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Numerical Analyses of CableRoof StructuresGunnar Tibert

TRITA-BKN. Bulletin 46, 1999ISSN 1103-4270ISRN KTH/BKN/EX--46--SE

Licentiate Thesis

Numerical Analyses

of

Cable Roof Structures

Gunnar Tibert

Department of Structural EngineeringRoyal Institute of TechnologySE-100 44 Stockholm, Sweden

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

Licentiate Thesis

c©Gunnar Tibert 1999KTH, TS–Hogskoletryckeriet, Stockholm 1999

Abstract

This thesis deals with the techniques used in the numerical analysis of cable roofstructures. These structures are usually very light and flexible and require analysismethods, which take their non-linear behaviour into account.

An extensive literature survey, concerned with both practical and theoretical aspectsof cable roofs, is presented. Some aspects included are: structural systems, rooferection 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, itcannot be described by simple geometrical models. Special iterative methods, usu-ally not familiar to the structural engineer, have to be utilised in order to find thepretensioned configuration of the roof. The simple force density method is presentedin detail and applied to a number of different types of cable roof structures. Themethod worked well for structures composed of only cables, but not for structureswith compression members.

Three analytical finite cable elements are presented. Two elements are mathemat-ically exact and can accurately model both taut and slack cables using only oneelement per cable. It is shown that the analytical elements are advantageous inmodelling 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 designof the same object. It was found that the bending moments in the supportingstructure—the concrete ring beam—were very sensitive to its shape. This explainedthe large discrepancy in the bending moment distribution between the analyses.Results from a simplified method, used for preliminary calculations, agreed wellwith those of the more accurate finite element calculations, for a studied symmetricload case.

Failure stage analysis of the class of self-stressed cable structures called tensegritystructures has been identified as an area of further research.

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

iii

Preface

The research work in this thesis was carried out at the Department of StructuralEngineering, Structural Mechanics Group, at the Royal Institute of Technology inStockholm, under the supervision of Professor Anders Eriksson. The work reportedin this thesis was financed through a personal grant from KTH.

First of all, I express my gratitude to my supervisor Professor Anders Eriksson forhis scientific guidance and valuable advice.

I also thank Docent Costin Pacoste for help with the selection of a suitable beamelement for the static analyses.

I would also like to thank Professor Emeritus Alf Samuelsson at Chalmers Uni-versity of Technology in Gothenburg and Mr. Nils Dahlstedt, Technical Managerat the Scandinavium Arena in Gothenburg, for the valuable information about theScandinavium Arena.

Finally, I am grateful to all people at the Department of Structural Engineering thathave 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

2.4.1 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.2 Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.3 Axial stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

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

2.5 Cladding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

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

vii

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

2.5.3 Panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.6 Structural details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6.1 End fittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6.2 Intermediate fittings . . . . . . . . . . . . . . . . . . . . . . . 25

2.6.3 Saddles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6.4 Anchorages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.7 Roof loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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

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

2.7.3 Earthquake load . . . . . . . . . . . . . . . . . . . . . . . . . 37

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

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

3 The initial equilibrium problem 41

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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

3.2 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

3.3 The force density method . . . . . . . . . . . . . . . . . . . . . . . . 56

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

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

3.4 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

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

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

4.2.3 Effect of cable bending stiffness . . . . . . . . . . . . . . . . . 92

4.3 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

4.4 Straight and parabolic elements . . . . . . . . . . . . . . . . . . . . . 99

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

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

4.5 Catenary elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

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

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

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

4.6 Comparison of elements . . . . . . . . . . . . . . . . . . . . . . . . . 113

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



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

5 Static analysis 123

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

ix

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

5.2 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

5.3 Comparison with a simplified method . . . . . . . . . . . . . . . . . . 154

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

6 Conclusions and further research 157

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

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

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

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

6.2 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 chaptersof 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 cross-sectional area, 45A0 cross-sectional area of core wire, 19Ai cross-sectional area of a wire in layer i, 19A equilibrium matrix, 52B compatibility matrix, 54C length of cable chord, 102Cp pressure coefficient, 30C connectivity matrix for free nodes, 56Cf connectivity matrix for fixed nodes, 56Cs connectivity matrix for all nodes, 56d vector of nodal displacements, 54E Young’s modulus, 45E0 Young’s modulus of core wire, 19Ei Young’s modulus of wires in layer i, 19e vector of bar elongations, 54F force in global coordinate system, 99F ′ component of cable force in local coordinate system, 87f vector of nodal loads, 52H horizontal component of the cable force T , 88h projection of cable profile on z′-axis, 87I moment of inertia, 87Ii moment of inertia of wire in layer (around its own centerline), 92K tangent stiffness matrix in global coordinate system, 101K′ tangent stiffness matrix in local coordinate system, 102K′

E elastic stiffness matrix in local coordinate system, 100KG geometric stiffness matrix in global coordinate system, 50L length, 43L0 unstrained length, 45l projection of cable profile on x′-axis, 87mi number of wires in layer i, 19n number of wire layers, 19p wind pressure at time t, 30q0 intensity of distributed load on cable, 87

xi

q vector of force densities, 58Ri wire radius in layer i, 19ri radius of wire centerline helix in layer i, 19s arc length (elastic cable), 87s0 arc length (inextensible cable), 88T cable force, 43Tb cable force at the base (s0 = 0), 91T transformation matrix, 100t vector of bar axial forces, 52U mean wind velocity, 29u turbulence component of the wind field in the x-direction, 29u vector of free x-coordinate differences, 58V total wind velocity, 30v turbulence component of the wind field in the y-direction, 29v vector of free y-coordinate differences, 58w turbulence component of the wind field in the z-direction, 29w vector of free z-coordinate differences, 58xg vector of nodal coordinates, 50x vector of free x-coordinates, 58xf vector of fixed x-coordinates, 58y vector of free y-coordinates, 58yf vector of fixed y-coordinates, 58z vector of free z-coordinates, 58zf vector of fixed z-coordinates, 58αi angle of wire centerline helix in layer i, 19β angle between the cable chord and horizontal, 102ν Poisson’s ratio, 92φ angle between x′- and x-axis, 103ρ air density, 30θ angle between tangent to cable profile and x′-axis, 87θm mean wind direction, 30θv azimuth angle of turbulent wind component v, 30θw elevation angle of turbulent wind component w, 30

xii

Chapter 1

Introduction

Tensile architecture represents the new trend in design: construction with the mini-mum amount of material. As is well-known, the primary advantage of tensile mem-bers over compression members is that they can be as light as the tensile strengthpermits. With new materials, such as high strength steel cables and silicone-coatedglass fibre membranes, larger distances can be spanned using the same amount ofmaterial as before.

Tensile structures have always fascinated architects and engineers, mainly becauseof the aesthetic shapes they produce. Despite this, very few tensile structures havebeen built. Why are they not more common, if they are both economic and beauti-ful? One answers might be that tent-like structures have always been thought of astemporary. Although, a probably more correct answer is that they are more difficultto 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 distri-bution of the pretensioning forces may lead to damage of the cladding under largeloads.

If the numerical analysis of building structures is concerned, the finite elementmethod is the dominating tool. In this method, the structural characteristics andexternal 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 struc-ture, which generally is known a priori. However, this is not the case for tensilestructures. Due to the negligible flexural stiffness of cables and membranes, theinitial configuration of these structures must be stressed, even if the self-weight isdisregarded. Thus, before the analysis of the behaviour of the structure to externalloads can be performed, the initial equilibrium configuration must be found. Theshape of a tensile structure, which very much depends on the internal forces, alsogoverns the load-bearing capacity of the structure. Therefore, the process of deter-mining the initial equilibrium configuration calls for the designer’s ability to findan optimum compromise between shape, load capacity and constructional require-ments. 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 finiteelement programs, e.g. ABAQUS 1 and are not familiar to the practising structuralengineer.

After the initial reference configuration has been determined, the structural membershave to be described by stiffness matrices and force vectors. Special elements forcables or chains are often not available in commercial finite element programs. Thesingle cable is instead modelled by one or several other elements depending onthe sag-to-span ratio. Nevertheless, this approach has problems such as numericalinstability of the solution algorithms. To avoid these problems it is desirable to haveat hand a robust element which accurately describes the behaviour of both taut andslack cables. Since cable structures in general are very flexible, a geometrically non-linear solution method has to be used. The most common is the Newton-Raphsonalgorithm, 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. Themagnitude and distribution of these loads is a constant source of research. Thepresent knowledge in the area is found in the national building codes, which aidthe engineers in their decisions. Tensile structures often have irregular shapes andlow self-weights which may give rise to unforeseen effects such as very high snowloads and flutter instability due to wind. To ensure the safety of the structure,experimental tests have to be undertaken together with statistical analyses to findthe 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 andgood engineering judgement are frequent characteristics among famous designersof tensile structures: Fritz Leonhardt, Jorg Schlaich, Frei Otto, Horst Berger andDavid 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 typestructures (roofs, cooling towers, etc.). The first part of the work is concerned withthe basic aspects of these specific types of structures. These include: the principalarrangements of the cables, pretensioning schemes necessary to obtain a prescribedshape, and the practical aspects of connections between and supports for the cables.Further, basic computational models are to be studied. These include, but are notlimited to, the methods for analysis and force distribution. The second part isconcerned with the formulation of suitable finite elements, which take into accountthe non-linear behaviour of a cable. Basic analyses are performed, and verified.

Theoretical and numerical studies are included in this thesis, but no experimental

1ABAQUS is a registered trademark of Hibbitt, Karlsson & Sorensen, Inc., 1080 Main Street,Pawtucket, RI 02860-4847, U.S.A. Internet: http://www.abaqus.com.

2

1.2. GENERAL STRUCTURE OF THESIS

work is conducted. In the discussed methods, only elastic structures and static loadsare considered.

All of the numerical calculations in this thesis has been coded in the Matlab2 lan-guage. Some expressions have been derived using the computer algebra packageMaple3

1.2 General structure of thesis

To get an overview of the structure of this thesis, the contents of the chapters arepresented 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. Amongthe practical aspects are: different structural systems, roof erection processes, dif-ferent 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 equilibriumconfiguration of cable structures and structures of mixed type (cables and stiff struc-tural members) are presented. One of the methods—the force density method—isfurther described in detail and coded. A variety of examples are analysed to illus-trate both the advantages and the drawbacks of the force density method.

In Chapter 4, the difficulties of modelling cable behaviour using finite elements basedon the conventional approach, i.e. using shape functions, are discussed. Further, fourfinite cable elements are presented: the straight bar, the parabolic cable, the elasticcatenary and the associate catenary. The internal force vectors and tangent stiffnessmatrices 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 programwritten by the author. The structure is the Scandinavium Arena in Gothenburg,which consists of a pretensioned cable net anchored in a nearly circular concretering beam. The results of the calculations are compared to the results from theinitial 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 furtherresearch are suggested.

In Appendix A, data used in the analysis of the Scandinavium Arena are presented.

2Matlab is a registered trademark of The MathWorks Inc., 24 Prime Park Way, Natick, MA01760-1500, U.S.A. Internet: http://www.mathworks.com.

3Maple is registered trademark of Waterloo Maple Inc., 57 Erb Street W., Waterloo, Ontario,Canada N2L 5J2. Internet: http://www.maplesoft.com.

3

Chapter 2

Literature review

2.1 Historical review

The first structures regarded as cable roofs are four pavilions with hanging roofsbuilt by the Russian engineer V. G. Shookhov at an exhibition in Nizjny-Novgorodin 1896. During the 1930’s a small number of roof structures of moderate sizes werebuilt 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 MatthewNowicki 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 continuedthrough the architect William Henry Deitrick and civil engineer Fred Severud andin 1953 the arena was completed [88].

(a) (b)

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

On an exchange visit to the U.S.A. in 1950 a German student in architecture, namedFrei Otto, previewed the drawings for the Raleigh Arena in the New York office ofFred 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 begana systematic investigation of suspended roofs. The investigation was presented inthe doctoral thesis Das Hangende Dach (The Suspended Roof), which was the firstcomprehensive documentation on the subject [31].

The thesis caught the attention of Peter Stromeyer of Stromeyer Company, one ofthe largest tent manufacturers in the world. Stromeyer contacted Otto and theybegan a fruitful cooperation. In 1957 Otto formed the Development Centre forLightweight Construction in Berlin in order to further increase the research abouttensile architecture. In 1964 he incorporated the centre into the Institute of LightSurface Structures at the University of Stuttgart. A massive research work was un-dertaken 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 moderntensile architecture. He was involved in the construction of many of the large tensilestructures during the mid 1960’s to early 1970’s. Among these was the first largecable net structure with fabric cladding, the German pavilion at the World’s fair inMontreal 1967 [10], Figure 2.2.

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

Another pioneering structure at this time was the large low-profile super ellipticair-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 theWorld’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 elegantdevelopment 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


Kenneth Snelson and Richard Buckminster Fuller. The first two domes were builtfor the 1988 Seoul Olympics. The latest and biggest, the Georgia Dome, was builtin Atlanta 1994, Figure 2.5.

Figure 2.5: The Georgia Dome in Atlanta, U.S.A., during construction. Reproducedfrom [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 largestdome 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. Reproducedfrom 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 fairlynew one. Each roof type is presented very briefly, but references to more informationare given.

Cable roofs can be divided into different categories depending upon the criterionused for classification. In accordance with how the cables are used, they can beclassified as [57]:

1. cable supported roofs, and

2. cable suspended roofs.

Cable supported roofs are, in principle, similar to cable-stayed bridges. In theseroofs, the cables only provide additional support for elements which themselvescarry a major part of the load. In cable suspended roofs the load is carried directlyby the cable system [57]. The cable supported roofs, for which the cables only havean 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-self-balancing. In a self-balancing structure, the forces in the cables are balanced inter-nally 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 increasethe stiffness of a roof. In the following, the traditional types of cable suspended roofswill be described. In each category, the structural systems are illustrated by a limitednumber of figures. More examples can be found in the references 16 and 57.

9


2.2.1 Simply suspended cable structures

The first type is the simply suspended roof. These roofs have a single curvature ora positive double curvature (like a bowl). Systems of this type have no stiffness. Toreduce the displacements caused by any form of applied loading, the roof claddingmust either be very heavy or stiff. Concrete is perhaps, therefore, the most suitableroofing 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 thebridge deck.

Figure 2.7: Simply suspended roof.

The simply suspended roofs, which are stiffened by the cladding material, will notbe considered in this thesis; only systems, which can be pretensioned before thecladding is applied, will be analysed.

2.2.2 Pretensioned cable trusses

Lighter and stiffer systems than the simply suspended systems can be achieved if asecond set of cables with reverse curvature is connected to the hanging cables. A ca-ble truss is quite stiff if it is tensioned to a level which ensures that both the hangingand the bracing cables remain in tension under any load case. The basic cable trussconfigurations with vertical connecting elements are shown in Figure 2.8. Anothersystem is the cable truss with diagonal ties, Figure 2.9, developed by the Swedishengineer David Jawerth. Generally, the cable trusses with the vertical connectingelements are structural mechanisms if they are considered as pin-jointed trusses.However, the cable truss with diagonal connecting elements is statically indetermi-nate [76]. Therefore, the Jawerth truss is stiffer than the other trusses [51]. Thecable 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


(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 roofdecking

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

11


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 audito-rium 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, thissurface must be anticlastic (saddle-shaped) at every point [16].

The stiffness of a cable net depends mainly on: the curvature of the net surfaceand the level of pretension. In order to minimise the material in both the net andthe supporting structure it is advantageous to have a surface with a relatively small

12


radius [31]. The prestressing force must not be exceeded by any type of loading, orthe cables become slack. Areas of slack cables may damage the cladding or give riseto the destructive phenomenon of flutter [16].

Cable nets can be designed with masts and edge cables or with stiff boundaries suchas beams, arches and rings, Figure 2.11. The first type is generally less stiff andmore complicated to construct than the latter ones. The cladding is often placeddirectly on the cable network [16,57].

Figure 2.11: A cable net structure—the Scandinavium Arena in Gothenburg, Swe-den.

More information on different types of cable nets and their properties can be foundin 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. Thekey feature of such structures is that they are self-stressed; no external devices toequilibrate the cable forces are needed. Tensegrity structures can be said to havebeen invented by Kenneth Snelson and Richard Buckminster Fuller [99].

Several new systems, based on the tensegrity principle, have been developed in recentyears. The most well-known of these new systems is the cable dome concept byDavid Geiger. The cable dome, Figure 2.12, is not a pure tensegrity structure sincea curved ring beam is used to balance the cable forces. The cable dome concept wasdeveloped as an economically equal alternative to air-supported structures, whichseveral 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


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

More about tensegrity structures and some other fairly new structural concepts canbe 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 differentshapes. In practice, the number of configurations is restricted, as shown in theprevious section. Of course, with the use of scaffolding, more shapes would bepossible, 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 easytask. Every step of the erection process must be computer controlled to avoid over-stressing of the supporting structures. It is important that the contractor responsiblefor the roof erection fully understands and exactly follows the erection plan specifiedby the designer [57].

Cable trusses have the easiest erection process among cable roof structures and maybe assembled in the air or on the ground, of which the latter is to prefer. Afterbeing assembled on the ground the truss is hoisted into position and prestressedby applying tension at both ends simultaneously. Depending upon whether thecentre of the truss needs to be lifted or lowered, tension is applied to the suspensionor prestressing cable. Since the cable trusses usually do not interact with eachother before the cladding is applied several trusses can be erected and prestressedsimultaneously to reduce overall construction time [16]. Double layer grids withradial symmetry can be erected in the same fashion as trusses, but care has to betaken to not over-stress the compression ring as bending moments are introducedwhen just a few trusses are tensioned. An erection scheme for radial double layergrids is given in [57].

14

2.4. CABLES

Cable nets may be preassembled on the ground and hoisted into position or as-sembled in the air. Nets with flexible boundaries (i.e. edge cables) are usually pre-assembled on the ground, but for nets with stiff boundaries either of the methodscan be used [16]. With use of computational methods (see Chapter 3) the shapeof a net and the corresponding cable forces can be very accurately determined. Toobtain the computed shape of the real net a high dimensional accuracy in fabricationis required. Small errors in unstrained length may cause large errors in force. Onemethod to achieve a high accuracy at a minimal cost is to specify a net with a squareunstrained mesh and uniform cable stresses. In this way, the same cable dimensioncan be used for the whole net (not the edges) and the equidistant cable-to-cableconnections can be factory-assembled. But, even with a high accuracy some ad-justment can be necessary after the net has been lifted into its final position. Thisadjustment is possible if tensioning devices (turnbuckles) are incorporated at theends of the cables [63].

For tensegrity structures, suitable methods for prestressing large tensegrity frame-works have not yet been developed. This is probably the main reason for the veryfew large tensegrity structures today. Nonetheless, one exception is the cable domesby David Geiger. These domes were developed as an economically equal alternativeto air-supported structures, but without the risk for deflation. From the economicpoint of view, it was necessary that the domes could be constructed without anyscaffolding. Figure 2.13 shows the steps of erection of a cable dome [99]. For acomplicated structure, the best way to plan the erection steps is to build a physicalmodel of the structure [99].

2.4 Cables

The main load carrying element in the structures considered in this thesis is thecable. In structural applications, the term ‘cable’ means a flexible tension member.However, a cable can have different configurations. In this section, the differenttypes 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 circularin cross section, with a diameter between 3 and 8 mm, but may be non-circular inlocked coil strands. The wire has a high tensile strength that is obtained by colddrawing or cold rolling [35].

A spiral strand, Figure 2.15(a), is an assembly of wires laid helically around a centralstraight wire. An assembly of a small number of wires is called a spiral strand andif there are more than three layers it is called a spiral bridge strand. The successivelayers are usually wound in opposite directions to get equal torsional stiffness inboth directions [35].

