRP lamella

155

Figure 6.62: Geometric model including boundary conditions

156

Figure 6.63: Finite element model of investigated thimble-clamp anchorage

156

Figure 6.64: Contour of von Mises stress of aluminium thimble (unit: MPa)

158

Figure 6.65: Contour of von Mises stress of steel plates (unit: MPa)

158

203 

 

Figure 6.66: Contour of maximum principal stress of epoxy adhesive layer gluing CFRP lamella (unit: MPa)

159

Figure 6.67: Contour of maximum principal stress of epoxy adhesive layers gluing steel plates to CFRP lamella (unit: MPa) Figure 6.68: Contour of fibre tension failure (failure index

159 = SDV9) = SDV10)

Figure 6.69: Contour of fibre compression failure (failure index

= SDV11)

Figure 6.70: Contour of matrix tension failure (failure index

= SDV12)

Figure 6.71: Contour of matrix compression failure (failure index

160 160 160 161

Figure 6.72: Load-displacement curve during numerical test

161

Figure 6.73: Experimental set-up

163

Figure 6.74: Final failure situation of specimen

164

Figure 6.75: Load-displacement curves of two specimens

164

Figure 7.1: Hand drawing of preliminary design of prototype CFRP spoked wheel cable roof

167

Figure 7.2: Rendering of prototype CFRP spoked wheel cable roof

168

Figure 7.3: Diagrammatic sketch of node at outer compression ring

168

Figure 7.4: Diagrammatic sketch of node at inner tension ring

168

Figure 7.5: Loop-shaped CFRP cable specimen

169

Figure 7.6: Geometry of CFRP cable specimen

169

Figure 7.7: Geometric model of pin-loaded CFRP cable specimen

171

Figure 7.8: Finite element model of pin-loaded CFRP cable specimen

171

Figure 7.9: Experimental set-up

172

Figure 7.10: Contour of fibre tension failure (failure index

= SDV9) = SDV10)

Figure 7.11: Contour of fibre compression failure (failure index Figure 7.12: Contour of matrix tension failure (failure index

= SDV11)

Figure 7.13: Contour of matrix compression failure (failure index

= SDV12)

172 173 173 173

Figure 7.14: Manufacture of CFRP inner tension ring

175

Figure 7.15: Manufacture of CFRP spoke cable

175

Figure 7.16: Photo of prototype CFRP spoke-wheel cable roof

176

204 

 

Figure 7.17: Geometry of prototype CFRP spoke-wheel cable roof

177

Figure 7.18: Details of nodes

178

Figure 8.1: Swimming pool cable roof with CFRP continuous band winding system

179

Figure 8.2: Swimming pool cable roof with CFRP continuous band winding system (including membrane)

180

Figure 8.3: CFRP continuous band anchored at both ends

180

Figure 8.4: Diagrammatic sketch of end node

180

Figure 8.5: Diagrammatic sketch of intermediate node

181

Figure 8.6: Winding-clamp anchorage for CFRP continuous band (a) assembled view (b) exploded view

181

Figure 8.7: Spoked wheel cable roof #1 using CFRP continuous band winding system 182 Figure 8.8: Spoked wheel cable roof #1 using CFRP continuous band winding system (including membrane)

182

Figure 8.9: CFRP continuous band and CFRP inner tension ring

182

Figure 8.10: Node at lower outer compression ring

183

Figure 8.11: Node at upper outer compression ring

183

Figure 8.12: Node at inner tension ring

183

Figure 8.13: Spoked wheel cable roof #2 using CFRP continuous band winding system 184 Figure 8.14: Spoked wheel cable roof #2 using CFRP continuous band winding system (including membrane)

184

Figure 8.15: CFRP continuous band and CFRP inner tension rings

185

Figure 8.16: Node at outer compression ring

185

Figure 8.17: Node at lower inner tension ring

185

Figure 8.18: Node at upper inner tension ring

186

Figure A.1: Solution process of Newton-Raphson method

208

Figure A.2: Double-curved cable net with control angle  = 89°

209

Figure A.3: Double-curved cable net with control angle  = 85°

212

Figure A.4: Double-curved cable net with control angle  = 81°

214

205 

 

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T.:

Verfahren

zur

Ausbildung

einer

Verankerung

eines

länglichen

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Appendix A – Numerical Analysis Procedure of Cable Structures

A.1 Introduction to Theory In the analysis of geometric non-linear structures, the equilibrium equations have to be established in the geometry after the deformation. The analysis method consists of formulation method and solution method. The total Lagrangian formulation and update Lagrangian formulation are two commonly used formulation methods. These two methods are both based on material coordinate, Green strain and 2nd Kirchhoff-Piola stress [Zhang, 2002]. The concepts of time and iteration in non-linear static numerical analysis should be presented. The time, represented as the symbol t, means the step t of the load increment or the displacement increment; usually, t = 0,1,2,…,n and

t = 1 (the step 0 means the

corresponding initial state). The iteration, expressed as the symbol i, represents the i time iteration in a step; usually, i = 0,1,2,…,n and

i = 1 (the iteration 0 means the

corresponding initial state). The difference between these two formulations is that: in the total Lagrangian formulation all variables (like stress and strain) at any time t or any iteration i are referred to the initial configuration, i.e. the configuration at time 0; in the update Lagrangian formulation, however, all variables at time t or iteration i are referred to the last known configuration [Bathe and Bolourchi, 1979]. Because in the update Lagrangian formulation the increments of stress and strain in a step (from t to t+ t) or an iteration (from i to i+1) are usually relatively small, the second-order term in the Green strain can be ignored and the linear strain and the corresponding Cauchy stress can be adopted, so as to increase computational speed [Zhang, 2002]. The solution methods include iterative method, incremental method and mixed method. The Newton-Raphson method is a commonly used solution method [Cajori, 1911], which is a mixed method based on the Newton method (an iterative method) and the incremental method.

