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SE-100 44 STOCKHOLM

DEPARTMENT OF STRUCTURAL ENGINEERING

ROYAL INSTITUTE OF TECHNOLOGY

Numerical Analyses of Cable Roof Structures Gunnar Tibert

KUNGL TEKNISKA HÖGSKOLAN INSTITUTIONEN FÖR BYGGKONSTRUKTION TRITA-BKN. Bulletin 46, 1999 ISSN 1103-4270 ISRN KTH/BKN/EX--46--SE Licentiate Thesis

Numerical Analyses of Cable Roof Structures

Gunnar Tibert

Department of Structural Engineering Royal Institute of Technology SE-100 44 Stockholm, Sweden

TRITA-BKN. Bulletin 46, 1999 ISSN 1103-4270 ISRN KTH/BKN/B--46--SE Licentiate Thesis

c Gunnar Tibert 1999 KTH, TS–H¨ogskoletryckeriet, Stockholm 1999

Abstract This thesis deals with the techniques used in the numerical analysis of cable roof structures. These structures are usually very light and ﬂexible and require analysis methods, which take their non-linear behaviour into account. An extensive literature survey, concerned with both practical and theoretical aspects of cable roofs, is presented. Some aspects included are: structural systems, roof erection procedures, diﬀerent cable types and their properties, structural details, roof loads and analysis methods. As the initial shape of a cable roof depends on the internal force distribution, it cannot be described by simple geometrical models. Special iterative methods, usually not familiar to the structural engineer, have to be utilised in order to ﬁnd the pretensioned conﬁguration of the roof. The simple force density method is presented in detail and applied to a number of diﬀerent types of cable roof structures. The method worked well for structures composed of only cables, but not for structures with compression members. Three analytical ﬁnite cable elements are presented. Two elements are mathematically exact and can accurately model both taut and slack cables using only one element per cable. It is shown that the analytical elements are advantageous in modelling cable behaviour. A static analysis of the Scandinavium Arena in Gothenburg has been performed. The results from this analysis were compared with results from the original design of the same object. It was found that the bending moments in the supporting structure—the concrete ring beam—were very sensitive to its shape. This explained the large discrepancy in the bending moment distribution between the analyses. Results from a simpliﬁed method, used for preliminary calculations, agreed well with those of the more accurate ﬁnite element calculations, for a studied symmetric load case. Failure stage analysis of the class of self-stressed cable structures called tensegrity structures has been identiﬁed as an area of further research.

Keywords: cable roof structures, loads, form-ﬁnding, force density method, ﬁnite cable elements, static analysis, the Scandinavium Arena.

iii

Preface The research work in this thesis was carried out at the Department of Structural Engineering, Structural Mechanics Group, at the Royal Institute of Technology in Stockholm, under the supervision of Professor Anders Eriksson. The work reported in this thesis was ďŹ nanced through a personal grant from KTH. First of all, I express my gratitude to my supervisor Professor Anders Eriksson for his scientiďŹ c guidance and valuable advice. I also thank Docent Costin Pacoste for help with the selection of a suitable beam element for the static analyses. I would also like to thank Professor Emeritus Alf Samuelsson at Chalmers University of Technology in Gothenburg and Mr. Nils Dahlstedt, Technical Manager at the Scandinavium Arena in Gothenburg, for the valuable information about the Scandinavium Arena. Finally, I am grateful to all people at the Department of Structural Engineering that have helped me in the work with this thesis.

Stockholm, April 1999 Gunnar Tibert

v

Contents Abstract

iii

Preface

v

List of symbols

xi

1 Introduction

1

1.1

Aims and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

General structure of thesis . . . . . . . . . . . . . . . . . . . . . . . .

3

2 Literature review

5

2.1

Historical review . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.2

Structural systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2.1

Simply suspended cable structures

. . . . . . . . . . . . . . . 10

2.2.2

Pretensioned cable trusses . . . . . . . . . . . . . . . . . . . . 10

2.2.3

Pretensioned cable net structures . . . . . . . . . . . . . . . . 12

2.2.4

Tensegrity systems . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3

Roof erection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4

Cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5

2.4.1

Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.2

Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.3

Axial stiďŹ&#x20AC;ness . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.4

Corrosion protection . . . . . . . . . . . . . . . . . . . . . . . 20

Cladding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5.1

Fabrics and foils . . . . . . . . . . . . . . . . . . . . . . . . . . 21

vii

2.6

2.7

2.8

2.5.2

Metal sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5.3

Panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Structural details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6.1

End ďŹ ttings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6.2

Intermediate ďŹ ttings . . . . . . . . . . . . . . . . . . . . . . . 25

2.6.3

Saddles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6.4

Anchorages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Roof loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.7.1

Wind load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7.2

Snow load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.7.3

Earthquake load

2.7.4

Other loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

. . . . . . . . . . . . . . . . . . . . . . . . . 37

Analysis methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 The initial equilibrium problem 3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1.1

3.2

3.3

3.4

41

Physical modelling . . . . . . . . . . . . . . . . . . . . . . . . 42

Literature review of initial equilibrium solution methods . . . . . . . 42 3.2.1

The non-linear displacement method . . . . . . . . . . . . . . 44

3.2.2

The grid method . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.3

The force density method . . . . . . . . . . . . . . . . . . . . 49

3.2.4

Least squares stress determination methods . . . . . . . . . . 51

3.2.5

A combined approach . . . . . . . . . . . . . . . . . . . . . . . 53

3.2.6

Initial equilibrium of tensegrity structures . . . . . . . . . . . 53

The force density method . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3.1

The linear force density method . . . . . . . . . . . . . . . . . 56

3.3.2

The non-linear force density method . . . . . . . . . . . . . . 61

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4.1

Smaller cable nets . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4.2

A large cable net . . . . . . . . . . . . . . . . . . . . . . . . . 73

viii

3.4.3

Cooling towers . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.4.4

A structure composed of both cables and struts . . . . . . . . 79

3.4.5

Cable dome . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.4.6

Tensegrity structures . . . . . . . . . . . . . . . . . . . . . . . 83

3.4.7

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4 Finite cable elements

85

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.2

Analytical cable solutions . . . . . . . . . . . . . . . . . . . . . . . . 85

4.3

4.4

4.5

4.6

4.2.1

The inextensible catenary . . . . . . . . . . . . . . . . . . . . 87

4.2.2

The elastic catenary . . . . . . . . . . . . . . . . . . . . . . . 90

4.2.3

EďŹ&#x20AC;ect of cable bending stiďŹ&#x20AC;ness . . . . . . . . . . . . . . . . . 92

Literature review of cable elements . . . . . . . . . . . . . . . . . . . 95 4.3.1

Elements based on polynomial interpolation functions . . . . . 95

4.3.2

Elements based on analytical functions . . . . . . . . . . . . . 97

Straight and parabolic elements . . . . . . . . . . . . . . . . . . . . . 99 4.4.1

Straight bar element . . . . . . . . . . . . . . . . . . . . . . . 99

4.4.2

Elastic parabolic element . . . . . . . . . . . . . . . . . . . . . 101

Catenary elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.5.1

Elastic catenary element . . . . . . . . . . . . . . . . . . . . . 105

4.5.2

Associate catenary element . . . . . . . . . . . . . . . . . . . . 107

4.5.3

Convergence of solution . . . . . . . . . . . . . . . . . . . . . 111

Comparison of elements . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.6.1

Comparison example 1 . . . . . . . . . . . . . . . . . . . . . . 113

4.6.2

Comparison example 2 . . . . . . . . . . . . . . . . . . . . . . 114

4.6.3

Comparison example 3 . . . . . . . . . . . . . . . . . . . . . . 115

4.6.4

Conclusions from the comparisons . . . . . . . . . . . . . . . . 117

5 Static analysis 5.1

123

Static analysis of the Scandinavium Arena . . . . . . . . . . . . . . . 123

ix

5.2

5.3

5.1.1

The Scandinavium Arena—background . . . . . . . . . . . . . 123

5.1.2

Prestressing forces . . . . . . . . . . . . . . . . . . . . . . . . 126

5.1.3

Finite element model . . . . . . . . . . . . . . . . . . . . . . . 130

5.1.4

Calculation results . . . . . . . . . . . . . . . . . . . . . . . . 135

5.1.5

Calculation results from 1972 . . . . . . . . . . . . . . . . . . 138

5.1.6

Comparison of the results . . . . . . . . . . . . . . . . . . . . 142

Sensitivity of bending moment to the shape of the ring beam . . . . . 142 5.2.1

Description of the structure . . . . . . . . . . . . . . . . . . . 142

5.2.2

Diﬀerent shapes of the ring beam . . . . . . . . . . . . . . . . 144

5.2.3

Results and discussion . . . . . . . . . . . . . . . . . . . . . . 145

Comparison with a simpliﬁed method . . . . . . . . . . . . . . . . . . 154 5.3.1

Results and discussion . . . . . . . . . . . . . . . . . . . . . . 154

6 Conclusions and further research 6.1

6.2

157

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.1.1

The initial equilibrium problem . . . . . . . . . . . . . . . . . 157

6.1.2

Finite cable elements . . . . . . . . . . . . . . . . . . . . . . . 158

6.1.3

Static analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Further research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.2.1

Failure analysis—background . . . . . . . . . . . . . . . . . . 159

6.2.2

Failure analysis—further research . . . . . . . . . . . . . . . . 163

Bibliography

165

A Numerical data for the Scandinavium Arena

175

x

List of symbols The following is a list of the most important symbols that appear in the chapters of the thesis. Symbols not included in this list are deﬁned when they ﬁrst appear. The number refer to the page where the symbol ﬁrst appear. A A0 Ai A B C Cp C Cf Cs d E E0 Ei e F F f H h I Ii K K KE KG L L0 l mi n p q0

cross-sectional area, 45 cross-sectional area of core wire, 19 cross-sectional area of a wire in layer i, 19 equilibrium matrix, 52 compatibility matrix, 54 length of cable chord, 102 pressure coeﬃcient, 30 connectivity matrix for free nodes, 56 connectivity matrix for ﬁxed nodes, 56 connectivity matrix for all nodes, 56 vector of nodal displacements, 54 Young’s modulus, 45 Young’s modulus of core wire, 19 Young’s modulus of wires in layer i, 19 vector of bar elongations, 54 force in global coordinate system, 99 component of cable force in local coordinate system, 87 vector of nodal loads, 52 horizontal component of the cable force T , 88 projection of cable proﬁle on z  -axis, 87 moment of inertia, 87 moment of inertia of wire in layer (around its own centerline), 92 tangent stiﬀness matrix in global coordinate system, 101 tangent stiﬀness matrix in local coordinate system, 102 elastic stiﬀness matrix in local coordinate system, 100 geometric stiﬀness matrix in global coordinate system, 50 length, 43 unstrained length, 45 projection of cable proﬁle on x -axis, 87 number of wires in layer i, 19 number of wire layers, 19 wind pressure at time t, 30 intensity of distributed load on cable, 87

xi

q Ri ri s s0 T Tb T t U u u V v v w w xg x xf y yf z zf αi β ν φ ρ θ θm θv θw

vector of force densities, 58 wire radius in layer i, 19 radius of wire centerline helix in layer i, 19 arc length (elastic cable), 87 arc length (inextensible cable), 88 cable force, 43 cable force at the base (s0 = 0), 91 transformation matrix, 100 vector of bar axial forces, 52 mean wind velocity, 29 turbulence component of the wind ﬁeld in the x-direction, 29 vector of free x-coordinate diﬀerences, 58 total wind velocity, 30 turbulence component of the wind ﬁeld in the y-direction, 29 vector of free y-coordinate diﬀerences, 58 turbulence component of the wind ﬁeld in the z-direction, 29 vector of free z-coordinate diﬀerences, 58 vector of nodal coordinates, 50 vector of free x-coordinates, 58 vector of ﬁxed x-coordinates, 58 vector of free y-coordinates, 58 vector of ﬁxed y-coordinates, 58 vector of free z-coordinates, 58 vector of ﬁxed z-coordinates, 58 angle of wire centerline helix in layer i, 19 angle between the cable chord and horizontal, 102 Poisson’s ratio, 92 angle between x - and x-axis, 103 air density, 30 angle between tangent to cable proﬁle and x -axis, 87 mean wind direction, 30 azimuth angle of turbulent wind component v, 30 elevation angle of turbulent wind component w, 30

xii

Chapter 1 Introduction Tensile architecture represents the new trend in design: construction with the minimum amount of material. As is well-known, the primary advantage of tensile members over compression members is that they can be as light as the tensile strength permits. With new materials, such as high strength steel cables and silicone-coated glass ﬁbre membranes, larger distances can be spanned using the same amount of material as before. Tensile structures have always fascinated architects and engineers, mainly because of the aesthetic shapes they produce. Despite this, very few tensile structures have been built. Why are they not more common, if they are both economic and beautiful? One answers might be that tent-like structures have always been thought of as temporary. Although, a probably more correct answer is that they are more diﬃcult to analyse and construct than traditional buildings. From a structural viewpoint, tension structures have several special features, such as light weight and ﬂexibility. These features require special care in the design; for example, an error in the distribution of the pretensioning forces may lead to damage of the cladding under large loads. If the numerical analysis of building structures is concerned, the ﬁnite element method is the dominating tool. In this method, the structural characteristics and external loads are described by matrices and vectors. The sought parameters, e.g. displacements and internal forces, are found by matrix operations. The ﬁrst step in the analysis process is the deﬁnition of the geometry of the structure, which generally is known a priori. However, this is not the case for tensile structures. Due to the negligible ﬂexural stiﬀness of cables and membranes, the initial conﬁguration of these structures must be stressed, even if the self-weight is disregarded. Thus, before the analysis of the behaviour of the structure to external loads can be performed, the initial equilibrium conﬁguration must be found. The shape of a tensile structure, which very much depends on the internal forces, also governs the load-bearing capacity of the structure. Therefore, the process of determining the initial equilibrium conﬁguration calls for the designer’s ability to ﬁnd an optimum compromise between shape, load capacity and constructional requirements. Several numerical methods, applicable to the initial equilibrium problem,

1

CHAPTER 1. INTRODUCTION

can be found in literature. Most of these methods are not included in general ﬁnite element programs, e.g. ABAQUS 1 and are not familiar to the practising structural engineer. After the initial reference conﬁguration has been determined, the structural members have to be described by stiﬀness matrices and force vectors. Special elements for cables or chains are often not available in commercial ﬁnite element programs. The single cable is instead modelled by one or several other elements depending on the sag-to-span ratio. Nevertheless, this approach has problems such as numerical instability of the solution algorithms. To avoid these problems it is desirable to have at hand a robust element which accurately describes the behaviour of both taut and slack cables. Since cable structures in general are very ﬂexible, a geometrically nonlinear solution method has to be used. The most common is the Newton-Raphson algorithm, embedded in more or less sophisticated load incrementation techniques. The ﬁnal step in the analysis process is to deﬁne the external loads on the structure. For civil engineering structures there are a number of loads that must be considered: self-weight, vehicles, wind, rain, snow, ice, earthquakes, temperature, etc. The magnitude and distribution of these loads is a constant source of research. The present knowledge in the area is found in the national building codes, which aid the engineers in their decisions. Tensile structures often have irregular shapes and low self-weights which may give rise to unforeseen eﬀects such as very high snow loads and ﬂutter instability due to wind. To ensure the safety of the structure, experimental tests have to be undertaken together with statistical analyses to ﬁnd the magnitudes of the snow and wind loads. Even with the right tools, the design of tensile structures will not be straightforward. Each new roof type has its own features. It is no surprise that experience and good engineering judgement are frequent characteristics among famous designers of tensile structures: Fritz Leonhardt, J¨org Schlaich, Frei Otto, Horst Berger and David Geiger, to mention a few.

1.1

Aims and scope

The aim of this work is to study the mechanical aspects of cable supported shell type structures (roofs, cooling towers, etc.). The ﬁrst part of the work is concerned with the basic aspects of these speciﬁc types of structures. These include: the principal arrangements of the cables, pretensioning schemes necessary to obtain a prescribed shape, and the practical aspects of connections between and supports for the cables. Further, basic computational models are to be studied. These include, but are not limited to, the methods for analysis and force distribution. The second part is concerned with the formulation of suitable ﬁnite elements, which take into account the non-linear behaviour of a cable. Basic analyses are performed, and veriﬁed. Theoretical and numerical studies are included in this thesis, but no experimental ABAQUS is a registered trademark of Hibbitt, Karlsson & Sorensen, Inc., 1080 Main Street, Pawtucket, RI 02860-4847, U.S.A. Internet: http://www.abaqus.com. 1

2

1.2. GENERAL STRUCTURE OF THESIS

work is conducted. In the discussed methods, only elastic structures and static loads are considered. All of the numerical calculations in this thesis has been coded in the Matlab 2 language. Some expressions have been derived using the computer algebra package Maple 3

1.2

General structure of thesis

To get an overview of the structure of this thesis, the contents of the chapters are presented below. In Chapter 2, an extensive literature study on cable roof structures is presented. The study includes both practical and theoretical aspects of cable roofs. Among the practical aspects are: diﬀerent structural systems, roof erection processes, different types of cables and their properties, rooﬁng materials, and structural details. Diﬀerent types of loads and their eﬀect on cable roofs are also presented. Finally, methods used to analyse the behaviour of cable roofs under loads are reviewed. In Chapter 3, a review of the numerical methods used to ﬁnd the initial equilibrium conﬁguration of cable structures and structures of mixed type (cables and stiﬀ structural members) are presented. One of the methods—the force density method—is further described in detail and coded. A variety of examples are analysed to illustrate both the advantages and the drawbacks of the force density method. In Chapter 4, the diﬃculties of modelling cable behaviour using ﬁnite elements based on the conventional approach, i.e. using shape functions, are discussed. Further, four ﬁnite cable elements are presented: the straight bar, the parabolic cable, the elastic catenary and the associate catenary. The internal force vectors and tangent stiﬀness matrices are presented and the elements are compared by some simple examples. In Chapter 5, an existing cable roof structure is analysed by a ﬁnite element program written by the author. The structure is the Scandinavium Arena in Gothenburg, which consists of a pretensioned cable net anchored in a nearly circular concrete ring beam. The results of the calculations are compared to the results from the initial design process and the reasons for discrepancies in the results are discussed. In addition, results from a simpliﬁed method, mainly used in preliminary design, are compared to the results from the ﬁnite element calculations. In Chapter 6, the conclusions of this study are stated and directions for further research are suggested. In Appendix A, data used in the analysis of the Scandinavium Arena are presented. Matlab is a registered trademark of The MathWorks Inc., 24 Prime Park Way, Natick, MA 01760-1500, U.S.A. Internet: http://www.mathworks.com. 3 Maple is registered trademark of Waterloo Maple Inc., 57 Erb Street W., Waterloo, Ontario, Canada N2L 5J2. Internet: http://www.maplesoft.com. 2

3

Chapter 2 Literature review 2.1

Historical review

The ﬁrst structures regarded as cable roofs are four pavilions with hanging roofs built by the Russian engineer V. G. Shookhov at an exhibition in Nizjny-Novgorod in 1896. During the 1930’s a small number of roof structures of moderate sizes were built in the U.S.A. and Europe, but none of major importance [88]. A big step in the development of suspended roofs came in 1950 when Matthew Nowicki designed the State Fair Arena, Figure 2.1, at Raleigh, North Carolina, USA. Sadly, Nowicki died that same year in a plane crash, but his work continued through the architect William Henry Deitrick and civil engineer Fred Severud and in 1953 the arena was completed [88].

(a)

(b)

Figure 2.1: The State Fair Arena at Raleigh, North Carolina, U.S.A., (a) Reproduced from [10], (b) Structural system, reproduced from [16].

On an exchange visit to the U.S.A. in 1950 a German student in architecture, named Frei Otto, previewed the drawings for the Raleigh Arena in the New York oﬃce of Fred Severud. Otto saw that the project embodied many of his own ideas about how

5

CHAPTER 2. LITERATURE REVIEW

to construct with minimal amount of material. After graduation in 1952 Otto began a systematic investigation of suspended roofs. The investigation was presented in the doctoral thesis Das H¨angende Dach (The Suspended Roof), which was the ﬁrst comprehensive documentation on the subject [31]. The thesis caught the attention of Peter Stromeyer of Stromeyer Company, one of the largest tent manufacturers in the world. Stromeyer contacted Otto and they began a fruitful cooperation. In 1957 Otto formed the Development Centre for Lightweight Construction in Berlin in order to further increase the research about tensile architecture. In 1964 he incorporated the centre into the Institute of Light Surface Structures at the University of Stuttgart. A massive research work was undertaken at the two institutes during 1957–1965 and published in Tensile Structures (two volumes) [31, 124]. Frei Otto is considered by many to be responsible for the development of modern tensile architecture. He was involved in the construction of many of the large tensile structures during the mid 1960’s to early 1970’s. Among these was the ﬁrst large cable net structure with fabric cladding, the German pavilion at the World’s fair in Montreal 1967 [10], Figure 2.2.

Figure 2.2: The German pavilion at the World’s fair in Montreal 1967. Reproduced from [10].

Another pioneering structure at this time was the large low-proﬁle super elliptic air-supported roof, Figure 2.3, with a membrane attached to a diagonal cable net. This structure was designed by David Geiger for the United States pavilion at the World’s fair in Osaka 1970 [10].

6

2.1. HISTORICAL REVIEW

Figure 2.3: David Geiger’s air-supported roof at the World’s fair in Osaka 1970. Reproduced from [10].

Following the success of the cable net in Montreal, Frei Otto produced a very elegant development of the Montreal design for the Olympic Stadium in Munich 1972 [87], Figure 2.4.

Figure 2.4: The Olympic Stadium in Munich. Reproduced from [31].

After the Osaka dome, several air-supported domes were built around the world, because they provided the economically best alternative to span large distances. However, several of them deﬂated due to heavy snow loads or compressor failure. To overcome the deﬂation problems, David Geiger invented another structure 1986— the cable dome. The cable dome concept was inspired by the tensegrity principle by

7

CHAPTER 2. LITERATURE REVIEW

Kenneth Snelson and Richard Buckminster Fuller. The ďŹ rst two domes were built for the 1988 Seoul Olympics. The latest and biggest, the Georgia Dome, was built in Atlanta 1994, Figure 2.5.

Figure 2.5: The Georgia Dome in Atlanta, U.S.A., during construction. Reproduced from [10].

In the year 2000, the Millennium Experience will be held in Greenwich, London, close to the Greenwich meridian. This exhibition will be held inside the largest dome ever. The diameter of the dome is 364 m and the height is 50 m [70].

Figure 2.6: The Millennium Dome in London, during construction. Reproduced from the cover of Bautechnik, Vol. 75, No. 11, 1998.

8

2.2. STRUCTURAL SYSTEMS

2.2

Structural systems

In this section, the traditional cable roof systems are presented together with a fairly new one. Each roof type is presented very brieﬂy, but references to more information are given. Cable roofs can be divided into diﬀerent categories depending upon the criterion used for classiﬁcation. In accordance with how the cables are used, they can be classiﬁed as [57]: 1. cable supported roofs, and 2. cable suspended roofs. Cable supported roofs are, in principle, similar to cable-stayed bridges. In these roofs, the cables only provide additional support for elements which themselves carry a major part of the load. In cable suspended roofs the load is carried directly by the cable system [57]. The cable supported roofs, for which the cables only have an auxiliary function, will not be considered in this thesis. The cable suspended roofs may be divided into the following categories [16]: 1. simply suspended cables, 2. pretensioned cable trusses, and 3. pretensioned cable nets. Further, the pretensioned cable structures may be either self-balancing or non-selfbalancing. In a self-balancing structure, the forces in the cables are balanced internally in the supporting structure, e.g. a ring beam. In a non-self-balancing structure, the cable forces are resisted by ground anchors [16]. In general, the stiﬀness of a pretensioned cable structure depends on [16]: • the curvature of the cable, • the cross-sectional areas of the cables, • the level of pretension, and • the stiﬀness of the supporting structure. The cladding will not, unless it is in the form of a concrete shell, signiﬁcantly increase the stiﬀness of a roof. In the following, the traditional types of cable suspended roofs will be described. In each category, the structural systems are illustrated by a limited number of ﬁgures. More examples can be found in the references 16 and 57.

9

CHAPTER 2. LITERATURE REVIEW

2.2.1

Simply suspended cable structures

The ﬁrst type is the simply suspended roof. These roofs have a single curvature or a positive double curvature (like a bowl). Systems of this type have no stiﬀness. To reduce the displacements caused by any form of applied loading, the roof cladding must either be very heavy or stiﬀ. Concrete is perhaps, therefore, the most suitable rooﬁng material; both prefabricated slabs and in situ cast concrete are used [16]. One can compare this roof type to a suspension bridge which is stiﬀened by the bridge deck.

Figure 2.7: Simply suspended roof.

The simply suspended roofs, which are stiﬀened by the cladding material, will not be considered in this thesis; only systems, which can be pretensioned before the cladding is applied, will be analysed.

2.2.2

Pretensioned cable trusses

Lighter and stiﬀer systems than the simply suspended systems can be achieved if a second set of cables with reverse curvature is connected to the hanging cables. A cable truss is quite stiﬀ if it is tensioned to a level which ensures that both the hanging and the bracing cables remain in tension under any load case. The basic cable truss conﬁgurations with vertical connecting elements are shown in Figure 2.8. Another system is the cable truss with diagonal ties, Figure 2.9, developed by the Swedish engineer David Jawerth. Generally, the cable trusses with the vertical connecting elements are structural mechanisms if they are considered as pin-jointed trusses. However, the cable truss with diagonal connecting elements is statically indeterminate [76]. Therefore, the Jawerth truss is stiﬀer than the other trusses [51]. The cable trusses may be arranged in parallel planes, Figure 2.8, or radially, Figure 2.10. A parallel Jawerth system was used in the Johanneshov Ice Stadium in Stockholm, Sweden. An extensive study of cable trusses is presented in reference 76.

10

2.2. STRUCTURAL SYSTEMS

(a) Convex cable truss structure with corrugated metal roof decking

(b) Concave cable truss structure with corrugated metal roof decking

(c) Convex-concave cable truss structure with corrugated metal roof decking

Figure 2.8: Cable trusses. Redrawn from [16].

11

CHAPTER 2. LITERATURE REVIEW

Figure 2.9: Cable truss system developed by the Swedish engineer David Jawerth. Redrawn from [16].

Figure 2.10: Radial cable truss structure—Lev Zetlin’s cable roof over the auditorium in the city of Utica, U.S.A. Reproduced from [10].

2.2.3

Pretensioned cable net structures

The third type of cable roof structures is that in which the hanging and bracing (pretensioning) cables all lie in one surface and form a net. To be pretensioned, this surface must be anticlastic (saddle-shaped) at every point [16]. The stiﬀness of a cable net depends mainly on: the curvature of the net surface and the level of pretension. In order to minimise the material in both the net and the supporting structure it is advantageous to have a surface with a relatively small

12

2.2. STRUCTURAL SYSTEMS

radius [31]. The prestressing force must not be exceeded by any type of loading, or the cables become slack. Areas of slack cables may damage the cladding or give rise to the destructive phenomenon of ﬂutter [16]. Cable nets can be designed with masts and edge cables or with stiﬀ boundaries such as beams, arches and rings, Figure 2.11. The ﬁrst type is generally less stiﬀ and more complicated to construct than the latter ones. The cladding is often placed directly on the cable network [16, 57].

Figure 2.11: A cable net structure—the Scandinavium Arena in Gothenburg, Sweden.

More information on diﬀerent types of cable nets and their properties can be found in references 16 and 76.

2.2.4

Tensegrity systems

A pure tensegrity structure is a structure composed of a relatively few non-touching, straight compression members which are suspended in a net of tension members. The key feature of such structures is that they are self-stressed; no external devices to equilibrate the cable forces are needed. Tensegrity structures can be said to have been invented by Kenneth Snelson and Richard Buckminster Fuller [99]. Several new systems, based on the tensegrity principle, have been developed in recent years. The most well-known of these new systems is the cable dome concept by David Geiger. The cable dome, Figure 2.12, is not a pure tensegrity structure since a curved ring beam is used to balance the cable forces. The cable dome concept was developed as an economically equal alternative to air-supported structures, which several times have deﬂated due to mechanical failure or excessive snow loads. Today, at least eight cable domes exist, but more will surely be build.

13

CHAPTER 2. LITERATURE REVIEW

Figure 2.12: The cable dome by David Geiger. Redrawn from [99].

More about tensegrity structures and some other fairly new structural concepts can be found in reference 99. Tensegrity structures are further discussed in Chapter 3.

2.3

Roof erection

Theoretically, cable and membrane structures can be given inﬁnitely many diﬀerent shapes. In practice, the number of conﬁgurations is restricted, as shown in the previous section. Of course, with the use of scaﬀolding, more shapes would be possible, but this eliminates some of the beneﬁts with cable structures. Generally, cable roof construction has two advantages over other forms of roof construction: very little or no scaﬀolding is required, and quite rapid erection process. However, these advantages do not indicate that the erection of a cable structure is an easy task. Every step of the erection process must be computer controlled to avoid overstressing of the supporting structures. It is important that the contractor responsible for the roof erection fully understands and exactly follows the erection plan speciﬁed by the designer [57]. Cable trusses have the easiest erection process among cable roof structures and may be assembled in the air or on the ground, of which the latter is to prefer. After being assembled on the ground the truss is hoisted into position and prestressed by applying tension at both ends simultaneously. Depending upon whether the centre of the truss needs to be lifted or lowered, tension is applied to the suspension or prestressing cable. Since the cable trusses usually do not interact with each other before the cladding is applied several trusses can be erected and prestressed simultaneously to reduce overall construction time [16]. Double layer grids with radial symmetry can be erected in the same fashion as trusses, but care has to be taken to not over-stress the compression ring as bending moments are introduced when just a few trusses are tensioned. An erection scheme for radial double layer grids is given in [57].

14

2.4. CABLES

Cable nets may be preassembled on the ground and hoisted into position or assembled in the air. Nets with ﬂexible boundaries (i.e. edge cables) are usually preassembled on the ground, but for nets with stiﬀ boundaries either of the methods can be used [16]. With use of computational methods (see Chapter 3) the shape of a net and the corresponding cable forces can be very accurately determined. To obtain the computed shape of the real net a high dimensional accuracy in fabrication is required. Small errors in unstrained length may cause large errors in force. One method to achieve a high accuracy at a minimal cost is to specify a net with a square unstrained mesh and uniform cable stresses. In this way, the same cable dimension can be used for the whole net (not the edges) and the equidistant cable-to-cable connections can be factory-assembled. But, even with a high accuracy some adjustment can be necessary after the net has been lifted into its ﬁnal position. This adjustment is possible if tensioning devices (turnbuckles) are incorporated at the ends of the cables [63]. For tensegrity structures, suitable methods for prestressing large tensegrity frameworks have not yet been developed. This is probably the main reason for the very few large tensegrity structures today. Nonetheless, one exception is the cable domes by David Geiger. These domes were developed as an economically equal alternative to air-supported structures, but without the risk for deﬂation. From the economic point of view, it was necessary that the domes could be constructed without any scaﬀolding. Figure 2.13 shows the steps of erection of a cable dome [99]. For a complicated structure, the best way to plan the erection steps is to build a physical model of the structure [99].

2.4

Cables

The main load carrying element in the structures considered in this thesis is the cable. In structural applications, the term ‘cable’ means a ﬂexible tension member. However, a cable can have diﬀerent conﬁgurations. In this section, the diﬀerent types of cables and their characteristics will be examined.

2.4.1

Products

The smallest single tension element in a cable is the steel wire. It is usually circular in cross section, with a diameter between 3 and 8 mm, but may be non-circular in locked coil strands. The wire has a high tensile strength that is obtained by cold drawing or cold rolling [35]. A spiral strand, Figure 2.15(a), is an assembly of wires laid helically around a central straight wire. An assembly of a small number of wires is called a spiral strand and if there are more than three layers it is called a spiral bridge strand. The successive layers are usually wound in opposite directions to get equal torsional stiﬀness in both directions [35].

15

CHAPTER 2. LITERATURE REVIEW

(a)

(b)

(c)

(d)

(e) Figure 2.13: Cable dome erection steps: (a) The upper cables are hung, then (b) a hoop and struts are hung, raising the inverted ridge cables. More hoops and struts, (c)â&#x20AC;&#x201C;(e), further raise and tension the ridge cables. Drawn from data given in [39].

16

2.4. CABLES

Figure 2.14: A wire rope and its parts. Reproduced from [29].

Locked coil strands, Figure 2.15(b), are similar to spiral strands but are composed of two types of helically laid wires: the core is a spiral strand with helically laid circular wires, and at least the two outer layers have wires with a special Z-shape that interlock with each other. The special shaped wires together with the selfcompacting eﬀect of the helical arrangement result in a tight surface and a low void ratio in the outer layers [42]. A wire rope, Figure 2.15(c), is an assembly of spiral strands that are laid helically around a central core that can be a strand or another independent wire rope. The spiral strands are usually laid in the opposite direction to the wires in the spiral strands (ordinary lay) but can be laid in the other direction (Lang’s lay) [35]. The helical lay of wires increases the ﬂexibility of the cable, but reduces the strength and stiﬀness. In some applications, particularly suspended bridges, a high strength and stiﬀness are more important than ﬂexibility and therefore products with parallel wires and strands have become popular. Other beneﬁts with parallel strands and wires are easier handling and transportation. In the last decade, parallel strands have also found use in roof construction. Parallel strand systems were used as the hoop and ridge cables in the cable domes by David Geiger [100]. The development of parallel products over recent years is reviewed by Walton [125].

17

CHAPTER 2. LITERATURE REVIEW

(a) Bridge strand

(b) Locked coil bridge strand

(c) Wire rope

Figure 2.15: Cable cross sections. Reproduced from [16].

2.4.2

Strength

For the wires commonly used in cables the guaranteed minimum tensile strength is 1570 MPa and the guaranteed 0.2 % proof stress is 1180 MPa. The limit of proportionality (0.01 % proof stress), which is the absolute upper limit for the stresses in the service condition, has a value of 65–70 % of the tensile strength. When deciding the allowable stress level, the eﬀect of relaxation must also be taken into account. Tests on steel wires show that the relaxation accelerates when the wire is held under a permanent stress larger than 50 % of the tensile strength. Therefore, the stresses from permanent loads should not exceed 45 % of the tensile strength [42].

2.4.3

Axial stiﬀness

For structural applications, the perhaps most important property of the cable, besides the tensile strength, is the axial stiﬀness. As mentioned above, a cable with helical wires has a lower stiﬀness than a cable with straight wires. In the design of cable structures, it is of cardinal importance to know the axial stiﬀness of the cables since the force distribution in, for example, a cable net is very sensitive to small errors in the cable properties (modulus and length). Several methods have been developed to calculate the axial stiﬀness of a helically wound cable, see for example reference 22. Most of these methods are based on contact theories and are, thus, very complex. Nevertheless, two simple and accurate methods have been found and will be presented in this section. For explanation of the notations see Figure 2.16.