15


(a)

(b)

(c)

(d)

(e)

Figure 2.13: Cable dome erection steps: (a) The upper cables are hung, then (b) ahoop and struts are hung, raising the inverted ridge cables. More hoopsand struts, (c)–(e), further raise and tension the ridge cables. Drawnfrom 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 composedof two types of helically laid wires: the core is a spiral strand with helically laidcircular wires, and at least the two outer layers have wires with a special Z-shapethat interlock with each other. The special shaped wires together with the self-compacting effect of the helical arrangement result in a tight surface and a low voidratio in the outer layers [42].

A wire rope, Figure 2.15(c), is an assembly of spiral strands that are laid helicallyaround a central core that can be a strand or another independent wire rope. Thespiral strands are usually laid in the opposite direction to the wires in the spiralstrands (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 strengthand stiffness. In some applications, particularly suspended bridges, a high strengthand stiffness are more important than flexibility and therefore products with parallelwires and strands have become popular. Other benefits with parallel strands andwires are easier handling and transportation. In the last decade, parallel strandshave also found use in roof construction. Parallel strand systems were used as thehoop and ridge cables in the cable domes by David Geiger [100]. The developmentof parallel products over recent years is reviewed by Walton [125].

17


(a) Bridge strand (b) Locked coil bridgestrand

(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 strengthis 1570 MPa and the guaranteed 0.2 % proof stress is 1180 MPa. The limit ofproportionality (0.01 % proof stress), which is the absolute upper limit for thestresses 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 takeninto account. Tests on steel wires show that the relaxation accelerates when the wireis 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, be-sides the tensile strength, is the axial stiffness. As mentioned above, a cable withhelical wires has a lower stiffness than a cable with straight wires. In the design ofcable structures, it is of cardinal importance to know the axial stiffness of the cablessince the force distribution in, for example, a cable net is very sensitive to smallerrors in the cable properties (modulus and length). Several methods have beendeveloped to calculate the axial stiffness of a helically wound cable, see for examplereference 22. Most of these methods are based on contact theories and are, thus,very complex. Nevertheless, two simple and accurate methods have been found andwill 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] andarrived at the following closed-form expression for the axial stiffness:

(AE)eq = A0E0 +n∑

i=1

miAiEi sin αi

[1 − (1 + ν)pi cos2 αi

], (2.1)

whereAi = πR2

i , (2.2)

and

pi =

(1 − ν

Ri

ri

cos2 αi

)[1 − R2

i

4r2i

(1 − ν

1 + νcos 2αi

)cos2 αi

]. (2.3)

Kumar and Cochran [58] also provide an even simpler expression for the equivalentaxial stiffness

(AE)eq = A0E0 +n∑

i=1

miAiEi sin3 αi

(1 − ν cot2 αi

). (2.4)

Another method, in which the wire layers are modelled as orthotropic sheets, hasbeen developed by Raoof [98]. The method is quite cumbersome and not suitablefor practical design work. Therefore, Raoof derived a simplified procedure, by para-metric studies of different cable dimensions. In that, Hruska’s1 parameter is firstcomputed as:

κ =n∑

i=1

miAi

AT

cos4 αi, (2.5)

in which

AT =n∑

i=1

miAi. (2.6)

1From F. H. Hruska, Calculation of stresses in wire ropes, Wire, Vol. 26, No. 9, 1951.

19


The relation between the full-slip modulus (no friction between wires) and steelmodulus is computed as:

Efull-slip

Es

= −0.26442 − 2.004046κ + 6.5735κ2 − 3.3068κ

3, (2.7)

Denoting Efull−slip/Es = ϑ, the no-slip modulus is found from

Eno-slip

Efull-slip

= 3.998 − 7.916ϑ + 7.238ϑ2 − 2.321ϑ3. (2.8)

The full-slip and no-slip axial stiffnesses are obtained by multiplying Efull-slip andEno-slip, respectively, with AT . The method by Raoof, equations (2.5)–(2.8), areincluded in Eurocode 3 [35]. Raoof’s method has been checked against experimentalresults in [47]. It was found that the experimental moduli of newly manufacturedcables agreed well with the theoretical full-slip modulus.

The methods presented above have also been compared to other analytical methodsand it is concluded that the overall elastic behaviour of helical cables under axialloading is well represented by the available mechanical models. Which model oneshould use is dependent on the size of the cable [22]. The expressions by Kumarand 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 axialstiffness, 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, loadedrepetitively to a load well within the elastic limit of the wire material. The purposeof this procedure is to remove the constructional stress and, thereby, obtain analmost linear stress-strain curve [16]. However, despite this linearising process, thecable stiffness will vary; it is lower when the cable is new and becomes higher duringthe useful life of cable [97].

2.4.4 Corrosion protection

Cables made of high strength steel wires are extremely vulnerable to phenomena suchas stress and fretting corrosion. Add to this that most of the wires will be inaccessiblefor inspection and maintenance in the completed cable and that numerous of cavitiesare present between wires, and one understands that it is essential to ensure thatthe corrosion protection is of highest quality, particularly in the regions of end orintermediate 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 techniquehas become the preferred method. There are different classes of coating thicknessdependent on the severity of the exposure conditions. The coating is usually of purezinc but zinc-aluminium alloys are also used. Hydrogen embrittlement of galvanizedsteel 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 offilling the interstices between the wires with a blocking material and coating theouter surface. The primary purpose of the blocking material is to prevent ingress ofmoisture [42]. Suitable blocking materials are synthetic waxes and compounds basedon petrolatum (petroleum jelly), which are hydrophobic and have good adherence.The final coating can be ordinary paint or, if necessary, a more displacement resistantcompound [125].

If the cable is exposed to an aggressive environment it is normally sheathed with atube made of steel or polyethylene. The space between the tube and the cable isfilled 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 consideredas impermeable. Materials used for sheathing must be ductile and if polyethylene isused it must be resistant to ultraviolet radiation. An alternative sheathing methodis 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 addany contribution to the structural stiffness. Some contribution will in any case beadded to the performance of the building, which cannot be neglected. Especially thedamping properties of the roof will be enhanced, which have significant importancefor the dynamic behaviour of the structure.

There are two main categories of cladding: continuous membranes and unit cov-erings. Membranes can be made of fabric, foil or metal sheet. Unit coverings arepanels of metal, wood or plastic [23]. The choice of cladding material depends onthe type of structure (e.g. its shape), the expected lifetime, static and dynamic be-haviour, security and maintenance. What type of cladding to be used should bedecided 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 tensionstructures. As a structural element, the fabric must have the strength to spanbetween 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 hasa negligible bending stiffness. The amount of prestress and the patterning of themembrane, i.e. how the membrane should be cut and assembled, is given by thestructural analysis of the roof. Besides the structural requirements, the fabric mustmeet the requirements which affect the environment inside the building; these areair tightness, water protection, fire resistance, heat insulation, light transmission,acoustic properties, maintenance and durability [10].

21


Fabric membranes are composite materials. Inside the membrane there are filamentfibre yarn, designed to resist tensile forces, woven in different directions formingan anisotropic surface. For permanent buildings with expected long lifetimes onlytwo types of fibres can be used: glass and aramid (Kevlar2) fibres, of which glassfibre is the most common. The mechanical properties of these fibres compared tothe properties of a steel wire are shown in Table 2.1. To protect the fibres fromenvironmental 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 Glass Aramid Steel(E-HTS glass) (Kevlar 49) (Cold drawn wire)

Density (g/cm3) 2.55 1.44 7.86Young’s modulus (GPa) 69 124 205Tensile strength (MPa) 2410 2760 1570Max. elongation (%) 3.5 2.5 4.0Temp. resistance (C) 350 250 500

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 withoutdamaging the coating. It is also resistant to abrasion and highly reflective, absorbinglittle light as well as heat. The fact that Teflon comes in two forms, PTFE andFEP5, with different melting points makes it possible to heat weld seams, whichenables 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 requiresconsiderable care in the packing, shipping and installation of panels, and it has littleelastic forgiveness and must therefore be accurately patterned [99].

Fibreglass coated with silicone is more flexible than PTFE-coated fibreglass, so it isless 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 artificiallightning during daytime can be almost eliminated. Fabric joints are chemicallybonded or glued. The self-cleaning properties of silicone rubber are not yet as goodas 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 withouta supporting cable net. One major disadvantage with fibres of Kevlar is that theyare highly susceptible to ultraviolet radiation and cannot be coated with translucentresin. The fibres must be shielded with an opaque carbon black coat. Due to thesensitivity to ultraviolet radiation the joints of Kevlar fabrics cannot be heat weldedwith clear Teflon. The seams must instead be sewed, but it is impossible to develop

2Kevlar and Teflon are registered trademarks of E. I. du Pont de Nemours and Company3Abbreviation for Polytetrafluoroethylene4Abbreviation for Polyvinyl Chloride5Abbreviation for Fluorinated Ethylene Propylene

22

2.5. CLADDING

the full strength of the fabric through the joints due to the high strength of the basematerial [40].

The newest membrane material is EFTE6 foil, which is not a woven fabric but apolymer film sheet. From a structural viewpoint, EFTE foil is interesting becauseof its high tear resistance. In addition to the structural properties, the foil hasmany properties that make it work well as enclosure material. For example, itcan be considered as incombustible, impervious to ultraviolet radiation and mostchemicals, 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 aluminiumor steel sheets with thicknesses of 1 to 5 mm are found to be suitable for this appli-cation. Due to the low bending stiffness of the sheets, it is necessary to prestress themembrane to prevent buckling. Prestressing is achieved by applying the membranebefore the roof is fully erected. When the roof is raised to the final position themembrane is pretensioned. The metal membrane is composed of small accuratelycut sections jointed by welding, gluing or bolting. Metal sheet membrane is a fea-sible choice for long-life structures and can be designed with openings covered withglass to provide natural lightning. Heat loss is prevented by attaching insulationmaterial internally [23]. In [131], Yeremeyv and Kiselev describe the manufacturingand erection of a number of large projects in Russia where metal sheets are used ascovering.

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 theywill conform to the shape of the structure. The panel system is most economical if itis made of light material, not to impose extra weight on the cable structure. Panelsof fibreboard, aluminium and plastic are appropriate to use for covering roofs [23].

For the Olympic Stadium in Munich, a system with translucent plastic panels (Plex-iglas7) with thickness of 4 mm and size of 2.90 m × 2.90 m was used. The panelswere fastened to the supporting cable net with shock absorbing flexible connectionsto prevent cracking of the panels under roof movements. The joints between thepanels 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], findthe Plexiglas cladding of the Olympic Stadium ugly. Therefore, it will probably notbe used again.

6Abbreviation for Tetrafluoroethylene7Plexiglas is a registered trademark of AtoHaas Americas Inc.

23


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

2.6 Structural details

Already at early stages of the design process the designer has to pay attentionto the design of the structural details. Structural details are fittings, saddles andanchorages. Fittings are attachments used to grip the cable at the ends or along itslength. They can be classified, in accordance with the type of application, as thefriction or clamp type, the pressed or swaged type, and the socketed type. Saddlesare 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 aconcrete ring or an arch. In other systems the cable forces are resisted by anchorsin the ground [57]. A comprehensive survey of structural details is given by Chaplinet al. [23].

2.6.1 End fittings

An end fitting (terminal) is an attachment, which transmits the cable force to thesupporting system. To be totally effective, the end fitting must withstand the fullbreaking force of the cable without significant yielding, endure dynamic loadingwithout risk of fatigue failure and not induce fatigue failure of the cable. For ap-plications where large forces are to be transmitted to the supporting structure twodifferent 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 socketedtype. It is manufactured by splaying the end of the cable a prescribed length andcleaning the individual wires. When the wires are cleaned and dried the conicalsocket of machined or casted steel is positioned on the splayed cable section. Thenmolten socketing material is poured into the socket, hardens and forms a cone, Fig-ure 2.18(a). As tension is applied to the cable the cone is drawn into the socketand wedging forces are developed which grip the wires. As socketing material eitherof zinc or resin is used. Pure zinc has been used for over a century and it offersa cathodic protection for the cable, but it is sometimes criticised for impairing thefatigue resistance of the cable in this region. Another, more important, disadvan-tage with sockets filled with pure zinc is that they are prone to creep effects underhigh 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 lowtemperature the fatigue resistance of the cable will not be impaired. Socketed endfittings can be used for all cable sizes but cables of smaller diameter, approximatelyless than 38 mm, can be terminated by means of hydraulically compacted fittingscalled swaged end fittings. Swaged end fittings are cheaper than socketed typesbut they are only guaranteed to resist 95 % of minimum breaking load of the ca-ble. All end fittings are manufactured, installed and rigorously tested by the cablemanufacturer [16,125].

2.6.2 Intermediate fittings

Intermediate fittings are used to connect cables to other cables. These fittings areusually not standard appliances and their behaviour depend on the frictional forcebetween the cable and the clamp. To prevent sliding of the clamp, the clamping forcemust be large and thereby high radial stresses are induced. Cables are more proneto 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 thecable. 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]. Whenthe cable is tensioned the diameter will decrease and consequently the clampingforce. It can therefore be necessary to retension the clamp bolts to prevent sliding.

25


To avoid abrasion between the clamp and cable under cable movements, which canresult in fatigue failure, the ends of the fittings must be radiused. Different types ofintermediate 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 structuraldetails, as the number of fittings in, for example, a cable net can be quite large. Away to achieve this is to use a fitting which can be adjusted for different anglesbetween cables. The fitting shown in Figure 2.19(b) can be mounted in a factoryand thereby it is possible to reach a high accuracy. As mentioned above, accurateassembly of the fittings is necessary in order to obtain the desired internal forcedistribution 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 onehas to take the bending stiffness of the cable into account. Two factors have to bechecked:

• 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 resistanceof the cable will be affected. The common rule is that the diameter of the saddleshould 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 structuresare usually rings, arches and masts made of concrete or steel. In open systemsthe cable forces are resisted by tension anchors in the ground. A survey of exist-ing tension anchors and methods for estimating their capacities for various groundconditions can be found in [16]. Which of the two anchorage alternatives that willbe most economical, if both are architecturally accepted, depends upon the groundconditions, 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 constantlybeing 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 theprescribed loads acting on the structure. Since most loads are environmental loadswith random distributions, durations and magnitudes, the ‘exact’ values will neverbe known. In an attempt to achieve higher accuracy in the results from a structuralanalysis 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 othertype of loads acting on more conventional buildings. However, it is well known thatnon-uniformly distributed loads are more dangerous to cable structures than uniformloads. Therefore, it is important to determine the ‘true’ load distribution on thestructure. Nonetheless, the unusual shape of these structures, together with theirlow weight and large scale, make this a difficult task. A further complication is thatpractically no guidance is available from codes of practice. This implies additional

27


costs to the project, because of the need for expertise. The latest methods fordetermining the loads on roofs of general shapes involve very sophisticated physicaland computational modelling techniques, which require expensive equipment andpowerful computers. In this section, these methods are reviewed. The loads areviewed 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 ofthe most important forms of loading. The variability and large number parametersinvolved in the determination of wind effects on structures make it a very complexproblem. Some undesirable effects and partial collapses have been caused by windon tension structures [16]. Among these can be mentioned the vibrations due towind on the roof of the Raleigh Arena, U.S.A., which made it necessary to insertsupplementary internal cables.

The nature of wind

Wind is initiated by pressure differences between points of equal elevation, causedby variable solar heating of the atmosphere of the earth. The motion of the air massis modified by the rotation of the earth and close to the ground the velocity of themoving air is reduced due to friction. At a certain height above the surface of theearth the effect of the surface friction becomes negligible. Above this boundary layera frictionless wind balance is established, and the wind flows with the gradient speedalong lines of equal barometric pressure. The height of the atmospheric boundarylayer normally ranges from a few hundred meters to several kilometres, dependingupon wind intensity, roughness of terrain, and angle of latitude [32].

Physically, the wind is composed of two different velocity components [16]. The firstcomponent is the velocity of a steady flow determined by the long-term pressurevariations (approximately four day periods). This component is called the meanwind velocity. The second velocity component, which is superimposed on the steadyflow, 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. Thetwo velocity components are clearly seen when the wind velocity is plotted in a vander Hoven power spectrum, Figure 2.22. This spectrum shows the variations of themean square of the amplitudes of the fluctuating components against the frequenciesof these components.

Hence, the analysis of linear structures can be divided into to two parts: the cal-culation of the quasi-static response due to the steady velocity component and theresponse caused by the turbulence components. As cable structures have a non-linear behaviour this division is generally not valid. Instead, the total wind loadmust be used in the dynamic analysis of cable structures. In the sequel to thissection 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. Reproducedfrom [26].

Mathematical description of natural wind

To describe the wind velocity mathematically a Cartesian coordinate system is ap-plied, with the x-axis in the direction of the mean wind velocity, the y-axis hori-zontal 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, areturbulence 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 abovethe ground. The turbulence components are treated mathematically as stationary,stochastic processes with a zero mean value. The mean wind velocity U(z) and theturbulence component u in the wind direction are often most important, as theyusually 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 Uvaries with height [32]. The first law is the power law, which has been adopted inmany codes. The second law is the logarithmic law, which is derived not only fromempirical data, but also from theoretical considerations. The Deaves and Harrismodel, which is the third law, is the most exact one since it is fitted to experimentaldata [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]. Thisdirectly 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 ischaotic, 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 beapplied [32].

29


Wind load on a structure

The earliest method for the assessment of the action of turbulent wind is the quasi-steady vector model [27]. It makes the simple, but inaccurate, assumption that thepressure fluctuations correspond exactly with the variations of the wind velocity.Other methods may be found in [27], but wind loads on buildings are determinedusing the quasi-steady model in many codes [64]. Therefore, a detailed descriptionof the quasi-steady theory will be given here. For a point on a surface, (x,y,z), theinstantaneous pressure, p, is given by [27,64]

p =1

2ρV 2Cp(θm + θv, θw), (2.10)

where V is the wind velocity given by equation (2.9). Cp(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 meanwind direction θm. The magnitude of the wind velocity is given by

V 2 = (U + u)2 + v2 + 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 andthe velocity magnitude reduces to

V 2 ≈ U2 + 2Uu. (2.14)

The fluctuating wind directions are assumed linear for small v and w, which gives

Cp(θm + θv, θw) ≈ Cp(θm) +( v

U

) ∂Cp(θm)

∂θv

+(w

U

) ∂Cp(θm)

∂θw

. (2.15)

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

p(t) ≈ 1

2ρ(U2 + 2Uu

)(Cp(θm) +

( v

U

) ∂Cp(θm)

∂θv

+(w

U

) ∂Cp(θm)

∂θw

). (2.16)

Dividing both sides of equation (2.16) by the mean dynamic pressure 12ρU2, ex-

panding and discarding small turbulent cross terms gives the instantaneous pressurecoefficient

Cp(θm + θv, θw, t) ≈ Cp(θm) + 2( u

U

)Cp(θm) +

( v

U

) ∂Cp(θm)

∂θv

+(w

U

) ∂Cp(θm)

∂θw

.

(2.17)

30

2.7. ROOF LOADS

Taking the time average of equation (2.17) leaves the expected result

Cp(θm + θv, θw, t) = Cp(θm). (2.18)

To evaluate the performance of the quasi-steady theory, full-scale wind velocity andpressure measurements were recently done on an full scale experimental building atthe Texas Tech Field Research Laboratory [64]. The results from that study showedthat the area-averaged pressures over a substantial area of the roof as well as rootmean square (rms) and peak pressure coefficients can be well predicted using thequasi-steady theory. However, the spectra of the pressure coefficients cannot bepredicted 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, surfacecondition, surroundings, wind velocity and wind direction [32]. Because of the com-plexity of these factors, the pressure coefficients must be determined by full-scalemeasurements 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 windtunnel tests. Hence, the most appropriate method for determining the wind loadis 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 bythe 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 pa-rameters. In wind engineering, the number of parameters is so large that it isimpossible to satisfy all the conditions simultaneously. Therefore, some parametersthat are of minor importance have to be disregarded. Besides the laws for the modelitself, the wind tunnel must also be able to simulate the wind climate in the atmo-spheric boundary layer for the site considered. A general description of model lawsand boundary-layer wind tunnels can be found in e.g. [32].