215 

 

Assume the external force is known, the process of using Newton-Raphson method to solve a cable structure can be explained as follows [Ypma, 1995]. First, the external force vector Fext is divided into n force increment vectors △F of the same size and the same direction: n

Fext   ΔF

(A.1)

Therefore, the applied external force vector at the t+ t step can be expressed as (set

=

0 when t = 0): t t t Fext  Fext  ΔF

(A.2)

The total stiffness equation of the structure from t to t+ t can be written as (also see Section 2.3):

KΔS  ΔF

(A.3)

where K is the structural stiffness matrix of the cable structure, which varies from t to t+ t with the iterations; △S is the displacement increment vector from t to t+ t; △F is the external force increment vector from t to t+ t. Substitute Equation (A.5) into Equation (A.3), one can obtain: t t t KΔS  Fext  Fext

Furthermore, because the structure achieves equilibrium at time t,

(A.4) is equal to

,

which means the external force is balanced by the internally generated force of the structure. Therefore, Equation (A.4) can also be written as: t t KΔS  Fext  Fintt

where,

is the internal force at time t, which is determined by the geometry of the

structure at time t, i.e. the displacement at time t: St.

216 

(A.5)

 

The equilibrium iterative process from t to t+ t can be expressed as follows: t t K i Si  Fext  Finti

(A.6)

Si 1  Si  Si

(A.7)

where i indicates the i time iteration at the step t; Ki is the structural stiffness matrix (i.e. the tangential stiffness matrix) of the i time iteration, which is determined by the geometry of the i time iteration, i.e. the displacement after i times of iteration: Si (when i = 0, Si = St);

is the internal force vector of the i time iteration, which is also determined by the

geometry of the i time iteration; △Si is the displacement increment vector of the i time iteration. The right side of Equation (A.6) represents the unbalanced force (Fun), which can be reduced to zero or a permissive range through the following iteration: (1)

Set the initial displacement Si (i = 0), which is equal to St, i.e. the displacement at

time t. If t is 0, Si (i = 0) is set to 0. (2)

Calculate Ki and

based on Si.

(3)

Calculate △Si according to Equation (A.6).

(4)

Calculate the new displacement Si+1 according to Equation (A.7).

(5)

Repeat the above steps from (2) to (4) until the unbalanced force Fun is zero or in a permissive range, i.e. the convergence is achieved.

After the convergence, △S can be calculated by totalising all △Si (n indicates the times of iteration): n

ΔS   ΔSi

(A.8)

i 0

Then, update the displacement:

217 

 

St t  St  ΔS

(A.9)

The final displacement can be calculated by: n

S   ΔS

(A.10)

The solution process of the Newton-Raphson method can also be shown as the figure bellow [Manie, 2009]:

Figure A.1: Solution process of Newton-Raphson method As can be seen from Figure A.1, the structural stiffness matrix is the slope of the iterative tangent line, which will be formed and decomposed at each iteration within a particular step. The Newton-Raphson method yields a quadratic convergence characteristic, which makes it have a high convergence rate and only need a few iterations to achieve the final convergence within a particular step [Manie, 2009].

A.2 Case Studies Three cases mentioned in Section 2.4, i.e. the double-curved cable net with the control angle  = 89°, 85° or 81°, are studied in this section to show the solution process of cable 218 

  structures as well as the changes of the elastic stiffness and geometric stiffness during the loading. Furthermore, other properties of the three investigated double-curved cable nets were set to the same: initial cable length L = 1 m; Young’s modulus E = 160 GPa; cross-sectional area A = 100 mm2; external load P = 10 kN; pre-tension force F = 100 kN. In the analysis, the formulation method was the update Lagrangian formulation and the solution method was the Newton-Raphson method. Moreover, the Green strain and the 2rd Kirchhoff-Piola stress were adopted. In addition, the △F used in the Newton-Raphson method was set to 1 kN for all the three cases. The solution was completed through programming in MS Excel. The results were compared with the FEM results from the SOFiSTiK [SOFiSTiK, 2012] and the ABAQUS [Dassault Systèmes, 2013]. A.2.1 Case A: initial control angle  = 89° The investigated cable structure is shown in Figure A.2.

Figure A.2: Double-curved cable net with control angle  = 89° The solution process and results are presented in Table A.1. It should be noted that the lengths and forces of the cable elements will change during the loading and the changes are different between the support cables, i.e. the cable elements between the node 1 and node 3 as well as between the node 1 and node 5 (see Figure A.2), and the tension cables, i.e. the cable elements between the node 1 and node 2 as well as between the node 1 and node 4 219 

 

(see Figure A.2). Therefore, the lengths of the support cables are represented with L1 and their cable forces are represented with f1; the lengths of the tension cables are represented with L2 and their cable forces are represented with f2. K, KE and KG were calculated according to Equation (2.69), (2.70) and (2.71), respectively. Furthermore,

, Fun and

i

△S are three vectors; the positive numbers indicate the positive direction of Z axis, while

the negative numbers indicate the negative direction of Z axis. Table A.1: Solution process and results in MS Excel (control angle  = 89°) (kN)

Fun (kN)

KE (kN/m)

KG (kN/m)

K (kN/m)

△Si (m)