18

2.4. CABLES

Figure 2.16: Geometry of a helically wound cable. Reproduced from [58].

In reference 58, Kumar and Cochran linearised the equations from Costello [29] and arrived at the following closed-form expression for the axial stiﬀness: (AE)eq = A0 E0 +

n 

  mi Ai Ei sin αi 1 − (1 + ν)pi cos2 αi ,

(2.1)

i=1

where Ai = πRi2 ,

(2.2)

    Ri2 ν Ri 2 2 (2.3) pi = 1 − ν cos αi 1 − 2 1 − cos 2αi cos αi . ri 4ri 1+ν Kumar and Cochran [58] also provide an even simpler expression for the equivalent axial stiﬀness n   (2.4) (AE)eq = A0 E0 + mi Ai Ei sin3 αi 1 − ν cot2 αi . and



i=1

Another method, in which the wire layers are modelled as orthotropic sheets, has been developed by Raoof [98]. The method is quite cumbersome and not suitable for practical design work. Therefore, Raoof derived a simpliﬁed procedure, by parametric studies of diﬀerent cable dimensions. In that, Hruska’s1 parameter is ﬁrst computed as: n  mi Ai cos4 αi , (2.5) κ= A T i=1 in which AT =

n 

mi Ai .

i=1 1

From F. H. Hruska, Calculation of stresses in wire ropes, Wire, Vol. 26, No. 9, 1951.

19

(2.6)

CHAPTER 2. LITERATURE REVIEW

The relation between the full-slip modulus (no friction between wires) and steel modulus is computed as: Efull-slip = −0.26442 − 2.004046κ + 6.5735κ 2 − 3.3068κ 3 , Es

(2.7)

Denoting Efull−slip /Es = ϑ, the no-slip modulus is found from Eno-slip = 3.998 − 7.916ϑ + 7.238ϑ2 − 2.321ϑ3 . Efull-slip

(2.8)

The full-slip and no-slip axial stiﬀnesses are obtained by multiplying Efull-slip and Eno-slip , respectively, with AT . The method by Raoof, equations (2.5)–(2.8), are included in Eurocode 3 [35]. Raoof’s method has been checked against experimental results in [47]. It was found that the experimental moduli of newly manufactured cables agreed well with the theoretical full-slip modulus. The methods presented above have also been compared to other analytical methods and it is concluded that the overall elastic behaviour of helical cables under axial loading is well represented by the available mechanical models. Which model one should use is dependent on the size of the cable [22]. The expressions by Kumar and Cochran is expected to yield higher accuracy for cables with few layers of wires, while the opposite can be said about the method by Raoof, [22]. Although any of the methods presented above gives an accurate value for the axial stiﬀness, a newly assembled cable does not have a linear stress-strain relationship. The reason is that a cable consists of moving parts which need a run-in period. In order to obtain a more linear behaviour the cable is, after the assembly, loaded repetitively to a load well within the elastic limit of the wire material. The purpose of this procedure is to remove the constructional stress and, thereby, obtain an almost linear stress-strain curve [16]. However, despite this linearising process, the cable stiﬀness will vary; it is lower when the cable is new and becomes higher during the useful life of cable [97].

2.4.4

Corrosion protection

Cables made of high strength steel wires are extremely vulnerable to phenomena such as stress and fretting corrosion. Add to this that most of the wires will be inaccessible for inspection and maintenance in the completed cable and that numerous of cavities are present between wires, and one understands that it is essential to ensure that the corrosion protection is of highest quality, particularly in the regions of end or intermediate ﬁttings [35, 42]. It is nowadays normal practice to protect the wires in a cable by galvanization. Both electrolytic and hot-dip techniques can be used, although the hot-dip technique has become the preferred method. There are diﬀerent classes of coating thickness dependent on the severity of the exposure conditions. The coating is usually of pure zinc but zinc-aluminium alloys are also used. Hydrogen embrittlement of galvanized steel is not recognised as a real problem with wire ropes and strands [125].

20

2.5. CLADDING

It is today generally agreed that cables should have two barriers against corrosion. For spiral strands, wire ropes and locked coil strands the second barrier consists of ﬁlling the interstices between the wires with a blocking material and coating the outer surface. The primary purpose of the blocking material is to prevent ingress of moisture [42]. Suitable blocking materials are synthetic waxes and compounds based on petrolatum (petroleum jelly), which are hydrophobic and have good adherence. The ﬁnal coating can be ordinary paint or, if necessary, a more displacement resistant compound [125]. If the cable is exposed to an aggressive environment it is normally sheathed with a tube made of steel or polyethylene. The space between the tube and the cable is ﬁlled with a suitable compound such as polymer cement grout or petroleum wax [42]. Sheathing is the most eﬀective method for corrosion protection and it is considered as impermeable. Materials used for sheathing must be ductile and if polyethylene is used it must be resistant to ultraviolet radiation. An alternative sheathing method is to extrude polyethylene directly onto the cables (no ﬁlling) [35].

2.5

Cladding

In analysis of a prestressed cable structure the cladding is usually assumed not to add any contribution to the structural stiﬀness. Some contribution will in any case be added to the performance of the building, which cannot be neglected. Especially the damping properties of the roof will be enhanced, which have signiﬁcant importance for the dynamic behaviour of the structure. There are two main categories of cladding: continuous membranes and unit coverings. Membranes can be made of fabric, foil or metal sheet. Unit coverings are panels of metal, wood or plastic [23]. The choice of cladding material depends on the type of structure (e.g. its shape), the expected lifetime, static and dynamic behaviour, security and maintenance. What type of cladding to be used should be decided upon at an early stage in the design process in order to avoid large changes, which might eﬀect the cable spacing and the design of structural details [16].

2.5.1

Fabrics and foils

Fabric is today the most common cladding material used for lightweight tension structures. As a structural element, the fabric must have the strength to span between supporting elements, carry wind and snow loads, and be safe to walk on. To comply with these requirements, the fabric must be prestressed, since it has a negligible bending stiﬀness. The amount of prestress and the patterning of the membrane, i.e. how the membrane should be cut and assembled, is given by the structural analysis of the roof. Besides the structural requirements, the fabric must meet the requirements which aﬀect the environment inside the building; these are air tightness, water protection, ﬁre resistance, heat insulation, light transmission, acoustic properties, maintenance and durability [10].

21

CHAPTER 2. LITERATURE REVIEW

Fabric membranes are composite materials. Inside the membrane there are ﬁlament ﬁbre yarn, designed to resist tensile forces, woven in diﬀerent directions forming an anisotropic surface. For permanent buildings with expected long lifetimes only two types of ﬁbres can be used: glass and aramid (Kevlar2 ) ﬁbres, of which glass ﬁbre is the most common. The mechanical properties of these ﬁbres compared to the properties of a steel wire are shown in Table 2.1. To protect the ﬁbres from environmental degradation, they are coated with some resin. Resin used are PTFE3 (Teﬂon2 ), silicone and PVC4 [99]. Table 2.1: Comparison of ﬁlament yarn characteristics [42, 130].

Property Density (g/cm3 ) Young’s modulus (GPa) Tensile strength (MPa) Max. elongation (%) Temp. resistance (◦ C)

Glass

Aramid

Steel

(E-HTS glass)

(Kevlar 49)

(Cold drawn wire)

2.55 69 2410 3.5 350

1.44 124 2760 2.5 250

7.86 205 1570 4.0 500

Fibreglass coated with PTFE has found the broadest use for permanent buildings. PTFE is a clear material which is chemically inert, so all dirt washes oﬀ without damaging the coating. It is also resistant to abrasion and highly reﬂective, absorbing little light as well as heat. The fact that Teﬂon comes in two forms, PTFE and FEP5 , with diﬀerent melting points makes it possible to heat weld seams, which enables a fast installation of the roof cladding. In addition to its high initial cost, PTFE-coated ﬁbreglass has two disadvantages: the material is brittle and requires considerable care in the packing, shipping and installation of panels, and it has little elastic forgiveness and must therefore be accurately patterned [99]. Fibreglass coated with silicone is more ﬂexible than PTFE-coated ﬁbreglass, so it is less likely to be damaged during shipment and installation. With a silicone coating, the fabric can be made more translucent than with PTFE and the need for artiﬁcial lightning during daytime can be almost eliminated. Fabric joints are chemically bonded or glued. The self-cleaning properties of silicone rubber are not yet as good as those of PTFE; it is recommended to clean the membrane once a year [99]. Fabrics of Kevlar have high tensile strength, high stiﬀness and very low weight. These properties make it possible to span large distances with Kevlar fabrics without a supporting cable net. One major disadvantage with ﬁbres of Kevlar is that they are highly susceptible to ultraviolet radiation and cannot be coated with translucent resin. The ﬁbres must be shielded with an opaque carbon black coat. Due to the sensitivity to ultraviolet radiation the joints of Kevlar fabrics cannot be heat welded with clear Teﬂon. The seams must instead be sewed, but it is impossible to develop Kevlar and Teﬂon are registered trademarks of E. I. du Pont de Nemours and Company Abbreviation for Polytetraﬂuoroethylene 4 Abbreviation for Polyvinyl Chloride 5 Abbreviation for Fluorinated Ethylene Propylene 2 3

22

2.5. CLADDING

the full strength of the fabric through the joints due to the high strength of the base material [40]. The newest membrane material is EFTE6 foil, which is not a woven fabric but a polymer ﬁlm sheet. From a structural viewpoint, EFTE foil is interesting because of its high tear resistance. In addition to the structural properties, the foil has many properties that make it work well as enclosure material. For example, it can be considered as incombustible, impervious to ultraviolet radiation and most chemicals, and it can be manufactured with a translucency of over 90 % [99].

2.5.2

Metal sheets

Instead of a fabric membrane a metal membrane can be chosen. Sheets of aluminium or steel sheets with thicknesses of 1 to 5 mm are found to be suitable for this application. Due to the low bending stiﬀness of the sheets, it is necessary to prestress the membrane to prevent buckling. Prestressing is achieved by applying the membrane before the roof is fully erected. When the roof is raised to the ﬁnal position the membrane is pretensioned. The metal membrane is composed of small accurately cut sections jointed by welding, gluing or bolting. Metal sheet membrane is a feasible choice for long-life structures and can be designed with openings covered with glass to provide natural lightning. Heat loss is prevented by attaching insulation material internally [23]. In [131], Yeremeyv and Kiselev describe the manufacturing and erection of a number of large projects in Russia where metal sheets are used as covering.

2.5.3

Panels

A cable net with cable spacing of around half a meter is ideal for small elements (panels). The elements are either shape-cutted or jointed in such a way so that they will conform to the shape of the structure. The panel system is most economical if it is made of light material, not to impose extra weight on the cable structure. Panels of ﬁbreboard, aluminium and plastic are appropriate to use for covering roofs [23]. For the Olympic Stadium in Munich, a system with translucent plastic panels (Plexiglas7 ) with thickness of 4 mm and size of 2.90 m × 2.90 m was used. The panels were fastened to the supporting cable net with shock absorbing ﬂexible connections to prevent cracking of the panels under roof movements. The joints between the panels were sealed with continuous neoprene proﬁles, as seen in Figure 2.17 [63]. However, it should be mentioned that many architects, e.g. Philip Drew [31], ﬁnd the Plexiglas cladding of the Olympic Stadium ugly. Therefore, it will probably not be used again. 6 7

Abbreviation for Tetraﬂuoroethylene Plexiglas is a registered trademark of AtoHaas Americas Inc.

23

CHAPTER 2. LITERATURE REVIEW

Figure 2.17: Acrylic panels for the Olympic stadium in Munich. from [57].

2.6

Reproduced

Structural details

Already at early stages of the design process the designer has to pay attention to the design of the structural details. Structural details are ﬁttings, saddles and anchorages. Fittings are attachments used to grip the cable at the ends or along its length. They can be classiﬁed, in accordance with the type of application, as the friction or clamp type, the pressed or swaged type, and the socketed type. Saddles are used when the cable has to run continuously over masts and other supports. In self-supporting systems, cables are anchored into structural members, such as a concrete ring or an arch. In other systems the cable forces are resisted by anchors in the ground [57]. A comprehensive survey of structural details is given by Chaplin et al. [23].

2.6.1

End ﬁttings

An end ﬁtting (terminal) is an attachment, which transmits the cable force to the supporting system. To be totally eﬀective, the end ﬁtting must withstand the full breaking force of the cable without signiﬁcant yielding, endure dynamic loading without risk of fatigue failure and not induce fatigue failure of the cable. For applications where large forces are to be transmitted to the supporting structure two diﬀerent end ﬁttings are accepted [125]: the socketed type and the swaged type, Figure 2.18.

24

2.6. STRUCTURAL DETAILS

(a) Socketed type

(b) Swaged type

Figure 2.18: Cable end ﬁttings with pin connectors. Reproduced from [16].

The most reliable, but also the most expensive, of the end ﬁttings is the socketed type. It is manufactured by splaying the end of the cable a prescribed length and cleaning the individual wires. When the wires are cleaned and dried the conical socket of machined or casted steel is positioned on the splayed cable section. Then molten socketing material is poured into the socket, hardens and forms a cone, Figure 2.18(a). As tension is applied to the cable the cone is drawn into the socket and wedging forces are developed which grip the wires. As socketing material either of zinc or resin is used. Pure zinc has been used for over a century and it oﬀers a cathodic protection for the cable, but it is sometimes criticised for impairing the fatigue resistance of the cable in this region. Another, more important, disadvantage with sockets ﬁlled with pure zinc is that they are prone to creep eﬀects under high stresses. Therefore zinc alloy, with improved creep resistance, is often used. Polyester or epoxy resin has better creep resistance. As the resin is casted at low temperature the fatigue resistance of the cable will not be impaired. Socketed end ﬁttings can be used for all cable sizes but cables of smaller diameter, approximately less than 38 mm, can be terminated by means of hydraulically compacted ﬁttings called swaged end ﬁttings. Swaged end ﬁttings are cheaper than socketed types but they are only guaranteed to resist 95 % of minimum breaking load of the cable. All end ﬁttings are manufactured, installed and rigorously tested by the cable manufacturer [16, 125].

2.6.2

Intermediate ﬁttings

Intermediate ﬁttings are used to connect cables to other cables. These ﬁttings are usually not standard appliances and their behaviour depend on the frictional force between the cable and the clamp. To prevent sliding of the clamp, the clamping force must be large and thereby high radial stresses are induced. Cables are more prone to fatigue when the pressure between adjacent wires is high and it is, therefore, important to use ﬁttings where the clamping force is evenly distributed over the cable. The resistance of a spiral strand and a locked coil strand to clamping forces, where the latter has the higher resistance, can be found in Eurocode 3 [35]. When the cable is tensioned the diameter will decrease and consequently the clamping force. It can therefore be necessary to retension the clamp bolts to prevent sliding.

25

CHAPTER 2. LITERATURE REVIEW

To avoid abrasion between the clamp and cable under cable movements, which can result in fatigue failure, the ends of the ﬁttings must be radiused. Diﬀerent types of intermediate ﬁttings are shown in Figures 2.19–2.20.

(a) Clamp connection

(b) Swaged clamp connection

Figure 2.19: Cable connections for dual-strand cable nets. Reproduced from [16].

(a) Single U bolt connection

(b) Double U bolt connection

Figure 2.20: Cable connections for two-way cable nets. Reproduced from [16].

In the search for the best economical solution one key is to use few types of structural details, as the number of ﬁttings in, for example, a cable net can be quite large. A way to achieve this is to use a ﬁtting which can be adjusted for diﬀerent angles between cables. The ﬁtting shown in Figure 2.19(b) can be mounted in a factory and thereby it is possible to reach a high accuracy. As mentioned above, accurate assembly of the ﬁttings is necessary in order to obtain the desired internal force distribution in a cable net.

2.6.3

Saddles

When the cables have to run continuously over supports like columns and masts, they have to be supported by saddles, Figure 2.21. When designing a saddle one has to take the bending stiﬀness of the cable into account. Two factors have to be checked: • the tensile stress in the outer wires, and • the pressure between the cable and the saddle.

26

2.7. ROOF LOADS

If the pressure between the cable and the saddles is too high the fatigue resistance of the cable will be aﬀected. The common rule is that the diameter of the saddle should not be less than 30d, where d is the diameter of the cable [16, 35].

Figure 2.21: Saddle. Reproduced from [16].

2.6.4

Anchorages

In self-supporting systems, the cables are anchored into the boundary structures, which resist the cable forces due to either geometry or self-weight. These structures are usually rings, arches and masts made of concrete or steel. In open systems the cable forces are resisted by tension anchors in the ground. A survey of existing tension anchors and methods for estimating their capacities for various ground conditions can be found in [16]. Which of the two anchorage alternatives that will be most economical, if both are architecturally accepted, depends upon the ground conditions, cost of material, and availability of expertise and labour skill.

2.7

Roof loads

Today, structural analyses are performed using commercial ﬁnite element programs, which contain elements for almost every application. New elements are constantly being developed and older reﬁned in an attempt to obtain more accurate results. Nevertheless, the accuracy of the results will mainly depend on the errors in the prescribed loads acting on the structure. Since most loads are environmental loads with random distributions, durations and magnitudes, the ‘exact’ values will never be known. In an attempt to achieve higher accuracy in the results from a structural analysis more reliable data on the extreme loads acting on buildings are needed. Apart from the prestress, the loads acting on cable roofs are the same as any other type of loads acting on more conventional buildings. However, it is well known that non-uniformly distributed loads are more dangerous to cable structures than uniform loads. Therefore, it is important to determine the ‘true’ load distribution on the structure. Nonetheless, the unusual shape of these structures, together with their low weight and large scale, make this a diﬃcult task. A further complication is that practically no guidance is available from codes of practice. This implies additional

27

CHAPTER 2. LITERATURE REVIEW

costs to the project, because of the need for expertise. The latest methods for determining the loads on roofs of general shapes involve very sophisticated physical and computational modelling techniques, which require expensive equipment and powerful computers. In this section, these methods are reviewed. The loads are viewed in order of their importance on the structural behaviour of tension structures.

2.7.1

Wind load

Due to the low weight of cable roofs with membrane cladding, wind pressure is one of the most important forms of loading. The variability and large number parameters involved in the determination of wind eﬀects on structures make it a very complex problem. Some undesirable eﬀects and partial collapses have been caused by wind on tension structures [16]. Among these can be mentioned the vibrations due to wind on the roof of the Raleigh Arena, U.S.A., which made it necessary to insert supplementary internal cables. The nature of wind Wind is initiated by pressure diﬀerences between points of equal elevation, caused by variable solar heating of the atmosphere of the earth. The motion of the air mass is modiﬁed by the rotation of the earth and close to the ground the velocity of the moving air is reduced due to friction. At a certain height above the surface of the earth the eﬀect of the surface friction becomes negligible. Above this boundary layer a frictionless wind balance is established, and the wind ﬂows with the gradient speed along lines of equal barometric pressure. The height of the atmospheric boundary layer normally ranges from a few hundred meters to several kilometres, depending upon wind intensity, roughness of terrain, and angle of latitude [32]. Physically, the wind is composed of two diﬀerent velocity components [16]. The ﬁrst component is the velocity of a steady ﬂow determined by the long-term pressure variations (approximately four day periods). This component is called the mean wind velocity. The second velocity component, which is superimposed on the steady ﬂow, is due to a turbulent ﬂuctuating system with high frequency components, which is caused by the friction between the air and the surface of the earth. The two velocity components are clearly seen when the wind velocity is plotted in a van der Hoven power spectrum, Figure 2.22. This spectrum shows the variations of the mean square of the amplitudes of the ﬂuctuating components against the frequencies of these components. Hence, the analysis of linear structures can be divided into to two parts: the calculation of the quasi-static response due to the steady velocity component and the response caused by the turbulence components. As cable structures have a nonlinear behaviour this division is generally not valid. Instead, the total wind load must be used in the dynamic analysis of cable structures. In the sequel to this section the common expressions for description of the wind load on buildings and

28

2.7. ROOF LOADS

ways to obtain the pressure distribution will be described in brief.

Figure 2.22: Spectrum of horizontal wind speed after van der Hoven. Reproduced from [26].

Mathematical description of natural wind To describe the wind velocity mathematically a Cartesian coordinate system is applied, with the x-axis in the direction of the mean wind velocity, the y-axis horizontal and the z-axis vertical, positive upwards. The total wind velocity at time t, V (x, y, z, t), is formulated as: V (x, y, z, t) = U (z) + u(x, y, z, t) + v(x, y, z, t) + w(x, y, z, t),

(2.9)

where U (z) is the mean wind velocity in the mean direction θm , u, v and w, are turbulence components of the wind ﬁeld in the x, y and z directions, respectively. It can be noted that the mean wind velocity U (z) only depends on the height above the ground. The turbulence components are treated mathematically as stationary, stochastic processes with a zero mean value. The mean wind velocity U (z) and the turbulence component u in the wind direction are often most important, as they usually give the main contributions to the wind forces on a structure [32]. Three laws have been proposed to describe the way in which the mean velocity U varies with height [32]. The ﬁrst law is the power law, which has been adopted in many codes. The second law is the logarithmic law, which is derived not only from empirical data, but also from theoretical considerations. The Deaves and Harris model, which is the third law, is the most exact one since it is ﬁtted to experimental data [16,26]. In urban areas, where stadiums and other large roofs usually are built, the terrain roughness might change if buildings are erected or demolished [32]. This directly aﬀects the mean wind velocity and has to be considered at the design stage. The wind in the boundary layer is always turbulent, which means that the ﬂow is chaotic, with random periods varying from fractions of a second to several minutes, Figure 2.22. In order to describe a turbulent ﬂow, statistical methods must be applied [32].

29

CHAPTER 2. LITERATURE REVIEW

Wind load on a structure The earliest method for the assessment of the action of turbulent wind is the quasisteady vector model [27]. It makes the simple, but inaccurate, assumption that the pressure ﬂuctuations correspond exactly with the variations of the wind velocity. Other methods may be found in [27], but wind loads on buildings are determined using the quasi-steady model in many codes [64]. Therefore, a detailed description of the quasi-steady theory will be given here. For a point on a surface, (x,y,z), the instantaneous pressure, p, is given by [27, 64] 1 p = ρV 2 C p (θm + θv , θw ), 2

(2.10)

where V is the wind velocity given by equation (2.9). C p (θm + θv , θw ) is the mean, with respect to time, pressure coeﬃcient for the instantaneous azimuth angle, θv and the elevation angle θw , of the wind velocity vector measured from the mean wind direction θm . The magnitude of the wind velocity is given by V 2 = (U + u)2 + v 2 + w2 .

(2.11)

The instantaneous azimuth angle θv is given by θv = tan−1

v . U +u

(2.12)

In the same way the vertical component θw can be expressed as θw = tan−1

w . U +u

(2.13)

By removing small second order terms, the full quasi-steady model is linearised and the velocity magnitude reduces to V 2 ≈ U 2 + 2U u.

(2.14)

The ﬂuctuating wind directions are assumed linear for small v and w, which gives C p (θm + θv , θw ) ≈ C p (θm ) +

v ∂C (θ ) w ∂C (θ ) p m p m + . U ∂θv U ∂θw

Substituting (2.14) and (2.15) into equation (2.10) yields 

v ∂C (θ ) w ∂C (θ )  1  2 p m p m . + p(t) ≈ ρ U + 2U u C p (θm ) + 2 U ∂θv U ∂θw

(2.15)

(2.16)

Dividing both sides of equation (2.16) by the mean dynamic pressure 12 ρU 2 , expanding and discarding small turbulent cross terms gives the instantaneous pressure coeﬃcient

u

v ∂C (θ ) w ∂C (θ ) p m p m C p (θm ) + + . Cp (θm + θv , θw , t) ≈ C p (θm ) + 2 U U ∂θv U ∂θw (2.17)

30

2.7. ROOF LOADS

Taking the time average of equation (2.17) leaves the expected result C p (θm + θv , θw , t) = C p (θm ).

(2.18)

To evaluate the performance of the quasi-steady theory, full-scale wind velocity and pressure measurements were recently done on an full scale experimental building at the Texas Tech Field Research Laboratory [64]. The results from that study showed that the area-averaged pressures over a substantial area of the roof as well as root mean square (rms) and peak pressure coeﬃcients can be well predicted using the quasi-steady theory. However, the spectra of the pressure coeﬃcients cannot be predicted at high frequencies using the quasi-steady model. Wind tunnel testing The value of the wind pressure coeﬃcient Cp is a function of shape, scale, surface condition, surroundings, wind velocity and wind direction [32]. Because of the complexity of these factors, the pressure coeﬃcients must be determined by full-scale measurements or wind tunnel tests. Full-scale measurements are the most accurate, but not possible in practice and is therefore only carried out to verify the wind tunnel tests. Hence, the most appropriate method for determining the wind load is to test a model of the structure in a wind tunnel. Surface pressure coeﬃcients, based on such tests, for traditional building shapes can be found in diﬀerent codes. However, as mentioned above, the shapes of tension structures are not covered by the codes [40]. To interpret the results from a model test, the model must satisfy several laws [32]. These model laws are formulated by introducing a number of non-dimensional parameters. In wind engineering, the number of parameters is so large that it is impossible to satisfy all the conditions simultaneously. Therefore, some parameters that are of minor importance have to be disregarded. Besides the laws for the model itself, the wind tunnel must also be able to simulate the wind climate in the atmospheric boundary layer for the site considered. A general description of model laws and boundary-layer wind tunnels can be found in e.g. [32]. Earlier, rigid models were used in wind tunnel studies to determine the pressure distribution on the exterior and interior surfaces of a building, for a variety of wind directions. The rigid model studies have been developed over many years of testing conventional buildings and are relatively straightforward. However, high wind speeds can change some of the pressure coeﬃcients when aeroelastic models are used [49]. These changes are attributed to roof deﬂections and to the non-linear stiﬀness of the roof. The conclusion is that the use of rigid pressure-tapped models can underestimate the pressure coeﬃcients for ﬂexible structures undergoing large displacements. Another reason, maybe the main one, for conducting wind tunnel experiments with aeroelastic models is to search for unforeseen aerodynamic instabilities, i.e. large amplitude vibrations [49]. In references 33 and 49 wind tunnel tests of tension roofs with aeroelastic models are performed, but no types of aerodynamic

31

CHAPTER 2. LITERATURE REVIEW

instabilities were found. Certain aspects that have to be taken into account when modelling cable and membrane structures in a wind tunnel are found in [121]. Methods of analysis In wind tunnel testing the pressure distribution is measured using electronically scanned multi-channel pressure systems, with up to 512 channels and sampling frequencies of up to 100 Hz [11]. Hence, the amount of data from one series of testing is enormous and in its raw state very diﬃcult to use for analysis. A method to describe the wind velocity proﬁle and wind pressure pattern was recently rediscovered in the ﬁeld of wind engineering. This method, often used for stochastic problems, is the proper orthogonal decomposition (POD), known also as the Karhunen-Loeve expansion8 . The POD method resembles the modal analysis used in structural dynamics [11, 12]. The main objective of the POD method is to ﬁnd a deterministic function Φ(x, y) which is best correlated with all the elements of a random ﬁeld [11]. The deterministic function Φ(x, y) is found through a maximisation of the projection of the random pressure ﬁeld p(x, y, t) on Φ(x, y)

p(x, y, t)Φ(x, y)dxdy

= max . (2.19) Φ2 (x, y)dxdy If the maximisation of equation (2.19) is performed in the mean-square sense for a discrete pressure ﬁeld, it leads to the following eigenvalue problem Rp Φ = λΦ,

(2.20)

where Rp is the covariance matrix of the pressure space, and Φ and λ are, respectively, a vector and a value, both to be determined. The eigenvectors Φn (xi , yj ) are base functions in a series expansion of the pressure ﬁeld  an (tk )Φn (xi , yj ), (2.21) p(xi , yj , tk ) = n

where the expansion coeﬃcients, i.e. modal amplitudes, an (tk ) are easily computed due to the orthogonality of the eigenfunctions Φn (xi , yj ) i j p(xi , yj , tk )Φn (xi , yj ) 2 an (tk ) = . (2.22) i j Φn (xi , yj ) The eigenvalue λk is the measure of the contribution of each eigenmode to the pressure mean squares [30]. Depending on the number of terms included in the expansion, equation (2.21), diﬀerent levels of accuracy are reached. In [12] about 30 % 8

The expansion was derived independently by a number of investigators; Karhunen in 1947, Loeve in 1948, and Kac and Siegert in 1947 (according to “Stochastic Finite Elements: A Spectral Approach” by Ghanem, R. G. and Spanos, P. D., which can be found at http://venus.ce.jhu.edu/book/)

32

2.7. ROOF LOADS

of the eigenvectors were required in the expansion to represent the peak pressure with an error of approximately 10 %. However, only one term, the ﬁrst eigenvector was needed to represent the mean point and area-average roof pressure with an error of approximately 1 %. A physical interpretation of the ﬁrst three eigenvectors can be provided by the quasi-steady theory [122]. It was shown in reference 122 that the ﬁrst eigenvector was closely related to the mean pressure distribution C p (θm ), the second and third eigenvectors were related to ∂C p (θm )/∂θv , and ∂C p (θm )/∂θw , respectively. These results, together with the full scale measurements by Letchford et al. [64] indicate that the pressure ﬁeld over certain roof types can be described by the quasi-steady theory and that the POD method and the quasi-steady theory in some sense are related to each other. A simpliﬁed method to calculate wind loads on tension structures with irregular shapes has been presented by Tabarrok and Qin [115]. The method simpliﬁes input data, because it does not need experimentally measured wind pressure coeﬃcients. In their method a membrane structure was discretized with constant strain shell elements. To calculate the wind load on each element, the designer speciﬁes a magnitude of the pressure coeﬃcient Cp that deﬁnes the wind pressure on a vertical surface normal to the wind direction, and a direction that deﬁnes the source of the wind. The wind pressure normal to each element is then computed by scaling the wind velocity by the cosine of the angle between the wind direction and the outward normal to the element. This means that the model gives zero pressure for surfaces parallel to the wind and suction on leeward surfaces. Computational wind engineering Wind tunnel testing, including model making, is expensive, tedious and in some cases inaccurate due to limitations associated with the boundary layer wind tunnel [109]. These limitations might be overcome if the pressure values could be derived by numerical computer simulations. Savings, in both time and money, would also be possible. Application of Computational Fluid Dynamics (CFD) to wind engineering problems means large computer memory and CPU time, because very ﬁne computational grids are needed to deal with the modelling of turbulence, complex building conﬁgurations and the large area of model domain [109]. Today, only smaller structures with coarse grids can be analysed. However, it is anticipated that current limitations due to long CPU time and large memory requirements will be overcome in the near future through new computational, parallel-processing based architectures, faster computer hardware and more eﬃcient computational algorithms [11]. The current state-of-the-art of Computational Wind Engineering (CWE) is reviewed by Stathopoulos [109]. Results from CFD simulations are compared with those obtained from wind tunnel or full scale experiments for buildings of diﬀerent shape and various wind directions. According to Stathopoulos: “Disagreements proved to be higher than what is tolerable, particularly for cases that require complex building shapes, surroundings and for results other than mean pressure coeﬃcients.” He

33

CHAPTER 2. LITERATURE REVIEW

further concludes that “at present time CWE may be used only for the assessment of the wind environment around buildings. For cases which involve mean values of the wind speed and pressure coeﬃcient the numerical results may be used for preliminary design purposes.” Several areas have to be improved before CFD can be used in design, which include: numerical accuracy, description of boundary conditions and reﬁnement of turbulence models [109]. The most optimal method today seems to be a hybrid analysis, where experimental wind tunnel data are combined with numerical simulations [11]. For engineering problems, a hybrid analysis could also involve combining the experimental data with those from available CFD software packages. To summarise wind loads on cable roof structures: • The mean and turbulent parts of the wind cannot be separated in analysis due to the ﬂexibility (geometric non-linearity) of the roofs. • Pressure coeﬃcients have to be determined by wind tunnel tests. The pressure distribution from the tests can be described mathematically using the POD method. • To be able to recognise any unforeseeable wind related instability, the wind tunnel model should be of the aeroelastic type. Although several methods are at hand for the wind engineer, the assessment of wind eﬀects on structures is certainly not easy. As mentioned before, this area is very complicated and for a more in-depth analysis one can refer to the many references given in this section, especially [26,27,32]. As the design recommendations concerning tension roof structures are non-existent, good engineering judgement and experience are important characteristics of designers of tensile structures.

2.7.2

Snow load

Apart from wind loads, snow loads play an important part in the design of structures. Many modern buildings have moved away from traditional shapes and their behaviour with respect to snow accumulation is not known well enough [41]. When constructing a tension roof, the load corresponding to the expected intensity of snow has to be considered. As for the wind loads this is more or less straightforward for ordinary types of roofs, and is found in national building codes. For cable and membrane roofs this is considerably more diﬃcult. Snow distribution The snow intensity is measured at meteorological weather stations as the ground snow depth. Prior to 1970, many buildings were designed and built assuming uniformly distributed snow loads [118]. After a number of failures, attention was given

34

2.7. ROOF LOADS

to unbalanced loads, due to snow drift. Therefore, surveys of actual snow loads were started to determine the diﬀerence between ground and roof snow load. The results from the surveys showed that, in cold and windy areas, the roof snow load was considerably lower than the ground snow load. Nonetheless, on certain parts of some roofs the load was signiﬁcantly higher. Today, building codes make provision for drifting of snow by specifying a number of snow load cases for the type of roof considered. Some shapes of roofs tend to accumulate more unbalanced loads than others, and the load cases try to cover the possible snow distributions over the roof. Unfortunately, the roof types covered by the codes are usually traditional. Tension structures, such as cable and membrane roofs, with sculptural forms are not covered by the codes. Due to the ﬂexibility of tension roofs, ponding of snow can occur in ﬂat areas or under heavy snow loads. This requires consideration in design and can only be analysed with the aid of wind tunnel or water ﬂume experiments. Wind tunnel and water ﬂume testing Like wind tunnel experiments, some model laws has to be followed when modelling the snow in air or water. In wind tunnels, granular materials, such as tea, glass, and nut shells, are used to simulate dry snow, while sand is used in water ﬂume experiments, Figure 2.23. One limitation in modelling, which cannot easily be overcome, concerns the great variation in snow properties. Common simulation materials cannot model sticky snow. For example, sand will not stay on steep surfaces which makes it diﬃcult to simulate snow accumulations on steep slopes where snow will accumulate before eventually sliding oﬀ. Another limitation in model studies is that only one wind direction is considered at a time but the overall seasonal environment consists of a sequence of snow storms and high winds from diﬀerent directions. In reality, the ﬁnal snow accumulation depends on the chronological order and duration of the storms and on temperature, sunshine, humidity, etc. Surrounding terrain may also aﬀect total snow accumulation and drift patterns on structures. Whether air or water is the medium, a model can provide a good simulation of the ﬂow around structures. However, the state-of-the-art of snow drift modelling prevents the measurement of quantitative results [50, 118]. Recently a water ﬂume test was used to determine the snow loads on a large tension roof at Denver International Airport, U.S.A. [10]. Denver is known for its heavy snow falls, and the shape of the roof leads to high snow load intensities being expected. The tests also showed that in the valleys of the roof the design snow intensity was very high, 3.8 kN/m2 , Figure 2.23. The predicted snow pattern from the model test agreed well to that seen on the roof after the ﬁrst snow falls, conﬁrming the reliability of the test.