Earlier, rigid models were used in wind tunnel studies to determine the pressuredistribution on the exterior and interior surfaces of a building, for a variety of winddirections. The rigid model studies have been developed over many years of test-ing conventional buildings and are relatively straightforward. However, high windspeeds can change some of the pressure coefficients when aeroelastic models areused [49]. These changes are attributed to roof deflections and to the non-linearstiffness of the roof. The conclusion is that the use of rigid pressure-tapped modelscan underestimate the pressure coefficients for flexible structures undergoing largedisplacements. Another reason, maybe the main one, for conducting wind tunnelexperiments with aeroelastic models is to search for unforeseen aerodynamic insta-bilities, i.e. large amplitude vibrations [49]. In references 33 and 49 wind tunnel testsof tension roofs with aeroelastic models are performed, but no types of aerodynamic

31


instabilities were found. Certain aspects that have to be taken into account whenmodelling 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 electronicallyscanned multi-channel pressure systems, with up to 512 channels and sampling fre-quencies of up to 100 Hz [11]. Hence, the amount of data from one series of testingis enormous and in its raw state very difficult to use for analysis. A method to de-scribe the wind velocity profile and wind pressure pattern was recently rediscoveredin the field of wind engineering. This method, often used for stochastic problems,is the proper orthogonal decomposition (POD), known also as the Karhunen-Loeveexpansion8. The POD method resembles the modal analysis used in structural dy-namics [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 deter-ministic function Φ(x, y) is found through a maximisation of the projection of therandom pressure field p(x, y, t) on Φ(x, y)∫

p(x, y, t)Φ(x, y)dxdy∫Φ2(x, y)dxdy

= max . (2.19)

If the maximisation of equation (2.19) is performed in the mean-square sense for adiscrete pressure field, it leads to the following eigenvalue problem

RpΦ = λΦ, (2.20)

where Rp is the covariance matrix of the pressure space, and Φ and λ are, respec-tively, a vector and a value, both to be determined. The eigenvectors Φn(xi, yj) arebase functions in a series expansion of the pressure field

p(xi, yj, tk) =∑

n

an(tk)Φn(xi, yj), (2.21)

where the expansion coefficients, i.e. modal amplitudes, an(tk) are easily computeddue to the orthogonality of the eigenfunctions Φn(xi, yj)

an(tk) =

∑i

∑j p(xi, yj, tk)Φn(xi, yj)∑

i

∑j Φ2

n(xi, yj). (2.22)

The eigenvalue λk is the measure of the contribution of each eigenmode to thepressure mean squares [30]. Depending on the number of terms included in the ex-pansion, equation (2.21), different levels of accuracy are reached. In [12] about 30 %

8The expansion was derived independently by a number of investigators; Karhunen in1947, Loeve in 1948, and Kac and Siegert in 1947 (according to “Stochastic Finite Ele-ments: A Spectral Approach” by Ghanem, R. G. and Spanos, P. D., which can be found athttp://venus.ce.jhu.edu/book/)

32

2.7. ROOF LOADS

of the eigenvectors were required in the expansion to represent the peak pressurewith an error of approximately 10 %. However, only one term, the first eigenvectorwas needed to represent the mean point and area-average roof pressure with an errorof approximately 1 %. A physical interpretation of the first three eigenvectors canbe provided by the quasi-steady theory [122]. It was shown in reference 122 thatthe first eigenvector was closely related to the mean pressure distribution Cp(θm),the second and third eigenvectors were related to ∂Cp(θm)/∂θv, and ∂Cp(θm)/∂θw,respectively. These results, together with the full scale measurements by Letchfordet al. [64] indicate that the pressure field over certain roof types can be describedby the quasi-steady theory and that the POD method and the quasi-steady theoryin some sense are related to each other.

A simplified method to calculate wind loads on tension structures with irregularshapes has been presented by Tabarrok and Qin [115]. The method simplifies inputdata, because it does not need experimentally measured wind pressure coefficients.In their method a membrane structure was discretized with constant strain shellelements. To calculate the wind load on each element, the designer specifies amagnitude of the pressure coefficient Cp that defines the wind pressure on a verticalsurface normal to the wind direction, and a direction that defines the source of thewind. The wind pressure normal to each element is then computed by scaling thewind velocity by the cosine of the angle between the wind direction and the outwardnormal to the element. This means that the model gives zero pressure for surfacesparallel to the wind and suction on leeward surfaces.

Computational wind engineering

Wind tunnel testing, including model making, is expensive, tedious and in some casesinaccurate due to limitations associated with the boundary layer wind tunnel [109].These limitations might be overcome if the pressure values could be derived bynumerical computer simulations. Savings, in both time and money, would also bepossible.

Application of Computational Fluid Dynamics (CFD) to wind engineering problemsmeans large computer memory and CPU time, because very fine computational gridsare needed to deal with the modelling of turbulence, complex building configurationsand the large area of model domain [109]. Today, only smaller structures with coarsegrids can be analysed. However, it is anticipated that current limitations due tolong CPU time and large memory requirements will be overcome in the near futurethrough new computational, parallel-processing based architectures, faster computerhardware and more efficient computational algorithms [11].

The current state-of-the-art of Computational Wind Engineering (CWE) is reviewedby Stathopoulos [109]. Results from CFD simulations are compared with thoseobtained from wind tunnel or full scale experiments for buildings of different shapeand various wind directions. According to Stathopoulos: “Disagreements proved tobe higher than what is tolerable, particularly for cases that require complex buildingshapes, surroundings and for results other than mean pressure coefficients.” He

33


further concludes that “at present time CWE may be used only for the assessment ofthe wind environment around buildings. For cases which involve mean values of thewind speed and pressure coefficient the numerical results may be used for preliminarydesign purposes.” Several areas have to be improved before CFD can be used indesign, which include: numerical accuracy, description of boundary conditions andrefinement of turbulence models [109].

The most optimal method today seems to be a hybrid analysis, where experimentalwind tunnel data are combined with numerical simulations [11]. For engineeringproblems, a hybrid analysis could also involve combining the experimental datawith 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 dueto the flexibility (geometric non-linearity) of the roofs.

• Pressure coefficients have to be determined by wind tunnel tests. The pressuredistribution from the tests can be described mathematically using the PODmethod.

• To be able to recognise any unforeseeable wind related instability, the windtunnel model should be of the aeroelastic type.

Although several methods are at hand for the wind engineer, the assessment ofwind effects on structures is certainly not easy. As mentioned before, this areais very complicated and for a more in-depth analysis one can refer to the manyreferences given in this section, especially [26,27,32]. As the design recommendationsconcerning tension roof structures are non-existent, good engineering judgement andexperience 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 struc-tures. Many modern buildings have moved away from traditional shapes and theirbehaviour with respect to snow accumulation is not known well enough [41]. Whenconstructing a tension roof, the load corresponding to the expected intensity of snowhas to be considered. As for the wind loads this is more or less straightforward forordinary types of roofs, and is found in national building codes. For cable andmembrane roofs this is considerably more difficult.

Snow distribution

The snow intensity is measured at meteorological weather stations as the groundsnow depth. Prior to 1970, many buildings were designed and built assuming uni-formly 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 loadswere started to determine the difference between ground and roof snow load. Theresults from the surveys showed that, in cold and windy areas, the roof snow loadwas considerably lower than the ground snow load. Nonetheless, on certain parts ofsome roofs the load was significantly higher.

Today, building codes make provision for drifting of snow by specifying a numberof snow load cases for the type of roof considered. Some shapes of roofs tend toaccumulate more unbalanced loads than others, and the load cases try to cover thepossible snow distributions over the roof. Unfortunately, the roof types covered bythe codes are usually traditional. Tension structures, such as cable and membraneroofs, with sculptural forms are not covered by the codes. Due to the flexibility oftension 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 windtunnel or water flume experiments.

Wind tunnel and water flume testing

Like wind tunnel experiments, some model laws has to be followed when modellingthe snow in air or water. In wind tunnels, granular materials, such as tea, glass, andnut shells, are used to simulate dry snow, while sand is used in water flume experi-ments, Figure 2.23. One limitation in modelling, which cannot easily be overcome,concerns the great variation in snow properties. Common simulation materials can-not model sticky snow. For example, sand will not stay on steep surfaces whichmakes it difficult to simulate snow accumulations on steep slopes where snow willaccumulate before eventually sliding off. Another limitation in model studies is thatonly one wind direction is considered at a time but the overall seasonal environmentconsists of a sequence of snow storms and high winds from different directions. Inreality, the final snow accumulation depends on the chronological order and dura-tion of the storms and on temperature, sunshine, humidity, etc. Surrounding terrainmay also affect total snow accumulation and drift patterns on structures. Whetherair or water is the medium, a model can provide a good simulation of the flowaround structures. However, the state-of-the-art of snow drift modelling preventsthe measurement of quantitative results [50,118].

Recently a water flume test was used to determine the snow loads on a large tensionroof at Denver International Airport, U.S.A. [10]. Denver is known for its heavy snowfalls, 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 wasvery high, 3.8 kN/m2, Figure 2.23. The predicted snow pattern from the modeltest agreed well to that seen on the roof after the first snow falls, confirming thereliability of the test.

35


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

Computer simulations

In the design of the tension roof of Denver International Airport, the water flumeexperiments were supplemented by a computer program based on the Finite AreaElement (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 ob-tained from wind tunnel experiments. A brief description of the FAE method andits properties is given below. First, the roof is divided into many area elements bya grid. The wind velocities are measured at grid intersection points. Time historiesof meteorological data concerning the wind direction and speed are used as inputfor the computations. Snow drift is computed using empirical relationships for snowflux versus wind velocity. By computing the mass fluxes into and out of each ele-ment, the rate of build up or depletion of snow mass in the element due to driftingis determined. The mass balance computations at each time step include the addi-tional mass from snow fall and the depletion due to melting. The method also takesinto consideration the less significant drift of snow that has been rained upon, orthat 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 itmelts). Therefore, the area elements are assigned with a certain storage capacity forsnow depending on surface roughness. Also included in the FAE method is a heatbalance used to calculate the melting rate of the snow pack inside each element, andthe 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 quantitativeresults can be obtained with higher accuracy [38].

A purely computational method for predicting snow accumulation, called SNOW-SIM, has been developed under a research project at Narvik Institute of Technologyin Norway [7]. The method includes a commercial CFD program, combined with asimplified drift-flux model to simulate snow drift. A computer simulation of snowdrift has the advantage over wind tunnel or water flume experiments that it canbe 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. Simulationsin three dimensions were presented, but due to limitations in computer power onlysmall buildings of regular shape with a coarse mesh resolution, and simulation timesof up to 50 seconds could be studied. Therefore, the simulations could only be re-garded as an indication of where the snow will deposit, not as an exact quantitycalculation. Compared with real measurements the simulations gave similar snowdrift patterns. Bang et al. [7] gave a number of problems that have to be resolvedbefore quantitative results can be available. These included the proper treatmentof turbulence in the CFD program, evaluation of different drift-flux models, andmodelling of structures with their local terrain. Thus, a complete computer simula-tion of snow drift magnitude is today not available even for buildings of traditionalshape [7].

As in the case with wind loads on tension structures with complex shapes, Tabarrokand Qin [115] have proposed a simplified method to calculate the snow load distri-bution. In their method, vertical snow loads are generated based on the horizontalprojection of each elemental area and a snow load magnitude per unit horizontalarea specified by the designer. This means that there is full snow load on a hor-izontal surface and zero load on a vertical surface. This method is of course veryapproximative as it cannot handle snow drift.

It has been seen that the determination of snow load magnitude and distribution is atask of equal difficulty as that for wind load. For a roof with a complex shape the onlyway to find the sought quantities, i.e. magnitude and distribution, is through modeltests. This procedure is expensive, time consuming and requires special knowledgeand experience.

2.7.3 Earthquake load

Another important form of loading, which has to be considered in certain parts ofthe world, is earthquake ground motion. Even smaller earthquakes may lead tocollapse of stiff structures. Many studies have been concerned with the earthquakeresponse of building structures but, like the wind and snow load studies, very fewhave included cable roof structures. Two works on the topic have been found andare briefly presented in the following.

In reference 78, a cable truss with diagonal ties (system Jawerth) is subjected tovertical and horizontal earthquake loadings. Both a linear and a non-linear analysiswas performed. The maximum displacements did not differ significantly betweenthe analyses. It was also found that under a horizontal earthquake, all the diagonalsbecame slack at many instances. As far as the diagonal forces are concerned, theresponse 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 stiffenedwith a tensegrity system. Tatemichi et al. conclude that the analysis “confirmedeffectiveness of the suspen-dome against earthquake motions, particularly vertical

37


motions.”

In general, the response of structures to dynamic loading is determined by a finiteelement analyses. Such analyses of cable roofs have shown that these structuresusually have a long period of vibration. In addition, the supporting structures arerelatively much stiffer and heavier than the cable system. Therefore, high-frequencycontents of the earthquake ground motion will be amplified by the supporting struc-tures. Conversely, low-frequency components will be reduced considerably by thetime they reach the cable system. Hence, the response of the structure is dependenton 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 alarge part of the total load. This is not the case for prestressed cable roofs. Forthese roofs the dead load consists of the weight of cladding, insulation, cables andfittings, etc., [57]. The magnitude of the dead load for a prestressed roof with fabriccladding is very low, values as low as 0.1 kN/m2 are common [99]. Hence, the deadload cannot be considered as important in ensuring the safety of correctly designedprestressed 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 itgoverns 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, butmust, due to stress relaxation, not be greater than 45 % of the breaking force of thecable [42], section 2.4.2. Apart from relaxation, loss of cable tension also occurs as aresult from creep in the supporting structure, slippage of cables at anchorage pointsand increase in temperature [57].

Live loads are usually taken into account by specifying the intensity of a uniformlydistributed load. Usually, cable roofs have curved shapes and may therefore beconsidered as inaccessible to people except for maintenance purposes. This justifiesthe 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 inthis section are not considered separately. Design load cases consist of combinationsof the different loads. The load combinations forming these cases are given in thenational building codes and there is no reason to expect that the cases will bedifferent 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 men-tioned. None of the numerical techniques described here are applicable to only cablestructures. Therefore, only a short historical review is considered necessary.

Early analyses were done by applying membrane shell theory to cable nets. Appli-cation of the membrane shell theory results in a set of differential equations. Exceptfor special cases, these equations are difficult to solve in closed form. In mostcases, the equations are solved by numerical techniques, such as the finite differencemethod [112]. Shore and Bathish [107] used double Fourier series to transform thedifferential equations to a system of algebraic equations. One flat and one hyperbolicparaboloid prestressed cable net, both square in plan, were analysed numerically andexperimentally. The agreement between the results was acceptable. Recently, Tarnoperformed a parametric study of saddle-shaped networks with elliptic plane layoutsand stiff contours [116]. Such a study serves as an aid in choosing the dimensionsof the roof and the structural elements. In general, membrane shell theory is lessaccurate if the cable mesh in a net is coarse; it is inadequate for complicated roofshapes.

Since the introduction of computers in the 1960’s, several numerical methods havebeen developed for the general analysis of structures. Among these methods, thestiffness technique (finite element method) have been widely adopted. Originally,the method was developed to analyse structures with small displacements. Underthe action of external loads cable nets undergo large displacements and it becameevident that the stiffness technique was not applicable to such structures in itsoriginal form. Therefore, the original method was modified and applicable structureswith geometrically non-linear characteristics. Several iterative methods have beenapplied to the non-linear stiffness method. The most popular is the Newton-Raphsontechnique, which has proven to be accurate, efficient and applicable to the majorityof cable structures [1]. A comprehensive description of the finite element method andthe 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 potentialenergy of the structure. The minimisation was done using the the conjugate gradientmethod. The dynamic relaxation technique has been used by several authors forboth form-finding [8, 66] and load analysis [68,69].

Approximate methods for the preliminary design of cable trusses and simple cablenets, can be found in references 16, 57 and 76. For elliptical cable nets the methoddescribed by Tarno [116] is recommended.

The following chapters will investigate some earlier reported analysis methods andsome new variants of them, aiming at accurate analyses of the form-finding andnormal usage stages of some cable roof structures. Failure stage analysis will beidentified 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 generallyknown in advance. This is not the case for tension structures, i.e. cable and mem-brane structures. Due to the low flexural stiffness of the cables and the fabric thesestructures have to be constructed so that they will experience a significant prestressat all times. Thus, there is no compatible unstressed configuration for a tensionstructure, even if no external loads are applied and its self-weight is neglected.Therefore, the designer must specify a reference configuration for the structure thatis stressed. The shape of the reference configuration depends upon the internalstresses and forces. Hence, the load bearing behaviour and the shape of the struc-ture cannot be separated and cannot be described by simple geometric models. Inaddition to satisfying the equilibrium conditions, the initial configuration must ac-commodate both architectural, structural and constructional requirements [44,114].Finding the stressed initial configuration is an inverse structural problem, in whichthe specified force distribution is the driving parameter in the process. This isinverse to standard problems where the forces are the structural response to thedeformations of the structure [13].

The problem of finding a configuration that satisfies the laws of equilibrium is usu-ally called form-finding or shape-finding. Haber and Abel [44] thought that thisnomenclature was inappropriate to use when describing methods in which variablesbesides the shape were adjusted to satisfy equilibrium. Therefore, they used theterm initial equilibrium problem instead. Throughout the present chapter and therest of the thesis this term will mainly be used.

The objectives of this chapter are to describe the initial equilibrium problem andreview the existing computer methods for solving it. All the methods that are tobe described are applicable to mainly cable structures, membrane structures andbar frameworks. Among the methods, one is especially interesting, namely the forcedensity method. This method will further on be described in detail and applied toa 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 oftension structures, physical modelling was the primary method to solve the initialequilibrium problem of tensile structures until 1969 [44], when the cable roofs for the1972 Munich Olympic Games were about to be built [62]. A pioneer in the physicalmodelling field was Frei Otto, who performed extensive experiments using a varietyof different media, including soap films, fabric, and wire models.

The most simple modelling tools are soap film models, which are obtained by dippingwire frames into soap water. These models are used as a first check if the curvature ofthe roof surface is appropriate. As is well known, a soap film always contracts to theminimal surface. The minimal surface may be the most aesthetic shape, but it is notalways the best structural form, as the minimal surface approach tends to producevery flat areas, which may induce flutter (see also section 3.2.1). After the soap filmmodels, working models of larger scale and of other materials were built for furtherprocessing within the design process [87]. The Institute for Lightweight Structuresin Stuttgart has worked with physical modelling techniques for several decades anda large number of structures with complex structural shapes of different scales havebeen realised during the years, e.g. [31] or [87]. The Institute was involved in theconstruction of the some of the largest and most complicated cable nets built in theworld: the German pavilion at the 1967 World’s fair in Montreal and the Olympiccable roofs in Munich.

Models give many useful insights into the the behaviour of tension structures. Alarge number of configurations can in a short time be studied if for example soap filmmodels are used. For practical design work, however, they do not provide sufficientaccuracy and have to be replaced by computational methods [31,44]. Nevertheless,physical models of tension structures will always be made, because of the excellentstructural visualisation they provide [77].

3.2 Literature review of initial equilibrium solu-

tion methods

In a numerical method a solution to the initial equilibrium problem consists of acombination of parameters describing an equilibrium configuration of the structure.One method of distinguishing the different solution methods is to indicate whichof the parameters are specified by the designer and which are treated as problemunknowns [44]. The parameters involved in the initial equilibrium problem are:

• Structural topologyThe 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 (sec-tion 3.2.1) or the connectivity matrix in the force density method (section 3.3).

• External loadsTwo types of loadings may act on the structure: body forces and surface trac-tions. Inclusion of these loads often complicate the initial equilibrium problemas the direction and magnitude of the loads may depend on the unknown ref-erence configuration.

• Structural geometryThe actual shape or surface geometry of the structure is one of two key parame-ters in the initial equilibrium problem. It plays a major role in determining thestresses that will act in the structure at various times. For a tension structure,curvature is the parameter that mostly affects the structural behaviour.

• Boundary conditionsIn methods where the geometry is treated as an unknown, it is necessary tointroduce some boundary conditions to ensure a unique solution.