100.000000

0.000000

-1.000000

19.493539

400.000000

419.493539

-0.002384

99.379805

1.000140

0.000140

19.856398

400.180597

420.036996

0.000000

100.711012

99.379885

1.000000

0.000000

19.856297

400.180547

420.036844

0.000000

0.999961

100.711012

99.379885

1.000000

-1.000000

19.856297

400.180547

420.036844

-0.002381

1.000094

0.999928

101.511871

98.851184

2.001419

0.001419

20.942846

400.721124

421.663970

0.000003

2

1.000094

0.999928

101.510675

98.851867

2.000000

0.000000

20.940800

400.720106

421.660906

0.000000

0

1.000094

0.999928

101.510675

98.851868

2.000000

-1.000000

20.940799

400.720106

421.660905

-0.002372

1

1.000150

0.999901

102.398389

98.415217

3.002660

0.002660

22.741655

401.616015

424.357670

0.000006

2

1.000150

0.999901

102.395924

98.416253

3.000001

0.000001

22.735948

401.613176

424.349124

0.000000

3

1.000150

0.999901

102.395924

98.416253

3.000000

0.000000

22.735946

401.613175

424.349121

0.000000

0

1.000150

0.999901

102.395924

98.416253

3.000000

-1.000000

22.735946

401.613175

424.349121

-0.002357

1

1.000210

0.999879

103.366790

98.071263

4.003843

0.003843

25.234640

402.856257

428.090898

0.000009

2

1.000210

0.999880

103.362924

98.072408

4.000002

0.000002

25.223778

402.850854

428.074632

0.000000

3

1.000210

0.999880

103.362922

98.072409

4.000000

0.000000

25.223774

402.850852

428.074626

0.000000

0

1.000210

0.999880

103.362922

98.072409

4.000000

-1.000000

25.223774

402.850852

428.074626

-0.002336

1

1.000276

0.999864

104.412547

97.817788

5.004947

0.004947

28.397448

404.429777

432.827225

0.000011

2

1.000275

0.999864

104.407200

97.818822

5.000002

0.000002

28.380224

404.421209

432.801433

0.000000

3

1.000275

0.999864

104.407197

97.818822

5.000000

0.000000

28.380216

404.421205

432.801421

0.000000

0

1.000275

0.999864

104.407197

97.818822

5.000000

-1.000000

28.380216

404.421205

432.801421

-0.002311

1

1.000346

0.999853

105.530642

97.652463

6.005960

0.005960

32.200535

406.321929

438.522464

0.000014

2

1.000345

0.999853

105.523785

97.653192

6.000003

0.000003

32.176072

406.309759

438.485831

0.000000

3

1.000345

0.999853

105.523781

97.653192

6.000000

0.000000

32.176059

406.309752

438.485811

0.000000

0

1.000345

0.999853

105.523781

97.653192

6.000000

-1.000000

32.176059

406.309752

438.485811

-0.002281

1

1.000420

0.999848

106.715716

97.572278

7.006870

0.006870

36.610275

408.516042

445.126317

0.000015

2

1.000419

0.999848

106.707372

97.572546

7.000004

0.000004

36.578039

408.500003

445.078042

0.000000

3

1.000419

0.999848

106.707367

97.572546

7.000000

0.000000

36.578020

408.499993

445.078013

0.000000

0

1.000419

0.999848

106.707367

97.572546

7.000000

-1.000000

36.578020

408.499993

445.078013

-0.002247

1

1.000498

0.999848

107.962228

97.573675

8.007671

0.007671

41.590102

410.993983

452.584085

0.000017

2

1.000497

0.999848

107.952461

97.573364

8.000005

0.000005

41.549888

410.973974

452.523861

0.000000

3

1.000497

0.999848

107.952455

97.573364

8.000000

0.000000

41.549860

410.973960

452.523820

0.000000

0

1.000497

0.999848

107.952455

97.573364

8.000000

-1.000000

41.549860

410.973960

452.523820

-0.002210

1

1.000579

0.999853

109.264586

97.652686

9.008359

0.008359

47.101607

413.736708

460.838314

0.000018

2

1.000578

0.999853

109.253499

97.651717

9.000006

0.000006

47.053505

413.712771

460.766276

0.000000

3

1.000578

0.999853

109.253490

97.651716

9.000000

0.000000

47.053468

413.712753

460.766221

0.000000

0

1.000578

0.999853

109.253490

97.651716

9.000000

-1.000000

47.053468

413.712753

460.766221

-0.002170

1

1.000664

0.999863

110.617272

97.805059

10.008938

0.008938

53.105543

416.724752

469.830295

0.000019

2

1.000663

0.999863

110.604993

97.803387

10.000008

0.000008

53.049893

416.697057

469.746950

0.000000

3

1.000663

0.999863

110.604983

97.803386

10.000000

0.000000

53.049845

416.697033

469.746878

0.000000

Step

Iteration 0

1.000000

1

1

1.000044

2

1.000044

0 2

3

4

5

6

7

8

9

10

L1 (m)

L2 (m)

f1 (kN)

f2 (kN)

1.000000

100.000000

0.999961

100.711117

0.999961

1.000044

1

As can be seen from Table A.1, L1 and f1 continuously increase to 1.000663 m and 110.604983 kN, respectively; however, from the step 1 to the step 7, L2 and f2 decrease, 220 

 

while from the step 8 to the step 10, they increase, which is because the sign of the slope of tension cables was changed during the loading; finally, L2 and f2 vary to 0.999863 m and 97.803386 kN, respectively.

eventually increase to 10 kN, which is equal to the

external load P, and the unbalanced force Fun is also reduced to 0. From the step 1 to the step 10, both KE and KG increase continuously, but KE increases faster than KG, which indicates that the elastic stiffness will take an increasingly large proportion in the total structural stiffness. Moreover, through solving Equations A.8 and A.10, the final displacement S can be obtained, which is equal to -0.022933 m. The cable forces and the final displacement are usually the most concerned results. Therefore, f1, f2 and S from Excel, SOFiSTiK and ABAQUS are compared in Table A.2. The error is the relative error of the Excel result to the SOFiSTiK result (error 1) or to the ABAQUS result (error 2). Table A.2: Comparison of results from Excel, SOFiSTiK and ABAQUS ( = 89°) Excel result

SOFiSTiK result

Error 1

ABAQUS result

Error 2

f1

110.604983 kN

110.666557 kN

-0.06%

110.612097 kN

-0.006%

f2

97.803386 kN

97.788719 kN

0.01%

97.804388 kN

-0.001%

S

-0.022933

-0.022922

0.05%

-0.022945

-0.05%

As can be seen from the above table, the errors between the excel results and the SOFiSTiK results or between the excel results and the ABAQUS results are very small. This proves the correctness of the solution in Excel. A.2.2 Case B: initial control angle  = 85° The investigated cable structure is shown in Figure A.3.