35

CHAPTER 2. LITERATURE REVIEW

Figure 2.23: Investigation of snow drift with the help of a model test, where water replaces air and sand represents snow. Reproduced from [10].

Computer simulations In the design of the tension roof of Denver International Airport, the water ﬂume experiments were supplemented by a computer program based on the Finite Area Element (FAE) method (not to be confused with the Finite Element Method) [38]. This is a so-called hybrid method, which means that the wind velocity ﬁeld is obtained from wind tunnel experiments. A brief description of the FAE method and its properties is given below. First, the roof is divided into many area elements by a grid. The wind velocities are measured at grid intersection points. Time histories of meteorological data concerning the wind direction and speed are used as input for the computations. Snow drift is computed using empirical relationships for snow ﬂux versus wind velocity. By computing the mass ﬂuxes into and out of each element, the rate of build up or depletion of snow mass in the element due to drifting is determined. The mass balance computations at each time step include the additional mass from snow fall and the depletion due to melting. The method also takes into consideration the less signiﬁcant drift of snow that has been rained upon, or that has experienced a melting episode. Some surfaces, which are rough or ribbed, have high snow storage capacities and can trap snow permanently (at least until it melts). Therefore, the area elements are assigned with a certain storage capacity for snow depending on surface roughness. Also included in the FAE method is a heat balance used to calculate the melting rate of the snow pack inside each element, and the ability of snow to store liquid water and thereby increasing the snow density. The FAE method has proved to be a good tool to supplement the model studies, and overcome the limitations associated with them. With this method quantitative results can be obtained with higher accuracy [38]. A purely computational method for predicting snow accumulation, called SNOWSIM, has been developed under a research project at Narvik Institute of Technology in Norway [7]. The method includes a commercial CFD program, combined with a simpliﬁed drift-ﬂux model to simulate snow drift. A computer simulation of snow drift has the advantage over wind tunnel or water ﬂume experiments that it can be more available and less expensive. Simulations can be done with snow drifts

36

2.7. ROOF LOADS

from diﬀerent directions and with variations in velocity and intensity. Simulations in three dimensions were presented, but due to limitations in computer power only small buildings of regular shape with a coarse mesh resolution, and simulation times of up to 50 seconds could be studied. Therefore, the simulations could only be regarded as an indication of where the snow will deposit, not as an exact quantity calculation. Compared with real measurements the simulations gave similar snow drift patterns. Bang et al. [7] gave a number of problems that have to be resolved before quantitative results can be available. These included the proper treatment of turbulence in the CFD program, evaluation of diﬀerent drift-ﬂux models, and modelling of structures with their local terrain. Thus, a complete computer simulation of snow drift magnitude is today not available even for buildings of traditional shape [7]. As in the case with wind loads on tension structures with complex shapes, Tabarrok and Qin [115] have proposed a simpliﬁed method to calculate the snow load distribution. In their method, vertical snow loads are generated based on the horizontal projection of each elemental area and a snow load magnitude per unit horizontal area speciﬁed by the designer. This means that there is full snow load on a horizontal surface and zero load on a vertical surface. This method is of course very approximative as it cannot handle snow drift. It has been seen that the determination of snow load magnitude and distribution is a task of equal diﬃculty as that for wind load. For a roof with a complex shape the only way to ﬁnd the sought quantities, i.e. magnitude and distribution, is through model tests. This procedure is expensive, time consuming and requires special knowledge and experience.

2.7.3

Earthquake load

Another important form of loading, which has to be considered in certain parts of the world, is earthquake ground motion. Even smaller earthquakes may lead to collapse of stiﬀ structures. Many studies have been concerned with the earthquake response of building structures but, like the wind and snow load studies, very few have included cable roof structures. Two works on the topic have been found and are brieﬂy presented in the following. In reference 78, a cable truss with diagonal ties (system Jawerth) is subjected to vertical and horizontal earthquake loadings. Both a linear and a non-linear analysis was performed. The maximum displacements did not diﬀer signiﬁcantly between the analyses. It was also found that under a horizontal earthquake, all the diagonals became slack at many instances. As far as the diagonal forces are concerned, the response was, according to Mote and Chu, “very erratic” [78]. An elastic earthquake response analysis of a type of cable dome—the suspen-dome— is presented in reference 117. The suspen-dome is a single-layer truss dome stiﬀened with a tensegrity system. Tatemichi et al. conclude that the analysis “conﬁrmed eﬀectiveness of the suspen-dome against earthquake motions, particularly vertical

37

CHAPTER 2. LITERATURE REVIEW

motions.” In general, the response of structures to dynamic loading is determined by a ﬁnite element analyses. Such analyses of cable roofs have shown that these structures usually have a long period of vibration. In addition, the supporting structures are relatively much stiﬀer and heavier than the cable system. Therefore, high-frequency contents of the earthquake ground motion will be ampliﬁed by the supporting structures. Conversely, low-frequency components will be reduced considerably by the time they reach the cable system. Hence, the response of the structure is dependent on the low-frequency content in the ground motion [57].

2.7.4

Other loads

For the majority of civil engineering structures, e.g. bridges, the dead load is a large part of the total load. This is not the case for prestressed cable roofs. For these roofs the dead load consists of the weight of cladding, insulation, cables and ﬁttings, etc., [57]. The magnitude of the dead load for a prestressed roof with fabric cladding is very low, values as low as 0.1 kN/m2 are common [99]. Hence, the dead load cannot be considered as important in ensuring the safety of correctly designed prestressed cable roofs. Nevertheless, wind suction may cause large deﬂections or, even worse, ﬂutter instability of a ﬂexible structure with a low dead load. Undoubtly, for suspended roofs the weight and stiﬀness of the cladding is more important, as it governs the stiﬀness of the roof. Of the permanent loads, the prestress is in many cases the most important one. The magnitude of the pretensioning force varies from structure to structure, but must, due to stress relaxation, not be greater than 45 % of the breaking force of the cable [42], section 2.4.2. Apart from relaxation, loss of cable tension also occurs as a result from creep in the supporting structure, slippage of cables at anchorage points and increase in temperature [57]. Live loads are usually taken into account by specifying the intensity of a uniformly distributed load. Usually, cable roofs have curved shapes and may therefore be considered as inaccessible to people except for maintenance purposes. This justiﬁes the use of a lighter design live load for the cable system and supporting structure, but for the cladding a normal design live load should be used [57]. Of course, all the diﬀerent loads (wind, snow, dead, live, temperature) presented in this section are not considered separately. Design load cases consist of combinations of the diﬀerent loads. The load combinations forming these cases are given in the national building codes and there is no reason to expect that the cases will be diﬀerent for cable roof structures.

38

2.8. ANALYSIS METHODS

2.8

Analysis methods

In this section, some techniques used for analysis of cable structures will be mentioned. None of the numerical techniques described here are applicable to only cable structures. Therefore, only a short historical review is considered necessary. Early analyses were done by applying membrane shell theory to cable nets. Application of the membrane shell theory results in a set of diﬀerential equations. Except for special cases, these equations are diﬃcult to solve in closed form. In most cases, the equations are solved by numerical techniques, such as the ﬁnite diﬀerence method [112]. Shore and Bathish [107] used double Fourier series to transform the diﬀerential equations to a system of algebraic equations. One ﬂat and one hyperbolic paraboloid prestressed cable net, both square in plan, were analysed numerically and experimentally. The agreement between the results was acceptable. Recently, T¨arno performed a parametric study of saddle-shaped networks with elliptic plane layouts and stiﬀ contours [116]. Such a study serves as an aid in choosing the dimensions of the roof and the structural elements. In general, membrane shell theory is less accurate if the cable mesh in a net is coarse; it is inadequate for complicated roof shapes. Since the introduction of computers in the 1960’s, several numerical methods have been developed for the general analysis of structures. Among these methods, the stiﬀness technique (ﬁnite element method) have been widely adopted. Originally, the method was developed to analyse structures with small displacements. Under the action of external loads cable nets undergo large displacements and it became evident that the stiﬀness technique was not applicable to such structures in its original form. Therefore, the original method was modiﬁed and applicable structures with geometrically non-linear characteristics. Several iterative methods have been applied to the non-linear stiﬀness method. The most popular is the Newton-Raphson technique, which has proven to be accurate, eﬃcient and applicable to the majority of cable structures [1]. A comprehensive description of the ﬁnite element method and the Newton-Raphson technique can be found in, for example, [28]. Other authors, e.g. [16, 111] have used a method based on the minimisation of the total potential energy of the structure. The minimisation was done using the the conjugate gradient method. The dynamic relaxation technique has been used by several authors for both form-ﬁnding [8, 66] and load analysis [68, 69]. Approximate methods for the preliminary design of cable trusses and simple cable nets, can be found in references 16, 57 and 76. For elliptical cable nets the method described by T¨arno [116] is recommended. The following chapters will investigate some earlier reported analysis methods and some new variants of them, aiming at accurate analyses of the form-ﬁnding and normal usage stages of some cable roof structures. Failure stage analysis will be identiﬁed as a topic for further research.

39

Chapter 3 The initial equilibrium problem 3.1

Introduction

For structural analysis, the equilibrium conﬁguration of a structure is generally known in advance. This is not the case for tension structures, i.e. cable and membrane structures. Due to the low ﬂexural stiﬀness of the cables and the fabric these structures have to be constructed so that they will experience a signiﬁcant prestress at all times. Thus, there is no compatible unstressed conﬁguration for a tension structure, even if no external loads are applied and its self-weight is neglected. Therefore, the designer must specify a reference conﬁguration for the structure that is stressed. The shape of the reference conﬁguration depends upon the internal stresses and forces. Hence, the load bearing behaviour and the shape of the structure cannot be separated and cannot be described by simple geometric models. In addition to satisfying the equilibrium conditions, the initial conﬁguration must accommodate both architectural, structural and constructional requirements [44,114]. Finding the stressed initial conﬁguration is an inverse structural problem, in which the speciﬁed force distribution is the driving parameter in the process. This is inverse to standard problems where the forces are the structural response to the deformations of the structure [13]. The problem of ﬁnding a conﬁguration that satisﬁes the laws of equilibrium is usually called form-ﬁnding or shape-ﬁnding. Haber and Abel [44] thought that this nomenclature was inappropriate to use when describing methods in which variables besides the shape were adjusted to satisfy equilibrium. Therefore, they used the term initial equilibrium problem instead. Throughout the present chapter and the rest of the thesis this term will mainly be used. The objectives of this chapter are to describe the initial equilibrium problem and review the existing computer methods for solving it. All the methods that are to be described are applicable to mainly cable structures, membrane structures and bar frameworks. Among the methods, one is especially interesting, namely the force density method. This method will further on be described in detail and applied to a number of diﬀerent problems. First, a brief description of the methods that were

41

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

used before computer methods were available will be given.

3.1.1

Physical modelling

Due to the mathematical diﬃculty and spatial complexity of the structural forms of tension structures, physical modelling was the primary method to solve the initial equilibrium problem of tensile structures until 1969 [44], when the cable roofs for the 1972 Munich Olympic Games were about to be built [62]. A pioneer in the physical modelling ﬁeld was Frei Otto, who performed extensive experiments using a variety of diﬀerent media, including soap ﬁlms, fabric, and wire models. The most simple modelling tools are soap ﬁlm models, which are obtained by dipping wire frames into soap water. These models are used as a ﬁrst check if the curvature of the roof surface is appropriate. As is well known, a soap ﬁlm always contracts to the minimal surface. The minimal surface may be the most aesthetic shape, but it is not always the best structural form, as the minimal surface approach tends to produce very ﬂat areas, which may induce ﬂutter (see also section 3.2.1). After the soap ﬁlm models, working models of larger scale and of other materials were built for further processing within the design process [87]. The Institute for Lightweight Structures in Stuttgart has worked with physical modelling techniques for several decades and a large number of structures with complex structural shapes of diﬀerent scales have been realised during the years, e.g. [31] or [87]. The Institute was involved in the construction of the some of the largest and most complicated cable nets built in the world: the German pavilion at the 1967 World’s fair in Montreal and the Olympic cable roofs in Munich. Models give many useful insights into the the behaviour of tension structures. A large number of conﬁgurations can in a short time be studied if for example soap ﬁlm models are used. For practical design work, however, they do not provide suﬃcient accuracy and have to be replaced by computational methods [31, 44]. Nevertheless, physical models of tension structures will always be made, because of the excellent structural visualisation they provide [77].

3.2

Literature review of initial equilibrium solution methods

In a numerical method a solution to the initial equilibrium problem consists of a combination of parameters describing an equilibrium conﬁguration of the structure. One method of distinguishing the diﬀerent solution methods is to indicate which of the parameters are speciﬁed by the designer and which are treated as problem unknowns [44]. The parameters involved in the initial equilibrium problem are: • Structural topology The structural topology deﬁnes the connectivity of the material of the struc-

42

3.2. LITERATURE REVIEW OF INITIAL EQUILIBRIUM SOLUTION METHODS

ture. This is done via the stiﬀness matrix in the ﬁnite element method (section 3.2.1) or the connectivity matrix in the force density method (section 3.3). • External loads Two types of loadings may act on the structure: body forces and surface tractions. Inclusion of these loads often complicate the initial equilibrium problem as the direction and magnitude of the loads may depend on the unknown reference conﬁguration. • Structural geometry The actual shape or surface geometry of the structure is one of two key parameters in the initial equilibrium problem. It plays a major role in determining the stresses that will act in the structure at various times. For a tension structure, curvature is the parameter that mostly aﬀects the structural behaviour. • Boundary conditions In methods where the geometry is treated as an unknown, it is necessary to introduce some boundary conditions to ensure a unique solution. • Internal force distribution The internal force distribution is the second key parameter. In order to obtain a safe and economical design, it is crucial to ﬁnd an appropriate force pattern. The initial equilibrium problem is a pure statics problem. Therefore, it is not necessary to introduce kinematic equations. However, some methods, e.g. the non-linear displacement method, are using kinematic equations to solve the initial equilibrium problem. This method requires material properties to be speciﬁed, although these need not be the actual properties. Fictitious material properties may be used to control the solution of the reference conﬁguration, [44]. As mentioned above, external loads may complicate the initial equilibrium problem. Therefore, it is assumed that the structural members are weight-less and that no loads act at the nodes. However, for completeness the external forces will still be present in many of the equations presented in this chapter, but the usual approach is to set them to zero. Initially, the only requirement put on the reference conﬁguration is that it should be in equilibrium. Consider a node i in a cable net where four cables meet, Figure 3.1. The equilibrium equations in the x-, y-, and z-directions at that node can be expressed as: x j − xi xk − xi x l − xi x m − xi + Tik + Til + Tim + Fxi = 0, Lij Lik Lil Lim yj − yi y k − yi yl − yi ym − yi Tij + Tik + Til + Tim + Fyi = 0, Lij Lik Lil Lim zj − zi zk − zi zl − zi zm − zi Tij + Tik + Til + Tim + Fzi = 0. Lij Lik Lil Lim

Tij

43

(3.1) (3.2) (3.3)

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

j

k i

m

l

z y

x

Figure 3.1: A connection in a cable net

Since the initial equilibrium is a static problem any conﬁguration where equations (3.1)–(3.3) are satisﬁed at each node is a solution to the present problem. Nevertheless, some solutions are better than others. The diﬀerent methods used to obtain these solutions will be described below. Their respective merits and drawbacks will be brought to light. At the end of each section, the features of each method are summarised.

3.2.1

The non-linear displacement method

Among the ﬁrst computer methods applied to the solution of the initial equilibrium problem was the non-linear displacement method, which is based on the large displacement ﬁnite element technique used for analysis of structural behaviour under external loads. As the same program can be used for both the initial equilibrium problem and the load analysis, this approach is quite common. Nonetheless, there are some serious disadvantages associated with this technique. This section will be divided into two subsections: the ﬁrst one dealing with cable nets and the second one with membranes. The non-linear displacement method may be summarised as follows. First, an element mesh in equilibrium with a prescribed force distribution is established in the horizontal plane. A three-dimensional form of the mesh is created by displacing the support points almost vertically until they attain their prescribed positions, Figure 3.2. An iterative algorithm, e.g. the Newton-Raphson method, is used to obtain the equilibrium conﬁguration of the deformed structure.

44

3.2. LITERATURE REVIEW OF INITIAL EQUILIBRIUM SOLUTION METHODS

Cable nets Argyris et al. [4] were among the ﬁrst to use the non-linear displacement method to solve the initial equilibrium problem for cable nets. Their method was developed in order to ﬁnd the form of the cable roofs at the 1972 Olympics in Munich. Straight bar elements were used to represent the cables. A method similar to the non-linear displacement method has been used by Barnes [8]. This method is an application of the dynamic relaxation method, where an initially out-of-balance structure is allowed to undergo damped vibrations until a steady equilibrium shape is obtained. The displacements of the ﬁxed nodes may give rise to an unfavourable force distribution in the net, when actual material properties are used. Therefore, when the ﬁxed nodes have reached their ﬁnal positions, a force adjustment procedure is applied to the net. In this procedure the original unstrained lengths of the elements are recomputed in such a manner that the desired force values are obtained. For a straight cable element satisfying Hooke’s law this is straightforward as the total lengths of the elements before and after adjustment must be the same, i.e. L0 + ∆L0 = L0 + ∆L0 . It leads to the following relation: L0 =

L0 + ∆L0 L0 + ∆L0 = . 1+ 1 + T /AE

(3.4)

After this adjustment step the structure is no longer in equilibrium. Therefore, some more iterations are needed to re-impose equilibrium. But, these iterations will not change the ﬁnal force distribution very much, so it will be close to the desired one. Another way to keep control over the forces is to use a very small modulus of elasticity for the cables, but then the control over the cable lengths is lost. With the procedure outlined above control of both the forces and cable lengths is possible.

Figure 3.2: The principles of the non-linear displacement method. from [16]

45

Reproduced

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

Membrane structures In principle, the application of the non-linear displacement method to membrane structures does not diﬀer very much from the case of cable nets. Some general approaches used for ﬁnding the reference conﬁguration of membranes will be discussed. First, a ﬁnite element suitable for membrane representation has to be selected. Owing to the great geometric non-linearity of membrane structures, it is preferable to use a dense mesh of primitive elements rather than a coarse mesh made up of higher order elements [77, 115]. Then, a stress distribution has to be chosen. Generally, two diﬀerent approaches are used: • the minimal surface approach, or • the nonuniform stress approach. The simplest choice for an initial equilibrium shape is the minimum surface conﬁguration, which is characterised by a state of isotropic tensile stress. In order to ﬁnd the minimal surface, it is assumed that the ﬂat membrane has a very small modulus of elasticity and is in the isotropic prestressed state [115]. For other members in the structure, such as beams or bars, actual material properties should be used [106]. Due to the small modulus used, the speciﬁed stresses in the membrane will only slightly change, even though large deformations occur during the displacements of the ﬁxed nodes [115]. The advantages of the minimum surface are its aesthetically pleasing shape and the associated uniform tensile stress. However, in some cases the minimal surface conﬁguration cannot satisfy all the architectural and structural requirements [115]. Since the mean curvature for minimum surfaces is zero, such surfaces are rather ﬂat and these have poor load bearing capacities. A nonuniform stress approach has to be used. In this, a very small modulus of elasticity is still used for the membrane, but as the name of the approach implies, the initial stresses are no longer speciﬁed uniformly. Following the same procedures as for the minimal surface approach, the ﬁnal conﬁguration should be in equilibrium with the nonuniform prestress. Several trial calculations are usually needed to ﬁnd a satisfactory equilibrium shape. Hence, it is not obvious how to choose the nonuniform stresses [114]. For structures, where it is diﬃcult to specify nonuniform initial stresses, an alternative approach can be used. This approach, which is based on elastic deformations, is similar to that of Argyris et al. [4] for cable nets. In this approach, the actual modulus of elasticity of the membrane is used. As for the cable net, the deformed equilibrium conﬁguration will have nonuniform and possibly large stresses. At this stage, the stresses due to deformation are removed by a stress adjustment algorithm and only the initial stresses are retained [115]. It was explained above that the way to keep the stresses within the elements constant during displacement is to assign a very small modulus of elasticity to the elements. However, in many cases the small elasticity has undesirable consequences, such as numerical instability and divergence of the solution [65]. These problems

46

3.2. LITERATURE REVIEW OF INITIAL EQUILIBRIUM SOLUTION METHODS

stem from the assumption of small strains made in the derivation of the membrane elements. For an ill-chosen initial surface, i.e. in most cases a horizontal surface, this assumption is often violated because gross changes in the element geometry take place during the displacements of the ﬁxed nodes. A way to avoid numerical instability is to choose a mathematically deﬁned initial surface close to the ﬁnal shape. Similar convergence problems were also reported in [36]. Another problem is that for surfaces which exhibit high curvatures, elements gather in certain regions of the surface and leave the remaining regions represented to a lesser accuracy. It is suggested that a suitable element arrangement in complex cases should be chosen with the aid of a physical model [66]. Recently, Bletzinger [13] used a method called the updated reference strategy, which is a numerical continuation method, to solve the initial equilibrium problem of membranes with minimal surfaces. This technique had to be used because of the occurrence of a singular stiﬀness matrix, which excludes the use of the ordinary NewtonRaphson algorithm. The stiﬀness matrix is singular when the nodal displacements are tangential to the membrane surface. To understand that, consider a plane, which obviously is a minimal surface. The surface area of that plane does not change if the geometry of discretization is changed, e.g. by small tangential displacements within the plane. Hence, the area variation of in-plane displacements is zero. A special case of the updated reference strategy is the force density method. As the updated reference strategy is claimed to be absolutely robust, it is perhaps the best non-linear displacement method available for the initial equilibrium problem. There are some drawbacks of the non-linear displacement method applied to the initial equilibrium problem. Both the ﬁnal shape and the stresses in the structure are diﬃcult for the designer to control. It is not an easy task to specify a desirable force distribution [44]. If actual material values are used it is possible for some elements in the structure to end up in compression [4]. The speciﬁcation of material properties (ﬁctitious or real) represents unnecessary additional decision making for the designer. In addition, the computations involved in this method are time consuming for large structures [44]. An advantage is that the program used to solve the initial equilibrium problem can also be used for further load analysis. The non-linear displacement method may be summarised as follows [44]. The variables speciﬁed by the designer are: • structural topology, • boundary conditions, and • material properties. The problem unknowns are: • structural geometry, and • internal force distribution.

47

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

The following additional constraint is placed on the solution: • an initial force distribution may be speciﬁed.

3.2.2

The grid method

Other methods for solving the initial equilibrium problem have been developed to overcome the problems associated with the non-linear displacement method. In many of these methods, a variety of limitations are imposed on the solution to transform the general non-linear problem into a linear one. The earliest method of this type was developed for orthogonal cable nets by Siev and Eidelmann in 1962 [44]. Their method use the equations (3.1)–(3.3). By placing restrictions on the cable net regarding the geometry, boundary conditions, and the internal stress distribution the remaining vertical equilibrium problem becomes linear [108]. How this is accomplished will be shown below. Siev and Eidelmann assumed that the horizontal

∆l ∆l ∆l

∆l ∆l ∆l

Figure 3.3: Cable net with orthogonal horizontal projection

projection of the cable net is orthogonal, i.e. xi = xk = xm and yi = yj = yl , with a grid size equal to ∆l, Figure 3.3. This gave the following modiﬁed equilibrium equations in the x- and y-directions (with zero external loads): ∆l ∆l + Til =0 Lij Lil ∆l ∆l + Tim =0 Tik Lik Lim Tij

48

(3.5) (3.6)

3.2. LITERATURE REVIEW OF INITIAL EQUILIBRIUM SOLUTION METHODS

Noting that Tij â&#x2C6;&#x2020;l/Lij and Til â&#x2C6;&#x2020;l/Lil are the horizontal components of the cable forces in the x-direction, and Tik â&#x2C6;&#x2020;l/Lik and Tim â&#x2C6;&#x2020;l/Lim the horizontal components in the y-direction, one can see that the horizontal force is constant in those directions. If the horizontal forces in the x- and y-directions at node i are denoted by Hix and Hiy , respectively, equation (3.3) can be written as: Hix (zj â&#x2C6;&#x2019; 2zi + zl ) + Hiy (zk â&#x2C6;&#x2019; 2zi + zm ) + Fiz = 0

(3.7)

If the horizontal components of the forces in the cables are speciďŹ ed, equation (3.7) becomes linear, and the only unknowns are the z-coordinates of the free nodes. Equation (3.7) is the discrete version of the vertical equilibrium equation of a shearfree membrane [119]: 2 2 x â&#x2C6;&#x201A; z + H  y â&#x2C6;&#x201A; z + Fz = 0, H (3.8) â&#x2C6;&#x201A;x2 â&#x2C6;&#x201A;y 2 x, H  y are the horizontal components of the prestressing force distribution where H (N/m) in x- and y-directions, respectively. Fz is the vertical (z-direction) load intensity (N/m2 ). The grid method may be summarised as follows [44]. The variables speciďŹ ed by the designer are: â&#x20AC;˘ structural topology, and â&#x20AC;˘ boundary conditions. The problem unknowns are: â&#x20AC;˘ structural geometry, and â&#x20AC;˘ internal force distribution. The following additional constraints are placed on the solution: â&#x20AC;˘ limited to line cable elements, â&#x20AC;˘ constant horizontal force along cables, and â&#x20AC;˘ limited to cable nets with straight-line plan projections.

3.2.3

The force density method

A linear solution to the initial equilibrium problem was derived in section 3.2.2 for orthogonal cable nets. However, because of the restrictions placed on the structure in that method, the resulting shapes are few. In this section, a more general method, called the force density method, will be presented.

49

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

The force density method is a strategy to solve the equations of equilibrium for a cable net without requiring any initial coordinates for the structure, just by taking advantage of a mathematical trick, [43]. Consider the equilibrium equations (3.1)– (3.3). These equations are non-linear since the element length L is a function of the node coordinates. If, instead of the element forces, the force-to-length ratios (denoted by q) for each element are speciﬁed (3.1)–(3.3) can be written as: qij (xj − xi ) + qik (xk − xi ) + qil (xl − xi ) + qim (xm − xi ) = 0, qij (yj − yi ) + qik (yk − yi ) + qil (yl − yi ) + qim (ym − yi ) = 0, qij (zj − zi ) + qik (zk − zi ) + qil (zl − zi ) + qim (zm − zi ) = 0.

(3.9) (3.10) (3.11)

It is obvious that the main advantage of using the force densities as description parameters for a cable net is that any state of equilibrium can be obtained by the solution of one system of linear equations. The equilibrium state so obtained has the prescribed force density in each element. No other conditions, such as equidistant meshes or constant element forces, are fulﬁlled. For a ﬁrst impression of the shape of the structure it is not necessary to consider auxiliary constraints, but in a detailed analysis these have to taken into account. For this purpose a nonlinear displacement method or the extended non-linear force density method can be used. The diﬀerence between these methods is that for the non-linear displacement method described in section 3.2.1 the number of equations is equal to the number of degrees of freedom, while for the non-linear force density it is equal to the number of additional constraints. In most cases the number of constraints is much smaller than the number of degrees of freedom [105]. Mollaert [75] applied the force density method to structures composed of both cables and compression members. To obtain a solution out of the plane of the ﬁxed nodes the tensile and compression parts of the structure were separated. At the common nodes the removed part was replaced by external forces. Both parts were then designed separately. In [77] the force density method was used together with a least squares minimisation approach presented in [43] to generate the cutting pattern for membrane structures. Although the problem of determining the membrane cutting pattern is outside the scope of this thesis it should be mentioned that the force density method can be used to solve also this problem. Due to the simple formulations of the force density method and the least squares minimisation technique, a solution can be obtained in short time although a very ﬁne mesh is used. These properties make the force density method a better choice than other methods, such as the dynamic relaxation method, at the patterning stage in the design of fabric structures [77]. As presented in [105] the force density method is limited to line cable elements. In reference 44 an extended version of the force density method, with curved cable and membrane elements, is presented. The method is based on assumed geometric stiﬀness matrices (3.12) KG xg = 0, where KG is the geometric stiﬀness matrix of the structure and xg the vector of nodal coordinates (x-, y- and z-coordinates). Equation (3.12) can be applied to any

50

3.2. LITERATURE REVIEW OF INITIAL EQUILIBRIUM SOLUTION METHODS

ﬁnite element structural model. Although (3.12) has the form of a standard stiﬀness equation, the unknowns are the nodal coordinates rather than nodal displacements. For structures composed of only straight bar elements, the set of equations in (3.12) are identical to the corresponding equations in the force density method. Nevertheless, if the choice of suitable force densities was quite easy, it is much more diﬃcult to choose the geometric stiﬀness matrices. For simple elements closed form expressions can be established, but for many elements the matrices have to be found by numerical integration. Even after the geometry has been solved, the determination of stresses for complex elements can be a problem [44]. Christou [24] implemented an elastic catenary element in the force density method to be able to take into account loads distributed along the cables. After the shape is found the force in each cable has to be found by iteration since the horizontal force is described by a non-linear equation. However, reﬁnement of the force density method to take into account distributed loads has less importance at the ‘form-ﬁnding’ stage since the loads are often neglected to simplify the problem. More recently Lai et al. [59] used the force density method to ﬁnd the form of a deployable reﬂector for space applications. They transformed the original membrane into an equivalent cable network and could, therefore, use the original equations of the force density method. This work shows that although the force density method was developed back in 1971 it ﬁnds new areas of application. The force density method may be summarised as follows [44]. The variables speciﬁed by the designer are: • structural topology, and • boundary conditions. The problem unknowns are: • structural geometry, and • internal stress distribution. The following additional constraints are placed on the solution: • limited to line cable elements, and • force density prescribed for each element.

3.2.4

Least squares stress determination methods

In all of the methods above the structural geometry is one of the problem unknowns. For structures where the geometry for some reasons is known the cable forces to

51

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

satisfy equilibrium has to be determined. In this section two methods that are appropriate for such cases will be presented. Both methods are derived from At = f ,

(3.13)

which is the matrix form of (3.1)–(3.3). Three cases can occur for (3.13) of which the following two are especially interesting: the over- and the under-determined cases [44]. The third case corresponds to a square matrix A. In the over-determined case there are more equilibrium equations than unknown cable forces. Because there is no exact solution to such a set of equations, it is necessary to seek an optimal set of forces which will only approximately equilibrate the loads on the structure. The optimal set of cable forces is selected by the least squares method (3.14) AT At = AT f . It should be emphasised that this method satisﬁes equilibrium in a least squares sense only. A disadvantage of this method is that the designer does not have much control over the force pattern. There is no restriction against compressive forces and the distribution of forces may be highly irregular. Some force control may be obtained by prescribing some of the forces. A major advantage of the method is that the solution is obtained by solving a set of symmetric linear equations [44]. The under-determined case occurs when there are more unknowns than equilibrium equations (membrane structures often fall into this category). For this case, there exists an inﬁnite number of exact equilibrium solutions for the forces. Therefore, an ideal force distribution t∗ has to be deﬁned and solved for. Generally, these ideal forces will not satisfy equilibrium. The actual forces are expressed in terms of the ideal forces and a set of deviations from the ideal force values, t = t∗ + ∆t.

(3.15)

Since the ideal forces are speciﬁed directly, the force deviations ∆t becomes the problem unknown. Equation (3.13) can now be written as: A∆t = f − At∗ .

(3.16)

The optimal solution to (3.16) is deﬁned by the set of force deviations that have the smallest Euclidean norm. This optimal solution is found by solving the following minimisation problem with Lagrange multipliers: ∆tT ∆t − 2kT [A∆t − (f − At∗ )] → min .

(3.17)

The solution to (3.17) is  −1 (f − At∗ ) . ∆t = AT AAT

(3.18)

The actual forces, obtained from (3.15), should satisfy equilibrium exactly. Since the force distribution has a minimal variation from the speciﬁed ideal forces it should be fairly smooth. However, large force deviations may occur if the structural geometry

52

3.2. LITERATURE REVIEW OF INITIAL EQUILIBRIUM SOLUTION METHODS

and the prescribed force distribution are incompatible. An advantage of the underdetermined least squares method is that the designer is given some control over the force distribution while the geometry may be speciﬁed exactly. As in the overdetermined case, the solution procedure only involves linear symmetric matrices [44]. The least squares stress determination methods may be summarised as follows, [44]: the variables speciﬁed directly by the designer are: • structural topology, • boundary conditions, and • structural geometry. The problem unknown is • internal force distribution.

3.2.5

A combined approach

No single solution method is optimal for all problems. It is possible to use several of the better solution methods in combinations to create more ﬂexible design tools. A combined approach lets the designer experiment with various methods to ﬁnd the optimal solution. Approximate results from one solution method may be used as input data to another method to obtain an improved solution [44]. For cable structures it seems that the best strategy is to ﬁrst use the force density method, which uses linear cable elements, and then take this solution to a non-linear ﬁnite element program, which uses more reﬁned cable formulations (see Chapter 4). Since the solution obtained by the force density method is approximate, but still very good, only a few iterations should be needed to ﬁnd the ‘true’ equilibrium conﬁguration.

3.2.6

Initial equilibrium of tensegrity structures

The most interesting class of space structures is that of the self-stressed systems, called tensegrity systems. Basically, tensegrity systems are composed of two sets of elements, a continuous set of cables, and a discontinuous set of rectilinear struts [81]. The self-stress makes these structures rigid without requiring any support to balance the stresses [80]. This property makes them very interesting from an economical point of view. Concerning initial equilibrium conﬁgurations of tensegrity structures, much research has been done on geometrical basis [81]. However, geometrical methods do not guarantee mechanical equilibrium and the solutions have to be checked by numerical techniques.

53

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

Pellegrino and Calladine have in a number of publications, [18–21, 92, 93], analysed statically and kinematically indeterminate frameworks with known geometries. In their analyses they used the equilibrium equation (3.13), repeated here, At = f ,

(3.19)

Bd = e.