• Internal force distributionThe internal force distribution is the second key parameter. In order to obtaina 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 neces-sary to introduce kinematic equations. However, some methods, e.g. the non-lineardisplacement method, are using kinematic equations to solve the initial equilibriumproblem. This method requires material properties to be specified, although theseneed not be the actual properties. Fictitious material properties may be used tocontrol 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 noloads act at the nodes. However, for completeness the external forces will still bepresent in many of the equations presented in this chapter, but the usual approachis to set them to zero.

Initially, the only requirement put on the reference configuration is that it shouldbe in equilibrium. Consider a node i in a cable net where four cables meet, Fig-ure 3.1. The equilibrium equations in the x-, y-, and z-directions at that node canbe expressed as:

Tijxj − xi

Lij

+ Tikxk − xi

Lik

+ Tilxl − xi

Lil

+ Timxm − xi

Lim

+ Fxi = 0, (3.1)

Tijyj − yi

Lij

+ Tikyk − yi

Lik

+ Tilyl − yi

Lil

+ Timym − yi

Lim

+ Fyi = 0, (3.2)

Tijzj − zi

Lij

+ Tikzk − zi

Lik

+ Tilzl − zi

Lil

+ Timzm − zi

Lim

+ Fzi = 0. (3.3)

43


xy

z

i

k

j

ml

Figure 3.1: A connection in a cable net

Since the initial equilibrium is a static problem any configuration where equa-tions (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 toobtain these solutions will be described below. Their respective merits and draw-backs will be brought to light. At the end of each section, the features of eachmethod are summarised.

3.2.1 The non-linear displacement method

Among the first computer methods applied to the solution of the initial equilibriumproblem was the non-linear displacement method, which is based on the large dis-placement finite element technique used for analysis of structural behaviour underexternal loads. As the same program can be used for both the initial equilibriumproblem and the load analysis, this approach is quite common. Nonetheless, thereare some serious disadvantages associated with this technique. This section will bedivided into two subsections: the first one dealing with cable nets and the secondone with membranes.

The non-linear displacement method may be summarised as follows. First, an ele-ment mesh in equilibrium with a prescribed force distribution is established in thehorizontal plane. A three-dimensional form of the mesh is created by displacing thesupport points almost vertically until they attain their prescribed positions, Fig-ure 3.2. An iterative algorithm, e.g. the Newton-Raphson method, is used to obtainthe equilibrium configuration of the deformed structure.

44


Cable nets

Argyris et al. [4] were among the first to use the non-linear displacement method tosolve the initial equilibrium problem for cable nets. Their method was developed inorder to find the form of the cable roofs at the 1972 Olympics in Munich. Straightbar 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 initiallyout-of-balance structure is allowed to undergo damped vibrations until a steadyequilibrium shape is obtained.

The displacements of the fixed nodes may give rise to an unfavourable force dis-tribution in the net, when actual material properties are used. Therefore, whenthe fixed nodes have reached their final positions, a force adjustment procedure isapplied to the net. In this procedure the original unstrained lengths of the ele-ments 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 thetotal 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

1 + ε=

L0 + ∆L0

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 willnot change the final force distribution very much, so it will be close to the desiredone. Another way to keep control over the forces is to use a very small modulus ofelasticity for the cables, but then the control over the cable lengths is lost. With theprocedure outlined above control of both the forces and cable lengths is possible.

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

45


Membrane structures

In principle, the application of the non-linear displacement method to membranestructures does not differ very much from the case of cable nets. Some general ap-proaches used for finding the reference configuration of membranes will be discussed.First, a finite element suitable for membrane representation has to be selected. Ow-ing to the great geometric non-linearity of membrane structures, it is preferable touse a dense mesh of primitive elements rather than a coarse mesh made up of higherorder 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 config-uration, which is characterised by a state of isotropic tensile stress. In order to findthe minimal surface, it is assumed that the flat membrane has a very small modulusof elasticity and is in the isotropic prestressed state [115]. For other members in thestructure, 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 onlyslightly change, even though large deformations occur during the displacements ofthe fixed nodes [115].

The advantages of the minimum surface are its aesthetically pleasing shape andthe associated uniform tensile stress. However, in some cases the minimal surfaceconfiguration cannot satisfy all the architectural and structural requirements [115].Since the mean curvature for minimum surfaces is zero, such surfaces are rather flatand these have poor load bearing capacities. A nonuniform stress approach has tobe 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 specifieduniformly. Following the same procedures as for the minimal surface approach, thefinal configuration should be in equilibrium with the nonuniform prestress. Severaltrial 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 alterna-tive 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 actualmodulus of elasticity of the membrane is used. As for the cable net, the deformedequilibrium configuration will have nonuniform and possibly large stresses. At thisstage, the stresses due to deformation are removed by a stress adjustment algorithmand only the initial stresses are retained [115].

It was explained above that the way to keep the stresses within the elements con-stant during displacement is to assign a very small modulus of elasticity to theelements. However, in many cases the small elasticity has undesirable consequences,such as numerical instability and divergence of the solution [65]. These problems

46


stem from the assumption of small strains made in the derivation of the membraneelements. 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 geometrytake place during the displacements of the fixed nodes. A way to avoid numericalinstability is to choose a mathematically defined initial surface close to the finalshape. Similar convergence problems were also reported in [36].

Another problem is that for surfaces which exhibit high curvatures, elements gatherin certain regions of the surface and leave the remaining regions represented to alesser accuracy. It is suggested that a suitable element arrangement in complex casesshould be chosen with the aid of a physical model [66].

Recently, Bletzinger [13] used a method called the updated reference strategy, whichis a numerical continuation method, to solve the initial equilibrium problem of mem-branes with minimal surfaces. This technique had to be used because of the occur-rence of a singular stiffness matrix, which excludes the use of the ordinary Newton-Raphson algorithm. The stiffness matrix is singular when the nodal displacementsare tangential to the membrane surface. To understand that, consider a plane, whichobviously is a minimal surface. The surface area of that plane does not change ifthe geometry of discretization is changed, e.g. by small tangential displacementswithin the plane. Hence, the area variation of in-plane displacements is zero. Aspecial case of the updated reference strategy is the force density method. As theupdated reference strategy is claimed to be absolutely robust, it is perhaps the bestnon-linear displacement method available for the initial equilibrium problem.

There are some drawbacks of the non-linear displacement method applied to theinitial equilibrium problem. Both the final shape and the stresses in the structureare difficult for the designer to control. It is not an easy task to specify a desirableforce distribution [44]. If actual material values are used it is possible for some el-ements in the structure to end up in compression [4]. The specification of materialproperties (fictitious or real) represents unnecessary additional decision making forthe designer. In addition, the computations involved in this method are time con-suming for large structures [44]. An advantage is that the program used to solvethe initial equilibrium problem can also be used for further load analysis.

The non-linear displacement method may be summarised as follows [44]. The vari-ables specified by the designer are:

• structural topology,

• boundary conditions, and

• material properties.

The problem unknowns are:

• structural geometry, and

• internal force distribution.

47


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 toovercome the problems associated with the non-linear displacement method. Inmany of these methods, a variety of limitations are imposed on the solution totransform the general non-linear problem into a linear one. The earliest methodof this type was developed for orthogonal cable nets by Siev and Eidelmann in1962 [44].

Their method use the equations (3.1)–(3.3). By placing restrictions on the cable netregarding the geometry, boundary conditions, and the internal stress distributionthe remaining vertical equilibrium problem becomes linear [108]. How this is ac-complished 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, witha grid size equal to ∆l, Figure 3.3. This gave the following modified equilibriumequations in the x- and y-directions (with zero external loads):

Tij∆l

Lij

+ Til∆l

Lil

= 0 (3.5)

Tik∆l

Lik

+ Tim∆l

Lim

= 0 (3.6)

48


Noting that Tij∆l/Lij and Til∆l/Lil are the horizontal components of the cableforces in the x-direction, and Tik∆l/Lik and Tim∆l/Lim the horizontal componentsin 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 andHiy, respectively, equation (3.3) can be written as:

Hix(zj − 2zi + zl) + Hiy(zk − 2zi + zm) + Fiz = 0 (3.7)

If the horizontal components of the forces in the cables are specified, 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 shear-free membrane [119]:

Hx∂2z

∂x2+ Hy

∂2z

∂y2+ Fz = 0, (3.8)

where Hx, Hy are the horizontal components of the prestressing force distribution

(N/m) in x- and y-directions, respectively. Fz is the vertical (z-direction) loadintensity (N/m2).

The grid method may be summarised as follows [44]. The variables specified by thedesigner are:

• structural topology, and

• boundary conditions.




The following additional constraints are placed on the solution:

• limited to line cable elements,

• constant horizontal force along cables, and

• 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 fororthogonal cable nets. However, because of the restrictions placed on the structurein that method, the resulting shapes are few. In this section, a more general method,called the force density method, will be presented.

49


The force density method is a strategy to solve the equations of equilibrium for acable net without requiring any initial coordinates for the structure, just by takingadvantage 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 ofthe 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, (3.9)

qij (yj − yi) + qik (yk − yi) + qil (yl − yi) + qim (ym − yi) = 0, (3.10)

qij (zj − zi) + qik (zk − zi) + qil (zl − zi) + qim (zm − zi) = 0. (3.11)

It is obvious that the main advantage of using the force densities as descriptionparameters for a cable net is that any state of equilibrium can be obtained bythe solution of one system of linear equations. The equilibrium state so obtainedhas the prescribed force density in each element. No other conditions, such asequidistant meshes or constant element forces, are fulfilled. For a first impressionof 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 non-linear displacement method or the extended non-linear force density method can beused. The difference between these methods is that for the non-linear displacementmethod described in section 3.2.1 the number of equations is equal to the number ofdegrees of freedom, while for the non-linear force density it is equal to the numberof additional constraints. In most cases the number of constraints is much smallerthan the number of degrees of freedom [105].

Mollaert [75] applied the force density method to structures composed of both cablesand compression members. To obtain a solution out of the plane of the fixed nodesthe tensile and compression parts of the structure were separated. At the commonnodes the removed part was replaced by external forces. Both parts were thendesigned separately.

In [77] the force density method was used together with a least squares minimisationapproach presented in [43] to generate the cutting pattern for membrane structures.Although the problem of determining the membrane cutting pattern is outside thescope of this thesis it should be mentioned that the force density method can beused to solve also this problem. Due to the simple formulations of the force densitymethod and the least squares minimisation technique, a solution can be obtainedin short time although a very fine mesh is used. These properties make the forcedensity method a better choice than other methods, such as the dynamic relaxationmethod, 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. Inreference 44 an extended version of the force density method, with curved cableand membrane elements, is presented. The method is based on assumed geometricstiffness matrices

KGxg = 0, (3.12)

where KG is the geometric stiffness matrix of the structure and xg the vector ofnodal coordinates (x-, y- and z-coordinates). Equation (3.12) can be applied to any

50


finite element structural model. Although (3.12) has the form of a standard stiffnessequation, 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. Neverthe-less, if the choice of suitable force densities was quite easy, it is much more difficultto choose the geometric stiffness matrices. For simple elements closed form expres-sions can be established, but for many elements the matrices have to be found bynumerical integration. Even after the geometry has been solved, the determinationof stresses for complex elements can be a problem [44].

Christou [24] implemented an elastic catenary element in the force density methodto be able to take into account loads distributed along the cables. After the shape isfound the force in each cable has to be found by iteration since the horizontal force isdescribed by a non-linear equation. However, refinement of the force density methodto take into account distributed loads has less importance at the ‘form-finding’ stagesince 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 adeployable reflector for space applications. They transformed the original membraneinto an equivalent cable network and could, therefore, use the original equations ofthe force density method. This work shows that although the force density methodwas developed back in 1971 it finds new areas of application.

The force density method may be summarised as follows [44]. The variables specifiedby the designer are:

• structural topology, and

• boundary conditions.



• 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


satisfy equilibrium has to be determined. In this section two methods that areappropriate 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 whichthe following two are especially interesting: the over- and the under-determinedcases [44]. The third case corresponds to a square matrix A.

In the over-determined case there are more equilibrium equations than unknowncable forces. Because there is no exact solution to such a set of equations, it isnecessary to seek an optimal set of forces which will only approximately equilibratethe loads on the structure. The optimal set of cable forces is selected by the leastsquares method

ATAt = AT f . (3.14)

It should be emphasised that this method satisfies equilibrium in a least squaressense only. A disadvantage of this method is that the designer does not have muchcontrol over the force pattern. There is no restriction against compressive forcesand the distribution of forces may be highly irregular. Some force control may beobtained by prescribing some of the forces. A major advantage of the method isthat the solution is obtained by solving a set of symmetric linear equations [44].

The under-determined case occurs when there are more unknowns than equilibriumequations (membrane structures often fall into this category). For this case, thereexists an infinite number of exact equilibrium solutions for the forces. Therefore, anideal force distribution t∗ has to be defined and solved for. Generally, these idealforces will not satisfy equilibrium. The actual forces are expressed in terms of theideal 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 theproblem 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 thesmallest Euclidean norm. This optimal solution is found by solving the followingminimisation problem with Lagrange multipliers:

∆tT ∆t − 2kT [A∆t − (f − At∗)] → min . (3.17)

The solution to (3.17) is

∆t = AT(AAT

)−1(f − At∗) . (3.18)

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

52


and the prescribed force distribution are incompatible. An advantage of the under-determined least squares method is that the designer is given some control overthe force distribution while the geometry may be specified exactly. As in the over-determined 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


3.2.5 A combined approach

No single solution method is optimal for all problems. It is possible to use severalof the better solution methods in combinations to create more flexible design tools.A combined approach lets the designer experiment with various methods to find theoptimal solution. Approximate results from one solution method may be used asinput 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 densitymethod, which uses linear cable elements, and then take this solution to a non-linearfinite element program, which uses more refined cable formulations (see Chapter 4).Since the solution obtained by the force density method is approximate, but stillvery good, only a few iterations should be needed to find the ‘true’ equilibriumconfiguration.

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 ofelements, 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 bal-ance the stresses [80]. This property makes them very interesting from an economicalpoint of view. Concerning initial equilibrium configurations of tensegrity structures,much research has been done on geometrical basis [81]. However, geometrical meth-ods do not guarantee mechanical equilibrium and the solutions have to be checkedby numerical techniques.

53


Pellegrino and Calladine have in a number of publications, [18–21, 92, 93], analysedstatically and kinematically indeterminate frameworks with known geometries. Intheir analyses they used the equilibrium equation (3.13), repeated here,

At = f , (3.19)

and the compatibility equationBd = e. (3.20)

By considering the four subspaces of the equilibrium matrix A (= BT ) useful infor-mation concerning the structure is obtained. The essence of their method is givenin Figure 3.4 and Table 3.1. From a structural view, the perhaps most importantresult from their studies is that only some kinematically indeterminate frameworkswith more than one independent state of self-stress (s > 1) can be stiffened to thefirst-order by a single state of prestress. This is crucial, because tensegrity systemsand cable nets must have first-order stiffness in all possible modes [21] to preventexcessive displacements, which may lead to collapse.

In most cases, the geometry of a tensegrity structure is unknown. To find the self-stressed configuration of a general tensegrity structure has proven to be much moredifficult than it is for cable and membrane structures, according to the few worksdealing with this problem.

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

Assembly type dimN (A) Static and kinematic featuresand

dimN (AT )I Statically determinate and s = 0 Both (3.19) and (3.20) have a

kinematically determinate m = 0 unique solution for any righthand side (r.h.s.).

II Statically determinate and s = 0 (3.19) has a unique solution forkinematically indeterminate m > 0 some particular r.h.s., but other-

wise no solution. (3.20) has aninfinite number of solutions forany r.h.s.

III Statically indeterminate and s > 0 (3.19) has a infinite number ofkinematically determinate m = 0 solutions for any r.h.s. (3.20)

has a unique solution for someparticular r.h.s., but otherwise.no solution.

IV Statically indeterminate and s > 0 Both (3.19) and (3.20) have ankinematically indeterminate m > 0 infinite number of solutions for

some particular r.h.s, but other-wise no solution.

54


Dim. Equilibrium A Compatibility B

r

Row space R(AT ):bar tensions inequilibrium with theloads in the columnspace.

=

Column space R(B):compatibility barelongations

Bar spaceRb ⊥ ⊥

s

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

Left null-space N (BT ):incompatible barelongations.

r

Column space R(A):loads which can beequilibrated in theinitial configuration.

=

Row space R(BT ):extensionaldisplacements.

Joint spaceR3j−c ⊥ ⊥

m

Left null-space N (AT ):loads which cannot beequilibrated in theinitial configuration

=

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

Figure 3.4: The four fundamental subspaces associated with the equilibrium matrixA and the compatibility matrix B(= AT ). The sign ‘=’ indicates thatthe two subspaces coincide, while ‘⊥’ indicates that they are orthogonalcomplements 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 iter-ations to find the form of double layer tensegrity dome. He also stated, contrary towhat is given in this chapter and in [82], that the initial equilibrium problem is kine-matic [45]: “The assumed geometry is, in general a mechanism, when constraintson strut and tendon elongations are considered. Shape finding consists essentiallyof activating the mechanisms, until a state is reached when only elastic deforma-tions are possible. This is the prestressable geometry.” But, an algorithm for thekinematic formulation was not presented in reference 45.

55


Motro et al. [82] used both the dynamic relaxation method and the force densitymethod to find the form of simple tensegrity systems. The results were compared toanalytical solutions. The dynamic relaxation technique worked well for systems withfew nodes but suffered from convergence problems when the number of nodes wasincreased. Although the force density method was applied only to small tensegritystructures, Motro et al. anticipated that the force density method could be conve-nient 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 structureresearchers. The method was developed by Linkwitz and Schek [71] for the initialequilibrium problem of the cable roofs at the 1972 Olympic Games in Munich. Itwas first published in [71] and later extended in [72] and [105]. The procedurein this section will mainly follow that in [105] but some additional comments andexplanations will be given. Throughout this section the following notation will beused: 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-jointedto each other or to the supporting structure [105]. First, a graph of a network isdrawn 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 theother n nodes are free. Thus, the total number of nodes is ns = n + nf . Then theconnectivity matrix Cs is constructed with the aid of the graph. Each element jhas the node numbers k and l (from k to l). The connectivity matrix Cs for thestructure is define by (i = 1, 2, ..., ns):

cs(j, i) =

+1 for i = k,

−1 for i = l,

0 in the other cases.

(3.21)

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 vectorscontaining the coordinates of the n free nodes x, y, z, and similarly for the nf fixednodes xf , yf , zf , the coordinate differences for each element can be written as:

u = Csxs = Cx + Cfxf , (3.23)

v = Csys = Cy + Cfyf , (3.24)

w = Cszs = Cz + Cfzf . (3.25)

56

3.3. THE FORCE DENSITY METHOD

Tab

le3.

2:Su

mm

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solu

tion

met

hods

(‘∗’

desi

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From

[44]

.

Des

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ural

Inte

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eria

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otes

topo

logy

load

sco

ndit

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geom

etry

forc

espr

oper

ties

The

non-

linea

r∗

∗∗

??

∗A

–Gdi

spla

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met

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The

grid

met

hod

∗∗

∗?

?−

H–K

The

forc

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?−

H,L

met

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(rel

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K:

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gid)

.L:

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may

beir

regu

lar.