221 

 

Figure A.3: Double-curved cable net with control angle  = 85° The solution process and results are presented in Table A.3. The meanings of the symbols in this table are the same as those in Table A.1.

222 

 

Table A.3: Solution process and results in MS Excel ( = 85°) Step

1

2

3

4

5

6

7

8

9

10

(kN)

Fun (kN)

KE (kN/m)

KG (kN/m)

K (kN/m)

△Si (m)

100.000000

0.000000

-1.000000

486.151922

400.000000

886.151922

-0.001128

98.436539

0.996616

-0.003384

486.228835

400.039883

886.268718

-0.000004

101.589230

98.431282

0.999988

-0.000012

486.229357

400.040149

886.269506

0.000000

0.999902

101.589248

98.431264

1.000000

0.000000

486.229359

400.040150

886.269508

0.000000

1.000099

0.999902

101.589248

98.431264

1.000000

-1.000000

486.229359

400.040150

886.269508

-0.001128

1

1.000200

0.999806

103.192996

96.888148

1.996745

-0.003255

486.460517

400.158804

886.619322

-0.000004

2

1.000200

0.999805

103.198247

96.883159

1.999989

-0.000011

486.461520

400.159317

886.620837

0.000000

3

1.000200

0.999805

103.198265

96.883142

2.000000

0.000000

486.461524

400.159318

886.620842

0.000000

0

1.000200

0.999805

103.198265

96.883142

2.000000

-1.000000

486.461524

400.159318

886.620842

-0.001128

1

1.000301

0.999710

104.821471

95.360771

2.996870

-0.003130

486.846740

400.356639

887.203380

-0.000004

2

1.000302

0.999710

104.826579

95.356041

2.999989

-0.000011

486.848186

400.357378

887.205564

0.000000

3

1.000302

0.999710

104.826596

95.356024

3.000000

0.000000

486.848191

400.357380

887.205571

0.000000

0

1.000302

0.999710

104.826596

95.356024

3.000000

-1.000000

486.848191

400.357380

887.205571

-0.001127

1

1.000404

0.999616

106.468788

93.854776

3.996992

-0.003008

487.387092

400.633179

888.020271

-0.000003

2

1.000405

0.999616

106.473753

93.850294

3.999990

-0.000010

487.388943

400.634124

888.023067

0.000000

3

1.000405

0.999616

106.473770

93.850278

4.000000

0.000000

487.388949

400.634127

888.023076

0.000000

0

1.000405

0.999616

106.473770

93.850278

4.000000

-1.000000

487.388949

400.634127

888.023077

-0.001126

1

1.000508

0.999523

108.134456

92.370505

4.997109

-0.002891

488.081000

400.988131

889.069131

-0.000003

2

1.000509

0.999523

108.139280

92.366260

4.999990

-0.000010

488.083220

400.989265

889.072485

0.000000

3

1.000509

0.999523

108.139297

92.366245

5.000000

0.000000

488.083228

400.989269

889.072496

0.000000

0

1.000509

0.999523

108.139297

92.366245

5.000000

-1.000000

488.083228

400.989269

889.072496

-0.001125

1

1.000614

0.999432

109.817969

90.908276

5.997221

-0.002779

488.927731

401.421125

890.348856

-0.000003

2

1.000614

0.999432

109.822654

90.904258

5.999990

-0.000010

488.930287

401.422430

890.352717

0.000000

3

1.000614

0.999432

109.822671

90.904244

6.000000

0.000000

488.930296

401.422435

890.352731

0.000000

0

1.000614

0.999432

109.822671

90.904244

6.000000

-1.000000

488.930296

401.422435

890.352731

-0.001123

1

1.000720

0.999342

111.518806

89.468383

6.997329

-0.002671

489.926399

401.931711

891.858109

-0.000003

2

1.000720

0.999342

111.523355

89.464580

6.999991

-0.000009

489.929259

401.933171

891.862430

0.000000

3

1.000720

0.999342

111.523371

89.464566

7.000000

0.000000

489.929269

401.933176

891.862445

0.000000

0

1.000720

0.999342

111.523371

89.464566

7.000000

-1.000000

489.929269

401.933176

891.862445

-0.001121

1

1.000828

0.999253

113.236432

88.051090

7.997431

-0.002569

491.075964

402.519361

893.595325

-0.000003

2

1.000828

0.999253

113.240849

88.047492

7.999991

-0.000009

491.079099

402.520962

893.600061

0.000000

3

1.000828

0.999253

113.240864

88.047479

8.000000

0.000000

491.079110

402.520968

893.600078

0.000000

0

1.000828

0.999253

113.240864

88.047479

8.000000

-1.000000

491.079110

402.520968

893.600078

-0.001119

1

1.000936

0.999166

114.970302

86.656640

8.997529

-0.002471

492.375243

403.183474

895.558717

-0.000003

2

1.000936

0.999166

114.974591

86.653234

8.999991

-0.000009

492.378625

403.185202

895.563827

0.000000

3

1.000936

0.999166

114.974606

86.653222

9.000000

0.000000

492.378638

403.185208

895.563846

0.000000

0

1.000936

0.999166

114.974606

86.653222

9.000000

-1.000000

492.378638

403.185208

895.563846

-0.001117

1

1.001045

0.999081

116.719861

85.285245

9.997621

-0.002379

493.822907

403.923378

897.746284

-0.000003

2

1.001046

0.999080

116.724026

85.282022

9.999991

-0.000009

493.826513

403.925219

897.751733

0.000000

3

1.001046

0.999080

116.724041

85.282010

10.000000

0.000000

493.826526

403.925226

897.751752

0.000000

Iteration

L1 (m)