(3.20)

and the compatibility equation By considering the four subspaces of the equilibrium matrix A (= BT ) useful information concerning the structure is obtained. The essence of their method is given in Figure 3.4 and Table 3.1. From a structural view, the perhaps most important result from their studies is that only some kinematically indeterminate frameworks with more than one independent state of self-stress (s > 1) can be stiﬀened to the ﬁrst-order by a single state of prestress. This is crucial, because tensegrity systems and cable nets must have ﬁrst-order stiﬀness in all possible modes [21] to prevent excessive displacements, which may lead to collapse. In most cases, the geometry of a tensegrity structure is unknown. To ﬁnd the selfstressed conﬁguration of a general tensegrity structure has proven to be much more diﬃcult than it is for cable and membrane structures, according to the few works dealing with this problem.

Table 3.1: Four diﬀerent types of structural assemblies. From [91].

I

Statically determinate and kinematically determinate

dim N (A) and dim N (AT ) s=0 m=0

II

Statically determinate and kinematically indeterminate

s=0 m>0

(3.19) has a unique solution for some particular r.h.s., but otherwise no solution. (3.20) has an inﬁnite number of solutions for any r.h.s.

III

Statically indeterminate and kinematically determinate

s>0 m=0

(3.19) has a inﬁnite number of solutions for any r.h.s. (3.20) has a unique solution for some particular r.h.s., but otherwise. no solution.

IV

Statically indeterminate and kinematically indeterminate

s>0 m>0

Both (3.19) and (3.20) have an inﬁnite number of solutions for some particular r.h.s, but otherwise no solution.

Assembly type

54

Static and kinematic features

Both (3.19) and (3.20) have a unique solution for any right hand side (r.h.s.).

3.2. LITERATURE REVIEW OF INITIAL EQUILIBRIUM SOLUTION METHODS

Dim.

r

Bar space Rb

Equilibrium A Row space R(AT ): bar tensions in equilibrium with the loads in the column space.

Compatibility B

=

s

r

Joint space R3j−c

Null-space N (A): states of self-stress. (Solutions of At = 0)

Column space R(A): loads which can be equilibrated in the initial conﬁguration.

=

Left null-space N (BT ): incompatible bar elongations.

=

Row space R(BT ): extensional displacements.

m

Column space R(B): compatibility bar elongations

Left null-space N (AT ): loads which cannot be equilibrated in the initial conﬁguration

=

Null-space N (B): inextensional displacements. (Solutions of Bd = 0)

Figure 3.4: The four fundamental subspaces associated with the equilibrium matrix A and the compatibility matrix B(= AT ). The sign ‘=’ indicates that the two subspaces coincide, while ‘⊥’ indicates that they are orthogonal complements of one another (note that s = b − r and m = 3j − c − r). Redrawn from [91].

Hanaor [45] used a non-linear displacement method based on Newton-Raphson iterations to ﬁnd the form of double layer tensegrity dome. He also stated, contrary to what is given in this chapter and in [82], that the initial equilibrium problem is kinematic [45]: “The assumed geometry is, in general a mechanism, when constraints on strut and tendon elongations are considered. Shape ﬁnding consists essentially of activating the mechanisms, until a state is reached when only elastic deformations are possible. This is the prestressable geometry.” But, an algorithm for the kinematic formulation was not presented in reference 45.

55

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

Motro et al. [82] used both the dynamic relaxation method and the force density method to ﬁnd the form of simple tensegrity systems. The results were compared to analytical solutions. The dynamic relaxation technique worked well for systems with few nodes but suﬀered from convergence problems when the number of nodes was increased. Although the force density method was applied only to small tensegrity structures, Motro et al. anticipated that the force density method could be convenient for the initial equilibrium solution of more complicated tensegrity systems [82].

3.3

The force density method

As shown in section 3.2.3, the force density method is popular among space structure researchers. The method was developed by Linkwitz and Schek [71] for the initial equilibrium problem of the cable roofs at the 1972 Olympic Games in Munich. It was ﬁrst published in [71] and later extended in [72] and [105]. The procedure in this section will mainly follow that in [105] but some additional comments and explanations will be given. Throughout this section the following notation will be used: V = diag (v), where v is a vector.

3.3.1

The linear force density method

In the force density method it is assumed that the cables are straight and pin-jointed to each other or to the supporting structure [105]. First, a graph of a network is drawn and all nodes are numbered from 1 to ns , all elements from 1 to m. The nf nodes which are to be ﬁxed points are taken at the end of the sequence. All the other n nodes are free. Thus, the total number of nodes is ns = n + nf . Then the connectivity matrix Cs is constructed with the aid of the graph. Each element j has the node numbers k and l (from k to l). The connectivity matrix Cs for the structure is deﬁne by (i = 1, 2, ..., ns ):   +1 for i = k, (3.21) cs (j, i) = −1 for i = l,   0 in the other cases. This connectivity matrix can be divided into two matrices   Cs = C Cf ,

(3.22)

where C and Cf contains the free and ﬁxed nodes, respectively. Denoting the vectors containing the coordinates of the n free nodes x, y, z, and similarly for the nf ﬁxed nodes xf , yf , zf , the coordinate diﬀerences for each element can be written as: u = Cs xs = Cx + Cf xf , v = Cs ys = Cy + Cf yf , w = Cs zs = Cz + Cf zf .

56

(3.23) (3.24) (3.25)

External loads ∗ ∗ ∗ ∗ ∗

Structural topology ∗ ∗ ∗ ∗ ∗

∗ ∗

Boundary conditions ∗

? ?

Structural geometry ?

?

?

? ?

Internal forces ?

Decription of notes (related to zero external loads) A: Requires trial surface geometry. B: Requires trial stress distribution. C: Requires material properties. D: Zero elastic modulus gives a conﬁguration in equilibrium with precribed forces. E: Large elastic modulus gives control over element length. F: Local compression may occur. G: Expensive nonlinear solution. H: Limited to line cable elements. I: Constant horizontal force along cables. J: Limited to structures with orthogonal straight-line horisontal projections. K: Edges must be ﬁxed (or rigid). L: A force density must be prescribed for each element. M: Equilibrium solution is approximate. N: Stresses may be constrained. O: The force distribution may be irregular.

The non-linear displacement method The grid method The force density method The over-determined least squares stress method The under-determined least squares stress method

Description

− −

Material properties ∗

B, N

F, M–O

H–K H, L

A–G

Notes

Table 3.2: Summary of initial equilibrium solution methods (‘∗’ designer speciﬁed, ‘?’ problem unknown, ‘−’ not included). From [44].

3.3. THE FORCE DENSITY METHOD

57

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

The equilibrium equations for the free nodes for the x-, y- and z-directions are written as: CT UL−1 t = fx ,

(3.26)

CT VL−1 t = fy ,

(3.27)

CT WL−1 t = fz .

(3.28)

By using the force-to-length ratios for the elements, i.e. the force densities, as description parameters, (3.26)–(3.28) are written as: CT Uq = fx ,

(3.29)

CT Vq = fy ,

(3.30)

T

C Wq = fz ,

(3.31)

where the vector q, of length m, is described as: q = L−1 t.

(3.32)

Using (3.23)–(3.25) and the following identities: Uq = Qu, Vq = Qv, Wq = Qw,

(3.33) (3.34) (3.35)

equations (3.29)–(3.31) are written as: CT QCx + CT QCf xf = fx ,

(3.36)

CT QCy + CT QCf yf = fy ,

(3.37)

CT QCz + CT QCf zf = fz .

(3.38)

By setting D = CT QC and Df = CT QCf , equations (3.36)–(3.38) can be written as: Dx = fx − Df xf , Dy = fy − Df yf , Dz = fz − Df zf .

(3.39) (3.40) (3.41)

Equations (3.39)–(3.41) are solved using elementary algebra: x = D−1 (fx − Df xf ) , y = D−1 (fy − Df yf ) , z = D−1 (fz − Df zf ) .

(3.42) (3.43) (3.44)

Two cases for matrix D can occur [82, 105]: 1. Determinant of D = 0 The matrix D has full rank and the form of the structure is governed by the

58

3.3. THE FORCE DENSITY METHOD

values chosen for the force densities. In the case of a prestressed cable net with a given connectivity (i.e. ﬁxed C and Cf ) the number of equilibrium shapes is identical to the number of vectors q. This justiﬁes the use of the force densities as description parameters for a cable net. Attention must be paid to the speciﬁc case which occurs when all ﬁxed nodes are coplanar, because then the solution will also be planar and without interest (see section 3.4.4). 2. Determinant of D = 0 The system can be solved for x only when vectors Df xf lie is the space spanned by the linearly independent vectors of matrix D (if the external loads are zero). Similarly for y and z. To illustrate the properties of the linear force densities a simple example will now be given. Consider the structure in Figure 3.5 with all ﬁxed nodes in the x–y plane.

6 2

1

1

2

3

6

5

4 3

7

8 8

9 10

7 4

5

y 12

11

z

x

9

Figure 3.5: A simple cable structure with zero external loads. The arrows indicate the directions of the elements.

59

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

The connectivity matrix Cs for this structure with, ďŹ rst, n = 5 free nodes and, then, nf = 4 ďŹ xed nodes, giving ns = 9 is written as (elements left out are zero): ďŁŽ

ďŁš

â&#x2C6;&#x2019;1

1

ďŁŻ ďŁş ďŁŻ ďŁş 1 â&#x2C6;&#x2019;1 ďŁŻ ďŁş ďŁŻ ďŁş â&#x2C6;&#x2019;1 ďŁŻ 1 ďŁş ďŁŻ ďŁş ďŁŻ 1 ďŁş â&#x2C6;&#x2019;1 ďŁŻ ďŁş ďŁŻ ďŁş ďŁŻ ďŁş 1 â&#x2C6;&#x2019;1 ďŁŻ ďŁş ďŁŻ ďŁş ďŁŻ ďŁş 1 â&#x2C6;&#x2019;1 ďŁŻ ďŁş Cs = ďŁŻ ďŁş. ďŁŻ ďŁş 1 â&#x2C6;&#x2019;1 ďŁŻ ďŁş ďŁŻ ďŁş ďŁŻ ďŁş 1 â&#x2C6;&#x2019;1 ďŁŻ ďŁş ďŁŻ ďŁş ďŁŻ ďŁş 1 â&#x2C6;&#x2019;1 ďŁŻ ďŁş ďŁŻ ďŁş 1 â&#x2C6;&#x2019;1 ďŁŻ ďŁş ďŁŻ ďŁş ďŁŻ 1 â&#x2C6;&#x2019;1 ďŁş ďŁ° ďŁť 1 â&#x2C6;&#x2019;1      C Cf

(3.45)

The matrices D and Df describe the equilibrium at the free and ďŹ xed nodes, spectively, and are written as: ďŁŽ 0 â&#x2C6;&#x2019;q4 0 0 q1 + q3 + q4 ďŁŻ ďŁŻ 0 q2 + q 5 + q 6 â&#x2C6;&#x2019;q5 0 0 ďŁŻ ďŁŻ â&#x2C6;&#x2019;q4 â&#x2C6;&#x2019;q5 q4 + q5 + q8 + q9 â&#x2C6;&#x2019;q8 â&#x2C6;&#x2019;q9 D=ďŁŻ ďŁŻ ďŁŻ 0 0 â&#x2C6;&#x2019;q8 q7 + q8 + q11 0 ďŁ° 0 and

â&#x2C6;&#x2019;q9

0 ďŁŽ

â&#x2C6;&#x2019;q1 â&#x2C6;&#x2019;q3

ďŁŻ ďŁŻ â&#x2C6;&#x2019;q2 0 ďŁŻ ďŁŻ 0 Df = ďŁŻ 0 ďŁŻ ďŁŻ 0 â&#x2C6;&#x2019;q 7 ďŁ° 0

0

0 0

0

â&#x2C6;&#x2019;q6

0

0

0

0

â&#x2C6;&#x2019;q11

reďŁš ďŁş ďŁş ďŁş ďŁş ďŁş ďŁş ďŁş ďŁť

q9 + q10 + q12 (3.46)

ďŁš ďŁş ďŁş ďŁş ďŁş ďŁş. ďŁş ďŁş ďŁť

(3.47)

â&#x2C6;&#x2019;q10 â&#x2C6;&#x2019;q12

Figure 3.6 shows the resulting shapes of this structure for diďŹ&#x20AC;erent force density values. It is seen that for a single element an increase in the force density relative to the others results in a contraction of that element. The opposite holds for a decrease in the force density, even more emphasised with negative values.

60

3.3. THE FORCE DENSITY METHOD

(a) Elements 1–12 have q = 1

(b) Elements 1–3, 5–12 have q = 1 and element 4 has q = 10

(c) Elements 1–3, 5–12 have q = 1 and element 4 has q = −0.1

(d) Interior elements have q = 1 and edge elements have q = 5

Figure 3.6: Diﬀerent equilibrium conﬁgurations for the plane structure in Figure 3.5.

3.3.2

The non-linear force density method

Multiple equilibrium shapes can be obtained with the linear force density method. However, these shapes may be unsatisfactory from a structural point of view. For example, the mesh may be irregular and the force distribution unsmooth. Therefore, it is necessary to ﬁnd a conﬁguration which is in equilibrium and which also satisﬁes additional conditions. These conditions are generally non-linear and so is the extended force density method. It is preferred to start the non-linear computations using the shape found with the linear method. In contrast to the non-linear

61

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

displacement method, the number of non-linear equations here is identical to the number of additional conditions and independent of the number of nodes. Thus, this non-linear approach is more eﬃcient than the non-linear displacement method. The additional constraints are put in the following form [105]: g (x, y, z, q) = 0,

(3.48)

where it is assumed that the constraints are functions of the coordinates and force densities. Since the coordinates also are functions of the force densities, (3.48) is written as: (3.49) g∗ (q) = g(x(q), y(q), z(q), q) = 0. Equation (3.49) is generally non-linear and has to be linearised to be solvable g∗ (q0 ) +

∂g∗ (q0 ) ∆q = 0. ∂q

(3.50)

Equation (3.50) can be written as: GT ∆q = r where

∂g∗ (q0 ) ∂q

(3.52)

r = −q∗ (q0 ).

(3.53)

GT = and

(3.51)

Equation (3.51) is similar to the non-linear Newton-Raphson equation used in ﬁnite element analysis. In many cases the number of constraints is less than the number of elements, i.e. m > r. This means that (3.51) is under-determined and has m − r linearly independent solutions. From these solutions a single solution can be chosen by considering the following variational principle [105]: ∆qT ∆q → min .

(3.54)

A Lagrange multiplier method is used to solve for the force density correction ∆q. The variations of the following functional:  (3.55) M = ∆qT ∆q − 2kT GT ∆q − r give the equations 1 ∂M = ∆q − Gk = 0, 2 ∂∆q 1 ∂M = GT ∆q − r = 0. 2 ∂k

(3.56) (3.57)

The solution to (3.56) and (3.57) is given by (3.62)–(3.64). A damped version of (3.51) is (3.58) GT ∆q = r + ω.

62

3.3. THE FORCE DENSITY METHOD

The minimum principle associated with (3.58) is ∆qT ∆q + ω T Pω → min .

(3.59)

Another approach, based upon modiﬁed damping, is GT ∆q = Ωr.

(3.60)

The associated minimum principle of (3.60) is ∆qT ∆q + (i − ω)T P (i − ω) → min .

(3.61)

where iT = (1, 1, ..., 1). According to Schek, [105], the approach given by (3.61) is useful if there are large changes in the force densities because the damped iterations converge without oscillation. Another advantage of this approach is that for constraints which cannot be satisﬁed the iterations may stop at a shape which fulﬁls the constraints as closely as possible. The Lagrange factors are given by k = T−1 r, where

 T  G G T = GT G + P−1   T G G + P−1 R2

in case (3.51), (3.54), in case (3.58), (3.59), in case (3.60), (3.61).

(3.62)

(3.63)

The solution to the minimisation problem is ∆q = Gk.

(3.64)

For the subsequent iterations, a new q is computed as: q1 = q0 + ∆q,

(3.65)

until convergence within a given tolerance. In [105] the following additional constraints are considered: • node distance, • element force, and • unstrained length. The Jacobian matrix GT for these constraints will now be given. Node distance constraints If elements with very large axial stiﬀnesses are used this kind of constraint may arise. This condition, with r prescribed node distances ls , is written as: gd = l − ls . The Jacobian matrix for the node distance constraint is written as: −1  GTd = −L U C D−1 CT U + V C D−1 CT V + W C D−1 CT W . This matrix is of dimension r × m.

63

(3.66)

(3.67)

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

Force constraints In many cases it is desirable to specify the forces for some elements. The force condition, with r prescribed forces tv , is written as: gf = L q − tv = Q l − tv The Jacobian matrix for this case is −1  GTf = L − Q L U C D−1 CT U + V C D−1 CT V + W C D−1 CT W .

(3.68)

(3.69)

In [105] no further comment to this equation is given. However, inspection of (3.69) reveals that the subtraction cannot be performed as the matrix L is of dimension r × r and the dimension of the resulting matrix after the minus sign is r × m. This problem can be solved if matrix L is expanded with m − r zero columns at the positions corresponding to the numbers of the unconstrained elements. Then the subtraction can be performed. Unstrained length constraints For fabrication purposes it may be necessary to prescribe the unstrained lengths of some elements (see also section 3.4.1). This condition, with r unstrained lengths luv , is written as: (3.70) gu = lu − luv . For a stressed straight cable element i, which satisﬁes Hooke’s law, the unstrained length is computed as: AEi li . (3.71) lui = AEi + Ti The Jacobian matrix for this unstrained length constraints is 2 −1 2 −3  GTu = −Lu K −Lu L U C D−1 CT U (3.72) + V C D−1 CT V + W C D−1 CT W . As for the force constraints, the subtraction to obtain GTu cannot be performed due to diﬀerent dimension of the matrices involved. The remedy is the same as 2 −1 above; after −Lu K is computed the resulting matrix is expanded with m − r zero columns. After considering the following limit case: lim GTu = GTd

AE→∞

(3.73)

a misprint in the expression for GTu was detected in [105]. Mixed constraints In many cases one may want to take into account more than one of the constraints given above. This can be accomplished by writing the total Jacobian matrix as:   G = Gd Gf Gu . (3.74)

64

3.4. EXAMPLES

Practical experience has shown that convergence is very slow if the magnitude of the numerical force values are much larger than the numerical values for the lengths. This occurs when an element is assigned with both a force and a length constraint. Nevertheless, the convergence rate can be signiﬁcantly improved if the force values are scaled down (divided by a scale parameter) so that they become of the same order as the lengths. No theoretical analysis has been done to justify this scaling procedure, but it has worked very well for the examples given in the next section.

3.4

Examples

Until now mainly theoretical issues concerning form-ﬁnding have been discussed. The force density method will be described by some additional examples in this section. Some of the examples that will be presented below are chosen to highlight problems that need special techniques and some just to show the versatility of the force density method in ﬁnding the shape of diﬀerent cable net structures. Finally, the applicability of the method for structures including both cables and struts will be checked. It should be emphasised that the aim of the following examples is to investigate the behaviour of the force density method when applied to diﬀerent types of cable structures. Numerical stability and convergence rate are in particular studied. No load bearing or architectural aspects are taken into account. In many of the examples the dimension of the structure and internal forces are chosen arbitrarily.

3.4.1

Smaller cable nets

To construct a cable net one must know how the cables are best arranged. Generally, two arrangements are worth considering: • Geodesic mesh, in which the cables run along the geodesic lines in the surface. A geodesic line is the shortest way between two points on a surface. This approach minimises the use of material but the manufacturing can be quite complicated. • Uniform mesh in the unstrained state. From a constructional point of view this approach is the best one. As, an error in length of 0.1 % can give rise to an error in force of about 50 %, accurate placing of the connections is crucial [63]. An equidistant mesh enables the mounting to be done in a factory. At the building site, the net can be assembled on the ground and hoisted into position. Figure 3.7 shows an example from [72], where certain elements are assigned with a constant unstrained length.

65

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

4

5

3

6 2

1 Figure 3.7: The uniform mesh approach. Elements on the sides of non-shaded areas should have equal unstrained lengths. Redrawn from [72].

Table 3.3: Coordinates in metres for the ﬁxed nodes of the structure in Figure 3.7.

Node 1 2 3 4 5 6

Coordinates x y 0.0000 0.0000 −6.4897 2.2285 −6.9932 9.3859 −2.4917 11.7810 3.8300 11.5901 7.0874 4.2153

z 0.0000 3.0000 0.0000 4.0000 0.0000 5.0000

Considering the magnitude of the prestressing force, the usual rule of thumb is that the magnitude should be such that no element goes slack under any load condition. However, according to Leonhardt and Schlaich [63] this rule “causes unacceptably high forces.” They further conclude that “it will never be possible to establish general rules for ﬁnding the shape of cable net structures.” This is due to the many factors which aﬀect the shape and load-bearing characteristics of a cable net. Among the factors are: prestress magnitude and distribution, mesh geometry of the net, edge rigidity (concrete ring or edge cables), angle between net cables and edge cables and stiﬀness ratios of the members [63]. In general, the magnitude of the prestressing force is determined by the allowable deformations and fatigue strength of the cables. However, an increase in prestressing force is not as eﬀective in reducing the deformations as an increase in curvature of the net [63]. Of course, the distribution of the prestressing force must be quite uniform.

66

3.4. EXAMPLES

Several conﬁgurations have to checked before ﬁnding one which best satisﬁes all requirements put on the structure. A common approach for prestressed cable nets is given in reference 72. In this approach, the interior cables are assigned with unstrained length constraints and the elements in contact with the edges are assigned with force constraints. In this way a good compromise between load-bearing behaviour and construction ease is reached. A special problem which may arise when using the outlined technique is that the elements connected to the edge cables may have too large angle changes, Figure 3.8. Since the real cable is continuous and has a ﬁnite bending stiﬀness, such angles cannot occur in practice. A technique to avoid these angle changes is to assign force constraints also to the interior cables. For a three-dimensional structure a conﬁguration that satisﬁes all the constraints exactly may not be found. But if the modiﬁed damped version of the non-linear force density method is used, the iterations stop at a satisfactory shape. To ﬁnd the shape of the structure in Figure 3.7, the following procedure was used: 1. All interior cable were assigned with a force density equal to 200 kN/m. The force density for the edge cables was 1200 kN/m. The interior cables had the following constraints: cable force equal to 200 kN and unstrained length of 1 m. No constraints were assigned to the edge cables. The heights of all the ﬁxed nodes are changed to z = 0. Thus, the cable net lies in the x–y plane. The stiﬀness of the cables was AE = 100 MN. 2. With the prescribed force densities, the linear force density method gave the shape shown in Figure 3.8. The net has a nearly square mesh, but some large distortions occur near the edges. 3. With all the ﬁxed nodes still in the same plane, 20 iterations with the nonlinear force density method gave a fairly smooth layout without large angle changes near the edges. Note that this conﬁguration does not satisfy the constraints. This step is an intermediate step to get rid of the irregularities in the edge areas and get a nearly equidistant interior net with a uniform force distribution. 4. In this, the last step, all the ﬁxed nodes have their original positions given by Table 3.3. The force constraints for all interior cables assigned with unstrained length constraints are removed. Only interior cables connected to the edge cables still have force constraints. The reason for this modiﬁcation is that for a three-dimensional cable net both force and unstrained length constraints cannot generally be satisﬁed within the net. However, for the plane conﬁguration in step 3 it is possible to satisfy both constraints. To obtain the ﬁnal shape shown in Figure 3.9 required 19 iterations. The reason for the somewhat slow convergence of the non-linear force density method, if one compares with the Newton-Raphson technique used in ﬁnite element analyses, is probably due to the highly distorted meshes in some parts of the net. If

67

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

large changes in the force densities of only a few elements are needed to satisfy the constraints, slow convergence follows.

Figure 3.8: Too large angles of the cables occur near the edges if only unstrained length constraints are used for interior cables. Dashed line = desired position of the cable. Redrawn from [72].

Figure 3.9: Three dimensional view of the equidistant cable net with smooth force distribution.

68

3.4. EXAMPLES

The next example will show that diﬀerent ﬁnal conﬁgurations will be obtained depending on the initial force density values of the edge cables. In the ﬁrst example the target shape is an interior mesh width of 1 metre and a prestressing force of 200 kN for each of the cables in the edge area. The stiﬀness of the cables was AE = 1000 MN. For both nets the starting value of the force density for each interior cable was qinterior = 200 kN/m. For the ﬁrst net, Figure 3.10, the force density in each of the edge cables was qedge = 5qinterior and for the second net, Figure 3.11, it was qedge = 50qinterior . The same procedure as for the previous example was used. To obtain the plane net, with force constraints for all interior cables, 8 iterations were required for net 1 and 7 iterations for net 2. This diﬀerence is probably due to the fact that the starting shape of net 2 is closer to the ﬁnal shape, see Figure 3.11(b). As above, the ﬁnal three-dimensional conﬁguration is obtained by removing the force constraints from interior cables not connected to the edge and ﬁxing the support nodes in their original positions. For both nets the ﬁnal shape was obtained with only 4 iterations. The solutions show that, depending on the starting values of the force densities in the edge cables, which are unconstrained, diﬀerent conﬁgurations and force values are obtained.

69

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

(a) Initial conﬁguration. qinterior = 200 kN/m and qedge = 1000 kN/m.

(b) Final conﬁguration.

( 6,6,4 )

3.976 4.025 3.858 3.944

( 0,12,0 )

3.789 (−6,6,4 )

3.916 3.789 3.944 3.858 4.025 3.976 ( 0,0,0 )

(c) Final conﬁguration. Coordinates of ﬁxed points (m) and edge cable forces (MN).

Figure 3.10: Hyperbolic paraboloid net 1.

70

3.4. EXAMPLES

(a) Initial conﬁguration. qinterior = 200 kN/m and qedge = 10000 kN/m.

(b) Final conﬁguration.

( 6,6,4 )

6.314 6.378 6.217 6.317

( 0,12,0 )

6.165 (−6,6,4 )

6.297 6.165 6.317 6.217 6.378 6.314 ( 0,0,0 )

(c) Final conﬁguration. Coordinates of ﬁxed points (m) and edge cable forces (MN).

Figure 3.11: Hyperbolic paraboloid net 2.

71

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

One of Frei Otto’s most famous membrane structures is the star-shaped pavilion over the dance fountain in Cologne. It was built for the 1957 Federal Garden Exhibition. The tent is still standing, although it was planned for a single summer [87]. This structure inspired the author in the next example—a star-shaped cable net. The net vertices lie on two circles with radii 14.421 m and 8.165 m, respectively. The height diﬀerence between the vertices is 4 m. All cables have the axial stiﬀness AE = 100 kN. The starting values of the force densities are: 200 N/m for the interior cables, 2000 N/m for the valley cables, 6000 N/m for the ridge cables and 1000 N/m for the edge cables. An unstrained length of 1.8 m are assigned to all net cables that are perpendicular to the ridge cables. Force constraints are assigned to all interior cables (not ridge or valley cables) and ridge cables. The force values are 200 N and 8000 N, respectively. To obtain the ﬁnal shape, shown in Figure 3.13, 10 iterations were required.

Figure 3.12: The pavilion over the Cologne Dance Fountain. Reproduced from Architectural Design, No. 117, “Tensile Structures”, 1995.

72

3.4. EXAMPLES

(a)

(b)

Figure 3.13: A star-shaped cable net structure inspired by Frei Otto’s pavilion over the Cologne Dance Fountain.

3.4.2

A large cable net

The previous examples, which had simple, symmetric conﬁgurations, show that the outlined form-ﬁnding procedure gave the desired shapes. In this section, a more complicated cable net will be analysed. The cable net of the German pavilion at the 1967 World’s fair in Montreal, Figure 3.14, inspired the author in the layout of the present example. As in the previous examples, a plane connectivity mesh with equidistant width is ﬁrst drawn, Figure 3.15. Then the elements are assigned with force density values of diﬀerent magnitudes so that the initial plane shape resembles that of the drawn mesh. The initial values of the force densities are: 67 kN/m for net cables, and 667 kN/m for edge cables (including the large cables within the

73

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

net, illustrated in Figure 3.15). The target shape is an unstrained mesh width of 3 m and a force of 200 kN in each of the net cables connected to the edges. The stiﬀness of each cable is 100 MN. Since the initial shape, shown in Figure 3.16(a), is highly distorted in some parts, more than 100 iterations were required to get a good plane initial shape. Nonetheless, a three-dimensional shape, which satisﬁes the constraints, could not be obtained. The reason for this is that the heights of the ﬁxed nodes are large relative to the other dimensions of roofs. Thus, an initial equidistant mesh in plane is incompatible with an equidistant mesh of the threedimensional structure. In some cases it might be possible to draw a slightly ﬁner mesh width than the target mesh width in space. However, this problem is not easily solved since quite detailed knowledge of the ﬁnal shape of the net is needed. One way to overcome this problem is to use a physical model to calculate the correct number of cables. For cable nets with smaller height-to-length ratios an equidistant initial mesh will work. Even though the desired shape could not be found, a three-dimensional shape was obtained by using the force densities of the plane structure in Figure 3.16(b) and the original coordinates for the ﬁxed points. As shown in Figure 3.17 the ‘ﬁnal’ shape has a smooth cable arrangement.

Figure 3.14: Aerial view of the German pavilion. Reproduced from Architectural Design, No. 117, “Tensile Structures”, 1995.

74

3.4. EXAMPLES

12 11

10

13 1

4 2

14 9

3

5

8 6 7

Figure 3.15: Mesh for a large cable net.

Table 3.4: Coordinates in metres for the ﬁxed nodes of the large cable net in Figure 3.15.

Node 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Coordinates x y 0.000 52.500 10.893 42.188 0.000 22.500 40.893 42.188 44.126 12.362 15.910 6.590 0.000 0.000 −18.106 9.142 −22.500 22.500 −14.317 56.974 −1.874 67.382 45.028 71.901 67.218 56.574 66.015 25.790

75

z 10.000 0.000 15.000 20.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

(a) Initial plane conﬁguration

(b) ‘Final’ plane conﬁguration

Figure 3.16: Equilibrium conﬁgurations in the horizontal plane.

76

3.4. EXAMPLES

Figure 3.17: Three-dimensional views of the ‘ﬁnal’ conﬁguration of the large cable net.

3.4.3

Cooling towers

Cooling towers enable a nuclear power plant to be built at any location independent of natural water supplies. The power station capacity is dependent on the cooling capacity. A high cooling capacity can be generated either by a pair of large or a number of small cooling towers. Due to inﬂuences concerning the air intake and wind forces when several towers are used, it is better to built only one or two [110]. Usually, the cooling towers are made of concrete. After the Munich Olympics in 1972, the structural engineers in Germany saw the potential of using a prestressed cable net shell for cooling towers instead of the ordinary concrete shell. The main advantage with the cable net tower is its high safety. Due to its ﬂexibility and lightness it is insensitive to earthquakes and settlements on bad ground, which would damage a concrete tower. If the pylon is made of steel

77

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

the safety is further enhanced [110]. The tower with the highest throughput of air per hour is that at SchmehausenUentrop. The height of its pylon is 180 m. Nevertheless, besides the economical and structural questions the one concerning the harmony of the landscape is a strong issue for not building cooling towers at all. The cable net cooling towers do not, like most other tension structures, ﬁt in the landscape [110]. The force density method can be used to ﬁnd the shape also for cooling towers with diﬀerent mesh conﬁgurations. For both the towers shown in Figure 3.18, the radius of the base is 5 m and the height of the pylon is 22 m. In the analyses, it is assumed that the pylon top is a ﬁxed node. The stiﬀness of the elements is AE = 100 MN. The diamond-shaped towers had the following starting values for the force densities: 200 kN/m for the net cables, 400 kN/m for the hangers and −2700 kN/m for the segments of the upper ring. For the rectangular net the values were: 1200 kN/m, 400 kN/m and −3300 kN/m, respectively. These values were chosen by studying the shapes of the nets for diﬀerent force density values (trial and error). Each tower consists of 16 hangers and 16 ring segments. The target shape was, for both towers, determined by the radius of the upper compression ring and the length of the hangers. Node distance constraints were assigned to the segments of the compression ring and the hangers. The strained lengths of each ring segment and hanger are 1.5607 m and 5.3852 m, respectively. This corresponds to a compression ring with radius of 4 m. For both nets, 5 iterations were required to obtain the desired shape. It should be noted that, in the present examples no attention has been paid to load bearing capacity or other requirements such as equidistant interior net.

78

3.4. EXAMPLES

(a) Diamond net

(b) Rectangular net

Figure 3.18: Two types of cooling towers.

3.4.4

A structure composed of both cables and struts

Until now structures with mainly cables has been studied in detail. Therefore, it is necessary to check the applicability of the force density method to structures composed of both cables and struts. Consider the simple structure in Figure 3.19. l

(x0 0 z0 ) z

y

x

Figure 3.19: A simple structure composed of both cables and a strut.

79

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

The governing matrices for this can be written as:   −qs 2qc + qs , D= −qs 2qc + qs   −qc −qc Df = . −qc −qc

(3.75)

(3.76)

If det(D) = 0 then the solutions, with the ﬁxed points given in Figure 3.19, are  xT = x0 + l/2 x0 + l/2 , (3.77)  T y = 0 0 , (3.78)  (3.79) zT = z0 z0 . Hence, the length of the strut is zero irrespective of the values chosen for the force densities as long as they do not give a singular matrix D. The elements for the structure do not have their speciﬁed force densities. Solutions ‘out-of-plane’ can be obtained by ﬁxing nodes that are not coplanar to the other nodes. However, then the equilibrium of that node is disturbed since any unbalanced load can be resolved by the ﬁxed support. One way to avoid this problem is to introduce more elements and connect one of the ends of the new elements to the free nodes. The other end is ﬁxed at the positions which the free nodes are supposed to have, Figure 3.20. Then, the new elements are assigned with constraints saying that the lengths of these elements should approach zero. If the force densities of the constrained elements do not change during the iterations, the element forces will also approach zero. This results in the desired shape with almost the prescribed element forces. Nevertheless, since the geometry for a prestressed structure, composed of both cables and struts, in most cases is speciﬁed the suggested approach is useless. Better methods are at hand, e.g. the subspace method by Pellegrino and Calladine [91]. For more complicated structures the approach suggested by Mollaert [75], where the compression members are replaced by external forces, may be used. No calculations have, however, been performed to check this approach.

z

y

x

Figure 3.20: The structure in Figure 3.19 with additional elements.

80

3.4. EXAMPLES

3.4.5

Cable dome

The cable domes by David Geiger represent the most advanced type of very large space structures. Pellegrino [92] has analysed a cut-down version of a larger dome (Figure 3.22). For this dome the degree of kinematic indeterminacy m, i.e. the number of independent mechanisms, is 13 and the degree of static indeterminacy s, i.e. the number of independent states of self-stress, is one. The mechanisms must be ﬁrst-order inﬁnitesimal, which they are if and only if there exists a state of selfstress which can impart positive ﬁrst-order stiﬀness to every mechanism. Structures with higher-order mechanisms are not stiﬀ enough to be used as real structures. Two of the thirteen inextensional mechanisms are shown in Figure 3.23. The single state of self-stress (s = 1) is obtained by computing the basis for the null-space of A. Whether this self-stress imparts the structure with ﬁrst-order stiﬀness can be checked by a method given in [20].