57


The equilibrium equations for the free nodes for the x-, y- and z-directions arewritten as:

CTUL−1t = fx, (3.26)

CTVL−1t = fy, (3.27)

CTWL−1t = fz. (3.28)

By using the force-to-length ratios for the elements, i.e. the force densities, as de-scription parameters, (3.26)–(3.28) are written as:

CTUq = fx, (3.29)

CTVq = fy, (3.30)

CTWq = fz, (3.31)

where the vector q, of length m, is described as:

q = L−1t. (3.32)

Using (3.23)–(3.25) and the following identities:

Uq = Qu, (3.33)

Vq = Qv, (3.34)

Wq = Qw, (3.35)

equations (3.29)–(3.31) are written as:

CTQCx + CTQCfxf = fx, (3.36)

CTQCy + CTQCfyf = fy, (3.37)

CTQCz + CTQCfzf = fz. (3.38)

By setting D = CTQC and Df = CTQCf , equations (3.36)–(3.38) can be writtenas:

Dx = fx − Dfxf , (3.39)

Dy = fy − Dfyf , (3.40)

Dz = fz − Dfzf . (3.41)

Equations (3.39)–(3.41) are solved using elementary algebra:

x = D−1 (fx − Dfxf ) , (3.42)

y = D−1 (fy − Dfyf ) , (3.43)

z = D−1 (fz − Dfzf ) . (3.44)

Two cases for matrix D can occur [82,105]:

1. Determinant of D = 0The matrix D has full rank and the form of the structure is governed by the

58


values chosen for the force densities. In the case of a prestressed cable net witha given connectivity (i.e. fixed C and Cf ) the number of equilibrium shapesis identical to the number of vectors q. This justifies the use of the forcedensities as description parameters for a cable net. Attention must be paid tothe specific case which occurs when all fixed nodes are coplanar, because thenthe solution will also be planar and without interest (see section 3.4.4).

2. Determinant of D = 0The system can be solved for x only when vectors Dfxf lie is the space spannedby 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 nowbe given. Consider the structure in Figure 3.5 with all fixed nodes in the x–y plane.

x

y

z

0

1

111

1

1

2

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

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

59


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

Cs =

1 −1

1 −1

1 −1

1 −1

1 −1

1 −1

1 −1

1 −1

1 −1

1 −1

1 −1

1 −1

.

︸︷︷︸C

︸︷︷︸Cf

(3.45)

The matrices D and Df describe the equilibrium at the free and fixed nodes, re-spectively, and are written as:

D=

q1 + q3 + q4 0 −q4 0 0

0 q2 + q5 + q6 −q5 0 0

−q4 −q5 q4 + q5 + q8 + q9 −q8 −q9

0 0 −q8 q7 + q8 + q11 0

0 0 −q9 0 q9 + q10 + q12

(3.46)

and

Df =

−q1 −q3 0 0

−q2 0 −q6 0

0 0 0 0

0 −q7 0 −q11

0 0 −q10 −q12

. (3.47)

Figure 3.6 shows the resulting shapes of this structure for different force densityvalues. It is seen that for a single element an increase in the force density relative tothe others results in a contraction of that element. The opposite holds for a decreasein the force density, even more emphasised with negative values.

60


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

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

(d) Interior elements have q = 1 and edgeelements 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. Forexample, the mesh may be irregular and the force distribution unsmooth. There-fore, it is necessary to find a configuration which is in equilibrium and which alsosatisfies additional conditions. These conditions are generally non-linear and so isthe extended force density method. It is preferred to start the non-linear computa-tions using the shape found with the linear method. In contrast to the non-linear

61


displacement method, the number of non-linear equations here is identical to thenumber 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 forcedensities. Since the coordinates also are functions of the force densities, (3.48) iswritten as:

g∗(q) = g(x(q),y(q), z(q),q) = 0. (3.49)

Equation (3.49) is generally non-linear and has to be linearised to be solvable

g∗(q0) +∂g∗(q0)

∂q∆q = 0. (3.50)

Equation (3.50) can be written as:

GT ∆q = r (3.51)

where

GT =∂g∗(q0)

∂q(3.52)

andr = −q∗(q0). (3.53)

Equation (3.51) is similar to the non-linear Newton-Raphson equation used in finiteelement analysis. In many cases the number of constraints is less than the numberof elements, i.e. m > r. This means that (3.51) is under-determined and has m − rlinearly independent solutions. From these solutions a single solution can be chosenby 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:

M = ∆qT ∆q − 2kT(GT ∆q − r

)(3.55)

give the equations

1

2

∂M

∂∆q= ∆q − Gk = 0, (3.56)

1

2

∂M

∂k= GT ∆q − r = 0. (3.57)

The solution to (3.56) and (3.57) is given by (3.62)–(3.64). A damped versionof (3.51) is

GT ∆q = r + ω. (3.58)

62


The minimum principle associated with (3.58) is

∆qT ∆q + ωTPω → 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) isuseful if there are large changes in the force densities because the damped iterationsconverge without oscillation. Another advantage of this approach is that for con-straints which cannot be satisfied the iterations may stop at a shape which fulfilsthe constraints as closely as possible. The Lagrange factors are given by

k = T−1r, (3.62)

where

T =

GTG in case (3.51), (3.54),

GTG + P−1 in case (3.58), (3.59),

GTG + P−1R2 in case (3.60), (3.61).

(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 mayarise. This condition, with r prescribed node distances ls, is written as:

gd = l − ls. (3.66)

The Jacobian matrix for the node distance constraint is written as:

GTd = −L

−1(UCD−1CTU + VCD−1CTV + WCD−1CTW

). (3.67)

This matrix is of dimension r × m.

63


Force constraints

In many cases it is desirable to specify the forces for some elements. The forcecondition, with r prescribed forces tv, is written as:

gf = Lq − tv = Ql − tv (3.68)

The Jacobian matrix for this case is

GTf = L − QL

−1(UCD−1CTU + VCD−1CTV + WCD−1CTW

). (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 dimensionr × r and the dimension of the resulting matrix after the minus sign is r ×m. Thisproblem can be solved if matrix L is expanded with m − r zero columns at thepositions corresponding to the numbers of the unconstrained elements. Then thesubtraction can be performed.

Unstrained length constraints

For fabrication purposes it may be necessary to prescribe the unstrained lengths ofsome elements (see also section 3.4.1). This condition, with r unstrained lengthsluv, is written as:

gu = lu − luv. (3.70)

For a stressed straight cable element i, which satisfies Hooke’s law, the unstrainedlength is computed as:

lui =AEi

AEi + Ti

li. (3.71)

The Jacobian matrix for this unstrained length constraints is

GTu = −L

2

uK−1−L

2

uL−3(

UCD−1CTU

+ VCD−1CTV + WCD−1CTW).

(3.72)

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

above; after −L2

uK−1

is computed the resulting matrix is expanded with m− r zerocolumns. After considering the following limit case:

limAE→∞

GTu = GT

d (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 constraintsgiven 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 ofthe 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 valuesare scaled down (divided by a scale parameter) so that they become of the sameorder as the lengths. No theoretical analysis has been done to justify this scalingprocedure, 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 thissection. Some of the examples that will be presented below are chosen to highlightproblems that need special techniques and some just to show the versatility of theforce density method in finding the shape of different cable net structures. Finally,the applicability of the method for structures including both cables and struts willbe checked. It should be emphasised that the aim of the following examples isto investigate the behaviour of the force density method when applied to differenttypes of cable structures. Numerical stability and convergence rate are in particularstudied. No load bearing or architectural aspects are taken into account. In many ofthe 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. Thisapproach minimises the use of material but the manufacturing can be quitecomplicated.

• Uniform mesh in the unstrained state. From a constructional point of viewthis approach is the best one. As, an error in length of 0.1 % can give rise to anerror 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 thebuilding 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 assignedwith a constant unstrained length.

65


1

2

3

4 5

6

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

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

NodeCoordinates

x y z

1 0.0000 0.0000 0.00002 −6.4897 2.2285 3.00003 −6.9932 9.3859 0.00004 −2.4917 11.7810 4.00005 3.8300 11.5901 0.00006 7.0874 4.2153 5.0000

Considering the magnitude of the prestressing force, the usual rule of thumb is thatthe magnitude should be such that no element goes slack under any load condition.However, according to Leonhardt and Schlaich [63] this rule “causes unacceptablyhigh forces.” They further conclude that “it will never be possible to establishgeneral rules for finding the shape of cable net structures.” This is due to themany factors which affect the shape and load-bearing characteristics of a cable net.Among the factors are: prestress magnitude and distribution, mesh geometry of thenet, edge rigidity (concrete ring or edge cables), angle between net cables and edgecables and stiffness ratios of the members [63].

In general, the magnitude of the prestressing force is determined by the allowabledeformations and fatigue strength of the cables. However, an increase in prestressingforce is not as effective in reducing the deformations as an increase in curvature of thenet [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 allrequirements put on the structure.

A common approach for prestressed cable nets is given in reference 72. In thisapproach, the interior cables are assigned with unstrained length constraints andthe elements in contact with the edges are assigned with force constraints. In thisway a good compromise between load-bearing behaviour and construction ease isreached. A special problem which may arise when using the outlined technique isthat 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 toassign force constraints also to the interior cables. For a three-dimensional structurea configuration that satisfies all the constraints exactly may not be found. But ifthe modified damped version of the non-linear force density method is used, theiterations 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. Theforce density for the edge cables was 1200 kN/m. The interior cables had thefollowing constraints: cable force equal to 200 kN and unstrained length of 1m. No constraints were assigned to the edge cables. The heights of all thefixed 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 theshape shown in Figure 3.8. The net has a nearly square mesh, but some largedistortions occur near the edges.

3. With all the fixed nodes still in the same plane, 20 iterations with the non-linear force density method gave a fairly smooth layout without large anglechanges near the edges. Note that this configuration does not satisfy theconstraints. This step is an intermediate step to get rid of the irregularities inthe edge areas and get a nearly equidistant interior net with a uniform forcedistribution.

4. In this, the last step, all the fixed nodes have their original positions given byTable 3.3. The force constraints for all interior cables assigned with unstrainedlength constraints are removed. Only interior cables connected to the edge ca-bles still have force constraints. The reason for this modification is that for athree-dimensional cable net both force and unstrained length constraints can-not generally be satisfied within the net. However, for the plane configurationin step 3 it is possible to satisfy both constraints. To obtain the final shapeshown 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 analy-ses, is probably due to the highly distorted meshes in some parts of the net. If

67


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

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

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

68

3.4. EXAMPLES

The next example will show that different final configurations will be obtained de-pending on the initial force density values of the edge cables. In the first example thetarget shape is an interior mesh width of 1 metre and a prestressing force of 200 kNfor each of the cables in the edge area. The stiffness of the cables was AE = 1000MN. For both nets the starting value of the force density for each interior cablewas qinterior = 200 kN/m. For the first net, Figure 3.10, the force density in eachof the edge cables was qedge = 5qinterior and for the second net, Figure 3.11, it wasqedge = 50qinterior. The same procedure as for the previous example was used. Toobtain the plane net, with force constraints for all interior cables, 8 iterations wererequired for net 1 and 7 iterations for net 2. This difference is probably due to thefact 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 forceconstraints from interior cables not connected to the edge and fixing the supportnodes in their original positions. For both nets the final shape was obtained withonly 4 iterations. The solutions show that, depending on the starting values of theforce densities in the edge cables, which are unconstrained, different configurationsand force values are obtained.

69


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

(b) Final configuration.

( , , )

( , , )

( , , )

( , , )

.

.

.

.

.

.

.

.

.

.

.

−

000

00

0

0

1

1

2

2

2

3

3

3

3

3

3

3

3

3

4

4

4

44

44

4

5

5

5

5

66

66

6

6

6

7

7

7

7

88

8

8

88

9

9

9

9

9

9

9

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

Figure 3.10: Hyperbolic paraboloid net 1.

70

3.4. EXAMPLES

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

(b) Final configuration.

( , , )

( , , )

( , , )

( , , )

.

.

.

.

.

.

.

.

.

.

.

−

000

00 1

1

1

1

1

1

1

1

1

2

2

2

2

3

3

3

3

3

3

4

4

4

4

5

5

66

66

6

6

6

6

66

6

66

6

6

6

6

7

7

7

7

7

7

7

8

8

9

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

Figure 3.11: Hyperbolic paraboloid net 2.

71


One of Frei Otto’s most famous membrane structures is the star-shaped pavilion overthe 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]. Thisstructure inspired the author in the next example—a star-shaped cable net. The netvertices lie on two circles with radii 14.421 m and 8.165 m, respectively. The heightdifference between the vertices is 4 m. All cables have the axial stiffness AE = 100kN. 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 forthe edge cables. An unstrained length of 1.8 m are assigned to all net cables thatare perpendicular to the ridge cables. Force constraints are assigned to all interiorcables (not ridge or valley cables) and ridge cables. The force values are 200 N and8000 N, respectively. To obtain the final shape, shown in Figure 3.13, 10 iterationswere required.

Figure 3.12: The pavilion over the Cologne Dance Fountain. Reproduced from Ar-chitectural 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 overthe Cologne Dance Fountain.

3.4.2 A large cable net

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

73


net, illustrated in Figure 3.15). The target shape is an unstrained mesh width of3 m and a force of 200 kN in each of the net cables connected to the edges. Thestiffness 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 agood plane initial shape. Nonetheless, a three-dimensional shape, which satisfiesthe constraints, could not be obtained. The reason for this is that the heights ofthe fixed nodes are large relative to the other dimensions of roofs. Thus, an initialequidistant mesh in plane is incompatible with an equidistant mesh of the three-dimensional structure. In some cases it might be possible to draw a slightly finermesh width than the target mesh width in space. However, this problem is noteasily 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 correctnumber of cables. For cable nets with smaller height-to-length ratios an equidistantinitial mesh will work.

Even though the desired shape could not be found, a three-dimensional shape wasobtained by using the force densities of the plane structure in Figure 3.16(b) and theoriginal coordinates for the fixed points. As shown in Figure 3.17 the ‘final’ shapehas a smooth cable arrangement.

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

74

3.4. EXAMPLES

0

1

1

1

11

11

2

2

3

34

4

5

6

7

8

9

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 Fig-ure 3.15.

NodeCoordinates

x y z

1 0.000 52.500 10.0002 10.893 42.188 0.0003 0.000 22.500 15.0004 40.893 42.188 20.0005 44.126 12.362 0.0006 15.910 6.590 0.0007 0.000 0.000 0.0008 −18.106 9.142 0.0009 −22.500 22.500 0.00010 −14.317 56.974 0.00011 −1.874 67.382 0.00012 45.028 71.901 0.00013 67.218 56.574 0.00014 66.015 25.790 0.000

75


(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 cablenet.

3.4.3 Cooling towers

Cooling towers enable a nuclear power plant to be built at any location independentof natural water supplies. The power station capacity is dependent on the coolingcapacity. A high cooling capacity can be generated either by a pair of large or anumber of small cooling towers. Due to influences concerning the air intake andwind 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 thepotential of using a prestressed cable net shell for cooling towers instead of the ordi-nary 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 settlementson bad ground, which would damage a concrete tower. If the pylon is made of steel

77


the safety is further enhanced [110].

The tower with the highest throughput of air per hour is that at Schmehausen-Uentrop. The height of its pylon is 180 m. Nevertheless, besides the economical andstructural questions the one concerning the harmony of the landscape is a strongissue for not building cooling towers at all. The cable net cooling towers do not, likemost other tension structures, fit in the landscape [110].

The force density method can be used to find the shape also for cooling towerswith different mesh configurations. For both the towers shown in Figure 3.18, theradius 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 isAE = 100 MN. The diamond-shaped towers had the following starting values for theforce densities: 200 kN/m for the net cables, 400 kN/m for the hangers and −2700kN/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 bystudying 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, forboth towers, determined by the radius of the upper compression ring and the lengthof the hangers. Node distance constraints were assigned to the segments of thecompression ring and the hangers. The strained lengths of each ring segment andhanger are 1.5607 m and 5.3852 m, respectively. This corresponds to a compressionring with radius of 4 m. For both nets, 5 iterations were required to obtain thedesired shape. It should be noted that, in the present examples no attention hasbeen paid to load bearing capacity or other requirements such as equidistant interiornet.

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, itis necessary to check the applicability of the force density method to structurescomposed of both cables and struts. Consider the simple structure in Figure 3.19.

xy

z

l

(x0 0 z0)

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

79


The governing matrices for this can be written as:

D =

[2qc + qs −qs

−qs 2qc + qs

], (3.75)

Df =

[ −qc −qc

−qc −qc

]. (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)

yT =(

0 0), (3.78)

zT =(

z0 z0

). (3.79)

Hence, the length of the strut is zero irrespective of the values chosen for the forcedensities as long as they do not give a singular matrix D. The elements for thestructure do not have their specified force densities. Solutions ‘out-of-plane’ can beobtained by fixing nodes that are not coplanar to the other nodes. However, then theequilibrium of that node is disturbed since any unbalanced load can be resolved bythe fixed support. One way to avoid this problem is to introduce more elements andconnect one of the ends of the new elements to the free nodes. The other end is fixedat the positions which the free nodes are supposed to have, Figure 3.20. Then, thenew elements are assigned with constraints saying that the lengths of these elementsshould approach zero. If the force densities of the constrained elements do not changeduring the iterations, the element forces will also approach zero. This results in thedesired shape with almost the prescribed element forces. Nevertheless, since thegeometry for a prestressed structure, composed of both cables and struts, in mostcases is specified the suggested approach is useless. Better methods are at hand,e.g. the subspace method by Pellegrino and Calladine [91]. For more complicatedstructures the approach suggested by Mollaert [75], where the compression membersare replaced by external forces, may be used. No calculations have, however, beenperformed to check this approach.

xy

z

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 largespace 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. thenumber 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 mustbe first-order infinitesimal, which they are if and only if there exists a state of self-stress which can impart positive first-order stiffness to every mechanism. Structureswith 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 singlestate of self-stress (s = 1) is obtained by computing the basis for the null-space ofA. Whether this self-stress imparts the structure with first-order stiffness can bechecked by a method given in [20].

Figure 3.21: David Geiger’s cable dome

81


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 config-uration of a framework with known geometry can be obtained. However, for morecomplicated structures, where the geometry is unknown, this approach cannot beused.

(a) Mechanism 1 (b) Mechanism 2

Figure 3.23: Two of the thirteen inextensional mechanisms of the cable dome. Re-drawn 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 oftensegrity structures. It was shown in section 3.4.4 that to avoid a coplanar solutionof a structure composed of cables and struts additional nodes must be fixed. Thisapproach was used by Motro et al. [82] to find the configuration of the structure inFigure 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 afixed node (support) in a framework always can be resolved. Some improvementscould perhaps be provided by the approach suggested by Mollaert [75]. Neverthe-less, it seems that a reliable method for the initial equilibrium problem of tensegritystructures is not available. Examples of some simple structures are shown in Fig-ure 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 elemen-tary self-stressed modules [79], Figure 3.24(a). This example shows the potentialof tensegrity structures as economical solutions for spanning large space. However,these ideas have yet to be realised at a larger scale.

83


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 ini-tial equilibrium configuration of different kinds of cable nets. However, for complexnets, physical models might be needed to construct the initial mesh.

The convergence rate of the non-linear force density method was relatively highwhen only one constraint was assigned to specific elements. When several elementswere assigned with two constraints, the convergence rate was much lower. For somenets it is impossible to satisfy two constraints, e.g. prescribed force and unstrainedlength for interior cables, and for them the iterations converged to a configurationclose to the final shape. It was also found that the starting shape had a considerableeffect on the number of iterations needed for convergence.

For structures composed of both cables and struts it is more difficult to controlthe final solution. For these structures, the most common approach is to specifythe geometry and then solve for the cable forces by analysing the subspaces of theequilibrium matrix A.

84

Chapter 4

Finite cable elements

4.1 Introduction

Today, in many technological fields, the finite element method is the dominatinganalysis tool. In structural analysis all members of a structure, such as plates,beams, bars, cables etc., need to be represented by suitable finite elements. Theseelements are formulated to accurately correspond to the behaviour of the real mem-ber. For beams, a number of finite elements are available for different applications.However, for cable problems special elements are rarely found in commercial finiteelement software, e.g. ABAQUS, and the common approach is to use straight barelements to model the cables. The lack of suitable cable elements stems from thehighly non-linear behaviour of a cable, which hardly can be modelled using the stan-dard Galerkin technique. This non-linearity, which is of the geometric rather thanmaterial 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 loseits stiffness. The steepness of the stiffness transition, shown in Figure 4.2, dependson the relation between the physical properties of the cable. Hence, it is the lowflexural rigidity that makes cable modelling difficult.