L2 (m)

f1 (kN)

f2 (kN)

0

1.000000

1.000000

100.000000

1

1.000099

0.999902

101.583836

2

1.000099

0.999902

3

1.000099

0

As can be seen from Table A.3, L1 and f1 increase to 1.001046 m and 116.724041 kN, respectively; however, L2 and f2 decrease to 0.999080 m and 85.282010 kN, respectively. eventually increase to 10 kN, which is equal to the external load P, and the unbalanced force Fun is also reduced to 0. Both KE and KG increase from the step 1 to the step 10, but KE increases faster than KG, which indicates that the elastic stiffness will take an increasingly large proportion in the total structural stiffness. Moreover, through solving Equations A.8 and A.10, the final displacement S can be obtained, which is equal to 0.011274 m.

223 

  The same as in Section A.2.1, f1, f2 and S from Excel, SOFiSTiK and ABAQUS are compared in Table A.4. The error is also the relative error of the Excel result to the SOFiSTiK result (error 1) or to the ABAQUS result (error 2). Table A.4: Comparison of results from Excel, SOFiSTiK and ABAQUS ( = 85°) Excel result

SOFiSTiK result

Error 1

ABAQUS result

Error 2

f1

116.724041 kN

116.773803 kN

-0.04%

116.785876 kN

-0.05%

f2

85.282010 kN

85.243240 kN

0.05%

85.231482 kN

0.06%

S

-0.011274

-0.011236

0.3%

-0.0113154

-0.4%

As can be seen from the above table, the errors between the excel results and the SOFiSTiK results or between the excel results and the ABAQUS results are very small. This proves the correctness of the solution in Excel. A.2.3 Case C: initial control angle  = 81° The investigated cable structure is shown in Figure A.4.

Figure A.4: Double-curved cable net with control angle  = 81° The solution process and results are presented in Table A.5. The meanings of the symbols in this table are the same as those in Table A.1.

224 

 

Table A.5: Solution process and results in MS Excel ( = 81°) Step

1

2

3

4

5

6

7

8

9

10

L1 (m)

L2 (m)

f1 (kN)

f2 (kN)

0

1.000000

1.000000

100.000000

100.000000

0.000000

-1.000000

1566.191511

400.000000

1966.191511

-0.000509

1

1.000080

0.999921

101.275064

98.729075

0.995025

-0.004975

1566.205137

400.007824

1966.212961

-0.000003

(kN)

Fun (kN)

KE (kN/m)

KG (kN/m)

K (kN/m)

△Si (m)