Figure 3.21: David Geiger’s cable dome

81

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

Figure 3.22: The cut-down version of a cable dome analysed by Pellegrino [92].

With the approach by Calladine and Pellegrino [93], properties of the initial conďŹ guration of a framework with known geometry can be obtained. However, for more complicated structures, where the geometry is unknown, this approach cannot be used.

(a) Mechanism 1

(b) Mechanism 2

Figure 3.23: Two of the thirteen inextensional mechanisms of the cable dome. Redrawn from [92].

82

3.4. EXAMPLES

3.4.6

Tensegrity structures

As mentioned in section 3.2.6 very little work has been done on form-ﬁnding of tensegrity structures. It was shown in section 3.4.4 that to avoid a coplanar solution of a structure composed of cables and struts additional nodes must be ﬁxed. This approach was used by Motro et al. [82] to ﬁnd the conﬁguration of the structure in Figure 3.24(a). Four of the eight nodes had to be ﬁxed to get a satisfactory solution. But is this solution in a state of self-stress? Remember that the equilibrium at a ﬁxed node (support) in a framework always can be resolved. Some improvements could perhaps be provided by the approach suggested by Mollaert [75]. Nevertheless, it seems that a reliable method for the initial equilibrium problem of tensegrity structures is not available. Examples of some simple structures are shown in Figure 3.24

(a) Skew 4-prismatic system

(b) Truncated tetrahedron

Figure 3.24: Some small tensegrity structures.

Larger tensegrity structures can be constructed, Figure 3.25, by assembling elementary self-stressed modules [79], Figure 3.24(a). This example shows the potential of tensegrity structures as economical solutions for spanning large space. However, these ideas have yet to be realised at a larger scale.

83

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

Figure 3.25: A double layer single curvature tensegrity system. Redrawn from [79].

3.4.7

Conclusions

The examples have shown that the force density method is well suited to ﬁnd the initial equilibrium conﬁguration of diﬀerent kinds of cable nets. However, for complex nets, physical models might be needed to construct the initial mesh. The convergence rate of the non-linear force density method was relatively high when only one constraint was assigned to speciﬁc elements. When several elements were assigned with two constraints, the convergence rate was much lower. For some nets it is impossible to satisfy two constraints, e.g. prescribed force and unstrained length for interior cables, and for them the iterations converged to a conﬁguration close to the ﬁnal shape. It was also found that the starting shape had a considerable eﬀect on the number of iterations needed for convergence. For structures composed of both cables and struts it is more diﬃcult to control the ﬁnal solution. For these structures, the most common approach is to specify the geometry and then solve for the cable forces by analysing the subspaces of the equilibrium matrix A.

84

Chapter 4 Finite cable elements 4.1

Introduction

Today, in many technological ﬁelds, the ﬁnite element method is the dominating analysis tool. In structural analysis all members of a structure, such as plates, beams, bars, cables etc., need to be represented by suitable ﬁnite elements. These elements are formulated to accurately correspond to the behaviour of the real member. For beams, a number of ﬁnite elements are available for diﬀerent applications. However, for cable problems special elements are rarely found in commercial ﬁnite element software, e.g. ABAQUS, and the common approach is to use straight bar elements to model the cables. The lack of suitable cable elements stems from the highly non-linear behaviour of a cable, which hardly can be modelled using the standard Galerkin technique. This non-linearity, which is of the geometric rather than material type, arises due to the very low bending stiﬀness of a cable, Figure 4.1. For very taut cables, the straight bar element is a good representation of the cable, but if the cable is subjected to a compressive force it will easily buckle and lose its stiﬀness. The steepness of the stiﬀness transition, shown in Figure 4.2, depends on the relation between the physical properties of the cable. Hence, it is the low ﬂexural rigidity that makes cable modelling diﬃcult. In this chapter, a number of ﬁnite elements which are applicable to general cable problems will be presented, but before starting with the elements it is necessary to ﬁrst present the analytical solutions for both the inextensible and extensible cables subjected to uniformly distributed loads.

4.2

Analytical cable solutions

A perfectly ﬂexible cable supported at its ends and acted upon by a uniform gravitational force assumes a curve called the catenary 1 . The equation of the catenary was ﬁrst obtained by Leibniz, Huygens and Johann Bernoulli in 1691. They were 1

The Latin word for chain

85

CHAPTER 4. FINITE CABLE ELEMENTS

0 −1

L0 = 13 m

z (m)

−3

−5

−7

−9 0

2

4

6

8

10

12

x (m) Figure 4.1: Various conﬁgurations of an inextensible cable (x = 2, 4, 6, 8, 10, 11.9, 11.99, 11.999 m; z = −5 m). 700

Horizontal stiﬀness (kN/m)

600

500

q0 = 800 N/m

400

300

200

L0 = 80 m AE = 51 MN

100

q0 = 80 N/m 0 75

76

77

78

80

79

81

82

83

84

85

Distance between supports (m) Figure 4.2: Comparison of the stiﬀness of an extensible cable and a straight bar with same properties.

86

4.2. ANALYTICAL CABLE SOLUTIONS

responding to a challenge put out by Johann’s brother Jacob to ﬁnd the equation of the chain curve [48]. In this section, the analytical solutions to the inextensible and extensible cables will be derived. These solutions can be found elsewhere, e.g. [48, 57, 61], but since diﬀerent notations have been used in these references they cannot easily be compared. For consistency and clarity, the derivation of the solutions will be repeated in this section. Finally, the validity of the assumption of zero bending stiﬀness of the cable will be discussed. z F6 θ2 j F4

h E, A q0 s

θ

F1 θ1

x

i F3 l

Figure 4.3: Extensible catenary element in x–z plane (x  x , z  z  ).

4.2.1

The inextensible catenary

To derive the equation of the inextensible catenary, certain assumptions have to be made about the properties of the cable. It is assumed that the cable is perfectly ﬂexible (EI ≡ 0), inextensible (AE → ∞), free of torsional rigidity and able to sustain only tensile forces. As a result of these assumptions, the cable force is tangential to the cable at every point. Consider a small segment of the extensible cable in Figure 4.3. Let AE → ∞ and the segment becomes inextensible, Figure 4.4. It may seem inconsistent to deﬁne the self-weight positive downwards, Figure 4.4, but

87

CHAPTER 4. FINITE CABLE ELEMENTS

since only loads acting downwards are considered in this chapter this convention is chosen (another argument is that this convention was chosen for the cable elements, which will be used later, cf. [1, 52]). Thus, in all the expressions below only the magnitude of the gravitational force, not its direction, needs to be inserted.   dz dz d T ∆s0 T + ds0 ds0 ds0

∆s 0 dx d T + ds0 ds0 T

dx ds0



dx T ds0

 ∆s0

limAE→∞ s = s0 T

q0 ∆s0

dz ds0

Figure 4.4: Segment of an inextensible cable

Returning to Figure 4.4, vertical equilibrium of the cable segment yields   dz d T = q0 , ds0 ds0

(4.1)

where

dz = sin θ. ds0 Horizontal equilibrium of the cable segment gives   d dx T = 0, ds0 ds0 where

dx = cos θ. ds0

(4.2)

(4.3)

(4.4)

Integrating (4.3) yields dx = H, (4.5) ds0 where H is the horizontal component of cable tension, which is constant along the cable since no horizontal loads are acting. Substituting equation (4.5) to equation (4.1) gives d2 z ds0 H 2 = q0 . (4.6) dx dx Using the following geometric constraint:  2  2 dx dz + = 1, (4.7) ds0 ds0 T

88

4.2. ANALYTICAL CABLE SOLUTIONS

equation (4.6) can be written as:

2

dz q0 = 1+ 2 dx H



dz dx

2 1/2 .

(4.8)

The solution to equation (4.8) with the boundary conditions x = 0, y = 0 and x = l, y = h, is

q x H 0 cosh z= + ζ − cosh ζ , (4.9) q0 H in which −1

ζ = sinh



q0 h 2H sinh η

 − η,

(4.10)

and

q0 l . (4.11) 2H Equation (4.9) is the inextensible catenary equation. Diﬀerentiating (4.9) gives the slope as:

q x dz 0 = sinh +ζ . (4.12) dx H The length of the inextensible catenary can be computed with the following equation [84]: l2 L2 = 2 sinh2 η + h2 . (4.13) η η=

An approximate equation for the inextensible catenary can be obtained for a shallow cable, that is a cable with a small sag-to-span ratio. In this case ds0 /dx ≈ 1 and equation (4.6) simpliﬁes to d2 z H 2 = q0 . (4.14) dx Using the same boundary conditions, the solution to (4.14) is the following parabolic equation:  

x x 2 x − z = ηl +h . (4.15) l l l Equation (4.15) is much easier to work with than (4.9). Therefore, the parabola has been used for many years as the cable equation in the development of approximate formulae for preliminary design of cable structures, see for example [42, 57]. As mentioned above, for a cable where only the self-weight is acting, the parabolic equation (4.15) is approximate and the error increases as the sag-to-span ratio increases. To obtain an estimation of the diﬀerence between the parabola and the catenary, we consider a cable with level supports, i.e. h = 0. For sag-to-span ratios larger than 0.2 the parabola is a rather crude representation of the catenary, Figure 4.5. Nevertheless, the use of the parabolic equation may in some cases be more correct, e.g. for a suspension bridge, where the cables sustain a load which is uniformly distributed along their span and much larger than the self-weights of the cables, that is the bridge deck [57].

89

CHAPTER 4. FINITE CABLE ELEMENTS

11 10 zcat (l/2) = 0.3l 9 8

Percent error

zcat (l/2) = 0.3l 7

zcat (x)−zpar (x) zcat (l/2)

6

Tcat (x)−Tpar (x) Tcat (x)

5

zcat (l/2) = 0.2l

4 3 2

zcat (l/2) = 0.2l

zcat (l/2) = 0.1l

1 zcat (l/2) = 0.1l 0 0

0. 05

0. 1

0. 15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

x/l Figure 4.5: Diﬀerence between sags and cable forces for the inextensible catenary and parabola with same spans l and horizontal forces H

4.2.2

The elastic catenary

A real cable is not inextensible as it has a ﬁnite axial ﬂexibility. The diﬀerential equations of equilibrium of a stretched chain, satisfying Hooke’s law, were derived by Jacob and Johann Bernoulli, but the solution was ﬁrst given in 1891 by Routh [48]. To derive the equations for the elastic catenary, the extensible cable segment in Figure 4.6 is considered. Horizontal and vertical equilibrium of the segment yields T

dx = H, ds

(4.16)

dz (4.17) = q0 s0 − F3 . ds It can be noted that due to the conservation of mass the total weight of the cable segment is unaﬀected by the elongation of the cable. For the elastic catenary, the geometric constraint to be satisﬁed is  2  2 dx dz + = 1. (4.18) ds ds T

It is assumed that the cable material satisﬁes Hooke’s law   ds T = AE −1 , ds0 where A is the uniform cross-sectional area in the unstrained proﬁle.

90

(4.19)

4.2. ANALYTICAL CABLE SOLUTIONS

T dz ds

T dx ds s H q0 s 0

F3

Figure 4.6: Segment of an extensible cable.

Then, (4.16) and (4.17) are squared, added and substituted into equation (4.18), which gives the cable force at any point s0 as [48]:  1/2 2 T (s0 ) = H 2 + (q0 s0 − F3 ) . (4.20) By noting that dx/ds0 = (dx/ds)(ds/ds0 ) and dz/ds0 = (dz/ds)(ds/ds0 ) the parametric solutions x(s0 ) and z(s0 ) are derived, since dx/ds, dz/ds and ds/ds0 are given by the equations (4.16), (4.17) and (4.19), respectively. The solutions are      F3 q0 s0 − F3 Hs0 H −1 −1 sinh + + sinh , (4.21) x(s0 ) = AE q0 H H !      2 "1/2   "1/2 !   2 q0 s0 − F3 F s0 q0 s0 H F3 . z(s0 ) = 3 −1 +  1+ − 1+  AE 2F3 q0 H H (4.22) It is obvious that the cable force increases with height above the lowest point of the cable. If the equations (4.20)–(4.22) are combined we ﬁnd the following interesting result [102]:  1/2 − AE, (4.23) T (z(s0 )) = (AE + Tb )2 + 2AEq0 z(s0 ) where Tb is the cable force at the base, i.e. at s0 = 0. Hence, the cable force at any value of z(s0 ) is independent of the span l. Practically, this result is important because the cable force at the base Tb is an input value in the design of some cable structures, for example guyed masts [102]. Recently, Russell and Lardner [102] did some experiments with a scale model of a guy cable, for which the elasticity is important. The purpose with their experiments

91

CHAPTER 4. FINITE CABLE ELEMENTS

was to compare the numerical predictions from the equations (4.20)–(4.22) of the elastic catenary with the measured values. The results show very good agreement between the experimental and theoretical cable force at the base, x = 0, for a number of horizontal spans l. An average error of 2.5 % below the theoretical values was reported.

4.2.3

Eﬀect of cable bending stiﬀness

Some authors, e.g. [1, 52], have termed the elastic catenary exact, which under the stated assumptions is correct. However, the elastic catenary is never exact in reality, because the assumptions are violated in many cases. One assumption which is more important than the others, and needs further studies, is that of the zero bending stiﬀness. For a real cable, as that in a suspension bridge, the magnitude of the ﬂexural rigidity can be very large. Therefore, the eﬀects of a non-zero ﬂexural rigidity on cable geometry and forces will be analysed in this section. First we need to know how large the bending stiﬀness of a cable actually can be. Calculating the bending stiﬀness is more complex than ﬁnding the axial stiﬀness, but if we think of how the cable is assembled, see section 2.4, two extreme cases can be distinguished: 1. All wires are stuck together. The cable has one neutral axis and the rigidity of the cable is similar to that of a beam. 2. The friction between the wires is zero, making them bend around their own neutral axis. The ﬁrst case represents the upper bound of the bending stiﬀness while the second represents the lower bound. Several methods to calculate the ﬂexural rigidity for helically wound cables have been suggested in literature [22]. Following the recommendations made by Cardou and Jolicoeur [22], a good estimate of the upper bound of the cable bending stiﬀness is obtained using the approach by Lanteigne [60]: (EI)max = E0 I0 +

n 

mi Ai Ei

i=1

(Ri2 /2 + ri2 ) 3 sin αi . 2

(4.24)

For the lower bound of the bending stiﬀness the frictionless model by Costello [29] should be used [22]: (EI)min = E0 I0 +

n 

mi Ei Ii

i=1

sin αi . (2 + ν cos2 αi )

(4.25)

The true bending stiﬀness is somewhere between the two extremes. Which of the extreme values that is closest to the real weakness depend from case to case. To further complicate things, experiments have shown that the bending stiﬀness is not constant; it varies along the length of the cable [22].

92

4.2. ANALYTICAL CABLE SOLUTIONS

The eﬀects of a non-zero bending stiﬀness on the cable geometry and forces has been analysed by Wang and Watson [127]. In their studies, they assumed that the cable was inextensible and used the elastica equation, EI

d2 θ = H sin θ − q0 s0 cos θ, ds20

(4.26)

ﬁrst formulated by Euler, to represent the cable, Figure 4.7. By dividing all forces by EI/S 2 and all lengths by S, where S = L/2, equation (4.26) is made dimensionless. Equation (4.26) then becomes d2 θ 3 2 = H sin θ − K s0 cos θ, d s0

(4.27)

where K = Sq0 (EI)−1/3 . K can be seen to represent the relative importance of density and length to ﬂexural rigidity or the ratio of half the cable length to the bending length (EI/q0 )1/3 [127]. Since no closed form solution to equation (4.27) exists, it has to be integrated numerically. However, due to the extreme numerical stiﬀness of (4.27) for large values of K, all classic numerical techniques, such as Newton’s method with the shooting algorithm or a ﬁnite diﬀerence approximation, fail to ﬁnd the correct solution [128]. Solutions for values of K in the whole range can only be provided by a sophisticated continuation method [127, 128]. 1/3

Some of the results from Wang and Watson’s studies are presented in Figures 4.8 and 4.9. For a > 0.8 the eﬀect of K on the shape can be neglected, Figure 4.8. For the maximum curvature shown in Figure 4.9(a), the eﬀect of K is considerable for a < 0.8. The horizontal force is very sensitive to K for all values of a, Figure 4.9(b). The following conclusions can be drawn from the studies by Wang and Watson: • The eﬀect of bending stiﬀness on the cable shape can be ignored for taut cables. • Ignoring the bending stiﬀness leads to a softer model, which in dynamic analyses gives lower eigenfrequencies. Still, the assumption of a constant bending stiﬀness along the length of the cable is not correct according to the experimental results reported in [22]. As a concluding remark one can say that for the quasi-static behaviour of a taut cable the bending stiﬀness is of less signiﬁcance.

93

CHAPTER 4. FINITE CABLE ELEMENTS

q0 S

z

T

a H

b

s0

θ x

Figure 4.7: Coordinate system for an inextensible cable with non-zero ﬂexural rigidity. Redrawn from [127].

Although the eﬀects of neglecting the bending stiﬀness of the cable have been analysed one assumption still remains: the zero torsional rigidity. To check the eﬀects of this assumption the cable model would have to be extended with rotational degrees of freedom, but this will not be done here. In their present form, the analytical solutions are of little use. For a general analysis of cable structures the cable must be represented by a ﬁnite element.

b

a Figure 4.8: The height b as a function of the span a for an inextensible cable with bending stiﬀness. Redrawn and slightly modiﬁed from [127].

94

4.3. LITERATURE REVIEW OF CABLE ELEMENTS

H

dθ (0) ds0

a

a (a) The maximum curvature

(b) The horizontal force

Figure 4.9: Some results for an inextensible cable with bending stiﬀness. Redrawn and slightly modiﬁed from [127].

4.3

Literature review of cable elements

The objective of this section has not been to give a historical review of all the developed cable elements. Instead, it is focused on diﬀerent formulations used to solve the cable problem. The literature review has indicated that generally two approaches for the development of ﬁnite cable elements are used. The ﬁrst approach is the use of polynomials to describe the shape and displacement ﬁeld, which is the ordinary approach in the development of ﬁnite element. In the second approach analytical expressions are used, which in a mathematical sense exactly describes the cable under certain load conditions.

4.3.1

Elements based on polynomial interpolation functions

In this class basically three types of elements are found in literature: • straight bar element, • curved isoparametric bar elements, and

95

CHAPTER 4. FINITE CABLE ELEMENTS

• curved elements with rotational degrees of freedom. Two-node straight bar element The straight bar element is the most common element used in the modelling of cables. A number of element formulations have been presented for geometrically non-linear analysis, see for example [5, 9, 14, 16]. However, in the end they should give similar results. Straight bar elements, which possess only axial stiﬀness, often provide suitable representation of highly pretensioned cables, such as those in cable nets and trusses. For slack cables with large curvature, the standard approach is to represent the cable by a large number of bar elements. This technique is ineﬃcient as the number of degrees of freedom drastically increases. Another drawback is the spurious slope discontinuities occurring at nodes where no concentrated loads act. These discontinuities are due to the straight element assumption and may lead to convergence problems in the analysis [37, 61]. One method often used to model slack cables is the equivalent modulus or stiﬀness approach. In this method, the cable is assigned a certain stiﬀness that take the cable sag into account. This equivalent stiﬀness was derived by Ernst [34]. By equating the stiﬀness of a straight bar element to that of a parabolic element, he obtained a simple expression for the equivalent axial stiﬀness of a slack cable. The equivalent stiﬀness is a function of the cable force, the self-weight of the cable, the length of the element and the axial stiﬀness of a straight cable. However, for cables with large sag a large number of elements is still needed. Multi-node isoparametric ﬁnite elements Instead of using many bar elements with linear interpolation functions one can use fewer elements with interpolation functions of higher order. By adding more nodes to the ﬁnite element, higher order polynomials for the shape and displacements of the element can be deﬁned [61]. Most common are the three and four node elements, which use parabolic and cubic interpolation functions, respectively. The tangent stiﬀness matrix and equivalent nodal forces are obtained using the isoparametric formulation. Due to the complex expressions involved in this formulation, the tangent stiﬀness matrix and equivalent nodal forces have to be found by numerical integration. These curved elements give accurate results for cables with small sag. For larger sag more element must be used. Between element nodes only the displacement continuity is enforced [89]. Curved elements with rotational degrees of freedom The continuity of the slopes can be enforced by adding rotational degrees of freedom to the nodes. Such an element was developed by Gambhir and Batchelor [37]. Cubic polynomials described the displacement ﬁeld and shape of the cable. Due to the

96

4.3. LITERATURE REVIEW OF CABLE ELEMENTS

simpliﬁcations made for this element it is applicable only for cables with small sagto-span ratios. To model a cable that has a large curvature more elements have to be used, but, contrary to the other elements presented above, no slope discontinuities will occur. It may be argued that a cable can be modelled using beam elements with a low bending stiﬀness. For cables where the bending stiﬀness is known or assumed this alternative can provide more reliable results. However, for very ﬂexible cables, a large number of beam elements with a very low bending stiﬀness are needed to model the cable [54]. Since rotational degrees of freedom are present in the beam elements, the total number of degrees of freedom will, therefore, substantially increase. Some advantages associated with the polynomial based cable elements are: • The formulation with polynomials is almost universal. • The out-of-plane response of the cable can be captured if the multi-node isoparametric formulation is used. • For dynamic analysis the mass matrices are consistent. The disadvantages are: • Without rotational degrees of freedom slope discontinuities occur, which may induce numerical convergence problems. • Many elements are needed to model slack cables with large sag-to-span ratios.

4.3.2

Elements based on analytical functions

The second class contains the analytical elements, which are based on analytical formulae to take into account the eﬀect of loading applied along the length of the cable. Three elements, each associated with a certain type of uniformly distributed loading, have been presented in literature. They are: • The parabolic element for which the load is uniformly distributed along the horizontal span of the cable. • The elastic catenary element for which the load is uniformly distributed along the unstrained length of the cable. • The associate catenary element for which the load is uniformly distributed along the strained length of the cable.

97

CHAPTER 4. FINITE CABLE ELEMENTS

Parabolic element As mentioned earlier the parabola has, because of its simpler form compared to the catenary, been more frequently used in the analysis of cable structures. One type of parabolic element, the three-node isoparametric element, has already been presented. Two other formulations, which have much in common, are presented in [1, 76]. Both formulations use shallow cable assumptions, which make them applicable only for cables with small sag-to-span ratios. For both elements, the cable force is obtained by solving a cubic equation. Elastic catenary element For a perfectly ﬂexible cable subjected to self-weight load only this element is exact. Basically, two elastic catenary elements have been developed. The ﬁrst element was presented by Peyrot and Goulois [94, 95]. They used the expressions by O’Brien et al. [53, 83, 84] to obtain the ﬂexibility matrix. The tangent stiﬀness matrix was derived by taking the inverse of the ﬂexibility matrix. In reference 94 the plane version of the element was used in the analysis of transmission lines, while in reference 95 three-dimensional structures were analysed. The tangent stiﬀness given in [95] is believed to be incorrect, because the out-of-plane stiﬀness, i.e. in the y  -direction, was set to zero. This was later corrected by Jayaraman and Knudson [52] who demonstrated the capability of the element for a number of examples. Irvine [48] derived a similar element, but reported that the stiﬀness matrix should be unsymmetric due to the geometric non-linearity. However, it is observed that the stiﬀness matrix in reference 48 in fact is symmetric, which it ought to be due to the conservative nature of the self-weight [3]. The second elastic catenary element was derived by Ahmadi-Kashani [1]. This element will be presented later in this chapter together with the last element in the class, the associate catenary element. Associate catenary element The last element in this section is a special one. For this element the load is distributed along the strained length of the element. A load of this type is the snow load. Under the action of the snow load the element stretches and thereby increases the available length for the snow ﬂakes to land on; it can be said that the element becomes heavier as it elongates. Thus, the total load is dependent on the displacements. In this case the loading is non-conservative and therefore the tangent stiﬀness matrix is unsymmetric. This element was derived in reference 1 and the equations expressing the nodal forces and the tangent stiﬀness matrix are utterly complex. The main advantage with the analytical elements is that only one element is needed to model a cable with a high degree of accuracy. More elements may be used in, for example, dynamic analysis and thanks to the exact analytical expressions no discontinuities occur across the element boundaries. Although these elements work very well there are some disadvantages:

98

4.4. STRAIGHT AND PARABOLIC ELEMENTS

• The equivalent nodal forces and the tangent stiﬀness matrix have to be found by iteration. • The use of trigonometric functions, such as the tangent function, in the formulations make them undeﬁned for certain angles or load cases. • Consistent mass matrices are not available. Finally, should be mentioned the general cable element in reference 1 which can be used for any type of loading along the length of the cable. For this element, a ﬁnite segment approach is used to calculate the equivalent nodal forces and tangent stiﬀness matrix. Using this general element only one element is needed to model a cable, irrespective of the variation of the load along the length of the cable. Thus, this element is very useful for the analysis of long guy cables or underwater cables, which exhibit varying wind and ﬂuid loads. From the elements presented in this review, four elements are selected and will be described in the following sections. These include: the straight bar, the elastic parabolic element, the elastic catenary element, and the associate catenary element. They are chosen because they all have six degrees of freedom and closed-form expressions for the equivalent nodal force vectors and tangent stiﬀness matrices, which means that no numerical integration is necessary to obtain them.

4.4

Straight and parabolic elements

These two elements are chosen to be presented in the same section because they are somewhat linked to each other by the equivalent stiﬀness by Ernst [34]. It will be shown that their tangent stiﬀness matrices have some similarities.

4.4.1

Straight bar element

This element, shown in Figure 4.10, is the most simple ﬁnite element for structural analysis. It has appeared in many forms, e.g. [5, 14, 16], but in the end practically all of them give the same results. The formulation used here is from [9], with no detailed discussion considered necessary. The vector of equivalent nodal forces in the local coordinate system is written as:  T (4.28) F 1 = −T 0 0 T 0 0 . The elastic stiﬀness sub-matrix in the local  1 AE   0 kE = L 0

99

coordinate system is given as:  0 0 0 0 . 0 0

(4.29)

CHAPTER 4. FINITE CABLE ELEMENTS

z

x

F6



j

z

y



F4

L

F1

F5

y i

F2 F3

x Figure 4.10: Straight bar element in space

Since the analysis of cable structures is non-linear the following geometric stiﬀness sub-matrix is needed:   1 0 0 T kG =  0 1 0  . (4.30) L 0 0 1 The total elastic matrix in local coordinate system tangent stiﬀness is written as:   kE −kE  , (4.31) KE = −kE kE while the geometric stiﬀness is given as:   kG −kG KG = . −kG kG

(4.32)

The transformation matrix from the local to the global system is   t1 0 , T1 = 0 t1 in which

x j − xi L yj − yi − Lxy

y j − yi L x j − xi Lxy

    t1 =    (x − x )(z − z ) (yj − yi )(zj − zi ) i j i  − j − LLxy LLxy

100

zj − zi   L   0  ,  Lxy   L

(4.33)

(4.34)

4.4. STRAIGHT AND PARABOLIC ELEMENTS

where

1/2  . Lxy = (xj − xi )2 + (yj − yi )2

(4.35)

Thus, the equivalent nodal forces and tangent stiﬀness matrix in global coordinates can now be written as [9]: (4.36) F1 = TT1 F1 , and

K1 = TT1 KE T1 + KG .

(4.37)

The straight bar element can be used in the modelling of slack cables if the axial stiﬀness AE/L is substituted by the equivalent axial stiﬀness by Ernst [34], which is given as:   AE AE . =  (4.38) q02 l2 L eq L 1+ AE 12T 3 For cables where the chord length is longer than the unstrained length the cable tension can be computed with the following well-known equation: T = AE

L − L0 . L0

(4.39)

When L ≤ L0 equation (4.39) yields a zero or compressive force. How to compute the cable force in those cases will be described in the next section.

4.4.2

Elastic parabolic element

As mentioned in section 4.3, for this type of element the load q0 is distributed along the horizontal span of the cable. Some authors [1, 76] have derived the nodal forces and element stiﬀness matrix for parabolic elements with small sag-to-span ratios. An elastic parabolic element for large sags has not yet been developed. The formulation for the elastic parabolic element presented in this section was given in [1]. The vector of equivalent nodal forces for a taut parabolic element is given as:  T (4.40) F 2 = F1 0 F3 F4 0 F6 , where F1 = −H,   h q0 l  F3 = −H − , l 2H F4 = −H,   h q0 l  F6 = H + , l 2H

101

(4.41) (4.42) (4.43) (4.44)

CHAPTER 4. FINITE CABLE ELEMENTS

from which we can obtain the end angles h q0 l − , l 2H h q0 l . tan θ2 = + l 2H

tan θ1 =

(4.45) (4.46)

The expressions for the equivalent nodal forces and end angles are applicable for elastic parabolic elements with large sag-to-span ratios. However, to obtain the tangent stiﬀness matrix some simpliﬁcations have to be introduced in order to solve the complex equations that arise in the formulations [1]. By introducing the assumption of small a sag-to-span ratio, these equations were simpliﬁed and solved in reference 1 resulting in the following local tangent stiﬀness matrix: 

k1

 k2 = α1  0

k2 in which k1 k2 k3 α1

 0 k2 H  0 , α1 l 0 k3

  L0 C − L0 q 2 l2 2 = cos β + sin2 β + 0 2 cos2 β sin2 β, C C 8T   2 2 q l 2L0 − 1 − 0 2 cos2 β sin β cos β, = C 8T   L0 C − L0 q 2 l2 2 = sin β + cos2 β + 0 2 cos4 β, C C 8T AE . =  2 2 q0 l 2 C 1+ AE cos β 12T 3

(4.47)

(4.48) (4.49) (4.50) (4.51)

The element matrix for the elastic parabolic element, in the local coordinate system, is written as:   k2 −k2  . (4.52) K2 = −k2 k2 The transformation matrix for the parabolic element is not the same as T1 . Since the x -axis is not oriented along the chord of the element, T1 cannot be used. However, since the z  -axis is vertical, which in the majority of cases is also the direction of the z-axis, the transformation matrix for the parabolic element is easily derived; for a cable loaded only in the z-direction the projection of the element is straight in the x–y plane. Thus, the rotation matrix is the same as the well-known rotation matrix for a plane beam element (if the rotational degrees of freedom are substituted by the translations in the z-direction)   t2 0 , (4.53) T2 = 0 t2

102

4.4. STRAIGHT AND PARABOLIC ELEMENTS

z

F6 j

h

F5 y F4

F1 F2

i

q0

F3

l x

Figure 4.11: Catenary element in x –z  plane

where

cos φ − sin φ cos φ t2 =  sin φ 0 0

 0 0 . 1

(4.54)

This transformation matrix is also compatible with the elastic and associate catenaries presented in section 4.5. The global nodal forces and tangent stiﬀness matrix for the parabolic element can now be written as: (4.55) F2 = TT2 F 2 , and

K2 = TT2 K2 T2 .

(4.56)

The tangent stiﬀness matrix is a function of the unknown cable force T . This force can be found from the following cubic equation:    22  C − L0 q0 l q 2 l2 1 3 2 4 T − T + cos β T − 0 cos2 β ≡ 0. (4.57) F (T ) = AE C 12AE 24 It is assumed that the contribution of the third term to the equation is negligible, [1], and the equation is simpliﬁed to   1 3 C − L0 q02 l2 2 F (T ) = (4.58) T − cos2 β ≡ 0. T − AE C 24

103

CHAPTER 4. FINITE CABLE ELEMENTS

z z F6

y



F5 F4

y F1

q0

F2

φ x

F3

x Figure 4.12: Rotation of catenary in the x–y plane

From Descartes’ rule of signs [96] it follows that there is only one positive root of equation (4.58). Equation (4.58) is in [1] solved with a Newton-Raphson technique. If the stiﬀness sub-matrix of the elastic parabolic element (4.47) is compared with that of the straight bar element, equation (4.37), it is noted that, apart from the cos2 β-term, α1 is identical to the equivalent axial stiﬀness given by Ernst, equation (4.38). It is claimed in [1] that the formulation presented here should provide a better approximation for the taut parabolic element than the bar element with an equivalent axial stiﬀness. However, with the load intensity q0 and angle β equal to zero, equation (4.47) becomes     L0 AE 0 0   C C   T   (4.59) k2 =  , 0 0   C    (C − L0 ) AE  0 0 C C where the following relation for the taut parabola is used: H T = . cos β

(4.60)

Comparison of (4.59) and the sub-matrix of (4.37) indicates that the geometric stiﬀness terms in the x- and z-direction are missing. This results in a softer behaviour

104

4.5. CATENARY ELEMENTS

of this element compared to a bar element. In addition to that, a small fraction, i.e. (C − L0 )/C, of the elastic stiﬀness AE/C also contributes to the stiﬀness in z-direction, which is peculiar. Hence, the elastic parabolic element is not identical to the bar element when the load q0 along the span is removed.

4.5

Catenary elements

In the preceding section two diﬀerent approaches were presented for a parabolic cable element, where one was said to be slightly more accurate than the other [1]. However, the accuracy of the results using a parabolic element to model a cable subjected to load uniformly distributed along its length is acceptable only for cables with small sag-to-span ratios. In addition to this, the ability of the straight bar element to take compressive forces may induce numerical instability in the analysis of, for example, cable nets. Therefore, a routine which eliminates the element from the model if a compressive force is detected, has to be used in the computer program [14, 16]. The element is eliminated by setting the axial stiﬀness to a very low value, say 10−20 . In this section elements are studied, which in a mathematical sense are exact for cables subjected to uniformly distributed loads along their lengths. These elements can be used for any sag-to-span ratio; both very slack and taut cables can be analysed with a high degree of accuracy.

4.5.1

Elastic catenary element

For this element the load is uniformly distributed along the unstrained length of the element (4.61) q(s0 ) = q0 . The self weight of the cable is a load of this type. Thus, this element gives the same results as the elastic catenary equations (4.20)–(4.22) and is therefore called elastic catenary element. In a mathematical sense this element is exact for a cable loaded by its self weight only. As mentioned in section 4.3, two diﬀerent formulations have been developed for this element. The ﬁrst formulation, which was given in [52, 95], was based on a ﬂexibility method by O’Brien et al. [53, 83, 84]. The second formulation, which will be presented here, was derived by Ahmadi-Kashani [1]. The equations that express the equivalent nodal forces and tangent stiﬀness matrix are written in terms of H, θ1 and θ2 , where θ1 and θ2 are the angles between the horizontal and and the tangent to the cable at the ends, Figure 4.3. The vector of nodal forces for the elastic catenary shown in Figure 4.3 can be written as:  T (4.62) F 3 = F1 0 F3 F4 0 F6 ,

105

CHAPTER 4. FINITE CABLE ELEMENTS

where F1 = −H,

  −1  F3 = −H sinh cosh

q 0 L0 2H sinh µ

F4 = H,

  −1  F6 = H sinh cosh

where

q0 µ= 2



q 0 L0 2H sinh µ

l L0 − H AE



(4.63)

 −µ ,

(4.64)



(4.65)



+µ ,

(4.66)

 .