In this chapter, a number of finite elements which are applicable to general cableproblems will be presented, but before starting with the elements it is necessary tofirst present the analytical solutions for both the inextensible and extensible cablessubjected to uniformly distributed loads.

4.2 Analytical cable solutions

A perfectly flexible cable supported at its ends and acted upon by a uniform grav-itational force assumes a curve called the catenary1. The equation of the catenarywas first obtained by Leibniz, Huygens and Johann Bernoulli in 1691. They were

1The Latin word for chain

85

CHAPTER 4. FINITE CABLE ELEMENTS

x (m)

z(m

)L0 = 13 m

0

00 1

1

1 22

3

4

5

6

7

89

−

−

−

−

−

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).

L0 = 80 mAE = 51 MN

q0 = 80 N/m

q0 = 800 N/m

Hor

izon

talst

iffne

ss(k

N/m

)

Distance between supports (m)

00

00

00

00

00

00

00

00

1

1

2

2

3

3

4

4

5

55

6

6

7

777777 8888888 9

Figure 4.2: Comparison of the stiffness of an extensible cable and a straight bar withsame properties.

86

4.2. ANALYTICAL CABLE SOLUTIONS

responding to a challenge put out by Johann’s brother Jacob to find the equationof the chain curve [48]. In this section, the analytical solutions to the inextensi-ble 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 theycannot easily be compared. For consistency and clarity, the derivation of the solu-tions will be repeated in this section. Finally, the validity of the assumption of zerobending stiffness of the cable will be discussed.

F ′1

F ′3

F ′4

F ′6

i

j

E, A

q0

z

x

h

l

θ

θ1

θ2

s

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 bemade about the properties of the cable. It is assumed that the cable is perfectlyflexible (EI ≡ 0), inextensible (AE → ∞), free of torsional rigidity and able tosustain only tensile forces. As a result of these assumptions, the cable force istangential to the cable at every point. Consider a small segment of the extensiblecable 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


since only loads acting downwards are considered in this chapter this convention ischosen (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 themagnitude of the gravitational force, not its direction, needs to be inserted.

Tdx

ds0

Tdz

ds0

Tdz

ds0

+d

ds0

(T

dz

ds0

)∆s0

Tdx

ds0

+d

ds0

(T

dx

ds0

)∆s0

limAE→∞ s = s0

q0∆s0

∆s0

Figure 4.4: Segment of an inextensible cable

Returning to Figure 4.4, vertical equilibrium of the cable segment yields

d

ds0

(T

dz

ds0

)= q0, (4.1)

wheredz

ds0

= sin θ. (4.2)

Horizontal equilibrium of the cable segment gives

d

ds0

(T

dx

ds0

)= 0, (4.3)

wheredx

ds0

= cos θ. (4.4)

Integrating (4.3) yields

Tdx

ds0

= H, (4.5)

where H is the horizontal component of cable tension, which is constant along the ca-ble since no horizontal loads are acting. Substituting equation (4.5) to equation (4.1)gives

Hd2z

dx2= q0

ds0

dx. (4.6)

Using the following geometric constraint:(dx

ds0

)2

+

(dz

ds0

)2

= 1, (4.7)

88


equation (4.6) can be written as:

d2z

dx2=

q0

H

[1 +

(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

z =H

q0

[cosh

(q0x

H+ ζ

)− cosh ζ

], (4.9)

in which

ζ = sinh−1

(q0h

2H sinh η

)− η, (4.10)

and

η =q0l

2H. (4.11)

Equation (4.9) is the inextensible catenary equation. Differentiating (4.9) gives theslope as:

dz

dx= sinh

(q0x

H+ ζ

). (4.12)

The length of the inextensible catenary can be computed with the following equa-tion [84]:

L2 =l2

η2sinh2 η + h2. (4.13)

An approximate equation for the inextensible catenary can be obtained for a shallowcable, that is a cable with a small sag-to-span ratio. In this case ds0/dx ≈ 1 andequation (4.6) simplifies to

Hd2z

dx2= q0. (4.14)

Using the same boundary conditions, the solution to (4.14) is the following parabolicequation:

z = ηl

[(x

l

)2

−(x

l

)]+ h

(x

l

). (4.15)

Equation (4.15) is much easier to work with than (4.9). Therefore, the parabolahas been used for many years as the cable equation in the development of approx-imate 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 parabolicequation (4.15) is approximate and the error increases as the sag-to-span ratio in-creases. To obtain an estimation of the difference between the parabola and thecatenary, we consider a cable with level supports, i.e. h = 0. For sag-to-span ra-tios 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 bemore correct, e.g. for a suspension bridge, where the cables sustain a load which isuniformly distributed along their span and much larger than the self-weights of thecables, that is the bridge deck [57].

89


. . . . . . . . . .

0

0000000000000

11

1

1

11

2

22

3

33

4

44

5

555555

6

7

8

9

zcat(x)−zpar(x)

zcat(l/2)

Tcat(x)−Tpar(x)

Tcat(x)

zcat(l/2) = 0.1l

zcat(l/2) = 0.2l

zcat(l/2) = 0.3l

zcat(l/2) = 0.1l

zcat(l/2) = 0.2l

zcat(l/2) = 0.3l

x/l

Per

cent

erro

r

Figure 4.5: Difference between sags and cable forces for the inextensible catenaryand 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 differentialequations of equilibrium of a stretched chain, satisfying Hooke’s law, were derived byJacob 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 inFigure 4.6 is considered. Horizontal and vertical equilibrium of the segment yields

Tdx

ds= H, (4.16)

Tdz

ds= q0s0 − F ′

3. (4.17)

It can be noted that due to the conservation of mass the total weight of the cablesegment is unaffected by the elongation of the cable. For the elastic catenary, thegeometric constraint to be satisfied is(

dx

ds

)2

+

(dz

ds

)2

= 1. (4.18)

It is assumed that the cable material satisfies Hooke’s law

T = AE

(ds

ds0

− 1

), (4.19)

where A is the uniform cross-sectional area in the unstrained profile.

90


H

F ′3

s

T dzds

T dxds

q0s0

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]:

T (s0) =[H2 + (q0s0 − F ′

3)2]1/2

. (4.20)

By noting that dx/ds0 = (dx/ds)(ds/ds0) and dz/ds0 = (dz/ds)(ds/ds0) the para-metric solutions x(s0) and z(s0) are derived, since dx/ds, dz/ds and ds/ds0 are givenby the equations (4.16), (4.17) and (4.19), respectively. The solutions are

x(s0) =Hs0

AE+

H

q0

[sinh−1

(F ′

3

H

)+ sinh−1

(q0s0 − F ′

3

H

)], (4.21)

z(s0) =F ′

3s0

AE

(q0s0

2F ′3

− 1

)+

H

q0

(1 +

(q0s0 − F ′

3

H

)2)1/2

−(

1 +

(F ′

3

H

)2)1/2

.

(4.22)

It is obvious that the cable force increases with height above the lowest point of thecable. If the equations (4.20)–(4.22) are combined we find the following interestingresult [102]:

T (z(s0)) =[(AE + Tb)

2 + 2AEq0z(s0)]1/2 − AE, (4.23)

where Tb is the cable force at the base, i.e. at s0 = 0. Hence, the cable force atany value of z(s0) is independent of the span l. Practically, this result is importantbecause the cable force at the base Tb is an input value in the design of some cablestructures, for example guyed masts [102].

Recently, Russell and Lardner [102] did some experiments with a scale model of aguy cable, for which the elasticity is important. The purpose with their experiments

91


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

4.2.3 Effect of cable bending stiffness

Some authors, e.g. [1, 52], have termed the elastic catenary exact, which under thestated assumptions is correct. However, the elastic catenary is never exact in reality,because the assumptions are violated in many cases. One assumption which is moreimportant than the others, and needs further studies, is that of the zero bendingstiffness. For a real cable, as that in a suspension bridge, the magnitude of theflexural rigidity can be very large. Therefore, the effects of a non-zero flexuralrigidity 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 canbe distinguished:

1. All wires are stuck together. The cable has one neutral axis and the rigidityof the cable is similar to that of a beam.

2. The friction between the wires is zero, making them bend around their ownneutral axis.

The first case represents the upper bound of the bending stiffness while the secondrepresents the lower bound. Several methods to calculate the flexural rigidity forhelically wound cables have been suggested in literature [22]. Following the recom-mendations made by Cardou and Jolicoeur [22], a good estimate of the upper boundof the cable bending stiffness is obtained using the approach by Lanteigne [60]:

(EI)max = E0I0 +n∑

i=1

miAiEi(R2

i /2 + r2i )

2sin3 αi. (4.24)

For the lower bound of the bending stiffness the frictionless model by Costello [29]should be used [22]:

(EI)min = E0I0 +n∑

i=1

miEiIisin αi

(2 + ν cos2 αi). (4.25)

The true bending stiffness is somewhere between the two extremes. Which of theextreme values that is closest to the real weakness depend from case to case. Tofurther complicate things, experiments have shown that the bending stiffness is notconstant; it varies along the length of the cable [22].

92


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

EId2θ

ds20

= H sin θ − q0s0 cos θ, (4.26)

first formulated by Euler, to represent the cable, Figure 4.7. By dividing all forces byEI/S2 and all lengths by S, where S = L/2, equation (4.26) is made dimensionless.Equation (4.26) then becomes

d2θ

d s20

= H sin θ − K3s0 cos θ, (4.27)

where K = Sq1/30 (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 thebending 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 numericalstiffness of (4.27) for large values of K, all classic numerical techniques, such asNewton’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 rangecan only be provided by a sophisticated continuation method [127,128].

Some of the results from Wang and Watson’s studies are presented in Figures 4.8and 4.9. For a > 0.8 the effect of K on the shape can be neglected, Figure 4.8. Forthe maximum curvature shown in Figure 4.9(a), the effect of K is considerable fora < 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 tautcables.

• Ignoring the bending stiffness leads to a softer model, which in dynamic anal-yses gives lower eigenfrequencies.

Still, the assumption of a constant bending stiffness along the length of the cable isnot correct according to the experimental results reported in [22]. As a concludingremark one can say that for the quasi-static behaviour of a taut cable the bendingstiffness is of less significance.

93


a

b

x

z

s0

H

q0S T

θ

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

Although the effects of neglecting the bending stiffness of the cable have been anal-ysed one assumption still remains: the zero torsional rigidity. To check the effects ofthis assumption the cable model would have to be extended with rotational degreesof freedom, but this will not be done here.

In their present form, the analytical solutions are of little use. For a general analysisof cable structures the cable must be represented by a finite element.

a

b

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

94

4.3. LITERATURE REVIEW OF CABLE ELEMENTS

a

dθds0

(0)

(a) The maximum curvature

a

H

(b) The horizontal force

Figure 4.9: Some results for an inextensible cable with bending stiffness. Redrawnand 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 thedeveloped cable elements. Instead, it is focused on different formulations used tosolve the cable problem. The literature review has indicated that generally twoapproaches for the development of finite cable elements are used. The first approachis the use of polynomials to describe the shape and displacement field, which is theordinary approach in the development of finite element. In the second approachanalytical expressions are used, which in a mathematical sense exactly describes thecable 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


• 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 ofcables. A number of element formulations have been presented for geometricallynon-linear analysis, see for example [5, 9, 14, 16]. However, in the end they shouldgive similar results. Straight bar elements, which possess only axial stiffness, oftenprovide suitable representation of highly pretensioned cables, such as those in cablenets and trusses. For slack cables with large curvature, the standard approach is torepresent the cable by a large number of bar elements. This technique is inefficientas the number of degrees of freedom drastically increases. Another drawback is thespurious slope discontinuities occurring at nodes where no concentrated loads act.These discontinuities are due to the straight element assumption and may lead toconvergence problems in the analysis [37,61].

One method often used to model slack cables is the equivalent modulus or stiffnessapproach. In this method, the cable is assigned a certain stiffness that take the cablesag into account. This equivalent stiffness was derived by Ernst [34]. By equatingthe stiffness of a straight bar element to that of a parabolic element, he obtained asimple expression for the equivalent axial stiffness of a slack cable. The equivalentstiffness is a function of the cable force, the self-weight of the cable, the length ofthe element and the axial stiffness of a straight cable. However, for cables with largesag 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 usefewer elements with interpolation functions of higher order. By adding more nodesto the finite element, higher order polynomials for the shape and displacements ofthe element can be defined [61]. Most common are the three and four node el-ements, which use parabolic and cubic interpolation functions, respectively. Thetangent stiffness matrix and equivalent nodal forces are obtained using the isopara-metric formulation. Due to the complex expressions involved in this formulation,the tangent stiffness matrix and equivalent nodal forces have to be found by numer-ical integration. These curved elements give accurate results for cables with smallsag. For larger sag more element must be used. Between element nodes only thedisplacement continuity is enforced [89].

Curved elements with rotational degrees of freedom

The continuity of the slopes can be enforced by adding rotational degrees of freedomto the nodes. Such an element was developed by Gambhir and Batchelor [37]. Cubicpolynomials 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 sag-to-span ratios. To model a cable that has a large curvature more elements have to beused, but, contrary to the other elements presented above, no slope discontinuitieswill occur.

It may be argued that a cable can be modelled using beam elements with a lowbending stiffness. For cables where the bending stiffness is known or assumed thisalternative can provide more reliable results. However, for very flexible cables, alarge number of beam elements with a very low bending stiffness are needed to modelthe 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-nodeisoparametric formulation is used.

• For dynamic analysis the mass matrices are consistent.

The disadvantages are:

• Without rotational degrees of freedom slope discontinuities occur, which mayinduce 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 analyticalformulae to take into account the effect of loading applied along the length of thecable. Three elements, each associated with a certain type of uniformly distributedloading, have been presented in literature. They are:

• The parabolic element for which the load is uniformly distributed along thehorizontal span of the cable.

• The elastic catenary element for which the load is uniformly distributed alongthe unstrained length of the cable.

• The associate catenary element for which the load is uniformly distributedalong the strained length of the cable.

97


Parabolic element

As mentioned earlier the parabola has, because of its simpler form compared tothe catenary, been more frequently used in the analysis of cable structures. Onetype of parabolic element, the three-node isoparametric element, has already beenpresented. Two other formulations, which have much in common, are presented in [1,76]. Both formulations use shallow cable assumptions, which make them applicableonly for cables with small sag-to-span ratios. For both elements, the cable force isobtained 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 waspresented by Peyrot and Goulois [94, 95]. They used the expressions by O’Brien etal. [53, 83, 84] to obtain the flexibility matrix. The tangent stiffness matrix was de-rived by taking the inverse of the flexibility matrix. In reference 94 the plane versionof the element was used in the analysis of transmission lines, while in reference 95three-dimensional structures were analysed. The tangent stiffness given in [95] is be-lieved to be incorrect, because the out-of-plane stiffness, i.e. in the y′-direction, wasset to zero. This was later corrected by Jayaraman and Knudson [52] who demon-strated the capability of the element for a number of examples. Irvine [48] deriveda similar element, but reported that the stiffness matrix should be unsymmetricdue to the geometric non-linearity. However, it is observed that the stiffness matrixin reference 48 in fact is symmetric, which it ought to be due to the conservativenature of the self-weight [3]. The second elastic catenary element was derived byAhmadi-Kashani [1]. This element will be presented later in this chapter togetherwith 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 dis-tributed along the strained length of the element. A load of this type is the snowload. Under the action of the snow load the element stretches and thereby increasesthe available length for the snow flakes to land on; it can be said that the elementbecomes heavier as it elongates. Thus, the total load is dependent on the displace-ments. In this case the loading is non-conservative and therefore the tangent stiffnessmatrix is unsymmetric. This element was derived in reference 1 and the equationsexpressing the nodal forces and the tangent stiffness matrix are utterly complex.

The main advantage with the analytical elements is that only one element is neededto 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 nodiscontinuities 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 foundby iteration.

• The use of trigonometric functions, such as the tangent function, in the for-mulations 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 canbe used for any type of loading along the length of the cable. For this element, afinite segment approach is used to calculate the equivalent nodal forces and tangentstiffness matrix. Using this general element only one element is needed to model acable, 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 bedescribed in the following sections. These include: the straight bar, the elasticparabolic element, the elastic catenary element, and the associate catenary element.They are chosen because they all have six degrees of freedom and closed-form ex-pressions for the equivalent nodal force vectors and tangent stiffness matrices, whichmeans 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 aresomewhat linked to each other by the equivalent stiffness by Ernst [34]. It will beshown 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 structuralanalysis. It has appeared in many forms, e.g. [5, 14, 16], but in the end practicallyall of them give the same results. The formulation used here is from [9], with nodetailed discussion considered necessary. The vector of equivalent nodal forces inthe local coordinate system is written as:

F′T1 =

( −T 0 0 T 0 0). (4.28)

The elastic stiffness sub-matrix in the local coordinate system is given as:

k′E =

AE

L

1 0 00 0 00 0 0

. (4.29)

99


F1

F2

F3

F4

F5

F6

x

y

z

x′

y′

z′

L

i

j

Figure 4.10: Straight bar element in space

Since the analysis of cable structures is non-linear the following geometric stiffnesssub-matrix is needed:

kG =T

L

1 0 00 1 00 0 1

. (4.30)

The total elastic matrix in local coordinate system tangent stiffness is written as:

K′E =

[k′

E −k′E

−k′E k′

E

], (4.31)

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 =

[t1 00 t1

], (4.33)

in which

t1 =

xj − xi

L

yj − yi

L

zj − zi

L

−yj − yi

Lxy

xj − xi

Lxy

0

−(xj − xi)(zj − zi)

LLxy

−(yj − yi)(zj − zi)

LLxy

Lxy

L

, (4.34)

100


whereLxy =

[(xj − xi)

2 + (yj − yi)2]1/2

. (4.35)

Thus, the equivalent nodal forces and tangent stiffness matrix in global coordinatescan now be written as [9]:

F1 = TT1 F′

1, (4.36)

andK1 = TT

1 K′ET1 + KG. (4.37)

The straight bar element can be used in the modelling of slack cables if the axialstiffness AE/L is substituted by the equivalent axial stiffness by Ernst [34], whichis given as: (

AE

L

)eq

=AE

L

(1 +

q20l

2

12T 3AE

) . (4.38)

For cables where the chord length is longer than the unstrained length the cabletension can be computed with the following well-known equation:

T = AEL − L0

L0

. (4.39)

When L ≤ L0 equation (4.39) yields a zero or compressive force. How to computethe 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 alongthe horizontal span of the cable. Some authors [1,76] have derived the nodal forcesand element stiffness matrix for parabolic elements with small sag-to-span ratios. Anelastic parabolic element for large sags has not yet been developed. The formulationfor 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:

F′T2 =

(F ′

1 0 F ′3 F ′

4 0 F ′6

), (4.40)

where

F ′1 = −H, (4.41)

F ′3 = −H

(h

l− q0l

2H

), (4.42)

F ′4 = −H, (4.43)

F ′6 = H

(h

l+

q0l

2H

), (4.44)

101


from which we can obtain the end angles

tan θ1 =h

l− q0l

2H, (4.45)

tan θ2 =h

l+

q0l

2H. (4.46)

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

k′2 = α1

k1 0 k2

0H

α1l0

k2 0 k3

, (4.47)

in which

k1 =L0

Ccos2 β +

(C − L0

C

)sin2 β +

q20l

2

8T 2cos2 β sin2 β, (4.48)

k2 =

(2L0

C− 1 − q2

0l2

8T 2cos2 β

)sin β cos β, (4.49)

k3 =L0

Csin2 β +

(C − L0

C

)cos2 β +

q20l

2

8T 2cos4 β, (4.50)

α1 =AE

C

(1 +

q20l

2

12T 3AE cos2 β

) . (4.51)

The element matrix for the elastic parabolic element, in the local coordinate system,is written as:

K′2 =

[k′

2 −k′2

−k′2 k′

2

]. (4.52)

The transformation matrix for the parabolic element is not the same as T1. Since thex′-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 thez-axis, the transformation matrix for the parabolic element is easily derived; for acable loaded only in the z-direction the projection of the element is straight in thex–y plane. Thus, the rotation matrix is the same as the well-known rotation matrixfor a plane beam element (if the rotational degrees of freedom are substituted bythe translations in the z-direction)

T2 =

[t2 00 t2

], (4.53)

102


F ′1

F ′2

F ′3

F ′4

F ′5

F ′6

i

j

q0

z′

x′

y′h

l

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

where

t2 =

cos φ − sin φ 0sin φ cos φ 0

0 0 1

. (4.54)

This transformation matrix is also compatible with the elastic and associate cate-naries presented in section 4.5.