Iteration

2

1.000080

0.999920

101.281416

98.722762

0.999975

-0.000025

1566.205272

400.007898

1966.213171

0.000000

3

1.000080

0.999920

101.281448

98.722730

1.000000

0.000000

1566.205273

400.007899

1966.213172

0.000000

0

1.000080

0.999920

101.281448

98.722730

1.000000

-1.000000

1566.205273

400.007899

1966.213172

-0.000509

1

1.000160

0.999841

102.560453

97.455776

1.995036

-0.004964

1566.246181

400.030636

1966.276817

-0.000003

2

1.000160

0.999841

102.566811

97.449495

1.999975

-0.000025

1566.246452

400.030784

1966.277235

0.000000

3

1.000160

0.999841

102.566843

97.449464

2.000000

0.000000

1566.246453

400.030784

1966.277237

0.000000

0

1.000160

0.999841

102.566843

97.449464

2.000000

-1.000000

1566.246453

400.030784

1966.277237

-0.000509

1

1.000241

0.999762

103.849758

96.186510

2.995045

-0.004955

1566.314675

400.068432

1966.383107

-0.000003

2

1.000241

0.999761

103.856124

96.180261

2.999975

-0.000025

1566.315080

400.068653

1966.383734

0.000000

3

1.000241

0.999761

103.856156

96.180230

3.000000

0.000000

1566.315083

400.068654

1966.383737

0.000000

0

1.000241

0.999761

103.856156

96.180230

3.000000

-1.000000

1566.315083

400.068654

1966.383737

-0.000509

1

1.000321

0.999683

105.142952

94.921304

3.995053

-0.004947

1566.410613

400.121210

1966.531822

-0.000003

2

1.000322

0.999682

105.149326

94.915087

3.999975

-0.000025

1566.411153

400.121505

1966.532657

0.000000

3

1.000322

0.999682

105.149358

94.915056

4.000000

0.000000

1566.411155

400.121506

1966.532661

0.000000

0

1.000322

0.999682

105.149358

94.915055

4.000000

-1.000000

1566.411155

400.121506

1966.532661

-0.000509

1

1.000403

0.999604

106.440004

93.660187

4.995061

-0.004939

1566.533986

400.188965

1966.722950

-0.000003

2

1.000403

0.999603

106.446387

93.653998

4.999975

-0.000025

1566.534660

400.189333

1966.723993

0.000000

3

1.000403

0.999603

106.446419

93.653967

5.000000

0.000000

1566.534663

400.189335

1966.723998

0.000000

0

1.000403

0.999603

106.446419

93.653967

5.000000

-1.000000

1566.534663

400.189335

1966.723998

-0.000508

1

1.000484

0.999525

107.740884

92.403183

5.995067

-0.004933

1566.684784

400.271691

1966.956475

-0.000003

2

1.000484

0.999525

107.747278

92.397023

5.999975

-0.000025

1566.685591

400.272132

1966.957724

0.000000

3

1.000484

0.999525

107.747310

92.396992

6.000000

0.000000

1566.685595

400.272134

1966.957730

0.000000

0

1.000484

0.999525

107.747310

92.396992

6.000000

-1.000000

1566.685595

400.272134

1966.957730

-0.000508

1

1.000565

0.999447

109.045563

91.150321

6.995072

-0.004928

1566.862993

400.369382

1967.232375

-0.000003

2

1.000566

0.999447

109.051969

91.144188

6.999975

-0.000025

1566.863934

400.369896

1967.233830

0.000000

3

1.000566

0.999447

109.052001

91.144157

7.000000

0.000000

1566.863939

400.369898

1967.233837

0.000000

0

1.000566

0.999447

109.052001

91.144157

7.000000

-1.000000

1566.863939

400.369898

1967.233837

-0.000508

1

1.000647

0.999369

110.354010

89.901625

7.995077

-0.004923

1567.068599

400.482029

1967.550627

-0.000003

2

1.000648

0.999369

110.360428

89.895518

7.999975

-0.000025

1567.069673

400.482615

1967.552288

0.000000

3

1.000648

0.999369

110.360461

89.895487

8.000000

0.000000

1567.069678

400.482618

1967.552297

0.000000

0

1.000648

0.999369

110.360461

89.895487

8.000000

-1.000000

1567.069679

400.482618

1967.552297

-0.000508

1

1.000729

0.999291

111.666195

88.657121

8.995080

-0.004920

1567.301582

400.609622

1967.911204

-0.000003

2

1.000730

0.999291

111.672627

88.651039

8.999975

-0.000025

1567.302790

400.610281

1967.913071

0.000000

3

1.000730

0.999291

111.672659

88.651008

9.000000

0.000000

1567.302796

400.610284

1967.913081

0.000000

0

1.000730

0.999291

111.672659

88.651008

9.000000

-1.000000

1567.302796

400.610284

1967.913081

-0.000508

1

1.000812

0.999214

112.982086

87.416835

9.995083

-0.004917

1567.561925

400.752150

1968.314075

-0.000002

2

1.000812

0.999213

112.988533

87.410776

9.999975

-0.000025

1567.563266

400.752882

1968.316148

0.000000

3

1.000812

0.999213

112.988565

87.410746

10.000000

0.000000

1567.563272

400.752886

1968.316158

0.000000

As can be seen from Table A.5, L1 and f1 increase to 1.000812 m and 112.988565 kN, respectively; however, L2 and f2 decrease to 0.999213 m and 87.410746 kN, respectively. eventually increase to 10 kN, which is equal to the external load P, and the unbalanced force Fun is also reduced to 0. Both KE and KG increase from the step 1 to the step 10, but KE increases faster than KG, which indicates that the elastic stiffness will take an increasingly large proportion in the total structural stiffness. Moreover, through solving Equations A.8 and A.10, the final displacement S can be obtained, which is equal to 0.005110 m.

225 

 

The same as in Section A.2.1, f1, f2 and S from Excel, SOFiSTiK and ABAQUS are compared in Table A.4. The error is also the relative error of the Excel result to the SOFiSTiK result (error 1) or to the ABAQUS result (error 2). Table A.4: Comparison of results from Excel, SOFiSTiK and ABAQUS ( = 81°) Excel result

SOFiSTiK result

Error 1

ABAQUS result

Error 2

f1

112.988565 kN

113.007667 kN

-0.02%

113.054382 kN

-0.06%

f2

87.410746 kN

87.398307 kN

0.01%

87.346942 kN

0.07%

S

-0.005110 m

-0.005084 m

0.5%

-0.005136 m

-0.5%

As can be seen from the above table, the errors between the excel results and the SOFiSTiK results or between the excel results and the ABAQUS results are very small. This proves the correctness of the solution in Excel.

226 

 

Appendix B – FORTRAN Source Code: ABAQUS UMAT Subroutine  

C UMAT FOR COMPOSITE MATERIAL C STRESS CALCULATION AND FAILURE NOTIFICATION C IT CAN ONLY BE USED IN 3D PROBLEM SUBROUTINE UMAT(STRESS,STATEV,DDSDDE,SSE,SPD,SCD,RPL,DDSDDT, 1 DRPLDE,DRPLDT,STRAN,DSTRAN,TIME,DTIME,TEMP,DTEMP,PREDEF,DPRED, 2 CMNAME,NDI,NSHR,NTENS,NSTATV,PROPS,NPROPS,COORDS,DROT, 3 PNEWDT,CELENT,DFGRD0,DFGRD1,NOEL,NPT,LAYER,KSPT,KSTEP,KINC) C INCLUDE 'aba_param.inc' C CHARACTER*80 CMNAME C DIMENSION STRESS(NTENS),STATEV(NSTATV),DDSDDE(NTENS,NTENS), 1 DDSDDT(NTENS),DRPLDE(NTENS),STRAN(NTENS),DSTRAN(NTENS), 2 TIME(2),PREDEF(1),DPRED(1),PROPS(NPROPS),COORDS(3),DROT(3,3), 3 DFGRD0(3,3),DFGRD1(3,3) DIMENSION CFULL(6,6),STRANT(6) C ---------------------------------------------------------------C PROPS(1) - E1(EL) C PROPS(2) - E2(ET) C PROPS(3) - P12(PLT) C PROPS(4) - P23(PTT) C PROPS(5) - G12(GLT) C PROPS(6) - G23(GTT) C ---------------------------------------------------------------C C ELASTIC PROPERTIES EL=PROPS(1) ET=PROPS(2) PLT=PROPS(3) PTT=PROPS(4)