(4.67)

The element stiﬀness matrix, in the local coordinate system, for the elastic catenary element is written as:   k3 −k3  , (4.68) K3 = −k3 k3 where

q 0 L0  AE + sin θ2 − sin θ1 0   H  0 k3 = α2  α2 l    0 cos θ1 − cos θ2 in which α2 =

where

 cos θ1 − cos θ2 0 q0 l − sin θ2 + sin θ1 H

    ,   

q0   , θ 2 − θ1 θ1 + θ2 θ 2 − θ1 q02 L0 l 4 sin µ cos − sin + 2 2 2 AEH     q 0 L0 −1 −µ , tan θ1 = sinh cosh 2H sinh µ     q 0 L0 −1 +µ . tan θ2 = sinh cosh 2H sinh µ

(4.69)

(4.70)

(4.71) (4.72)

It should be emphasised that the expressions for the nodal forces and tangent stiﬀness matrix are applicable only if node j is above or at the same level as node i, i.e. zj ≥ zi . However, this does not cause any serious problem for the implementation; it is just a matter of switching the directions of some of the stiﬀness coeﬃcients and the horizontal forces at the nodes and then change the rows and columns of the tangent stiﬀness matrix and the force vector. One particular case, which deserves attention, concerns an elastic catenary element where the intensity of the loading q0 approaches zero. For this case the tangent

106

4.5. CATENARY ELEMENTS

stiﬀness matrix (4.69) becomes  T AE 2  L cos θ1 + L     0 lim k3 =  q0 →0   AE  sin θ1 cos θ1 L

0 T L 0

 AE sin θ1 cos θ1  L    0 .   AE T  2 sin θ1 + L L

(4.73)

where the following relation between the unstrained length L0 and the strained length L for a straight elastic element is used:   T . (4.74) L = L0 1 + AE It is noted, that the stiﬀness sub-matrix in equation (4.73) is identical to that of a straight bar element rotated an angle θ1 . Hence, the elastic catenary element can be used to represent weight-less cables if, to avoid numerical instabilities, a very small value for q0 is speciﬁed [2]. It is seen that the equivalent nodal forces and the tangent stiﬀness matrix are functions of only one variable, i.e. the horizontal force H. For a given geometric conﬁguration, H is found by the following equation: F (H) =

4H 2 h2 2 2 sinh µ +    2 − L0 ≡ 0. q02 q 0 L0 1+ coth µ 2AE

(4.75)

Equation (4.75) is solved using the well-known Newton-Raphson algorithm for nonlinear equations of one variable, where the derivative of (4.75) is given as: 2  dF 8H hq0 2l lL0 2 = 2 sinh µ − sinh 2µ − . (4.76)    3 dH q0 q0 H sinh µ q 0 L0 2AE 1 + coth µ 2AE In the case of an inextensible cable element, where AE → ∞, (4.75) simpliﬁes to L20 =

4H 2 q0 l sinh2 + h2 , 2 q0 2H

(4.77)

which is identical to equation (4.13).

4.5.2

Associate catenary element

To understand the formulation of this element one can think of a long elastic cable which is supported at its ends and placed outdoors. One example of such a cable is the conductor in a transmission structure. Assume that outdoors there is a light wind and after a while it starts to snow. Every snow ﬂake that lands on the cable

107

CHAPTER 4. FINITE CABLE ELEMENTS

stretches it. The snow will be uniformly distributed along the strained length of the cable. Thus, the total load on the cable is dependent on the deformation of the cable. In this case the displacement dependent load is also non-conservative. In other words, the load along this element, in terms of the unstrained length, is not constant. Instead, the load along the unstrained length is expressed as: q(s0 ) = q0

ds , ds0

(4.78)

where q0 is the constant load per unit strained length of the cable. An element of this type was developed by Ahmadi-Kashani [1], who gave it the name associate catenary element. The vector of nodal forces for this element is  T (4.79) F 4 = F1 0 F3 F4 0 F6 , where F1 = −H,



F3 = −H sinh sinh−1



q0 h 2H sinh η

F4 = H,

  −1  F6 = H sinh sinh

q0 h 2H sinh η







(4.80)

−η ,

(4.81)



(4.82)

+η ,

(4.83) (4.84)

where η is given by (4.11). From the expressions for the equivalent nodal forces, the end slopes are found as:     q0 h −1 tan θ1 = sinh sinh −η , (4.85) 2H sinh η     q0 h −1 +η . (4.86) tan θ2 = sinh sinh 2H sinh η It can be seen that these end slopes are identical to those of the inextensible catenary, equation (4.12). Hence, the shape of the associate catenary is exactly the same as that of the inextensible catenary. Considering the properties of the associate catenary element, this result is not a surprise. Despite the elongation of the element the load intensity per actual length unit is always q0 , which is also the case for the inextensible catenary. For a suspended elastic catenary the load intensity per strained length unit is always less than q0 . The tangent stiﬀness matrix of the associate catenary element in the local coordinate system, as shown in Figure 4.11,

108

4.5. CATENARY ELEMENTS

is written as: 

a

0

  H  0  α3 l   0  λ1 a − λ2 c  K4 = α3  −a 0    H  0 −  α3 l   −λ3 a + λ4 c 0

−a

0

0

0

H − α3 l

λ1 b + λ2 d

−λ1 a + λ2 c

0

−b

a

0

0

0

H α3 l

−λ3 b − λ4 d − e

λ3 a − λ 4 c

0

b

−b

   0    −λ1 b − λ2 d   , b     0    λ3 b + λ 4 d + e (4.87)

where ψ , ∆ H γ= , AE

α3 = −

(4.88) (4.89)

a = sin θ2 − sin θ1 +

q0 L , AE

b = cos θ1 − cos θ2 , q0 h + ξ tan θ2 , c= H q0 l d= + ξ, H q0 , e= α3 sin θ2 λ1 = tan θ1 , cos θ1 + γ λ2 = , cos θ1 q0 h λ3 = tan θ2 − , H sin θ2 sin θ1 λ4 = λ 2 , sin θ2 H , ψ= q0 cos θ1 (cos θ1 + γ) ψ ∆ = − [ξL (cos θ1 + γ) + al + bh] , H     q 0 l q 0 L0 2 ξ = 1 − γ (cos θ2 + γ) γ − H AE   sin θ1 sin θ2 − . − cos θ2 + γ cos θ1 + γ

(4.90) (4.91) (4.92) (4.93) (4.94) (4.95) (4.96) (4.97) (4.98) (4.99) (4.100)

(4.101)

It is seen that the tangent stiﬀness matrix in (4.87) is not symmetric. This is due to the non-conservative nature of the load acting on the element. As in the case of the elastic catenary element, the equivalent nodal forces and the tangent stiﬀness matrix of the associate catenary element are functions of the sole

109

CHAPTER 4. FINITE CABLE ELEMENTS

variable H. To obtain H, the following non-linear equation has to be solved, using the Newton-Raphson technique outlined in section 4.5.3, # # AE ## (m + p) (m − n) ## AEl ln − ≡ 0, (4.102) F (H) = L0 − mq0 # (m − p) (m + n) # H where the derivative of (4.102) is  dF H 2AE sgn(κ) (m − n) [(m2 − 1) + r(p − 1)] =− ln |κ| − dH AEq0 m3 Hq0 m |κ| (m − p)(m + n) 2 (m + p) [(m − 1) − t(n − 1)] (m + p)(m − n) [(m2 − 1) − r(p − 1)] − + (m − p)(m + n) (m − p)2 (m + n)  AEl (m + p)(m − n) [(m2 − 1) + t(n − 1)] + 2 − 2 (m − p)(m + n) H (4.103) in which (m + p)(m − n) , (m − p)(m + n)  2 1/2  H , m= 1− AE κ=

H a1 e , AE H b1 p=1+ e , AE

n=1+

 q0 h a1 = sinh − η, 2H sinh η   q0 h −1 + η, b1 = sinh 2H sinh η h t = 1 + (η coth η − 1) + η, L h r = 1 + (η coth η − 1) − η. L −1

(4.104) (4.105) (4.106) (4.107) (4.108) (4.109) (4.110) (4.111)

The strained length L of the associate cable element is given by (4.13), but can be calculated by H (tan θ2 − tan θ1 ) . (4.112) L= q0 In references 1 and 3 the derivative of (4.102) is not given as in (4.103). When the expression for dF/dH given in these references was used in the Newton-Raphson algorithm the iterations did not converge. Therefore, equation (4.102) was checked with the mathematical symbolic software Maple. The diﬀerentiation of (4.102) thereby yielded (4.103). It should be mentioned that in practice this element may not be very useful. To explain the reason for this one can think of a load case where both wind and snow

110

4.5. CATENARY ELEMENTS

loads are acting. In the load analysis the design snow load is already on the cable when the wind load is applied. If we apply the wind load at the nodes, the cable will stretch and the total load on it will grow although the snowing has stopped. This behaviour is due to the formulation of the element, and from the discussion here it has less physical justiﬁcation.

4.5.3

Convergence of solution

In order to obtain the equivalent nodal forces and tangent stiﬀness matrix for the elastic and associate catenary elements, either equation (4.75) or (4.102) is solved by the classic Newton-Raphson method [3]. Due to the nature of this iterative technique, convergence to the correct solution cannot be guaranteed unless the original algorithm is modiﬁed. How this modiﬁcation should be formulated is determined by the behaviour of the function considered. The relationship between the horizontal force H and the unstrained length L0 , given by equation (4.102), is shown in Figure 4.13. It can be seen that for an element where the unstrained length L0 is shorter than the chord length C, there is a unique relation between H and L0 . However, for a slack element where L0 > C, three diﬀerent values for H can be found, each representing a certain equilibrium state. These three values are indicated by H1 , H2 and H3 in Figure 4.13. Only one value, that is H = H1 , corresponds to the correct solution. For the other two extraneous solutions H is negative and requires the element to be in compression, Figure 4.14. Thus, for a taut element convergence is always to the correct solution, while for a slack element the solution may converge to any of the three solutions of which only one is correct [1]. By studying the relation between H and L0 shown in Figure 4.13, Ahmadi-Kashani suggests the following modiﬁed Newton-Raphson algorithm  \$  dF  H − F (H ) for Hi+1 > 0, i i dH i (4.113) Hi+1 =  Hi /2 for Hi+1 < 0, in order to avoid the unwanted solutions. The objective of this modiﬁcation is to change an initial overestimate of H, which may cause convergence to a wrong solution, to an initial underestimate of H. This is due to the fact that an initial underestimate of H will always converge to the correct solution [1]. Although this algorithm will converge to the correct solution, the number of iterations needed depends on the starting value for H. Ahmadi-Kashani [1, 3] presents two initial estimates depending on the element being slack or taut: Case 1. In this case it is assumed that the unstrained length of the cable is longer than its chord length. If the small eﬀect of elasticity is ignored, the length of the cable is described by l2 (4.114) L20 = 2 sinh2 η + h2 . η 111

CHAPTER 4. FINITE CABLE ELEMENTS

L0 (m) 90 COMPRESSION

80

TENSION

L0 > C

70 60 50 L0 = C 40 30 H3 −105

−104

H2

20

H1

10−1 100 −103 −102 −101 −100 −10−1 H (kN)

101

102

103

104

105

Figure 4.13: Three possible solutions for catenary elements. Redrawn from [1].

H1

H3

H2 (a)

H3

H2

H1

(b)

(c)

Figure 4.14: Element conﬁgurations for the three diﬀerent solutions: (a) H = H1 , (b) H = H2 , and (c) H = H3 . Redrawn from [1].

A non-dimensional geometric parameter δ is deﬁned as: 1/2  2 L0 − h2 . δ= l2

(4.115)

Using this parameter it is shown in [1] that the following expressions provide a good estimate for the horizontal force H 

1/2   (120δ − 20)1/2 − 10 for 1 < δ ≤ 3.67, η≈  2.337 + 1.095 ln δ − 0.00473 (7.909 − ln δ)2.46 for 3.67 < δ < 4.5 · 105 . (4.116) After η has been obtained from (4.116), H can be calculated by (4.11). Case 2. In this case it is assumed that the unstrained length of the cable is shorter

112

4.6. COMPARISON OF ELEMENTS

than its chord length. The eﬀect of element elasticity cannot be neglected and, therefore, equation (4.116) is not applicable. An initial estimate for the cable force T is in this case provided by the following equation [1, 3]: % 1/3 1/3 b2 + a2 /3 for b2 > a2 , (4.117) T ≈ 1/3 a2 + b2 /2a22 for b2 ≤ a2 , 

where a2 = AE

C − L0 C

 (4.118)

and

q02 L20 cos2 β. (4.119) 24 Once T has been obtained a starting value for H is found from the following relation: b2 = AE

H=

Tl . C

(4.120)

It should be noted that for the particular case δ = 1, which, for example, occurs when C = l = L0 and h = 0, an initial estimate cannot be found using the suggested expressions. In that case a good initial value for H can be found by putting δ = 1.001 in (4.116). The use of the modiﬁed Newton-Raphson algorithm and the initial estimates given above ensure a fast convergence to the correct solution.

4.6

Comparison of elements

Most of the elements presented in this chapter are developed by one author only. Therefore, to conﬁrm the reliability of the ﬁnite cable elements and the present implementation of those it is necessary to analyse some simple structures using the presented elements and compare the results to those by other authors. Due to the complexity of the expressions for the nodal forces and tangent stiﬀness matrices, given in the preceding sections, diﬀerences between the equations are diﬃcult to distinguish. Therefore, a numerical comparison between the nodal forces and some selected stiﬀness components will be presented for a single cable suspended at different heights, Figure 4.15. Included in this comparison are: all elements in this chapter and the elastic catenary element by Jayaraman and Knudson [52]. In order to get a veriﬁcation of the present computer implementation of the elements, the cable in Figure 4.15, which was presented in [3], will be used in the force and stiﬀness comparison studies. Furthermore, two simple examples, which have been studied by many authors [52, 74, 84, 103, 129] are analysed.

4.6.1

Comparison example 1

In this example, the nodal forces and stiﬀness components of the tangent stiﬀness matrices will be compared. For the comparison, the single suspended cable shown

113

CHAPTER 4. FINITE CABLE ELEMENTS

in Figure 4.15 will be used. The following values are used: AE = 51 MN, q0 = 0.04 kN/m and L0 = 80.0 m. Following reference 3, the nodal forces and stiﬀness components will be presented for four diﬀerent chord lengths C and six angles θ. A comparison of nodal forces was not made in [3], but for the stiﬀness components the values obtained here will be compared with the values obtained by Ahmadi-Kashani. The results are presented in the Tables 4.5–4.8 on the pages 118–121.

z

j C

β

i

x

Figure 4.15: Conﬁguration for the cable used in the comparison of nodal forces and stiﬀness components.

4.6.2

Comparison example 2

This problem, which consists of a suspended cable subjected to uniform and concentrated loads, was ﬁrst considered by Michalos and Birnstiel [74] and later analysed by O’Brien and Francis [84], Saafan [103], and Jayaraman and Knudson [52]. The initial conﬁguration and the data for this structure are found in Figure 4.16 and Table 4.1, respectively. The results from the computations are shown in Table 4.2 together with the results obtained by other authors. z 1

x

3 8 kip 400 ft

100 ft 2

500 ft 1000 ft Figure 4.16: Prestressed cable under self-weight and concentrated load. Redrawn from [52].

114

4.6. COMPARISON OF ELEMENTS

Table 4.1: Initial data for the structure in Figure 4.16.

Description Magnitude Cross-sectional area of cable 0.85 in2 Equivalent modulus of elasticity 19000 kips/in2 Self weight of cable 3.16 lb/ft Segment 1–2 412.8837 ft Unstrained length Segment 2–3 613.0422 ft

Table 4.2: Displacement at load point for the structure in Figure 4.16.

Investigator

Element type

Saafan [103] O’Brien & Francis [84] Michalos & Birnstiel [74] Jayaraman & Knudson [52] Jayaraman & Knudson [52] Present Present Present Present

Straight bar Elastic catenary Straight bar Straight bar Elastic catenary Elastic parabola Elastic catenary Associate catenary Elastic catenarya

a

Displacements (ft) Vertical Horisontal −17.954 −2.774 −18.460 −2.820 −17.953 −2.773 −17.951 −2.772 −18.458 −2.819 −18.377 −2.842 −18.457 −2.819 −18.555 −2.820 −18.457 −2.819

Elastic catenary by Jayaraman and Knudson [52]

4.6.3

Comparison example 3

Here a slightly more complex structure will be studied. The structure considered is the prestressed cable net shown in Figure 4.17. This structure was ﬁrst studied by Saafan [103] and subsequently analysed by West and Kar [129], and Jayaraman and Knudson [52]. Initial data are given in Table 4.3. It seems that these values were chosen arbitrarily and therefore the structure is not in equilibrium in the assumed, prestressed conﬁguration. The results for this example are shown in Table 4.4 together with the results by other authors.

115

CHAPTER 4. FINITE CABLE ELEMENTS

z 100 ft

y

x 1

2 f

100 ft

f

f

3

4

f

100 ft Assumed initial conďŹ guration 100 ft

100 ft

100 ft

f = 30 ft

Figure 4.17: Prestressed cable net under vertical loads. Redrawn from [52].

Table 4.3: Data for the assumed conďŹ guration structure shown in Figure 4.17.

Description Cross-sectional area of cables Equivalent modulus of elasticity Self weight of cablea Horizontal members Prestressing force Inclined members Load acting vertically downward at nodes 1, 2, 3 and 4 a

A small self weight is assumed for the analytical cable elements

116

Magnitude 0.227 in2 12000 kips/in2 0.0001 kip/ft 5.459 kips 5.325 kips 8.0 kips

4.6. COMPARISON OF ELEMENTS

Table 4.4: Displacement for node 4 under concentrated loads for the structure in Figure 4.17.

Investigator

Element type

Saafan [103] Straight bar West & Kar [129] Straight bar Jayaraman & Knudson [52] Elastic catenary Jayaraman & Knudson [52] Straight bar Present Straight bar Present Elastic parabola Present Elastic catenary Present Associate catenary Present Elastic catenarya a

Displacements of node 1 (ft) x-dir. y-dir. z-dir. −0.1324 −0.1324 −1.4707 −0.1325 −0.1324 −1.4698 −0.1300 −0.1319 −1.4643 −0.1322 −0.1322 −1.4707 −0.1322 −0.1322 −1.4707 −0.1338 −0.1338 −1.4873 −0.1328 −0.1328 −1.4764 −0.1338 −0.1338 −1.4874 −0.1328 −0.1325 −1.4765

Elastic catenary by Jayaraman and Knudson [52]

4.6.4

Conclusions from the comparisons

From the element comparison the following observations are made: • The diﬀerence between horizontal and maximum cable forces for the associate and elastic catenary elements is limited when the cable is slack. H and T for the two diﬀerent formulations for the elastic catenaries are exactly the same. • Like the forces, the stiﬀness components for the two formulations for the elastic catenaries give identical results in almost all cases. Small diﬀerences occur for C = 80 and 80.7 m. For the associate catenary the diﬀerences are more frequent but still very small. The reason for this might be the errors in some of the equations in [2], which were reported in section 4.5.2 or the chosen tolerance for which the computations were discontinued. • The numerical results above demonstrate that the elastic catenary can simulate a weight-less cable very well.

117

118 – 0.000562 0.023131 0.023133 0.023131

Straight bar, Eeq Parabolic Elastic catenary Associate catenary Elastic catenary

65

85

b

Horizontal force H for chord length C Maximum cable force T for chord length C c Elastic catenary by Jayaraman and Knudson [52]

a

– 0.064049 0.209004 0.209007 0.209004

Straight bar, Eeq Parabolic Elastic catenary Associate catenary Elastic catenary

45

25

– 0.299996 0.483408 0.483415 0.483408

– 0.631662 0.744692 0.744706 0.744692

Straight bar, Eeq Parabolic Elastic catenary Associate catenary Elastic catenary

5

Straight bar, Eeq Parabolic Elastic catenary Associate catenary Elastic catenary

– 1.494822 1.532164 1.532215 1.532164

– 0.838857 0.881930 0.881949 0.881930

Straight bar, Eeq Parabolic Elastic catenary Associate catenary Elastic catenary

0

– 0.001001 0.031125 0.031126 0.031125

– 0.114140 0.325844 0.325849 0.325844

– 0.534606 0.807754 0.807772 0.807754

– 1.125619 1.281188 1.281227 1.281188

– 1.512016 1.543304 1.543356 1.543304

– 0.848507 0.888008 0.888027 0.888008

Straight bar, Eeq Parabolic Elastic catenary Associate catenary Elastic catenaryc

H(70.0)

Element type

β (deg)

H(60.0)a

– 0.094025 0.475646 0.474635 0.475646

– 3.741958 6.638973 6.641730 6.638973

– 12.436112 15.659681 15.667895 15.659681

– 22.191992 23.684242 23.680705 23.684242

– 27.670871 27.727619 27.741897 27.727619

– 27.918128 27.904380 28.012831 27.904380

H(80.0)

38.893250 38.555887 38.893159 38.555923 38.893159

188.593399 186.959046 188.600989 186.961717 188.600989

315.546401 312.829318 315.584355 312.828976 315.584355

404.439850 401.000859 404.520855 400.973085 404.520855

444.551884 440.806749 444.659844 440.750759 444.659844

446.250000 442.492270 446.359216 442.434778 446.359216

H(80.7)

– 0.111009 2.795775 2.795714 2.795775

– 0.647669 2.712710 2.712782 2.712710

– 1.187058 2.547063 2.547130 2.547063

– 1.519621 2.289398 2.289457 2.289398

– 1.521051 1.932461 1.932508 1.932461

– 1.469682 1.829907 1.829950 1.829907

T (60.0)b

– 0.133465 2.995888 2.995938 2.995888

– 0.844192 2.955816 2.955904 2.955816

– 1.615572 2.891783 2.891876 2.891783

– 2.117649 2.703852 2.703945 2.703852

– 2.135766 2.340404 2.340485 2.340404

– 2.060629 2.223013 2.223089 2.223013

T (70.0)

Table 4.5: Horizontal force H and maximum cable force T (kN)

– 1.217796 7.207200 7.196160 7.207200

– 9.471374 17.217899 17.224896 17.217899

– 18.404714 23.325135 23.337268 23.325135

– 25.133374 26.854539 26.851040 26.854539

– 27.960610 28.018741 28.033176 28.018741

– 27.963939 27.950214 28.058537 27.950214

T (80.0)

446.250000 442.519323 447.844763 443.989433 447.844763

446.250000 443.001083 447.720119 443.854031 447.720119

446.250000 443.215200 447.437426 443.550704 447.437426

446.250000 443.075675 447.018261 443.109723 447.018261

446.250000 442.633591 446.500682 442.577961 446.500682

446.250000 442.495213 446.362084 442.437722 446.362084

T (80.7)

CHAPTER 4. FINITE CABLE ELEMENTS

119 – 0.002977 0.013870 0.013870 0.013870 – 0.000108 0.005679 0.005680 0.005679

Straight bar, Eeq Parabolic Elastic catenary Associate catenary Elastic catenary

Straight bar, Eeq Parabolic Elastic catenary Associate catenary Elastic catenary

45

65

85

– – 0.000000 −0.000001 –

– – 0.000000 0.000000 –

– – 0.000000 −0.000001 –

– 0.000168 0.006847 0.006847 0.006847

– 0.005925 0.023058 0.023058 0.023058

– 0.026998 0.052044 0.052046 0.052044

– 0.061449 0.085890 0.085895 0.085890

– 0.085230 0.105809 0.105816 0.105809

– 0.086374 0.106723 0.106730 0.106723

K(70.0)

– – 0.000000 0.000000 –

– – 0.000000 0.000000 –

– – 0.000000 −0.000002 –

– – 0.000000 −0.000007 –

– – 0.000000 −0.000009 –

– – 0.000000 −0.000009 –

∆K(70.0)

b

Stiﬀness component K for chord length C Diﬀerence in stiﬀness components deﬁned as: K from [3] subtracted by K in this study c Elastic catenary by Jayaraman and Knudson [52]

a

– 0.010606 0.024590 0.024591 0.024590

25

Straight bar, Eeq Parabolic Elastic catenary Associate catenary Elastic catenary

– 0.021156 0.035543 0.035544 0.035543

Straight bar, Eeq Parabolic Elastic catenary Associate catenary Elastic catenary

5

– – 0.000000 −0.000001 –

– – 0.000000 −0.000002 –

– 0.027960 0.041654 0.041656 0.041654

Straight bar, Eeq Parabolic Elastic catenary Associate catenary Elastic catenary

0

∆K(60.0)b – – 0.000000 −0.000002 –

K(60.0) – 0.028281 0.041931 0.041932 0.041931

Element type

– 1.627559 1.700314 1.693119 1.700310

– 38.044727 38.179209 38.201664 38.179207

– 106.359921 106.508412 106.573848 106.507574

– 174.600851 174.781096 174.620453 174.777391

– 210.888461 211.096344 211.160614 211.096325

– 212.500000 212.709289 214.204762 212.709265

K(80.0)

Table 4.6: Stiﬀness component K44 (kN/m)

Straight bar, Eeq Parabolic Elastic catenary Associate catenary Elastic catenaryc

β (deg)

a

– – 0.000000 0.007082 –

– – 0.000000 −0.031623 –

– – 0.000000 −0.111539 –

– – 0.000000 0.052123 –

– – 0.000000 −0.217011 –

– – 0.000000 −1.650363 –

∆K(80.0)

10.288241 10.199018 10.330274 10.280973 10.330293

117.405852 116.395631 118.394574 117.980625 118.394531

318.671286 315.945260 321.438571 318.624227 321.438570

519.871708 515.395564 524.416773 516.995976 524.413889

626.901728 621.458426 632.392428 621.634540 632.392428

631.655484 626.168427 637.188184 626.267917 637.188184

K(80.7)

−5.487735 – 0.000000 0.047929 –

−4.542093 – 0.000000 0.044986 –

−2.764841 – 0.000000 0.025888 –

−0.987519 – 0.000000 −0.010611 –

−0.041780 – −0.000001 −0.037004 –

0.000231 – 0.000000 −0.038286 –

∆K(80.7)

4.6. COMPARISON OF ELEMENTS

120 – 0.004601 0.021216 0.021218 0.021216 – 0.000214 0.020019 0.020021 0.020019

Straight bar, Eeq Parabolic Elastic catenary Associate catenary Elastic catenary

Straight bar, Eeq Parabolic Elastic catenary Associate catenary Elastic catenary

45

65

85

– – 0.000000 0.000000 –

– – 0.000000 0.000000 –

– – 0.000000 0.000000 –

– 0.000653 0.020149 0.020150 0.020149

– 0.013365 0.028604 0.028606 0.028604

– 0.026998 0.037513 0.037516 0.037513

– 0.027246 0.033981 0.033984 0.033981

– 0.021924 0.028103 0.028105 0.028103

– 0.021600 0.027786 0.027788 0.027786

K(70.0)

– – 0.000000 0.000000 –

– – 0.000000 −0.000002 –

– – 0.000000 −0.000002 –

– – 0.000000 −0.000001 –

– – 0.000000 −0.000001 –

– – 0.078937 0.078932 –

∆K(70.0)

b

Stiﬀness component K for chord length C Diﬀerence in stiﬀness components deﬁned as: K from [3] subtracted by K in this study c Elastic catenary by Jayaraman and Knudson [52]

a

– 0.010606 0.023252 0.023253 0.023252

25

Straight bar, Eeq Parabolic Elastic catenary Associate catenary Elastic catenary

– 0.013690 0.023531 0.023532 0.023531

Straight bar, Eeq Parabolic Elastic catenary Associate catenary Elastic catenary

5

– – −0.000100 0.000000 –

– – 0.000000 0.000000 –

– 0.014141 0.022912 0.022913 0.022912

Straight bar, Eeq Parabolic Elastic catenary Associate catenary Elastic catenary

0

∆K(60.0)b – – 0.019058 0.019056 –

K(60.0) – 0.014142 0.022873 0.022874 0.022873

Element type

– 210.885926 201.455127 200.541388 201.454638

– 174.565951 173.673255 173.782197 173.673245

– 106.359921 106.139936 106.210122 106.139098

– 38.205226 38.166263 38.134140 38.165457

– 1.958746 1.957877 1.959130 1.957877

– 0.348977 0.349186 0.350541 0.349186

K(80.0)

Table 4.7: Stiﬀness component K66 (kN/m)

Straight bar, Eeq Parabolic Elastic catenary Associate catenary Elastic catenaryc

β (deg)

a

– – 0.000000 0.904867 –

– – 0.000000 −0.145141 –

– – 0.000000 −0.111544 –

– – 0.000000 0.011373 –

– – 0.000000 -0.001839 –

– – 0.000000 −0.001354 –

∆K(80.0)

627.209367 621.771224 632.697081 632.586334 632.697508

520.037904 515.564451 524.579643 522.944803 524.579484

318.671286 315.945260 321.435840 318.682722 321.435840

117.369714 116.359613 118.357913 116.731535 118.357310

10.285886 10.197993 10.329228 10.201084 10.329228

5.529740 5.483176 5.531117 5.482487 5.531117

K(80.7)

−0.042005 – 0.000000 0.047290 –

−0.987640 – 0.000000 0.035348 –

−2.099717 – −2.484440 −0.901773 –

−4.510313 – −3.113978 −2.170844 –

−5.491003 – −3.145456 −3.051004 –

−5.529740 – 0.000000 0.048630 –

∆K(80.7)

CHAPTER 4. FINITE CABLE ELEMENTS

121

– 0.000000 0.000000 0.000000 0.000001 0.000000 – 0.001218 0.001319 0.001319 0.001320 0.001319 – 0.004449 0.005233 0.005233 0.005234 0.005233 – 0.003535 0.005163 0.005163 0.005164 0.005163 – 0.000967 0.002254 0.002254 0.002254 0.002254 – 0.000009 0.000139 0.000139 0.000139 0.000139

Straight bar, Eeq Parabolic Elastic catenary As. cat. K46 As. cat. K64 Elastic catenary Straight bar, Eeq Parabolic Elastic catenary As. cat. K46 As. cat. K64 Elastic catenary Straight bar, Eeq Parabolic Elastic catenary As. cat. K46 As. cat. K64 Elastic catenary Straight bar, Eeq Parabolic Elastic catenary As. cat. K46 As. cat. K64 Elastic catenary

K(60.0)

Straight bar, Eeq Parabolic Elastic catenary As. cat. K46 As. cat. K64 Elastic catenaryc Straight bar, Eeq Parabolic Elastic catenary As. cat. K46 As. cat. K64 Elastic catenary

Element type

– – 0.000000 −0.000001 −0.000001 – – – 0.000000 0.000000 0.000000 – – – 0.000000 −0.000001 0.000000 – – – 0.000000 0.000000 0.000000 –

– – 0.000000 0.000000 −0.000001 0.000000 – – 0.000000 0.000000 0.000000 –

∆K(60.0)b

– 0.020381 0.022315 0.022316 0.022319 0.022315 – 0.016198 0.022156 0.022157 0.022159 0.022156 – 0.004433 0.009515 0.009515 0.009516 0.009515 – 0.000043 0.000483 0.000483 0.000483 0.000483

– 0.000000 0.000000 0.000000 0.000003 0.000000 – 0.005581 0.005605 0.005606 0.005609 0.005605

K(70.0)

– – 0.000000 −0.000002 −0.000001 – – – 0.000000 −0.000002 −0.000002 – – – 0.000000 −0.000001 −0.000001 – – – 0.000000 0.000000 0.000000 –

– – 0.000000 0.000000 −0.000003 0.000000 – – 0.000000 −0.000001 −0.000001 –

∆K(70.0)

b

– 81.274988 81.247867 81.172982 81.179087 81.246142 – 106.140079 106.046946 106.112197 106.117036 106.046109 – 81.349830 81.173395 81.221285 81.224258 81.173390 – 18.448948 18.129526 18.048903 18.049581 18.129481

– 0.000000 0.000004 0.000000 0.006724 0.000000 – 18.419973 18.417802 18.423413 18.430077 18.417800

K(80.0)

Table 4.8: Stiﬀness component K46 (kN/m)

Stiﬀness component K for chord length C Diﬀerence in stiﬀness components deﬁned as: K from [3] subtracted by K in this study c Elastic catenary by Jayaraman and Knudson [52]

a

85

65

45

25

5

0

β (deg)

a

– – −0.000001 0.024283 0.024284 – – – 0.000000 −0.111394 −0.060671 – – – 0.000000 −0.067641 −0.067644 – – – 0.000000 0.079253 0.079256 –

– – 0.000949 0.000000 −0.000052 0.000000 – – 0.000000 −0.018971 −0.018977 –

∆K(80.0)

239.841599 237.776264 241.959426 238.557555 238.539786 241.958136 313.141547 310.463135 315.906789 313.187830 313.154858 315.906789 239.919097 237.855438 242.037885 241.289523 241.261681 242.037804 54.389920 53.918340 54.870271 54.863038 54.856529 54.870397

0.000000 0.000000 0.000000 0.000000 0.019990 0.000000 54.363005 53.890853 54.843022 53.914194 53.925997 54.843022

K(80.7)

2.118072 – 0.000000 −0.026916 −0.027548 – 3.430022 – 0.666610 2.200997 2.239225 – 2.118016 – 0.000000 −0.005948 −0.006585 – 0.480114 – 0.000000 −0.000062 −0.000207 –

0.000000 – 0.001250 0.000000 0.000000 0.000000 0.480134 – 0.000000 −0.007344 −0.007487 –

∆K(80.7)

4.6. COMPARISON OF ELEMENTS

Chapter 5 Static analysis In this chapter, it will be demonstrated how a large prestressed cable roof structure is analysed using the ﬁnite element method. The structure chosen for the analysis is the Scandinavium Arena in Gothenburg. It was intentioned to also analyse another large Swedish cable roof structure—the Johanneshov Ice Stadium in Stockholm. However, it was found that this pioneering cable truss structure by David Jawerth had already been extensively analysed under diﬀerent loadings, see for example references 1, 76 and 78.