The global nodal forces and tangent stiffness matrix for the parabolic element cannow be written as:

F2 = TT2 F′

2, (4.55)

andK2 = TT

2 K′2T2. (4.56)

The tangent stiffness matrix is a function of the unknown cable force T . This forcecan be found from the following cubic equation:

F (T ) =1

AET 3 −

(C − L0

C

)T 2 +

(q20l

2

12AEcos4 β

)T − q2

0l2

24cos2 β ≡ 0. (4.57)

It is assumed that the contribution of the third term to the equation is negligible, [1],and the equation is simplified to

F (T ) =1

AET 3 −

(C − L0

C

)T 2 − q2

0l2

24cos2 β ≡ 0. (4.58)

103


F1

F2

F3

F4

F5

F6

x

y

z

x′

y′

z′

φ

q0

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 ofequation (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 withthat of the straight bar element, equation (4.37), it is noted that, apart from thecos2 β-term, α1 is identical to the equivalent axial stiffness given by Ernst, equa-tion (4.38). It is claimed in [1] that the formulation presented here should provide abetter approximation for the taut parabolic element than the bar element with anequivalent axial stiffness. However, with the load intensity q0 and angle β equal tozero, equation (4.47) becomes

k′2 =

L0

C

(AE

C

)0 0

0T

C0

0 0(C − L0)

C

(AE

C

) , (4.59)

where the following relation for the taut parabola is used:

T =H

cos β. (4.60)

Comparison of (4.59) and the sub-matrix of (4.37) indicates that the geometricstiffness 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 inz-direction, which is peculiar. Hence, the elastic parabolic element is not identicalto 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 cableelement, 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 toload uniformly distributed along its length is acceptable only for cables with smallsag-to-span ratios. In addition to this, the ability of the straight bar element to takecompressive forces may induce numerical instability in the analysis of, for example,cable nets. Therefore, a routine which eliminates the element from the model if acompressive force is detected, has to be used in the computer program [14,16]. Theelement 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 forcables subjected to uniformly distributed loads along their lengths. These elementscan be used for any sag-to-span ratio; both very slack and taut cables can be analysedwith a high degree of accuracy.

4.5.1 Elastic catenary element

For this element the load is uniformly distributed along the unstrained length of theelement

q(s0) = q0. (4.61)

The self weight of the cable is a load of this type. Thus, this element gives the sameresults as the elastic catenary equations (4.20)–(4.22) and is therefore called elasticcatenary element. In a mathematical sense this element is exact for a cable loaded byits self weight only. As mentioned in section 4.3, two different formulations have beendeveloped for this element. The first formulation, which was given in [52, 95], wasbased 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 equationsthat express the equivalent nodal forces and tangent stiffness matrix are written interms of H, θ1 and θ2, where θ1 and θ2 are the angles between the horizontal andand the tangent to the cable at the ends, Figure 4.3. The vector of nodal forces forthe elastic catenary shown in Figure 4.3 can be written as:

F′T3 =

(F ′

1 0 F ′3 F ′

4 0 F ′6

), (4.62)

105


where

F ′1 = −H, (4.63)

F ′3 = −H sinh

[cosh−1

(q0L0

2H sinh µ

)− µ

], (4.64)

F ′4 = H, (4.65)

F ′6 = H sinh

[cosh−1

(q0L0

2H sinh µ

)+ µ

], (4.66)

where

µ =q0

2

(l

H− L0

AE

). (4.67)

The element stiffness matrix, in the local coordinate system, for the elastic catenaryelement is written as:

K′3 =

[k′

3 −k′3

−k′3 k′

3

], (4.68)

where

k′3 = α2

q0L0

AE+ sin θ2 − sin θ1 0 cos θ1 − cos θ2

0H

α2l0

cos θ1 − cos θ2 0q0l

H− sin θ2 + sin θ1

, (4.69)

in whichα2 =

q0

4 sinθ2 − θ1

2

[µ cos

θ1 + θ2

2− sin

θ2 − θ1

2

]+

q20L0l

AEH

, (4.70)

where

tan θ1 = sinh

[cosh−1

(q0L0

2H sinh µ

)− µ

], (4.71)

tan θ2 = sinh

[cosh−1

(q0L0

2H sinh µ

)+ µ

]. (4.72)

It should be emphasised that the expressions for the nodal forces and tangent stiff-ness 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 coefficientsand the horizontal forces at the nodes and then change the rows and columns of thetangent stiffness matrix and the force vector.

One particular case, which deserves attention, concerns an elastic catenary elementwhere the intensity of the loading q0 approaches zero. For this case the tangent

106


stiffness matrix (4.69) becomes

limq0→0

k′3 =

AE

Lcos2 θ1 +

T

L0

AE

Lsin θ1 cos θ1

0T

L0

AE

Lsin θ1 cos θ1 0

AE

Lsin2 θ1 +

T

L

. (4.73)

where the following relation between the unstrained length L0 and the strainedlength L for a straight elastic element is used:

L = L0

(1 +

T

AE

). (4.74)

It is noted, that the stiffness sub-matrix in equation (4.73) is identical to that of astraight bar element rotated an angle θ1. Hence, the elastic catenary element can beused to represent weight-less cables if, to avoid numerical instabilities, a very smallvalue for q0 is specified [2].

It is seen that the equivalent nodal forces and the tangent stiffness matrix are func-tions of only one variable, i.e. the horizontal force H. For a given geometric config-uration, H is found by the following equation:

F (H) =4H2

q20

sinh2 µ +h2[

1 +

(q0L0

2AE

)coth µ

]2 − L20 ≡ 0. (4.75)

Equation (4.75) is solved using the well-known Newton-Raphson algorithm for non-linear equations of one variable, where the derivative of (4.75) is given as:

dF

dH=

8H

q20

sinh2 µ− 2l

q0

sinh 2µ− lL0

2AE

[1 +

(q0L0

2AE

)coth µ

]3

(hq0

H sinh µ

)2

. (4.76)

In the case of an inextensible cable element, where AE → ∞, (4.75) simplifies to

L20 =

4H2

q20

sinh2 q0l

2H+ h2, (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 cablewhich is supported at its ends and placed outdoors. One example of such a cableis the conductor in a transmission structure. Assume that outdoors there is a lightwind and after a while it starts to snow. Every snow flake that lands on the cable

107


stretches it. The snow will be uniformly distributed along the strained length ofthe cable. Thus, the total load on the cable is dependent on the deformation of thecable. 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 notconstant. Instead, the load along the unstrained length is expressed as:

q(s0) = q0ds

ds0

, (4.78)

where q0 is the constant load per unit strained length of the cable. An element ofthis type was developed by Ahmadi-Kashani [1], who gave it the name associatecatenary element. The vector of nodal forces for this element is

F′T4 =

(F ′

1 0 F ′3 F ′

4 0 F ′6

), (4.79)

where

F ′1 = −H, (4.80)

F ′3 = −H sinh

[sinh−1

(q0h

2H sinh η

)− η

], (4.81)

F ′4 = H, (4.82)

F ′6 = H sinh

[sinh−1

(q0h

2H sinh η

)+ η

], (4.83)

(4.84)

where η is given by (4.11). From the expressions for the equivalent nodal forces, theend slopes are found as:

tan θ1 = sinh

[sinh−1

(q0h

2H sinh η

)− η

], (4.85)

tan θ2 = sinh

[sinh−1

(q0h

2H sinh η

)+ η

]. (4.86)

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 sameas that of the inextensible catenary. Considering the properties of the associatecatenary element, this result is not a surprise. Despite the elongation of the elementthe load intensity per actual length unit is always q0, which is also the case forthe inextensible catenary. For a suspended elastic catenary the load intensity perstrained length unit is always less than q0. The tangent stiffness matrix of theassociate catenary element in the local coordinate system, as shown in Figure 4.11,

108


is written as:

K′4 = α3

a 0 b −a 0 −b

0H

α3l0 0 − H

α3l0

λ1a − λ2c 0 λ1b + λ2d −λ1a + λ2c 0 −λ1b − λ2d

−a 0 −b a 0 b

0 − H

α3l0 0

H

α3l0

−λ3a + λ4c 0 −λ3b − λ4d − e λ3a − λ4c 0 λ3b + λ4d + e

,

(4.87)where

α3 = −ψ

∆, (4.88)

γ =H

AE, (4.89)

a = sin θ2 − sin θ1 +q0L

AE, (4.90)

b = cos θ1 − cos θ2, (4.91)

c =q0h

H+ ξ tan θ2, (4.92)

d =q0l

H+ ξ, (4.93)

e =q0

α3 sin θ2

, (4.94)

λ1 = tan θ1, (4.95)

λ2 =cos θ1 + γ

cos θ1

, (4.96)

λ3 = tan θ2 − q0h

H sin θ2

, (4.97)

λ4 = λ2sin θ1

sin θ2

, (4.98)

ψ =H

q0 cos θ1 (cos θ1 + γ), (4.99)

∆ = − ψ

H[ξL (cos θ1 + γ) + al + bh] , (4.100)

ξ =(1 − γ2

)(cos θ2 + γ)

[γ

(q0l

H− q0L0

AE

)−

(sin θ2

cos θ2 + γ− sin θ1

cos θ1 + γ

)].

(4.101)

It is seen that the tangent stiffness matrix in (4.87) is not symmetric. This is dueto 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 thetangent stiffness matrix of the associate catenary element are functions of the sole

109


variable H. To obtain H, the following non-linear equation has to be solved, usingthe Newton-Raphson technique outlined in section 4.5.3,

F (H) = L0 − AE

mq0

ln

∣∣∣∣(m + p) (m − n)

(m − p) (m + n)

∣∣∣∣− AEl

H≡ 0, (4.102)

where the derivative of (4.102) is

dF

dH= − H

AEq0m3ln |κ| − 2AE

Hq0m

sgn(κ)

|κ|[(m − n) [(m2 − 1) + r(p − 1)]

(m − p)(m + n)

+(m + p) [(m2 − 1) − t(n − 1)]

(m − p)(m + n)− (m + p)(m − n) [(m2 − 1) − r(p − 1)]

(m − p)2(m + n)

− (m + p)(m − n) [(m2 − 1) + t(n − 1)]

(m − p)(m + n)2

]+

AEl

H2

(4.103)

in which

κ =(m + p)(m − n)

(m − p)(m + n), (4.104)

m =

[1 −

(H

AE

)2]1/2

, (4.105)

n = 1 +H

AEea1 , (4.106)

p = 1 +H

AEeb1 , (4.107)

a1 = sinh−1

(q0h

2H sinh η

)− η, (4.108)

b1 = sinh−1

(q0h

2H sinh η

)+ η, (4.109)

t = 1 + (η coth η − 1)h

L+ η, (4.110)

r = 1 + (η coth η − 1)h

L− η. (4.111)

The strained length L of the associate cable element is given by (4.13), but can becalculated by

L =H

q0

(tan θ2 − tan θ1) . (4.112)

In references 1 and 3 the derivative of (4.102) is not given as in (4.103). When the ex-pression for dF/dH given in these references was used in the Newton-Raphson algo-rithm the iterations did not converge. Therefore, equation (4.102) was checked withthe mathematical symbolic software Maple. The differentiation of (4.102) therebyyielded (4.103).

It should be mentioned that in practice this element may not be very useful. Toexplain the reason for this one can think of a load case where both wind and snow

110


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

4.5.3 Convergence of solution

In order to obtain the equivalent nodal forces and tangent stiffness matrix for theelastic and associate catenary elements, either equation (4.75) or (4.102) is solvedby the classic Newton-Raphson method [3]. Due to the nature of this iterative tech-nique, convergence to the correct solution cannot be guaranteed unless the originalalgorithm is modified. How this modification should be formulated is determinedby the behaviour of the function considered.

The relationship between the horizontal force H and the unstrained length L0, givenby equation (4.102), is shown in Figure 4.13. It can be seen that for an elementwhere the unstrained length L0 is shorter than the chord length C, there is a uniquerelation between H and L0. However, for a slack element where L0 > C, threedifferent 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 extraneoussolutions 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 aslack element the solution may converge to any of the three solutions of which onlyone is correct [1].

By studying the relation between H and L0 shown in Figure 4.13, Ahmadi-Kashanisuggests the following modified Newton-Raphson algorithm

Hi+1 =

Hi − F (Hi)/(

dF

dH

)i

for Hi+1 > 0,

Hi/2 for Hi+1 < 0,

(4.113)

in order to avoid the unwanted solutions. The objective of this modification isto change an initial overestimate of H, which may cause convergence to a wrongsolution, to an initial underestimate of H. This is due to the fact that an initialunderestimate of H will always converge to the correct solution [1].

Although this algorithm will converge to the correct solution, the number of itera-tions needed depends on the starting value for H. Ahmadi-Kashani [1, 3] presentstwo 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 longerthan its chord length. If the small effect of elasticity is ignored, the length of thecable is described by

L20 =

l2

η2sinh2 η + h2. (4.114)

111


COMPRESSION TENSION

H3 H2 H1

L0 = C

L0 > C

L0 (m)

H (kN)

20

30

40

50

60

70

80

90

−105 −104 −103 −102 −101 −100 −10−1 10−1 100 101 102 103 104 105

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

H1

H1

H2

H2

H3

H3

(a) (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:

δ =

(L2

0 − h2

l2

)1/2

. (4.115)

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

η ≈

((120δ − 20)1/2 − 10

)1/2

for 1 < δ ≤ 3.67,

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 forceT is in this case provided by the following equation [1, 3]:

T ≈

b1/32 + a2/3 for b

1/32 > a2,

a2 + b2/2a22 for b

1/32 ≤ a2,

(4.117)

where

a2 = AE

(C − L0

C

)(4.118)

and

b2 = AEq20L

20

24cos2 β. (4.119)

Once T has been obtained a starting value for H is found from the following relation:

H =T l

C. (4.120)

It should be noted that for the particular case δ = 1, which, for example, occurswhen C = l = L0 and h = 0, an initial estimate cannot be found using the suggestedexpressions. In that case a good initial value for H can be found by putting δ = 1.001in (4.116). The use of the modified Newton-Raphson algorithm and the initialestimates 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 im-plementation of those it is necessary to analyse some simple structures using thepresented elements and compare the results to those by other authors. Due to thecomplexity of the expressions for the nodal forces and tangent stiffness matrices,given in the preceding sections, differences between the equations are difficult todistinguish. Therefore, a numerical comparison between the nodal forces and someselected stiffness components will be presented for a single cable suspended at dif-ferent heights, Figure 4.15. Included in this comparison are: all elements in thischapter and the elastic catenary element by Jayaraman and Knudson [52]. In orderto get a verification of the present computer implementation of the elements, thecable in Figure 4.15, which was presented in [3], will be used in the force and stiffnesscomparison studies. Furthermore, two simple examples, which have been studied bymany 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 stiffnessmatrices will be compared. For the comparison, the single suspended cable shown

113


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 stiffnesscomponents will be presented for four different chord lengths C and six angles θ. Acomparison of nodal forces was not made in [3], but for the stiffness components thevalues 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.

C

βi

j

x

z

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


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

1

2

3x

z

500 ft

400 ft

1000 ft

8 kip 100 ft

Figure 4.16: Prestressed cable under self-weight and concentrated load. Redrawnfrom [52].

114


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

Unstrained lengthSegment 1–2 412.8837 ftSegment 2–3 613.0422 ft

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

Investigator Element typeDisplacements (ft)

Vertical Horisontal

Saafan [103] Straight bar −17.954 −2.774O’Brien & Francis [84] Elastic catenary −18.460 −2.820Michalos & Birnstiel [74] Straight bar −17.953 −2.773Jayaraman & Knudson [52] Straight bar −17.951 −2.772Jayaraman & Knudson [52] Elastic catenary −18.458 −2.819Present Elastic parabola −18.377 −2.842Present Elastic catenary −18.457 −2.819Present Associate catenary −18.555 −2.820Present Elastic catenarya −18.457 −2.819

aElastic catenary by Jayaraman and Knudson [52]


Here a slightly more complex structure will be studied. The structure considered isthe prestressed cable net shown in Figure 4.17. This structure was first studied bySaafan [103] and subsequently analysed by West and Kar [129], and Jayaraman andKnudson [52]. Initial data are given in Table 4.3. It seems that these values werechosen 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 to-gether with the results by other authors.

115


100 ft100 ft100 ft

100 ft

100 ft

100 ft

1 2

3 4

f

f

f

f

x

yz

Assumed initial configuration

f = 30 ft

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

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

Description Magnitude

Cross-sectional area of cables 0.227 in2

Equivalent modulus of elasticity 12000 kips/in2

Self weight of cablea 0.0001 kip/ft

Prestressing forceHorizontal members 5.459 kipsInclined members 5.325 kips

Load acting vertically downward at nodes 1, 2, 3 and 4 8.0 kipsaA small self weight is assumed for the analytical cable elements

116


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

Investigator Element typeDisplacements of node 1 (ft)

x-dir. y-dir. z-dir.

Saafan [103] Straight bar −0.1324 −0.1324 −1.4707West & Kar [129] Straight bar −0.1325 −0.1324 −1.4698Jayaraman & Knudson [52] Elastic catenary −0.1300 −0.1319 −1.4643Jayaraman & Knudson [52] Straight bar −0.1322 −0.1322 −1.4707Present Straight bar −0.1322 −0.1322 −1.4707Present Elastic parabola −0.1338 −0.1338 −1.4873Present Elastic catenary −0.1328 −0.1328 −1.4764Present Associate catenary −0.1338 −0.1338 −1.4874Present Elastic catenarya −0.1328 −0.1325 −1.4765

aElastic 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 associateand elastic catenary elements is limited when the cable is slack. H and T forthe two different formulations for the elastic catenaries are exactly the same.

• Like the forces, the stiffness components for the two formulations for the elasticcatenaries give identical results in almost all cases. Small differences occur forC = 80 and 80.7 m. For the associate catenary the differences are morefrequent but still very small. The reason for this might be the errors in someof the equations in [2], which were reported in section 4.5.2 or the chosentolerance for which the computations were discontinued.

• The numerical results above demonstrate that the elastic catenary can simulatea weight-less cable very well.

117


Tab

le4.

5:H

oriz

onta

lfo

rce

Han

dm

axim

umca

ble

forc

eT

(kN

)β

(deg

)E

lem

ent

type

H(6

0.0)

aH

(70.

0)H

(80.

0)H

(80.

7)T

(60.

0)b

T(7

0.0)

T(8

0.0)

T(8

0.7)

0

Stra

ight

bar,

Eeq

––

–44

6.25

0000

––

–44

6.25

0000

Par

abol

ic0.

8485

071.

5120

1627

.918

128

442.

4922

701.

4696

822.

0606

2927

.963

939

442.

4952

13E

last

icca

tena

ry0.

8880

081.

5433

0427

.904

380

446.

3592

161.

8299

072.

2230

1327

.950

214

446.

3620

84A

ssoc

iate

cate

nary

0.88

8027

1.54

3356

28.0

1283

144

2.43

4778

1.82

9950

2.22

3089

28.0

5853

744

2.43

7722

Ela

stic

cate

nary

c0.

8880

081.

5433

0427

.904

380

446.

3592

161.

8299

072.

2230

1327

.950

214

446.

3620

84

5

Stra

ight

bar,

Eeq

––

–44

4.55

1884

––

–44

6.25

0000

Par

abol

ic0.

8388

571.

4948

2227

.670

871

440.

8067

491.

5210

512.

1357

6627

.960

610

442.