227 

 

GLT=PROPS(5) GTT=PROPS(6) PTL=PLT/EL*ET C C FAILURE PROPERTIES STRESS FSTL=PROPS(7) FSCL=PROPS(8) FSTT=PROPS(9) FSCT=PROPS(10) FSHLT=PROPS(11) FSHTT=PROPS(12) C C CALCULATE THE TOTAL STRAIN DO I=1, NTENS STRANT(I)=STRAN(I)+DSTRAN(I) END DO C C FORM THE STIFFNESS MATRIX DO I = 1, 6 DO J = 1, 6 CFULL(I,J)=0 END DO END DO ES=1-2*PLT*PTL-PTT**2-2*PLT*PTL*PTT CFULL(1,1)=EL*(1-PTT**2)/ES CFULL(2,2)=ET*(1-PLT*PTL)/ES CFULL(3,3)=CFULL(2,2) CFULL(1,2)=ET*(PLT+PLT*PTT)/ES CFULL(1,3)=CFULL(1,2) CFULL(2,3)=ET*(PTT+PLT*PTL)/ES CFULL(4,4)=GLT CFULL(5,5)=GLT CFULL(6,6)=GTT DO I = 2, 6 DO J = 1, I-1

228 

 

CFULL(I,J)=CFULL(J,I) END DO END DO C FAILURE PROPERTIES STRAIN FETL=FSTL/CFULL(1,1) FECL=FSCL/CFULL(1,1) FETT=FSTT/CFULL(2,2) FECT=FSCT/CFULL(2,2) FEHLT=FSHLT/CFULL(4,4) FEHTT=FSHTT/CFULL(6,6) C C THE INITIAL FAILURE CONDITION C

XMSOLD=STATEV(1)

C C CALCULATE STRESS DO I=1, NTENS STRESS(I)=0 DO J=1, NTENS STRESS(I)=STRESS(I)+CFULL(I,J)*STRANT(J) END DO END DO C C CHECK THE FAILURE CONDITION (MAXIMUM STRESS) IF (STRESS(1).GE.0) THEN X1=FSTL ELSEIF (STRESS(1).LT.0) THEN X1=FSCL ENDIF IF (STRESS(2).GE.0) THEN X2=FSTT ELSEIF (STRESS(2).LT.0) THEN X2=FSCT ENDIF IF (STRESS(3).GE.0) THEN X3=FSTT

229 

 

ELSEIF (STRESS(3).LT.0) THEN X3=FSCT ENDIF S12=FSHLT S13=FSHLT S23=FSHTT TERM1=STRESS(1)/X1 TERM2=STRESS(2)/X2 TERM3=STRESS(3)/X3 TERM4=ABS(STRESS(4)/S12) TERM5=ABS(STRESS(5)/S13) TERM6=ABS(STRESS(6)/S23) XMS=MAX(TERM1,TERM2,TERM3,TERM4,TERM5,TERM6) STATEV(1)=XMS C C CHECK THE FAILURE CONDITION (MAXIMUM STRAIN) IF (STRESS(1).GE.0) THEN EX1=FETL ELSEIF (STRESS(1).LT.0) THEN EX1=FECL ENDIF IF (STRESS(2).GE.0) THEN EX2=FETT ELSEIF (STRESS(2).LT.0) THEN EX2=FECT ENDIF IF (STRESS(3).GE.0) THEN EX3=FETT ELSEIF (STRESS(3).LT.0) THEN EX3=FECT ENDIF ES12=FEHLT ES13=FEHLT ES23=FEHTT ETERM1=STRANT(1)/EX1

230 

 

ETERM2=STRANT(2)/EX2 ETERM3=STRANT(3)/EX3 ETERM4=ABS(STRANT(4)/ES12) ETERM5=ABS(STRANT(5)/ES13) ETERM6=ABS(STRANT(6)/ES23) EXMS=MAX(ETERM1,ETERM2,ETERM3,ETERM4,ETERM5,ETERM6) C UPDATE THE STATE VARIABLE STATEV(2)=EXMS C C CHECK THE FAILURE CONDITION (TSAI-WU STRESS) XT=FSTL XC=FSCL YT=FSTT YC=FSCT SLT=FSHLT STT=FSHTT F1=1/XT+1/XC F2=1/YT+1/YC F11=-1/(XT*XC) F22=-1/(YT*YC) F12=-1/(2*SQRT(XT*XC*YT*YC)) F23=-1/(2*SQRT(YT*YC*YT*YC)) F44=1/STT**2 F66=1/SLT**2 FIELDA=F11*STRESS(1)**2+F22*(STRESS(2)**2+STRESS(3)**2) 2 +2*F12*(STRESS(1)*STRESS(2)+STRESS(1)*STRESS(3)) 3 +2*F23*STRESS(2)*STRESS(3)+F44*STRESS(6)**2 4 +F66*(STRESS(4)**2+STRESS(5)**2) FIELDB=F1*STRESS(1)+F2*(STRESS(2)+STRESS(3)) TSW=1/(-FIELDB/2/FIELDA+SQRT((FIELDB/2/FIELDA)**2+1/FIELDA)) C C UPDATE THE STATE VARIABLE STATEV(3)=TSW C C CHECK THE FAILURE CONDITION (TSAI-WU STRAIN)

231 

 