5.1

Static analysis of the Scandinavium Arena

The analysis of the Scandinavium Arena was performed using a geometrically nonlinear ﬁnite element method. Data about the Scandinavium Arena were obtained from construction drawings, articles [55, 56], a Master’s thesis [86] and unpublished material [104]. Further, the results from the calculations were compared to the results from previous calculations [86], and results from a simpliﬁed method recently presented at the Department of Structural Engineering at the Royal Institute of Technology in Stockholm, Sweden.

5.1.1

The Scandinavium Arena—background

Before the analysis begins some background information on the Scandinavium Arena will be given. An extensive description can be found in reference 55. In 1948 an architect competition concerning an indoor sports building in central Gothenburg was announced. It was won by a working group led by the architect Poul Hultberg. In 1962 the preliminary design works started and a ﬁnal decision concerning the realisation of the structure was taken in June 1969. In May 1971 the Scandinavium Arena was completed. With space for 14000 spectators it was at the time the largest covered arena in northern Europe [56]. The arena has been and is still used for activities such as concerts, theatre shows, ice-hockey, soccer,

123

CHAPTER 5. STATIC ANALYSIS

Figure 5.1: The Scandinavium Arena after completion. A pylon is seen almost in the middle of the view. Reproduced from [55].

swimming, etc. Roof structure. The roof consists of a prestressed cable net cladded with thermal and water insulated corrugated steel plates. All cables are anchored in a spacecurved reinforced concrete ring. The concrete ring is supported by four stiﬀ pylons and 40 circular columns. The surface of the roof conforms nearly with a hyperbolic paraboloid. From the centre point of the roof the hanging cables rise 10 m to the top and the bracing cables fall 4 m to the valley of the ring. The cable spacing is nearly constant and equals to 4 m in both directions. Foundation. The building is supported partly by rock and partly by concrete piles. Two of the pylons are supported by concrete foundations that rest on 115 piles. The large number of piles needed is due to the horizontal forces that occur at the connections between the ring beam and the pylons. Ring beam. The ring beam has a rectangular cross-section with a width of 3.5 m and a height of 1.2 m. An alternative solution with a ring beam made as a hollow steel box was investigated during the design work, but it was found to be too expensive. Columns and pylons. The circular columns are cast in place and designed to carry mainly axial forces. The pylons consist of radially oriented concrete walls, with a side length of 3.5 m connected by beams, Figure 5.2. The space between the walls is approximately 3.5 m wide and ﬁlled with ventilation equipment. The pylons are relatively stiﬀ and can take large horizontal forces. Therefore, the ring beam is discontinuous at the top of the pylons which aﬀects the prestressing forces in the cable in the areas between the pylons and the top of the ring beam. The forces in the bracing cables are there signiﬁcantly smaller than in other parts of the roof. Tension rods. The colour telecasting (remember that the arena was built in 1971) required that the light and sound systems had to be stable. Therefore, it was not considered suitable to attach the systems directly to the roof. Instead, the light

124

5.1. STATIC ANALYSIS OF THE SCANDINAVIUM ARENA

Figure 5.2: The arena after completed erection of the cables. Reproduced from [55].

and sound systems are suspended in a cable system supported by the pylons. This radially oriented cable system also serves as tension rods for the ring beam, which is discontinuous at the pylons, Figure 5.5. Original calculations. Preliminary dimensions of the ring beam and the cables were estimated by a simpliﬁed method in which the cable net was represented by a shearfree membrane. The stiﬀness of the ring beam was taken as the stiﬀness of a plane ring beam with the same dimensions as the real one and supported at four stiﬀ pylons [56]. The membrane stresses were approximated by sectionally constant values in each direction and the deﬂection of the roof by polynomials [56, 104]. The unknowns were determined from equations expressing the vertical equilibrium of the membrane and the compatibility between the membrane and the ring beam. Axial forces and bending moments in the ring beam due to snow and wind loads were modiﬁed with respect to the inclination of the ring beam. Accurate values of the twisting moment could not be obtained by the simpliﬁed method. Thereafter, a more accurate analysis was performed using a mixed ﬁnite element method. In that method, the structure was divided into two substructures, the cable net and the ring beam on columns. The two substructures were analysed separately by the stiﬀness method and then connected by compatibility and equilibrium expressions. Since the analysis was non-linear, the substructures had to be iteratively connected. The maximum deﬂection of the roof surface under full snow load was found to be 64 cm. Comparison with the simpliﬁed method showed a diﬀerence of at most 10 % in bending moments in the ring [56]. More results from these ﬁnite element calculations will be presented in section 5.1.5.

125

CHAPTER 5. STATIC ANALYSIS

Figure 5.3: Erecting the sheet rooﬁng. Reproduced from [55].

5.1.2

Prestressing forces

A cable net with a ﬁne mesh can be assumed to behave like a membrane free of shear stresses. In that case, it is possible to obtain an analytical solution by introducing a number of assumptions concerning the load distribution and behaviour of the cables and ring beam, see for example [116]. In this section, a simpliﬁed approach will be used to determine the magnitude of the initial prestressing forces in the cables. According to reference 86, the roof surface of the Scandinavium Arena is described by the following equation:  x 2

y 2  − fy for x ≤ xp , fx R R

x 2

y 2 3  x − x 2 (5.1) z= p  fx − fy − fy for x ≥ xp , R R 4 R where xp is the x-coordinate of the pylon, and fx , fy are deﬁning height measures. R is the radius of the horizontal projection of the ring beam.

126

Figure 5.4: Construction drawing K27:1 for the Scandinavium Arena, showing primarily the dimensions of the concrete ring beam.

5.1. STATIC ANALYSIS OF THE SCANDINAVIUM ARENA

127

Figure 5.5: Construction drawing K27:3 for the Scandinavium Arena, showing primarily the conďŹ guration of the cable net (horizontal projection).

CHAPTER 5. STATIC ANALYSIS

128

5.1. STATIC ANALYSIS OF THE SCANDINAVIUM ARENA

z R fx

y

fy

x

Figure 5.6: A hyperbolic paraboloid with a circular plane projection.

In the original preliminary calculations, it is assumed that the shape of the roof surface is described by the ďŹ rst expression in (5.1), i.e. the equation of the hyperbolic paraboloid [104]:

x 2

y 2 â&#x2C6;&#x2019; fy . (5.2) z = fx R R Using polar coordinates the equation of the ring beam can be described as: ďŁą ďŁ´ ďŁ˛xring = R cos Î¸, (5.3) yring = R sin Î¸, ďŁ´ ďŁł 2 2 zring = fx cos Î¸ â&#x2C6;&#x2019; fy sin Î¸. The equilibrium equations in x-, y- and z-directions for a membrane free of shear forces, can be stated as [119]: x â&#x2C6;&#x201A;H + Fx = 0, â&#x2C6;&#x201A;x y â&#x2C6;&#x201A;H + Fy = 0, â&#x2C6;&#x201A;y     â&#x2C6;&#x201A; â&#x2C6;&#x201A; â&#x2C6;&#x201A;z â&#x2C6;&#x201A;z   Hx + Hy + Fz = 0, â&#x2C6;&#x201A;x â&#x2C6;&#x201A;x â&#x2C6;&#x201A;y â&#x2C6;&#x201A;y

(5.4) (5.5) (5.6)

x, H  y are the horizontal components of the prestressing force distribution where H (N/m) in x- and y-directions, respectively. Fx , Fy , Fz are the load intensities (N/m2 ) is the x-, y- and z-directions, respectively. With only vertical loads, (5.4)â&#x20AC;&#x201C;(5.6)

129

CHAPTER 5. STATIC ANALYSIS

simplify to: x â&#x2C6;&#x201A;H = 0, â&#x2C6;&#x201A;x y â&#x2C6;&#x201A;H = 0, â&#x2C6;&#x201A;y 2 2 x â&#x2C6;&#x201A; z + H  y â&#x2C6;&#x201A; z + Fz = 0. H â&#x2C6;&#x201A;x2 â&#x2C6;&#x201A;y 2

(5.7) (5.8) (5.9)

x = H y = H  0 yields [104]: Inserting (5.2) into (5.9) and prescribing H 0 = â&#x2C6;&#x2019; H

Fz R2 2(fx â&#x2C6;&#x2019; fy )

(5.10)

 0 increases as fy increases. For the Scandinavium Equation (5.10) shows that H 2 Arena Fz = â&#x2C6;&#x2019;0.6 kN/m (the combined weight of the cladding and cables), fx = 10  0 = 145.8 kN/m. The horizontal m, fy = 4 m, and R = 54 m. These values gives H  0 with the component of the force in a single cable is obtained by multiplying H  cable spacing. For non-equidistant cable spacing, H0 is multiplied with the sum of half the distance between adjacent cables, e.g. for a cable i in the y-direction it is (xi+1 â&#x2C6;&#x2019; xiâ&#x2C6;&#x2019;1 )/2.

5.1.3

Finite element model

The Scandinavium Arena will in this section be analysed with ďŹ nite elements, as a comparison to the original calculations. As a demonstration, the comparison will be performed for only one load case: uniformly distributed dead load of â&#x2C6;&#x2019;0.6 kN/m2 and snow load of â&#x2C6;&#x2019;0.75 kN/m2 on the whole roof. Due to symmetry in both structure and load case only a quarter of the structure had to be modelled for this case, Figure 5.7.

x

y

Figure 5.7: One quarter of the Scandinavium Arena.

The ďŹ nite element model of a quarter of the structure is shown in Figure 5.8. The beam nodes on the symmetry lines x = 0 and y = 0 have the following boundary

130

5.1. STATIC ANALYSIS OF THE SCANDINAVIUM ARENA

conditions: θy = 0, θz = 0, and ux = 0. Cable nodes on x = 0 are prevented to move in the y-direction and vice versa for the other symmetry line. All columns are pin-jointed to the ground and to the ring beam, while the pylon is rigidly connected at both ends.

Figure 5.8: Element model.

All cables are modelled using the elastic catenary element presented in Chapter 4. This element was chosen instead of the bar element to avoid the problems with local mechanisms and numerical instability due to slackening cables. The ring beam and the pylon were modelled by non-linear three-dimensional beam elements based on cubic shape functions. The co-rotational approach is used to compute the tangent stiﬀness matrix and internal force vector [90]. The Matlab routines for the beam element have been developed by Costin Pacoste and have been used to analyse other structures [85]. Since the ring beam is relatively stiﬀ, the results using the non-linear beam elements were compared with results using linear three-dimensional beam elements. The diﬀerences in the results were very small. Despite the small diﬀerence, the non-linear beam element was used for the analyses in this chapter. It should be mentioned that some of the beam elements are very short and stiﬀ, and therefore do not ﬁt into the beam assumption. Still, the beam model is used for these elements. A more accurate analysis, which avoids the short elements but keeps the cable spacing would require the use of solid elements. However, some diﬃculties in connecting the solid elements and the cable elements may arise since the solids cannot cope with high concentrated load in the same way as the beam elements. The columns were modelled with straight bar elements. The magnitudes of the prestressing forces are computed according to subsection 5.1.2. The radially oriented tension rod, shown in Figure 5.5, was not included in the ﬁnite element model of the structure. As mentioned earlier, the main diﬀerence in analysis between cable structures and other structures, such as frames and trusses, is that the initial conﬁguration is generally unknown for cable structures. According to Møllmann [76], the following iterative procedure is used for a cable structure with an elastic boundary structure (arches or beams):

131

CHAPTER 5. STATIC ANALYSIS

1. Assuming that the boundary joints are fixed in the positions corresponding to the unstressed state of the arch, the shape of the cable net is determined corresponding to cables in vertical planes. 2. The cable forces at the boundary joints obtained from the previous stage are now regarded as external loads acting on the arch. The arch is then analysed separately for these forces and for the weight of the arch members. 3. Keeping the boundary joints fixed in the positions obtained from stage 2, the shape of the net is now recalculated (cables in vertical planes or geodesic net). 4. Return to 2.

With this procedure, the boundary structure and the cable net are calculated separately until the displacement changes of the joint coordinates of the cable net and boundary are suﬃciently small. A somewhat similar procedure was used in the analysis of the Scandinavium Arena. The only diﬀerence is that at step 2 the pretensioned cable net is numerically attached to the unstressed ring beam. This means that the stiﬀness of the whole structure is used to compute the displacements of the ring beam. Some cables will be unloaded during step 2, but since the ring beam is quite stiﬀ only 3–4 iterations are needed to get a deviation in the horizontal component of the cable forces of a most 0.5 %. What this error corresponds to in the unstrained length of a cable will now be checked. For a bar the following equation holds: L − L0 ∆L = AE . (5.11) T = AE L0 L0 This equation can be written as: L0 =

L . T /AE + 1

(5.12)

The cables for the Scandinavium Arena have AE = 343 MN, T ≈ 145.8·4 = 583 kN. Assuming a cable length of 108 m, the error in unstrained length is 0.92 mm. This tolerance can not be reached in practice. However, one should bear in mind that a small error in unstrained length may give large errors in force; for example, an error in unstrained length of 0.05 % gives an error in force of 30 % with the cable data given here. For the form-ﬁnding of the cable net the grid method (section 3.2.2) was used. Nevertheless, equation (3.7) cannot be used in this case due to a nonequidistant mesh. Instead, the following equation expressing the vertical equilibrium is used:     zj − zi zk − zi zi − zl zi − zm + Hiy + Fiz = 0. − − (5.13) Hix xj − xi xi − xl y k − yi y i − ym Equation (3.7) is a special case of (5.13). Note that yl = yi = yj and xm = xi = xk in this case, Figure 3.1. Material and cross-sectional properties for pylons, the ring beam, columns and cables are given in Figure 5.10. More data, including coordinates for beam elements, node loads, prestressing forces, etc., can be found in Appendix A.

132

5.1. STATIC ANALYSIS OF THE SCANDINAVIUM ARENA

(a) Initial equilibrium procedure—step 1

(b) Initial equilibrium procedure—step 2

(c) Initial equilibrium procedure—step 3

Figure 5.9: Numerical procedure to ﬁnd the initial equilibrium conﬁguration suggested by Møllmann [76]: (a) the shape of the cable net is obtained by assuming ﬁxed nodes, (b) the cable forces are regarded as external loads on the ring beam (step 1 and 2 are repeated until convergence), and (c) the two structures are connected and the whole structure should now be in equilibrium.

133

CHAPTER 5. STATIC ANALYSIS

z

3500

650

x



y

Pylon E = 32 GPa G = 12.8 GPa A = 4.55 m2 Ix = 0.566 m4 Iy = 4.645 m4 Iz = 17.021 m4

y

Ring beam E = 32 GPa G = 12.8 GPa A = 4.2 m2 Ix = 1.581 m4 Iy = 0.504 m4 Iz = 4.288 m4

4500

1200

z

x



3500

Column E = 32 GPa G = 12.8 GPa A = 0.503 m2 Ix = 0.040 m4 Iy = 0.020 m4 Iz = 0.020 m4

800

Cable E = 162 GPa A = 2.12 · 10−3 m2

Figure 5.10: Cross-sectional and material properties for ﬁnite elements. Drawn dimensions in millimetres.

134

5.1. STATIC ANALYSIS OF THE SCANDINAVIUM ARENA

5.1.4

Calculation results

Some results from the present analysis of the Scandinavium Arena are presented in Figures 5.11–5.16. The maximum deﬂection of the net due to the snow load is 54 cm. Figure 5.11 shows that the contour curves of the net displacement are almost circular. All cable forces increase from the pretension values due to the snow load. The forces in the bracing cables increase as a result of the outwards movement of the valley of the ring beam. The axial force diagram indicates that the pylon signiﬁcantly reduces the axial force in the lower part of the ring beam between the pylons. Luckily, the bending moments in these parts are not that large. A discontinuity in the bending moment diagram occurs due to eﬀects from the pylon. The ‘height’ of this discontinuity depends on the relation between the bending stiﬀness of the ring beam and the torsional stiﬀness of the pylon. The twisting moments are much lower than the bending moments. As for the bending moment diagram, the pylon aﬀects the twisting moment distribution. Some more comments concerning the results are given in section 5.1.6, where the present results are compared with previous results. Note that in all bending moment diagrams in this chapter, a positive moment corresponds to tension at the outer side of the ring beam. (0 , 5 2 )

55

(− 1 ,5 1 ) (− 2 ,4 8 ) (− 4 ,4 3 )

50

(− 1 1 ,3 4 )

45

40

(− 2 3 ,2 4 )

−200 35

y (m)

(− 3 6 ,1 4 ) 30 −300

−100

25

(− 5 0 ,6 )

−400 20

(− 6 2 ,2 )

15

−500

(− 7 1 ,0 ) 10

5

(− 7 5 ,0 )

0 0

5

10

15

20

25

30

35

40

45

50

55

x (m) Figure 5.11: Contour lines of net displacement and ring beam displacements in x–y plane due to snow load (mm).

135

CHAPTER 5. STATIC ANALYSIS

400

380

379

378

376

373

373

371

368

362

350

351 322

Increase in force (kN)

300 248

250

200

150

100

50

0 0

10

5

15

20

25

30

40

35

50

45

y (m) Figure 5.12: Increase in the horizontal component of the cable forces for cables in the x-direction (hanging cables) due to snow load.

80 75 71

70

67

65 60

58

Increase in force (kN)

54 50

49

48 43

41

45 41

40

30

20

10

0 0

5

10

15

20

25

30

35

40

45

50

x (m) Figure 5.13: Increase in the horizontal component of the cable forces for cables in the y-direction (bracing cables) due to snow load.

136

5.1. STATIC ANALYSIS OF THE SCANDINAVIUM ARENA

Column Pylon

Axial force (MN)

0

−5

−10

−8.1

−7.8

−7.7

−8.6 −9.3 −8.9

−13.1 −13 −13 −13.9 −14.3 −14.1

−15 60

−12−11.6 −11.2

−10.8

50

60

40

50 30

y (m)

40 30

20

20

10

x (m)

10 0

0

Figure 5.14: Axial force in ring beam under snow load.

Bending moment (kNm)

Column Pylon 12988 0 1 7 93 10204

15000 10000

7148

5312

5037

5000

3897 2927

0

−729

−5000

−4535 −7066

−10000

−8208 −8762 −9881

−15000 60 50

60

40

y (m)

50 30

40 30

20

20

10 0

10

x (m)

0

Figure 5.15: Bending moment around the local z-axis (stiﬀ direction) for beam elements under snow load.

137

CHAPTER 5. STATIC ANALYSIS

Column Pylon Twisting moment (kNm)

2000 1515 1500 856

1000 472

500 209 0

0

−500

−595 −585

−687

−1000 60

0 −716

−647 −533 −384

50

60

40

50 30

y (m)

40 30

20

20

10

10 0

0

x (m)

Figure 5.16: Twisting moment around local x-axis for beam elements under snow load.

5.1.5

Calculation results from 1972

In reference 86, the Scandinavium Arena is analysed with the ﬁnite element method. Like the analysis above, the calculations were done on one quarter of the structure. The ring beam was modelled with linear three-dimensional beam element and the cables with straight bar elements. As mentioned above, the ring beam and the cable net were analysed separately with the stiﬀness method and connected by a ﬂexibility approach. Two load cases were considered: snow load (−0.75 kN/m2 ) on the whole roof and wind load (0.4 kN/m2 ) on the whole roof. The maximum deﬂection under full snow load was 64 cm. The ﬁnite element model in [86] diﬀers from the model analysed in the previous section. The most important diﬀerences between the two structural models are: • Cable spacing. While the cable spacing according to the drawings was used in the present analysis of the Scandinavium Arena, a non-equidistant mesh of cables was used in [86]. The cables are connected to the joints between the beam elements, to which also the columns and the pylon are connected, Figure 5.17. It appears that this mesh has been chosen to reduce the number of unknowns. • Contour shape. No information on the x- and y-coordinates of the ring could be obtained from any of the references 56, 86 and 104. Nonetheless, the ring

138

5.1. STATIC ANALYSIS OF THE SCANDINAVIUM ARENA

appears to be circular in the ﬁgures in [86]. In the present analysis the coordinates of the ring beam were obtained from the construction drawing in Figure 5.4. It is important to know the ‘exact’shape of the ring beam as it considerably aﬀects the distribution of the bending moment, see section 5.2. • Cable forces. As mentioned above, the cable forces in the upper bracing cables are lower than those in the lower bracing cables. In addition, the ﬁve lowest bracing cables were also post-tensioned by introducing a gap between the ring beam and the cable net. The ﬁnal cable forces after the post-tensioning were not given. Therefore, a uniform cable force distribution was used in the present analysis. • Young’s modulus. No information concerning the modulus of elasticity for the concrete is given in [86]. According to the drawings, concrete K400 (old notation) was used for the ring beam. A characteristic value of 32 GPa for the modulus of elasticity for this concrete class is found in the Swedish building codes. This value was used in the present analysis. The shear modulus is 0.4E, which corresponds to Poisson’s ratio of 0.2. Some of the results from reference 86 are given in Figures 5.17–5.21 to facilitate comparison. y (−1,85 )

(−3,79 ) (−6,68 )

−100

(−13,54 )

−200

(−24,39 )

−300

(−39,27 )

−400

(−54,17 )

−500

(−72,7 )

(−87,2 )

−600

(−99,1 )

(−105,0 )

−600

−400

−500

−300

−200 −100

x

Figure 5.17: Contour lines of net displacement and ring beam displacements in x–y plane due to snow load (mm). Redrawn from [86].

139

CHAPTER 5. STATIC ANALYSIS

900 845 815 800

769

750

Increase in force (kN)

700

659

600

560

500 400 335 300

277

200 100 45 0 0

7 5

10

15

20

30

25

35

40

45

50

55

60

y (m) Figure 5.18: Increase in the horizontal component of the force for cables in the xdirection (hanging cables) due to snow load. Redrawn from [86].

700 612 600 558 490

Increase in force (kN)

500

472 426

400

300

200

100

0 −25 −100

0

5

10

15

20

25

30

35

40

−14 −10 −9 −1 45

50

55

60

x (m) Figure 5.19: Increase in the horizontal component of the force for cables in the ydirection (bracing cables) due to snow load. Redrawn from [86].

140

5.1. STATIC ANALYSIS OF THE SCANDINAVIUM ARENA

Column Pylon Bending moment (kNm)

15000 10000

10787 10003

8649

7659

6649

5000

2452 1648

2962

0 −2844

−3344

−5000

−6374

−10000 60

−9238 −7639 −9807 60 50 40

50 40 30

y (m)

30

20

20

10 0

x (m)

10 0

Figure 5.20: Bending moment around the local z-axis (stiﬀ direction) for beam elements under snow load. Redrawn from [86].

Column Pylon 1393

Twisting moment (kNm)

1500

1265 943

1000 504

500 0

121

−500

−618

−248 −633 −625 −611 −485

−57

−1000 60 50

60

40

50 30

y (m)

40 30

20

20

10 0

10 0

x (m)

Figure 5.21: Twisting moment around local x-axis for beam elements under snow load. Redrawn from [86].

141

CHAPTER 5. STATIC ANALYSIS

5.1.6

Comparison of the results

The diﬀerences in midpoint deﬂection and in-plane displacements are probably due to a higher Young’s modulus of the concrete in the present analysis and diﬀerent initial cable forces in the analyses. For both analyses, both the cable forces in the x- and y-directions increase. This is due to the outwards displacement of the valley of the ring beam. A detailed comparison of the increase in the cable forces can not be done since the intial forces diﬀer much in some parts of the roof. No axial forces were presented in [86]. However, Figure 5.14 clearly shows that the pylon takes a large horizontal force. Concerning the twisting moments, they agree very well; the distributions of the twisting moments are similar and the values do not diﬀer very much. In the case with the bending moments, their distributions diﬀers qualitatively very much. The maximum magnitude of the bending moments is about the same for the two analyses. Thus, it can be concluded that all the results except the bending moment distribution have a satisfactory agreement. The disagreement in the bending moment distribution will be explained by another example in the next section.

5.2

Sensitivity of bending moment to the shape of the ring beam

In order to explain the discrepancies in the bending moment distributions for the calculations of the Scandinavium Arena and to verify the ﬁnite element program another structure will now be analysed. The structure shown in Figure 5.22 has been analysed by Møllmann [76] and was chosen because of its similarities with the Scandinavium Arena.

5.2.1

Description of the structure

The system used for the calculation consists of nine hanging and nine bracing cables in the net, and 28 straight space beam elements which form the ring beam. The joints between the beam elements coincide with the joints where the cables are attached. The ring beam is supported by vertical columns at all joints and it is assumed that vertical displacement is prevented at these joints. There are three main supports: joints 8, 15 and 22, Figure 5.22. In addition to the vertical constraints due to the columns, joints 8 and 15 are prevented from moving in the y-direction and joint 15 cannot move in the x-direction.

142

5.2. SENSITIVITY OF BENDING MOMENT TO THE SHAPE OF THE RING BEAM

y

z

1 8.60 5.40

4

26

x

5

25

52.00 8

22

11

19

x

52.00

Dimensions in m z 8.60 5.40

y

12

18 15

52.00

52.00

Figure 5.22: Geometry of the cable structure by Møllmann. Redrawn from [76].

All cables have the cross-sectional area A = 5.2 · 10−3 m2 and Young’s modulus E = 160 GPa. The concrete ring beam is assumed to be elastic with the material and cross-sectional properties given in Figure 5.23. The cross-section of the ring beam twists along the perimeter. The longer side of the rectangle is parallel to the tangent of the roof surface in the direction normal to the boundary. In the initial state, the cables are in vertical planes and the cable joints are very nearly located on a hyperbolic paraboloid, Figure 5.22. The initial state is the equilibrium conﬁguration where the cables are pretensioned and the combined weight of the cables and the cladding acts on the net. The initial equilibrium conﬁguration is determined according to the procedure described in section 5.1.3. The horizontal components of the cable forces in the initial state are 2600 kN for both the hanging and the bracing cables.

1300

z

y

x

Ring beam E = 20 GPa G = 10 GPa (ν = 0) A = 2.6 m2 Ix = 0.877 m4 Iy = 0.366 m4 Iz = 0.867 m4

2000 Figure 5.23: Cross-sectional and material properties. Drawn dimensions in millimetres.

143

CHAPTER 5. STATIC ANALYSIS

In reference 76, the following four load cases were analysed: 1. Uniformly distributed dead load plus snow load over the whole roof. 2. Uniformly distributed dead load on the whole roof plus snow load on half the roof (x > 0). 3. Uniformly distributed dead load on the whole roof plus snow load on half the roof (y > 0). 4. Uniformly distributed dead load plus wind load over the whole roof. Only the ﬁrst load case will be considered in this section. The vertical loads on the cable net (measured per unit horizontal area in the initial state) are: dead load (weight of cables and roof cladding)= −0.6 kN/m2 and snow load= −0.75 kN/m2 . In the present analysis, the whole structure (375 degrees of freedom) was modelled using the same program and ﬁnite elements as for the analysis of the Scandinavium Arena.

5.2.2

Diﬀerent shapes of the ring beam

In Figure 5.22 the ring beam is drawn as a circle. However, in [76] the boundary arcs 26–1–4, 5–8–11, 12–15–18 and 19–22–25 were replaced by parabolas with the same rise as the corresponding circular arcs. This was done in order to make the projected boundary curve conincide with the line of compression for the projected cable forces in the initial state. As is seen in Figure 5.22, along these arcs the ring beam is in the plane loaded only in one direction (either the x- or the y-direction). To investigate the eﬀects of a change of the shape of the ring beam arcs, the structure studied by Møllmann will be analysed for three diﬀerent shapes of the arcs: circular, parabolic, and cosine shape. These shapes are described by the following equations for the arc 26–1–4 (−3R/5 ≤ x ≤ 3R/5): ycircle = (R2 − x2 )1/2 , 5x2 yparabola = − + R, 9R    4 5x −1 ycosine = R cos cos . 3R 5 The y-coordinates for the diﬀerent shapes are shown in Figure 5.24.

144

(5.14) (5.15) (5.16)

5.2. SENSITIVITY OF BENDING MOMENT TO THE SHAPE OF THE RING BEAM

1

2 3

4

5

Node

x (m)

2 3

10.400 20.800

y (m) Parabola 50.844 47.378

Circle 50.949 47.659

Cosine 50.808 47.288

z (m) −4.840 −3.160

Figure 5.24: Diﬀerence between some ring beam shapes: circle and parabola— visually, circle, parabola and cosine—quantitatively.

5.2.3

Results and discussion

Some of the results from the calculations are given in Table 5.1, but the most interesting results are shown in Figures 5.26–5.38 on pages 147–153. Table 5.1: Midpoint and ring beam displacements under full snow load.

Description Midpoint displacement (m) Ring beam displacement in y-direction at (x = 0, y = R) (m) Ring beam displacement in x-direction at (x = R, y = 0) (m)

Shape of ring beam arcs Circle Parabola Cosine −1.174 −1.176 −1.177 0.189 0.190 0.190 −0.222

−0.222

−0.222

Ref. 76 −1.171 0.188 −0.221

It is shown in Table 5.1, Figures 5.26–5.29 that the shape of the ring beam has very small eﬀects on the displacements, the cable forces and the axial force in the ring beam. In addition, for these quantities the current results agree very well with the results given in reference 76. However, regarding the bending moment, large changes occur when the shape is varied, see Figures 5.30–5.35. Also the twisting moment

145

CHAPTER 5. STATIC ANALYSIS

changes, but not as much as the bending moment. Nevertheless, the twisting moment is not so important in this case since its maximum value is less than 10 % of the maximum value of the bending moment. The reason for the large diﬀerence in the bending moment distribution will be discussed below. Structures with circular or parabolic shape are optimised for a certain load case (cf. a bicycle wheel or a stone arch). If the load case changes, the structure may undergo large changes in displacements and force distribution. Anyone who has broken a spoke in a well-built bicycle wheel will agree that the structure adjust itself to the new load case and for some structures the adjustment might be large. The reverse must also hold: change the shape of the structure but keep the load distribution and possibly large changes in some quantities will follow. For the present example, consider the two statically equivalent systems in Figure 5.25. The eccentricity e = M/N can be considered as a measure of the importance of the bending moment compared to the normal force. Remember, in the present example, the axial force N changed very little when the shape of the ring beam was varied. This means that if the shape of the ring beam is changed, the eccentricities of the axial forces are changed.

N1

M1

N2 (a)

M2

(b)

N2

e1 N1 e2

Figure 5.25: A segment of the ring beam. The systems (a) and (b) are statically equivalent if e1 = M1 /N1 and e2 = M2 /N2 .

For the parabolic arcs, the eccentricities in the initial state for nodes 2 and 3 are e2 = 227/12431 = 0.018 m and e3 = 189/13700 = 0.014 m. The coordinate changes for these nodes are ∆y2 = 0.105 m and ∆y3 = 0.281 m. Hence, if the coordinate changes are larger than the eccentricities one can expect signiﬁcant changes of the bending moment diagram, Figures 5.30 and 5.31. If the axial force is much larger than the bending moment, i.e. a small eccentricity e,

146

5.2. SENSITIVITY OF BENDING MOMENT TO THE SHAPE OF THE RING BEAM

the moment is considered as ‘small’. This means that the whole cross-section of the ring beam will be subjected to compressive stresses only. It is possible to calculate the eccentricity e0 that, for a rectangular section of height h and width b, gives zero stress at the outermost ﬁber: σ=−

N 6M 6N e0 N =0 + 2 =− + bh bh bh bh2

(5.17)

Equation 5.17 yields e0 = h/6. If the eccentricity e is smaller than e0 no tensile stresses will occur. However, the compressive stresses on the other side of the crosssection can be very high and may crush the concrete. It is therefore desirable to have a small eccentricity in the whole ring beam. It is concluded that the bending moment distribution is strongly dependent on the shape of the ring beam. The shape of the ring beam in the present analysis of the Scandinavium Arena and in the previous analysis was diﬀerent. Thus, the bending moment distributions from the two studies cannot be compared. Also for the example by Møllmann, the present analysis gave bending moments values that diﬀered quite much from those given in [76], see Figures 5.31 and 5.34. As no coordinate values for the ring beam were given in reference [76], a detailed comparison of the bending moments is not possible. 3500

3000

3171 (3190) [3195]

3343 (3346) [3347]

3384

3357

3339

3357

3384

(3382)

(3346)

(3331)

(3346)

(3382)

[3382]

(3346)

[3342]

[3342]

[3382]

3171

[3327]

[3347]

(3190)

{3370}

3343

[3195]

{3370}

Cable force (kN)

2500

2000

1500

1000

500

0 − 50

− 40

− 30

20 −

10 −

0

10

20

30

40

50

y (m) Figure 5.26: Forces in the hanging cables under full snow load. The following notation is used: ·=Circle, (·)=Parabola, [·]=Cosine, {·}=Møllmann [76].

147

CHAPTER 5. STATIC ANALYSIS

3500 3380

3347

3347

3380

(3354)

3322

(3351)

(3342)

(3342)

(3354)

(3351)

[3356]

[3344]

(3336)

[3340]

[3340]

[3340]

[3344]

[3356]

3329 3108

3000

(3119) [3123]

{3340}

3329

{3340}

3108 (3119) [3123]

Cable force (kN)

2500

2000

1500

1000

500

0 50 −

40 −

30 −

20 −

10 −

0

20

10

30

40

50

x (m) Figure 5.27: Forces in the bracing cables under full snow load. The following notation is used: ·=Circle, (·)=Parabola, [·]=Cosine, {·}=Møllmann [76].

0

Axial force (kN)

−2000 −4000 −6000

{−11770}

−11740 −12429 (−11746) (−12431) [−11747] [−12431]

−8000

−10000

{−13860}

−13271 (−13270) [−13269]

−13685 (−13700) [−13701]

−12000

−13674

−12421

(−13696) (−12427) [−13700] [−12428]

−11729

(−11741) [−11743]

−14000 60 50

60

40

50 30

y (m)

40 30

20

20

10 0

10 0

x (m)

Figure 5.28: Axial force in beam elements in the initial state. The following notation is used: ·=Circle, (·)=Parabola, [·]=Cosine, {·}=Møllmann [76].

148

5.2. SENSITIVITY OF BENDING MOMENT TO THE SHAPE OF THE RING BEAM

0 −2000

Axial force (kN)

−4000 −6000 −8000

{−14930}

−10000

−15027 −15921 (−15033) (−15918) [−15032] [−15916]

−12000 −14000

{−17590}

−17500

−17176

−17567 (−17587) [−17586]

−15842

(−17505) (−15844) [−17500] [−15840]

(−17170) [−17164]

−14939

(−14949) [−14947]

−16000 −18000 60 50

60

40

y (m)

50 30

40 30

20

20

10

x (m)

10

0

0

Figure 5.29: Axial force in beam elements under full snow load. The following notation is used: ·=Circle, (·)=Parabola, [·]=Cosine, {·}=Møllmann [76].

Bending moment (kNm)

2500

2288

2000

1913

1500 1000 500 139

0 −500 −1000 −1500

−391

−1074

−1225

1109

−2000 60

−1697

50 40

60 50

30

y (m)

40 30

20

20

10 0

10 0

x (m)

Figure 5.30: Bending moment around local z-axis for beam elements in the initial state. Circular shape of ring segments.

149

CHAPTER 5. STATIC ANALYSIS

Bending moment (kNm)

300

{770}

227

245

200

189 109

100 0 −49

−100 −200

−181

−300 −283

−400 60

−310

{−1150}

50

60

40

50 30

y (m)

40 30

20

20

10

x (m)

10

0

0

Figure 5.31: Bending moment around local z-axis for beam elements in the initial state. Parabolic shape of ring segments. {·}=Møllmann [76].

Bending moment (kNm)

800 600

693

497

400

340 240

200 0

137

−200 −400

−270

−472

−600 −800

−843

−1000 60 50

60

40

50 30

y (m)

40 30

20

20

10 0

10

x (m)

0

Figure 5.32: Bending moment around local z-axis for beam elements in the initial state. Cosine shape of ring segments.

150

5.2. SENSITIVITY OF BENDING MOMENT TO THE SHAPE OF THE RING BEAM

Bending moment (kNm)

6000

5038

4000 2468

2000 0

648 980

−114 −1323

−2000 −4000

−4468

−6000 −8000 60

−6547

50

60

40

50

y (m)

30

40 30

20

20

10 0

x (m)

10 0

Figure 5.33: Bending moment around local z-axis for beam elements under snow load. Circular shape of ring segments.

Bending moment (kNm)

3000 2000

{3030}

2482

2555

2258

2020

1000 219 0 −1000 −2000 −3000

−2648

−4000 −5000 60

−4107

{−5570}

−4543

50 40

y (m)

60 50

30

40 30

20

20

10 0

10 0

x (m)

Figure 5.34: Bending moment around local z-axis for beam elements under snow load. Parabolic shape of ring segments. {·}=Møllmann [76].

151

CHAPTER 5. STATIC ANALYSIS

Bending moment (kNm)

4000 3000

3109

2504

2499

2000

1434 706

1000 0 −1000 −2000 −3000

−3483

−4000 −5000 60

−4064

−3928

50

60

40

50 30

y (m)

40 30

20

20

10 0

x (m)

10 0

Twisting moment (kNm)

Figure 5.35: Bending moment around local z-axis for beam elements under snow load. Cosine shape of ring segments.

200

169

100 21

10

0 −30

−100

−133 −200 −153 −300 60

−285

50

60

40

50 30

y (m)

40 30

20

20

10 0

10 0

x (m)

Figure 5.36: Twisting moment around local x-axis for beam elements under snow load. Circular shape of ring segments.

152

Twisting moment (kNm)

5.2. SENSITIVITY OF BENDING MOMENT TO THE SHAPE OF THE RING BEAM

200 109 100 21 0

0 −51

−100

−83

−200 −300 −330

−400 60

−255

50

60

40

y (m)

50 30

40 30

20

20

10 0

x (m)

10 0

Twisting moment (kNm)

Figure 5.37: Twisting moment around local x-axis for beam elements under snow load. Parabolic shape of ring segments.

80

100 34 14

0

−74

−100

−54 −200 −230

−300 −400 60

−390 50

60

40

y (m)

50 30

40 30

20

20

10 0

10

x (m)

0

Figure 5.38: Twisting moment around local x-axis for beam elements under snow load. Cosine shape of ring segments.

153

CHAPTER 5. STATIC ANALYSIS

5.3

Comparison with a simpliďŹ ed method

In reference 116, a simpliďŹ ed method to analyse a prestressed cable net anchored in an elliptic contour beam is presented. The simpliďŹ ed method is well suited for preliminary design, where the dimensions of the structure and its structural members are to be decided. However, in further design work a more accurate ďŹ nite element method is often needed. In this section the reliability and limitations of the simpliďŹ ed method will be investigated. In the simpliďŹ ed method, a number of assumptions concerning the behaviour of the structure and load cases are imposed to obtain closed-form solutions. The most important assumptions are: â&#x20AC;˘ The cable net is substituted by a continuous shear-free membrane. This can be considered to be valid for a net with a dense mesh. â&#x20AC;˘ The projection of the ring beam in the horizontal plane is an ellipse. â&#x20AC;˘ The ring beam is assumed to deform like a linear plane beam. Thus, the space-curved shape of the ring beam is not taken into consideration. â&#x20AC;˘ The ring beam is supported by a continuous wall which permits the ring to move freely in the horizontal plane. â&#x20AC;˘ In the initial state, the roof is uncladded. This is in contrast to the initial states of the Scandinavium Arena and MĂ¸llmannâ&#x20AC;&#x2122;s structure. The complete list, which includes 16 assumptions is given on the pages 34â&#x20AC;&#x201C;36 in reference 116. First, the example by MĂ¸llmann will be investigated. The geometry of the roof will diďŹ&#x20AC;er from that shown in Figure 5.22 due to the last assumption given above. For  0 in both directions and Fz = 0, equation (5.9) can the same prestressing intensity H be written as: 2 2  0 â&#x2C6;&#x201A; z = 0. 0 â&#x2C6;&#x201A; z + H (5.18) H â&#x2C6;&#x201A;x2 â&#x2C6;&#x201A;y 2 Solving (5.18) yields fx = fy as expected. The heights fx and fy will be diďŹ&#x20AC;erent also for the Scandinavium Arena as the self-weight is zero in the initial conďŹ guration.

5.3.1

Results and discussion

The results for MĂ¸llmannâ&#x20AC;&#x2122;s example and the Scandinavium Arena are presented in Tables 5.2 and 5.3. MĂ¸llmannâ&#x20AC;&#x2122;s example were calculated with both circular and parabolic shapes of the contour arcs.

154

5.3. COMPARISON WITH A SIMPLIFIED METHOD

Table 5.2: Comparison between the simpliﬁed method and the ﬁnite element method for Møllmann’s example. Load case: uniformly distributed load equal to −0.75 kN/m2 on the whole roof (N.B. zero load in the initial state).

Method Finite element Circular Parabolic 7.002 7.002 6.998 6.998 −1.689 −1.692 3009 2996

Description

fx (m) fy (m) Midpoint displacement (m) Force in the mid cable in the x-direction (kN) Force in the mid cable 2974 in the y-direction (kN) Axial force at (x = 0, y = R) (kN) −13538 Axial force at (x = R, y = 0) (kN) −13464 Bending moment at (x = 0, y = R) (kN) 2877 Bending moment at (x = R, y = 0) (kN) −7154

Simpliﬁed 7.000 7.000 −1.634 3196

2991

3107

−13549 −13471 4238 −5237

−15430 −15001 5577 −5577

For Møllmann’s example, the midpoint displacement agree very well for both the circular and parabolic shapes of the contour arcs. The same holds for the cable forces. For the axial forces and bending moments, however, the diﬀerences are larger. It should be mentioned that axial forces given for the ﬁnite element calculation are not the maximum axial forces, cf. Figure 5.29. The maximum axial force for the circular and parabolic shapes are −15785 kN and −15816 kN respectively. Nevertheless, the maximum positive and negative moments occur at the bottom and the top of the ring beam. The bending moments for the ring beam with a parabolic shape agree best with the simpliﬁed method. The diﬀerence in the positive bending moment may be a result of the assumption of a plane ring beam in the simpliﬁed method. The diﬀerence in axial force in the ring beam follows from the diﬀerences in the bending moments since the structure must be in equilibrium.

155

CHAPTER 5. STATIC ANALYSIS

Table 5.3: Comparison between the simpliﬁed method and the ﬁnite element method for the Scandinavium Arena (without the pylon). Load case: uniformly distributed load equal to −0.75 kN/m2 on the whole roof (N.B. zero load in the initial state).

Description fx (m) fy (m) Midpoint displacement (m) Force in the mid cable in the x-direction (kN) Force in the mid cable in the y-direction (kN) Axial force at (x = 0, y = R) (kN) Axial force at (x = R, y = 0) (kN) Bending moment at (x = 0, y = R) (kN) Bending moment at (x = R, y = 0) (kN)

Method Finite element Simpliﬁed 6.611 6.703 6.795 6.703 −1.999 −1.821 891 932 720

707

−10423 −8797 24985 −40350

−12197 −9250 39729 −39729

All comments on Møllmann’s example also hold for the Scandinavium Arena: midpoint deﬂection and cable forces show good agreement but the diﬀerences in axial forces and bending moments are larger. It should be pointed out that the simpliﬁed method assumes an elliptic contour beam. As the cable net is assumed to behave like a continuous membrane, the forces at every point on the ring beam act in both the x- and y-direction. In practice, some sections of the ring beam are subjected to forces in only one direction. In such cases these sections should have a parabolic shape otherwise the bending moment distribution will not be smooth as assumed in the simpliﬁed method. Nevertheless, it can be concluded that the simpliﬁed method is well suited for preliminary design works. Since the bending moment is very sensitive to the shape of the ring beam, analysis with a ﬁnite element program must be used in detailed design work.

156

Chapter 6 Conclusions and further research 6.1

Conclusions

The conclusions from the study are divided into three subsections, each having the same name as the chapter from which the respective conclusions are drawn.

6.1.1

The initial equilibrium problem

The conclusions from Chapter 3 are: • In the form-ﬁnding of cable structures, the optimal strategy seems to be a combined approach. A simple method is ﬁrst used to obtain a starting shape that is close to the ﬁnal shape. Then, a more accurate, iterative method is used to obtain the ﬁnal shape. • The linear version of the force density method is very simple to use, but it is not simple to specify force densities that give the desired force distribution or shape. Therefore, the linear force density method is used to obtain a starting shape for the non-linear force density method. • The linear force density method is not suitable for a cable net with an orthogonal projection in the horizontal plane. A better method for this case is the grid method. • For a structure composed of only cables, the force density stiﬀness matrix D is positive deﬁnite and, thus, the solution is unique. However, this matrix can be singular for structures composed of both tension and compression members. Therefore, the force density method, as presented in this thesis, is not suitable for such structures. • No suitable method to ﬁnd the initial equilibrium conﬁguration of pure tensegrity structures has been found. For a tensegrity structure with known geometry, the independent states of prestress and the number of inextensional mech-

157

CHAPTER 6. CONCLUSIONS AND FURTHER RESEARCH

anisms are given by the dimensions of the subspaces of the equilibrium matrix A.

6.1.2

Finite cable elements

The conclusions from Chapter 4 are: • The elastic and associate catenary elements are mathematically exact under the assumption that the cable is perfectly ﬂexible. Only one element per cable is necessary to model the static behaviour of both slack and taut cables subjected to uniformly distributed loads. • In the case when the self-weight of the cable approaches zero the tangent stiﬀness matrix of the elastic cable element approaches that of the straight bar element. Hence, both light and heavy cables can be modelled using the elastic catenary element. • The parabolic element is not mathematically exact but yields extremely accurate results for cables with small sag-to-span ratios. • The tangent stiﬀness matrices of the analytical cable elements are functions of the horizontal force and have to be obtained by iteration. Good starting values and a modiﬁed Newton-Raphson method ensure that the correct horizontal force is found with only 3–4 iterations. • Comparisons between the ﬁnite cable elements show, as expected, similar results for taut elements. For slack elements, however, the discrepancies are larger and are not negligible. • A real cable has a bending stiﬀness, but this can be neglected for very taut cables. However, for span-to-length ratios less than 0.8 the bending stiﬀness cannot be disregarded.

6.1.3

Static analysis

The conclusions from Chapter 5 are: • In the comparison between the new and old results for the Scandinavium Arena, the maximum vertical net displacement, in-plane ring beam displacements, cable forces and twisting moments show good agreement. The diﬀerences can be explained by diﬀerences in cable spacing, initial cable forces and modulus of elasticity for the concrete. However, the large discrepancies in the bending moment distribution cannot be explained in this way. • The bending moment distribution is found to be very sensitive to the shape of the elastic ring beam. A circular ring beam yields an irregular bending moment

158

6.2. FURTHER RESEARCH

distribution in both the initial and loaded states. A ring beam consisting of parabolic in-plane arcs is shown to provide a smoother moment distribution with lower values. This can be explained in that the parabolic shape is closer to the line of compression for the projected cable forces in the initial state. • The twisting moments in the ring are much lower than the bending moments and do not seem to be a problem in design. • The pylons stiﬀen the ring beam signiﬁcantly; much larger bending moments and net displacements are obtained if the pylons are not present. The pylons also considerably reduce the axial compressive force in the lower parts of the ring beam. With a low compressive force in the concrete ring beam, the importance of the bending moment increases as large tension stresses may arise. • Most of the results from a simpliﬁed method, which assumes that the cable net behaves like a shear-free membrane, agree well with the results from the ﬁnite element calculations. However, due to the many assumptions, the simpliﬁed method can only be recommended for preliminary calculations. Since the bending moments of the accurate and simpliﬁed methods diﬀer more than the other quantities, the most important of the assumptions in the simpliﬁed method seems to be that of a plane ring beam.

6.2

Further research

Static analyses of elastic cable nets and cable trusses are ubiquitous in literature which are shown by the published monographs, e.g. [16, 48, 57, 61, 113], related to this topic. The author, therefore, does not consider it necessary to do more work on elastic static analyses of these two types. Also their dynamic behaviour in the elastic range is extensively described in several of these references.

6.2.1

Failure analysis—background

The current trend in structural design is to optimise the load-bearing to weight ratio of structures, and cable structures are very much involved in this trend. Structural optimisation leads to non-linear phenomena and parameter sensitivity. For safety reasons, it is very important to know the behaviour of a structure under large loads. Generally, a structure has diﬀerent modes of failure: elastic or plastic instability, and material failure at ultimate strength. The ﬁrst failure type—the elastic instability phenomenon—has been extensively analysed for bar, beam and shell structures at the Department of Structural Engineering at the Royal Institute of Technology, Stockholm. An analysis tool has been developed for these structures, and this tool can also be used to analyse the stability of cable structures. In the following sections the other aspects related to failure analysis of cable structures are reviewed. In the last section, a structural concept is discussed.

159

CHAPTER 6. CONCLUSIONS AND FURTHER RESEARCH

Plasticity The high strength steel used in modern cables exhibits linear stress-strain characteristics over only a portion of its usable strength. Therefore, in ultimate load analyses of general cable structures, the resulting formulations must consider material nonlinearity. Under large external loads, some cables will go into the plastic range while some will stay in the elastic zone and other will lose their pretension. Hence, the cable elements must handle all three cases. In their present form, the elements in this thesis are only valid for the elastic and the slack cases and therefore have to be extended to consider plasticity. In reference 1 the general cable element was extended to include material non-linearity. Since the general cable element forms the basis in the derivation of the analytical cable elements, it is anticipated that it is possible to include material non-linearity also in the analytical elements. Some works have been concerned with the plastic analysis of cable structures. For example, Ma et al. [73] used four-node isoparametric cable elements in the plastic dynamic analysis of a saddle-shaped cable net. More recently, Atai and Mioduchowski [6] derived conditions for the stability of cable structures in the plastic state and addressed important issues, such as load history, path-dependency of solution and unloading of a cable from a plastic to a slack state. From these works some important conclusions have been drawn and they ought to provide a good starting point for further improvement of the plastic analysis of cable structures. Parameter and imperfection sensitivity The load-bearing capacity of general cable structures depends on several parameters, of which the level of prestressing is the most important. For ultimate load analysis it is necessary to determine the sensitivity of the structure to changes in some of these parameters. Also diﬀerent types of imperfections, e.g. non-straight compression elements or misplaced cables, may have a serious impact on the maximum allowable loads on the structure. Two works that can be referred to in this context are presented below. Lewis et al. [67] analysed the cladding stiﬀening eﬀects in prestressed cable roofs. A parametric study with diﬀerent cladding-to-net stiﬀness ratios showed that for ratios found in practice the cladding signiﬁcantly contributes to the net stiﬀness. As a result of the cladding, the prestressing forces could be lowered. However, in this case the composite action between the cladding and cable net must be assured throughout the lifetime of the structure. The connection between the cladding and net must be able to transmit large shear forces, which may give rise to higher cost and thereby decreases the beneﬁts of the cladding-net interaction. In general, space trusses are regarded as highly redundant structures with the ability to survive the loss of several members without losing overall stability. These structures also have the property to be very sensitive to imperfections. Wada and Wang [123] have investigated the eﬀect that diﬀerent types of imperfections have on the load-bearing capacity of a double layer space truss. Their investigation included

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variation of member strength, initial imperfection of member length, and errors in the assembly process. The conclusions were that the fabrication errors have minor inﬂuence on the capacity of the truss, but the human errors, like assembly errors, have enormous inﬂuence on the mechanical behaviour of the structure. For the space truss analysed, if two or more members out of 288 members had errors the structure had a large probability of collapsing. A reduction of the load-bearing capacity may also occur in cable structures if cable connections are not assembled in their right positions. Analysis of tensegrity structures During the last decade new structural principles have been developed, the structural behaviour of which are not yet fully understood. The most interesting of the new structural principles is that of self-stressed systems, also known as tensegrity systems. Self-stressing is interesting as it allows cable-strut structures to be built without the need for supporting structures to equilibrate the stresses in the initial conﬁguration. In addition to this advantage, tensegrity structures have other beneﬁts that make them interesting for research; for example, they are lightweight and earthquake resistant. There is also a strong desire from architects to implement the tensegrity principle in buildings because of the pure shapes it produces. Although the research activity in this area is high, see for example references 46, 80, 81, 126, there are still several questions that need to be answered before full scale application can be a reality. In this section, analysis aspects of tensegrity structures will be discussed. Morphology studies. From the invention of the tensegrity concept in the late 1940’s until the beginning of the 1990’s most research projects have dealt with the geometrical shapes of tensegrity networks. According to Hanaor [46] “It appears that the morphological study of tensegrity networks has reached a degree of saturation, whereby the range of conceivable patterns exceeds by far the likely range of applications.” Nevertheless, double-layer tensegrity grids have been developed by joining tensegrity simplexes [79]. Wang [126] concludes that “the future work will be concentrated on applying other simplexes in space structures.” Initial equilibrium conﬁgurations. Several diﬀerent shapes of tensegrity structures are available from the morphology studies. But, form-ﬁnding with geometrical methods does not guarantee mechanical equilibrium and solutions must be checked with a numerical method [81]. Motro et al. [82] applied both the dynamic relaxation method and the force density method to solve the initial equilibrium problem. They anticipated that the latter method is more suitable for large systems, but that more work has to be done to check its eﬃciency. One possible way to modify the force density method to apply to tensegrity systems might be to adopt Mollaert’s approach given in reference 75. In that approach, the compression and tension members are separated to ensure a solution out of the plane. Recently, Bruno [15] used a method based on the minimisation of the potential energy to ﬁnd the equilibrium conﬁguration of simple two-dimensional tensegrity systems. A more mathematical

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approach was developed by Roth and Whiteley [101] and extended by Connelly and Whiteley [25]. This approach has been used by Burkhardt [17] to analyse quite complicated tensegrity domes. Mechanism elimination. Finding a conﬁguration which satisﬁes equilibrium is not enough to solve the initial equilibrium problem for a tensegrity system; the internal mechanisms for that conﬁguration must be identiﬁed, classiﬁed and, if possible, eliminated by prestressing [80]. The method by Calladine and Pellegrino [20, 93] can be used to ﬁnd the mechanisms and determine the stability of the initial conﬁguration. Recently, Tomka [120] introduced a technique called the method of stabilising force to analyse the stability of cable structures. The advantage of this method is that in addition to the qualitative result (stable or unstable), obtained by the method by Calladine and Pellegrino, quantitative conclusions can also be drawn concerning the measure of stability. After an acceptable solution has been found, the behaviour of the tensegrity system has to be studied under the eﬀect of external loads. In particular, the stability of the self-stressing conﬁgurations should be studied [80]. It is of cardinal importance to know if mechanisms can reappear as a result of external loadings. According to Motro [80] this subject “still remains a fairly open matter.” Construction. Besides the theoretical aspects mentioned above, the analysed structures must be possible to build. In the present state, there has not been much application of the tensegrity principle in the construction ﬁeld. The reason for this is that several fundamental technical problems still need to be solved [46, 80]. The main problems are: • suitable prestressing procedures, • eﬃcient node systems, and • incorporation of cladding. Finding a suitable construction and prestressing procedure is quite diﬃcult. These procedures tend to be cumbersome and uneconomic because general tensegrity systems are geometrically complex and lack rigidity prior to prestressing [80]. The prestressing methods must be reliable and assure the level and permanence of the tension that has been put in. The eﬃciency of the node systems is very much related to the construction procedure and the prime objective is to have compact connections. Concerning the roof covering, a ﬂexible membrane is preferable because of the ﬂexibility of the tensegrity frameworks. It is important that the membrane forms an integral part of the design, as it is not a trivial matter to obtain a correct stress distribution in the membrane [46]. According to Hanaor [46] “none of the studies carried out to date, consider the surface membrane.”

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6.2.2

Failure analysis—further research

Because the tensegrity structures are made stiﬀ by prestressing, the eﬀects from loss of the prestress are more severe in these structures than in other cable structures. Cable structures relying on supporting structures or foundations to equilibrate the unbalanced loads have generally a higher safety to failure (if the supporting structures themselves are stable). Tensegrity structures are in many aspects very interesting, but also very complex. From the review in the previous section, the following research directions are suggested: • Analysis of the elastic stability of tensegrity structures under external loads. Most tensegrity structures are kinematically indeterminate, but in many cases it is possible to stabilise the mechanisms to the ﬁrst order by prestressing. However, it is of great importance to know if any of the mechanisms can reappear under external loads due to the loss of the prestress. First, the stability of an ideal conﬁguration composed of only cables and struts should be analysed. Then, it would be of interest to study what additional eﬀects the membrane cover has upon the overall stability. • The eﬀects of imperfections on the elastic stability of tensegrity structures. The stability of the ideal conﬁguration represents the theoretical upper bound of loading. Real structures have imperfections of diﬀerent kinds and in many cases, these greatly aﬀect the stability of the structures. A sensitivity analysis identiﬁes those parameters that signiﬁcantly aﬀect the behaviour of the structure. More or less automatic procedures for these analyses are highly desirable. • Analysis of the plastic stability of tensegrity structures under external loads. A further step would be to include also material non-linearities in the analysis. These non-linearities introduce several new problems such as load history and path-dependency of the solution. To simplify the analyses certain assumptions may be introduced to avoid some of these problems, for example that only linearly increasing loads are considered.

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174

Appendix A Numerical data for the Scandinavium Arena 150

152

275

155 280

157 159 285 163

290 167

120 125 112

Zoom

103

170 295

93 174

83 72

300

61

176

49 37

178

25

36

13

24

1

12

305

307

181

183

Figure A.1: Computational model for the Scandinavium Arena. A circle indicates a node. UnďŹ lled circles in the cable net are nodes which are loaded.

175

APPENDIX A. NUMERICAL DATA FOR THE SCANDINAVIUM ARENA

167

170

174

176

178 181

163 159 157 155

312

313

314

315

316

311

152

317

310

318

309 308

188

187

189

186

190

191

192

185

193 194

184

Figure A.2: Element model for the ring beam and columns. Node and element numbers.

z z x

y 2

y

1

3

x Figure A.3: Beam element in space with local and global coordinate systems.

176

Table A.1: Initial coordinates for beam elements (unstressed conﬁguration). Node numbers according to Figures A.1 and A.2.

Node number 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194

Node coordinates x y 0.000 53.927 2.000 53.927 3.550 53.927 6.000 53.629 10.000 53.143 10.671 53.062 14.000 52.237 18.000 51.229 21.300 49.944 25.202 48.424 26.000 47.942 29.180 46.000 29.500 45.811 32.436 44.017 33.900 42.570 34.462 42.000 37.194 39.294 38.504 38.000 41.585 34.000 42.500 32.822 43.222 31.883 44.376 30.000 46.600 26.377 46.834 26.000 47.437 25.015 48.817 22.000 50.612 18.000 51.835 14.000 52.733 11.058 52.894 10.000 53.506 6.000 53.855 3.710 53.855 2.000 53.855 0.000 3.550 53.927 10.671 53.062 18.000 51.229 25.202 48.424 32.436 44.017 38.504 38.000 43.222 31.883 47.437 25.015 50.612 18.000 52.733 11.058 53.855 3.710

177

(m) z −3.948 −3.948 −3.948 −3.785 −3.520 −3.475 −3.031 −2.490 −1.827 −1.044 −0.826 0.048 0.133 0.941 1.452 1.650 2.605 3.064 4.257 4.610 4.892 5.367 6.284 6.379 6.629 7.225 8.001 8.547 8.947 9.020 9.299 9.458 9.458 9.458 −19.500 −19.500 −19.500 −19.500 −19.500 −19.500 −19.500 −19.500 −19.500 −19.500 −19.500

APPENDIX A. NUMERICAL DATA FOR THE SCANDINAVIUM ARENA

Table A.2: Third point in x –y  plane for beam and bar elements. Node and element numbers according to Figures A.1 and A.2.

Element 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318

Node number 1 2 150 151 151 152 152 153 153 154 154 155 155 156 156 157 157 158 158 159 159 160 160 161 161 161 162 163 163 164 164 165 165 166 166 167 167 168 168 169 169 170 170 171 171 172 172 173 173 174 174 175 175 176 176 177 177 178 178 179 179 180 180 181 181 182 182 183 152 184 155 185 157 186 159 187 163 188 167 189 170 190 174 191 176 192 178 193 181 194

Coordinates x 0.000 3.653 6.223 10.223 11.006 14.440 18.527 21.951 25.947 26.908 30.085 30.403 33.473 35.095 35.662 38.390 39.826 42.918 43.828 44.642 45.807 48.024 48.255 48.949 50.334 52.196 53.425 54.370 54.544 55.150 55.521 55.521 55.521 4.550 11.671 19.000 26.202 30.851 39.504 44.222 48.437 51.612 53.733 54.855

178

for third point (m) y z 55.659 −4.202 55.659 −4.202 55.347 −4.026 54.863 −3.753 54.769 −3.699 53.920 −3.228 52.893 −2.660 51.562 −1.951 50.010 −1.126 49.425 −0.854 47.491 0.045 47.297 0.135 45.429 1.003 43.819 1.584 43.257 1.787 40.548 2.764 39.124 3.279 35.106 4.510 33.924 4.872 32.868 5.196 30.961 5.689 27.334 6.628 26.955 6.725 25.820 7.019 22.769 7.634 18.601 8.462 14.547 9.026 11.431 9.460 10.286 9.541 6.285 9.825 3.837 9.999 2.000 9.999 0.000 9.999 53.927 0.000 53.062 0.000 51.229 0.000 48.424 0.000 44.235 0.941 38.000 0.000 31.883 0.000 25.015 0.000 18.000 0.000 11.058 0.000 3.710 0.000

Table A.3: Load area for the loaded nodes according to Figure A.1. Nodal load = (load area)×(load intensity).

Nodes 1–48 49–60 61–71 72–82 83–92 93–102 103–111 112–119 120–125

Load area (m2 ) 16.0 14.6 16.0 16.4 15.8 15.4 17.2 18.8 16.2

Table A.4: Horizontal component of the initial cable force. Cables in y-direction are numbered from left to right, and in x-direction from down to up.

Cable number 1 2 3 4 5 6 7 8 9 10 11 12

Force x-dir. 583.20 583.20 583.20 583.20 583.20 583.20 583.20 583.20 583.20 583.20 583.20 583.20

179

(kN) y-dir. 583.20 583.20 583.20 583.20 532.17 583.20 597.78 575.91 561.33 626.94 685.26 590.49

List of Bulletins from the Department of Structural Engineering, The Royal Institute of Technology, Stockholm

TRITA-BKN. Bulletin Pacoste, C., On the Application of Catastrophe Theory to Stability Analyses of Elastic Structures. Doctoral Thesis, 1993. Bulletin 1. Stenmark, A-K., D¨ ampning av 13 m l˚ ang st˚ albalk—“Ullevibalken”. Utprovning av d¨ ampmassor och fasts¨attning av motbalk samt experimentell best¨ amning av modformer och f¨ orlustfaktorer. Vibration tests of full-scale steel girder to determine optimum passive control. Licentiatavhandling, 1993. Bulletin 2. Silfwerbrand, J., Renovering av asfaltgolv med cementbundna plastmodiﬁerade avj¨ amningsmassor. 1993. Bulletin 3. Norlin, B., Two-Layered Composite Beams with Nonlinear Connectors and Geometry—Tests and Theory. Doctoral Thesis, 1993. Bulletin 4. Habtezion, T., On the Behaviour of Equilibrium Near Critical States. Licentiate Thesis, 1993. Bulletin 5. Krus, J., H˚ allfasthet hos frostnedbruten betong. Licentiatavhandling, 1993. Bulletin 6. Wiberg, U., Material Characterization and Defect Detection by Quantitative Ultrasonics. Doctoral Thesis, 1993. Bulletin 7. Lidstr¨ om, T., Finite Element Modelling Supported by Object Oriented Methods. Licentiate Thesis, 1993. Bulletin 8. Hallgren, M., Flexural and Shear Capacity of Reinforced High Strength Concrete Beams without Stirrups. Licentiate Thesis, 1994. Bulletin 9. Krus, J., Betongbalkars lastkapacitet efter milj¨ obelastning. 1994. Bulletin 10. Sandahl, P., Analysis Sensitivity for Wind-related Fatigue in Lattice Structures. Licentiate Thesis, 1994. Bulletin 11. Sanne, L., Information Transfer Analysis and Modelling of the Structural Steel Construction Process. Licentiate Thesis, 1994. Bulletin 12. Zhitao, H., Inﬂuence of Web Buckling on Fatigue Life of Thin-Walled Columns. Doctoral Thesis, 1994. Bulletin 13. Kj¨ orling, M., Dynamic response of railway track components. Measurements during train passage and dynamic laboratory loading. Licentiate Thesis, 1995. Bulletin 14. Yang, L., On Analysis Methods for Reinforced Concrete Structures. Doctoral Thesis, 1995. Bulletin 15. ¨ Svensk metod f¨ Petersson, O., or dimensionering av betongv¨ agar. Licentiatavhandling, 1996. Bulletin 16. Lidstr¨ om, T., Computational Methods for Finite Element Instability Analyses. Doctoral Thesis, 1996. Bulletin 17. Krus, J., Environment- and Function-induced Degradation of Concrete Structures. Doctoral Thesis, 1996. Bulletin 18. Editor, Silfwerbrand, J., Structural Loadings in the 21st Century. Sven Sahlin Workshop, June 1996. Proceedings. Bulletin 19.

Ansell, A., Frequency Dependent Matrices for Dynamic Analysis of Frame Type Structures. Licentiate Thesis, 1996. Bulletin 20. Troive, S., Optimering av ˚ atg¨arder f¨ or ¨okad livsl¨angd hos infrastrukturkonstruktioner. Licentiatavhandling, 1996. Bulletin 21. Karoumi, R., Dynamic Response of Cable-Stayed Bridges Subjected to Moving Vehicles. Licentiate Thesis, 1996. Bulletin 22. Hallgren, M., Punching Shear Capacity of Reinforced High Strength Concrete Slabs. Doctoral Thesis, 1996. Bulletin 23. Hellgren, M., Strength of Bolt-Channel and Screw-Groove Joints in Aluminium Extrusions. Licentiate Thesis, 1996. Bulletin 24. Yagi, T., Wind-induced Instabilities of Structures. Doctoral Thesis, 1997. Bulletin 25. Eriksson, A., and Sandberg, G., (editors), Engineering Structures and Extreme Events—proceedings from a symposium, May 1997. Bulletin 26. Paulsson, J., Eﬀects of Repairs on the Remaining Life of Concrete Bridge Decks. Licentiate Thesis, 1997. Bulletin 27. Olsson, A., Object-oriented ﬁnite element algorithms. Licentiate Thesis, 1997. Bulletin 28. Yunhua, L., On Shear Locking in Finite Elements. Licentiate Thesis, 1997. Bulletin 29. Ekman, M., Sprickor i betongkonstruktioner och dess inverkan p˚ a best¨andigheten. Licentiate Thesis, 1997. Bulletin 30. Karawajczyk, E., Finite Element Approach to the Mechanics of Track-Deck Systems. Licentiate Thesis, 1997. Bulletin 31. Fransson, H., Rotation Capacity of Reinforced High Strength Concrete Beams. Licentiate Thesis, 1997. Bulletin 32. Edlund, S., Arbitrary Thin-Walled Cross Sections. Theory and Computer Implementation. Licentiate Thesis, 1997. Bulletin 33. Forsell, K., Dynamic analyses of static instability phenomena. Licentiate Thesis, 1997. Bulletin 34. Ik¨aheimonen, J., Construction Loads on Shores and Stability of Horizontal Formworks. Doctoral Thesis, 1997. Bulletin 35. Racutanu, G., Konstbyggnaders reella livsl¨ angd. Licentiatavhandling, 1997. Bulletin 36. Appelqvist, I., Sammanbyggnad. Datastrukturer och utveckling av ett IT-st¨ od f¨ or byggprocessen. Licentiatavhandling, 1997. Bulletin 37. Alavizadeh-Farhang, A., Plain and Steel Fibre Reinforced Concrete Beams Subjected to Combined Mechanical and Thermal Loading. Licentiate Thesis, 1998. Bulletin 38. Eriksson, A. and Pacoste, C., (editors), Proceedings of the NSCM-11: Nordic Seminar on Computational Mechanics, October 1998. Bulletin 39. Luo, Y., On some Finite Element Formulations in Structural Mechanics. Doctoral Thesis, 1998. Bulletin 40. Troive, S., Structural LCC Design of Concrete Bridges. Doctoral Thesis, 1998. Bulletin 41.

T¨arno, I., Eﬀects of Contour Ellipticity upon Structural Behaviour of Hyparform Suspended Roofs. Licentiate Thesis, 1998. Bulletin 42. Hassanzadeh, G., Betongplattor p˚ a pelare. F¨ orst¨arkningsmetoder och dimensioneringsmetoder f¨ or plattor med icke vidh¨ aftande sp¨ annarmering. Licentiatavhandling, 1998. Bulletin 43. Karoumi, R., Response of Cable-Stayed and Suspension Bridges to Moving Vehicles. Analysis methods and practical modeling techniques. Doctoral Thesis, 1998. Bulletin 44. Johnson, R., Progression of the Dynamic Properties of Large Suspension Bridges during Construction—A Case Study of the H¨ oga Kusten Bridge. Licentiate Thesis, 1999. Bulletin 45. Tibert, G., Numerical Analyses of Cable Roof Structures. Licentiate Thesis, 1999. Bulletin 46. The bulletins enumerated above, with the exception for those which are out of print, may be purchased from the Department of Structural Engineering, The Royal Institute of Technology, S-100 44 Stockholm, Sweden. The department also publishes other series. For full information see our homepage http://www.struct.kth.se