EXT=FETL EXC=FECL EYT=FETT EYC=FECT ESLT=FEHLT ESTT=FEHTT EF1=1/EXT+1/EXC EF2=1/EYT+1/EYC EF11=-1/(EXT*EXC) EF22=-1/(EYT*EYC) EF12=-1/(2*SQRT(EXT*EXC*EYT*EYC)) EF23=-1/(2*SQRT(EYT*EYC*EYT*EYC)) EF44=1/ESLT**2 EF66=1/ESTT**2 EIELDA=EF11*STRANT(1)**2+EF22*(STRANT(2)**2+STRANT(3)**2) 2 +2*EF12*(STRANT(1)*STRANT(2)+STRANT(1)*STRANT(3)) 3 +2*EF23*STRANT(2)*STRANT(3)+EF44*(STRANT(4)**2+STRANT(5)**2) 4 +EF66*STRANT(6)**2 EIELDB=EF1*STRANT(1)+EF2*(STRANT(2)+STRANT(3)) ETSW=1/(-EIELDB/2/EIELDA+SQRT((EIELDB/2/EIELDA)**2+1/EIELDA)) C C UPDATE THE STATE VARIABLE STATEV(4)=ETSW C C CHECK THE FAILURE CONDITION (Hashin Stress FEA K) DFT1=0 DFC1=0 DMT1=0 DMC1=0 IF (STRESS(1) .GE. 0) THEN DFT1=SQRT(ABS((STRESS(1)/XT)**2+ 1

(STRESS(4)**2+STRESS(5)**2)/SLT**2))

ELSEIF (STRESS(1) .LT. 0) THEN DFC1=SQRT(ABS((STRESS(1)/XT)**2)) ENDIF

232 

 

IF ((STRESS(2)+STRESS(3)) .GE. 0) THEN DMT1=SQRT(ABS((STRESS(2)+STRESS(3))**2/YT**2+ 1

(STRESS(6)**2-STRESS(2)*STRESS(3))/STT**2+

2

(STRESS(4)**2+STRESS(5)**2)/SLT**2))

ELSEIF ((STRESS(2)+STRESS(3)) .LT. 0) THEN DMC1=SQRT(ABS(((YC/2/STT)**2-1)*((STRESS(2)+STRESS(3))/YC)+ 1

(STRESS(2)+STRESS(3))**2/4/STT**2+

2

(STRESS(6)**2-STRESS(2)*STRESS(3))/STT**2+

3

(STRESS(4)**2+STRESS(5)**2)/SLT**2))

ENDIF C C UPDATE THE STATE VARIABLE STATEV(5)=DFT1 STATEV(6)=DFC1 STATEV(7)=DMT1 STATEV(8)=DMC1 C C CHECK THE FAILURE CONDITION (Hashin Stress k) DFT2=0 DFC2=0 DMT2=0 DMC2=0 IF (STRESS(1) .GE. 0) THEN DFT2=SQRT((STRESS(1)/XT)**2+ 1

(STRESS(4)**2+STRESS(5)**2)/SLT**2)

ELSEIF (STRESS(1) .LT. 0) THEN DFC2=SQRT((STRESS(1)/XT)**2) ENDIF IF ((STRESS(2)+STRESS(3)) .GE. 0) THEN DMT2=SQRT((STRESS(2)+STRESS(3))**2/YT**2+ 1

(STRESS(6)**2-STRESS(2)*STRESS(3))/STT**2+

2

(STRESS(4)**2+STRESS(5)**2)/SLT**2)

ELSEIF ((STRESS(2)+STRESS(3)) .LT. 0) THEN DMCA=(STRESS(2)+STRESS(3))**2/4/STT**2+ 1

(STRESS(6)**2-STRESS(2)*STRESS(3))/STT**2+

233 

 

2

(STRESS(4)**2+STRESS(5)**2)/SLT**2 DMCB=((YC/2/STT)**2-1)*((STRESS(2)+STRESS(3))/YC) DMC2=1/((-DMCB+SQRT(DMCB**2+4*DMCA))/2/DMCA)

ENDIF C C UPDATE THE STATE VARIABLE STATEV(9)=DFT2 STATEV(10)=DFC2 STATEV(11)=DMT2 STATEV(12)=DMC2 C C CALCULATE JACOBIAN MATRIX DO I=1, NTENS DO J=1, NTENS DDSDDE(J,I)=CFULL(J,I) END DO END DO C RETURN END  

234 

With the advantages of high strength, lightweight, no corrosion and excellent fatigue resistance, Carbon Fibre Reinforced Polymer (CFRP) cables have the potential to replace steel cables in a broad range of applications. The ideal structures for such cables are highly pre-tensioned cable systems that are loaded orthogonally to their cable axes. This type of structures with CFRP cables, such as cable net facades, spoked wheel cable roofs and stressed-ribbon bridges, can be built economically with large or small spans. This book is the first in the world to demonstrate the advantages of using CFRP cables in orthogonally loaded cable structures, including detailed analyses of mechanical properties and economic efficiencies. Furthermore, in order to solve the anchorage problem which hinders the application of CFRP cables, two new CFRP cable anchorages, especially suitable for orthogonally loaded cable structures, are proposed in this book. In addition, a prototype of CFRP spoked wheel cable roof built by the author is presented to show the feasibility of CFRP orthogonally loaded cable structures based on the present technology; a novel design, i.e. the CFRP Continuous Band Winding System, is also conceptually introduced, so as to show a feasible form of CFRP orthogonally loaded cable structures in the future. This book is written to encourage the use of CFRP cables and show that CFRP cable structures are feasible and have advantages over steel cable structures. It will be read by researchers of structural engineering and by consulting engineers.

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ISBN 978-3-8325-4128-6