6335

91E

last

icca

tena

ry0.

8819

301.

5321

6427

.727

619

444.

6598

441.

9324

612.

3404

0428

.018

741

446.

5006

82A

ssoc

iate

cate

nary

0.88

1949

1.53

2215

27.7

4189

744

0.75

0759

1.93

2508

2.34

0485

28.0

3317

644

2.57

7961

Ela

stic

cate

nary

0.88

1930

1.53

2164

27.7

2761

944

4.65

9844

1.93

2461

2.34

0404

28.0

1874

144

6.50

0682

25

Stra

ight

bar,

Eeq

––

–40

4.43

9850

––

–44

6.25

0000

Par

abol

ic0.

6316

621.

1256

1922

.191

992

401.

0008

591.

5196

212.

1176

4925

.133

374

443.

0756

75E

last

icca

tena

ry0.

7446

921.

2811

8823

.684

242

404.

5208

552.

2893

982.

7038

5226

.854

539

447.

0182

61A

ssoc

iate

cate

nary

0.74

4706

1.28

1227

23.6

8070

540

0.97

3085

2.28

9457

2.70

3945

26.8

5104

044

3.10

9723

Ela

stic

cate

nary

0.74

4692

1.28

1188

23.6

8424

240

4.52

0855

2.28

9398

2.70

3852

26.8

5453

944

7.01

8261

45

Stra

ight

bar,

Eeq

––

–31

5.54

6401

––

–44

6.25

0000

Par

abol

ic0.

2999

960.

5346

0612

.436

112

312.

8293

181.

1870

581.

6155

7218

.404

714

443.

2152

00E

last

icca

tena

ry0.

4834

080.

8077

5415

.659

681

315.

5843

552.

5470

632.

8917

8323

.325

135

447.

4374

26A

ssoc

iate

cate

nary

0.48

3415

0.80

7772

15.6

6789

531

2.82

8976

2.54

7130

2.89

1876

23.3

3726

844

3.55

0704

Ela

stic

cate

nary

0.48

3408

0.80

7754

15.6

5968

131

5.58

4355

2.54

7063

2.89

1783

23.3

2513

544

7.43

7426

65

Stra

ight

bar,

Eeq

––

–18

8.59

3399

––

–44

6.25

0000

Par

abol

ic0.

0640

490.

1141

403.

7419

5818

6.95

9046

0.64

7669

0.84

4192

9.47

1374

443.

0010

83E

last

icca

tena

ry0.

2090

040.

3258

446.

6389

7318

8.60

0989

2.71

2710

2.95

5816

17.2

1789

944

7.72

0119

Ass

ocia

teca

tena

ry0.

2090

070.

3258

496.

6417

3018

6.96

1717

2.71

2782

2.95

5904

17.2

2489

644

3.85

4031

Ela

stic

cate

nary

0.20

9004

0.32

5844

6.63

8973

188.

6009

892.

7127

102.

9558

1617

.217

899

447.

7201

19

85

Stra

ight

bar,

Eeq

––

–38

.893

250

––

–44

6.25

0000

Par

abol

ic0.

0005

620.

0010

010.

0940

2538

.555

887

0.11

1009

0.13

3465

1.21

7796

442.

5193

23E

last

icca

tena

ry0.

0231

310.

0311

250.

4756

4638

.893

159

2.79

5775

2.99

5888

7.20

7200

447.

8447

63A

ssoc

iate

cate

nary

0.02

3133

0.03

1126

0.47

4635

38.5

5592

32.

7957

142.

9959

387.

1961

6044

3.98

9433

Ela

stic

cate

nary

0.02

3131

0.03

1125

0.47

5646

38.8

9315

92.

7957

752.

9958

887.

2072

0044

7.84

4763

aH

oriz

onta

lfo

rce

Hfo

rch

ord

leng

thC

bM

axim

umca

ble

forc

eT

for

chor

dle

ngth

CcE

last

icca

tena

ryby

Jaya

ram

anan

dK

nuds

on[5

2]

118


Tab

le4.

6:St

iffne

ssco

mpo

nent

K44

(kN

/m)

β(d

eg)

Ele

men

tty

peK

(60.

0)a

∆K

(60.

0)b

K(7

0.0)

∆K

(70.

0)K

(80.

0)∆

K(8

0.0)

K(8

0.7)

∆K

(80.

7)

0

Stra

ight

bar,

Eeq

––

––

––

631.

6554

840.

0002

31Par

abol

ic0.

0282

81–

0.08

6374

–21

2.50

0000

–62

6.16

8427

–E

last

icca

tena

ry0.

0419

310.

0000

000.

1067

230.

0000

0021

2.70

9289

0.00

0000

637.

1881

840.

0000

00A

ssoc

iate

cate

nary

0.04

1932

−0.0

0000

20.

1067

30−0

.000

009

214.

2047

62−1

.650

363

626.

2679

17−0

.038

286

Ela

stic

cate

nary

c0.

0419

31–

0.10

6723

–21

2.70

9265

–63

7.18

8184

–

5

Stra

ight

bar,

Eeq

––

––

––

626.

9017

28−0

.041

780

Par

abol

ic0.

0279

60–

0.08

5230

–21

0.88

8461

–62

1.45

8426

–E

last

icca

tena

ry0.

0416

540.

0000

000.

1058

090.

0000

0021

1.09

6344

0.00

0000

632.

3924

28−0

.000

001

Ass

ocia

teca

tena

ry0.

0416

56−0

.000

002

0.10

5816

−0.0

0000

921

1.16

0614

−0.2

1701

162

1.63

4540

−0.0

3700

4E

last

icca

tena

ry0.

0416

54–

0.10

5809

–21

1.09

6325

–63

2.39

2428

–

25

Stra

ight

bar,

Eeq

––

––

––

519.

8717

08−0

.987

519

Par

abol

ic0.

0211

56–

0.06

1449

–17

4.60

0851

–51

5.39

5564

–E

last

icca

tena

ry0.

0355

430.

0000

000.

0858

900.

0000

0017

4.78

1096

0.00

0000

524.

4167

730.

0000

00A

ssoc

iate

cate

nary

0.03

5544

−0.0

0000

10.

0858

95−0

.000

007

174.

6204

530.

0521

2351

6.99

5976

−0.0

1061

1E

last

icca

tena

ry0.

0355

43–

0.08

5890

–17

4.77

7391

–52

4.41

3889

–

45

Stra

ight

bar,

Eeq

––

––

––

318.

6712

86−2

.764

841

Par

abol

ic0.

0106

06–

0.02

6998

–10

6.35

9921

–31

5.94

5260

–E

last

icca

tena

ry0.

0245

900.

0000

000.

0520

440.

0000

0010

6.50

8412

0.00

0000

321.

4385

710.

0000

00A

ssoc

iate

cate

nary

0.02

4591

−0.0

0000

10.

0520

46−0

.000

002

106.

5738

48−0

.111

539

318.

6242

270.

0258

88E

last

icca

tena

ry0.

0245

90–

0.05

2044

–10

6.50

7574

–32

1.43

8570

–

65

Stra

ight

bar,

Eeq

––

––

––

117.

4058

52−4

.542

093

Par

abol

ic0.

0029

77–

0.00

5925

–38

.044

727

–11

6.39

5631

–E

last

icca

tena

ry0.

0138

700.

0000

000.

0230

580.

0000

0038

.179

209

0.00

0000

118.

3945

740.

0000

00A

ssoc

iate

cate

nary

0.01

3870

0.00

0000

0.02

3058

0.00

0000

38.2

0166

4−0

.031

623

117.

9806

250.

0449

86E

last

icca

tena

ry0.

0138

70–

0.02

3058

–38

.179

207

–11

8.39

4531

–

85

Stra

ight

bar,

Eeq

––

––

––

10.2

8824

1−5

.487

735

Par

abol

ic0.

0001

08–

0.00

0168

–1.

6275

59–

10.1

9901

8–

Ela

stic

cate

nary

0.00

5679

0.00

0000

0.00

6847

0.00

0000

1.70

0314

0.00

0000

10.3

3027

40.

0000

00A

ssoc

iate

cate

nary

0.00

5680

−0.0

0000

10.

0068

470.

0000

001.

6931

190.

0070

8210

.280

973

0.04

7929

Ela

stic

cate

nary

0.00

5679

–0.

0068

47–

1.70

0310

–10

.330

293

–

aSt

iffne

ssco

mpo

nent

Kfo

rch

ord

leng

thC

bD

iffer

ence

inst

iffne

ssco

mpo

nent

sde

fined

as:

Kfr

om[3

]su

btra

cted

byK

inth

isst

udy

cE

last

icca

tena

ryby

Jaya

ram

anan

dK

nuds

on[5

2]

119


Tab

le4.

7:St

iffne

ssco

mpo

nent

K66

(kN

/m)

β(d

eg)

Ele

men

tty

peK

(60.

0)a

∆K

(60.

0)b

K(7

0.0)

∆K

(70.

0)K

(80.

0)∆

K(8

0.0)

K(8

0.7)

∆K

(80.

7)

0

Stra

ight

bar,

Eeq

––

––

––

5.52

9740

−5.5

2974

0Par

abol

ic0.

0141

42–

0.02

1600

–0.

3489

77–

5.48

3176

–E

last

icca

tena

ry0.

0228

730.

0190

580.

0277

860.

0789

370.

3491

860.

0000

005.

5311

170.

0000

00A

ssoc

iate

cate

nary

0.02

2874

0.01

9056

0.02

7788

0.07

8932

0.35

0541

−0.0

0135

45.

4824

870.

0486

30E

last

icca

tena

ryc

0.02

2873

–0.

0277

86–

0.34

9186

–5.

5311

17–

5

Stra

ight

bar,

Eeq

––

––

––

10.2

8588

6−5

.491

003

Par

abol

ic0.

0141

41–

0.02

1924

–1.

9587

46–

10.1

9799

3–

Ela

stic

cate

nary

0.02

2912

0.00

0000

0.02

8103

0.00

0000

1.95

7877

0.00

0000

10.3

2922

8−3

.145

456

Ass

ocia

teca

tena

ry0.

0229

130.

0000

000.

0281

05−0

.000

001

1.95

9130

-0.0

0183

910

.201

084

−3.0

5100

4E

last

icca

tena

ry0.

0229

12–

0.02

8103

–1.

9578

77–

10.3

2922

8–

25

Stra

ight

bar,

Eeq

––

––

––

117.

3697

14−4

.510

313

Par

abol

ic0.

0136

90–

0.02

7246

–38

.205

226

–11

6.35

9613

–E

last

icca

tena

ry0.

0235

31−0

.000

100

0.03

3981

0.00

0000

38.1

6626

30.

0000

0011

8.35

7913

−3.1

1397

8A

ssoc

iate

cate

nary

0.02

3532

0.00

0000

0.03

3984

−0.0

0000

138

.134

140

0.01

1373

116.

7315

35−2

.170

844

Ela

stic

cate

nary

0.02

3531

–0.

0339

81–

38.1

6545

7–

118.

3573

10–

45

Stra

ight

bar,

Eeq

––

––

––

318.

6712

86−2

.099

717

Par

abol

ic0.

0106

06–

0.02

6998

–10

6.35

9921

–31

5.94

5260

–E

last

icca

tena

ry0.

0232

520.

0000

000.

0375

130.

0000

0010

6.13

9936

0.00

0000

321.

4358

40−2

.484

440

Ass

ocia

teca

tena

ry0.

0232

530.

0000

000.

0375

16−0

.000

002

106.

2101

22−0

.111

544

318.

6827

22−0

.901

773

Ela

stic

cate

nary

0.02

3252

–0.

0375

13–

106.

1390

98–

321.

4358

40–

65

Stra

ight

bar,

Eeq

––

––

––

520.

0379

04−0

.987

640

Par

abol

ic0.

0046

01–

0.01

3365

–17

4.56

5951

–51

5.56

4451

–E

last

icca

tena

ry0.

0212

160.

0000

000.

0286

040.

0000

0017

3.67

3255

0.00

0000

524.

5796

430.

0000

00A

ssoc

iate

cate

nary

0.02

1218

0.00

0000

0.02

8606

−0.0

0000

217

3.78

2197

−0.1

4514

152

2.94

4803

0.03

5348

Ela

stic

cate

nary

0.02

1216

–0.

0286

04–

173.

6732

45–

524.

5794

84–

85

Stra

ight

bar,

Eeq

––

––

––

627.

2093

67−0

.042

005

Par

abol

ic0.

0002

14–

0.00

0653

–21

0.88

5926

–62

1.77

1224

–E

last

icca

tena

ry0.

0200

190.

0000

000.

0201

490.

0000

0020

1.45

5127

0.00

0000

632.

6970

810.

0000

00A

ssoc

iate

cate

nary

0.02

0021

0.00

0000

0.02

0150

0.00

0000

200.

5413

880.

9048

6763

2.58

6334

0.04

7290

Ela

stic

cate

nary

0.02

0019

–0.

0201

49–

201.

4546

38–

632.

6975

08–

aSt

iffne

ssco

mpo

nent

Kfo

rch

ord

leng

thC

bD

iffer

ence

inst

iffne

ssco

mpo

nent

sde

fined

as:

Kfr

om[3

]su

btra

cted

byK

inth

isst

udy

cE

last

icca

tena

ryby

Jaya

ram

anan

dK

nuds

on[5

2]

120


Tab

le4.

8:St

iffne

ssco

mpo

nent

K46

(kN

/m)

β(d

eg)

Ele

men

tty

peK

(60.

0)a

∆K

(60.

0)b

K(7

0.0)

∆K

(70.

0)K

(80.

0)∆

K(8

0.0)

K(8

0.7)

∆K

(80.

7)

0

Stra

ight

bar,

Eeq

––

––

––

0.00

0000

0.00

0000

Par

abol

ic0.

0000

00–

0.00

0000

–0.

0000

00–

0.00

0000

–E

last

icca

tena

ry0.

0000

000.

0000

000.

0000

000.

0000

000.

0000

040.

0009

490.

0000

000.

0012

50A

s.ca

t.K

46

0.00

0000

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121

Chapter 5

Static analysis

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

5.1 Static analysis of the Scandinavium Arena

The analysis of the Scandinavium Arena was performed using a geometrically non-linear finite element method. Data about the Scandinavium Arena were obtainedfrom construction drawings, articles [55,56], a Master’s thesis [86] and unpublishedmaterial [104]. Further, the results from the calculations were compared to theresults from previous calculations [86], and results from a simplified method recentlypresented at the Department of Structural Engineering at the Royal Institute ofTechnology in Stockholm, Sweden.

5.1.1 The Scandinavium Arena—background

Before the analysis begins some background information on the Scandinavium Arenawill be given. An extensive description can be found in reference 55.

In 1948 an architect competition concerning an indoor sports building in centralGothenburg was announced. It was won by a working group led by the architectPoul Hultberg. In 1962 the preliminary design works started and a final decisionconcerning the realisation of the structure was taken in June 1969. In May 1971the Scandinavium Arena was completed. With space for 14000 spectators it wasat the time the largest covered arena in northern Europe [56]. The arena has beenand 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 inthe middle of the view. Reproduced from [55].

swimming, etc.

Roof structure. The roof consists of a prestressed cable net cladded with thermaland water insulated corrugated steel plates. All cables are anchored in a space-curved reinforced concrete ring. The concrete ring is supported by four stiff pylonsand 40 circular columns. The surface of the roof conforms nearly with a hyperbolicparaboloid. From the centre point of the roof the hanging cables rise 10 m to thetop and the bracing cables fall 4 m to the valley of the ring. The cable spacing isnearly constant and equals to 4 m in both directions. Foundation. The buildingis supported partly by rock and partly by concrete piles. Two of the pylons aresupported by concrete foundations that rest on 115 piles. The large number of pilesneeded is due to the horizontal forces that occur at the connections between the ringbeam and the pylons.

Ring beam. The ring beam has a rectangular cross-section with a width of 3.5 m anda height of 1.2 m. An alternative solution with a ring beam made as a hollow steelbox 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 carrymainly axial forces. The pylons consist of radially oriented concrete walls, with aside length of 3.5 m connected by beams, Figure 5.2. The space between the wallsis approximately 3.5 m wide and filled with ventilation equipment. The pylons arerelatively stiff and can take large horizontal forces. Therefore, the ring beam isdiscontinuous at the top of the pylons which affects the prestressing forces in thecable in the areas between the pylons and the top of the ring beam. The forcesin 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 notconsidered 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. Thisradially oriented cable system also serves as tension rods for the ring beam, whichis discontinuous at the pylons, Figure 5.5.

Original calculations. Preliminary dimensions of the ring beam and the cables wereestimated by a simplified method in which the cable net was represented by a shear-free membrane. The stiffness of the ring beam was taken as the stiffness of a planering beam with the same dimensions as the real one and supported at four stiffpylons [56]. The membrane stresses were approximated by sectionally constantvalues in each direction and the deflection of the roof by polynomials [56,104]. Theunknowns were determined from equations expressing the vertical equilibrium of themembrane and the compatibility between the membrane and the ring beam. Axialforces and bending moments in the ring beam due to snow and wind loads weremodified with respect to the inclination of the ring beam. Accurate values of thetwisting moment could not be obtained by the simplified method.

Thereafter, a more accurate analysis was performed using a mixed finite elementmethod. In that method, the structure was divided into two substructures, thecable net and the ring beam on columns. The two substructures were analysed sep-arately by the stiffness method and then connected by compatibility and equilibriumexpressions. Since the analysis was non-linear, the substructures had to be itera-tively connected. The maximum deflection of the roof surface under full snow loadwas found to be 64 cm. Comparison with the simplified method showed a differenceof at most 10 % in bending moments in the ring [56]. More results from these finiteelement calculations will be presented in section 5.1.5.

125


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 shearstresses. In that case, it is possible to obtain an analytical solution by introducing anumber of assumptions concerning the load distribution and behaviour of the cablesand ring beam, see for example [116]. In this section, a simplified approach will beused to determine the magnitude of the initial prestressing forces in the cables.

According to reference 86, the roof surface of the Scandinavium Arena is describedby the following equation:

z =

fx

( x

R

)2

− fy

( y

R

)2

for x ≤ xp,

fx

( x

R

)2

− fy

( y

R

)2

− 3

4fy

(x − xp

R

)2

for x ≥ xp,(5.1)

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


Fig

ure

5.4:

Con

stru

ctio

ndr

awin

gK

27:1

for

the

Scan

dina

vium

Are

na,sh

owin

gpr

imar

ilyth

edi

men

sion

sof

the

conc

rete

ring

beam

.

127


Fig

ure

5.5:

Con

stru

ctio

ndr

awin

gK

27:3

for

the

Scan

dina

vium

Are

na,sh

owin

gpr

imar

ilyth

eco

nfigu

rati

onof

the

cabl

ene

t(h

oriz

onta

lpr

ojec

tion

).

128


xy

z

R

fx

fy

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

In the original preliminary calculations, it is assumed that the shape of the roofsurface is described by the first expression in (5.1), i.e. the equation of the hyperbolicparaboloid [104]:

z = fx

( x

R

)2

− fy

( y

R

)2

. (5.2)

Using polar coordinates the equation of the ring beam can be described as:xring = R cos θ,

yring = R sin θ,

zring = fx cos2 θ − fy sin2 θ.

(5.3)

The equilibrium equations in x-, y- and z-directions for a membrane free of shearforces, can be stated as [119]:

∂Hx

∂x+ Fx = 0, (5.4)

∂Hy

∂y+ Fy = 0, (5.5)

∂

∂x

(Hx

∂z

∂x

)+

∂

∂y

(Hy

∂z

∂y

)+ Fz = 0, (5.6)

where Hx, Hy are the horizontal components of the prestressing force distribution

(N/m) in x- and y-directions, respectively. Fx, Fy, Fz are the load intensities (N/m2)is the x-, y- and z-directions, respectively. With only vertical loads, (5.4)–(5.6)

129


simplify to:

∂Hx

∂x= 0, (5.7)

∂Hy

∂y= 0, (5.8)

Hx∂2z

∂x2+ Hy

∂2z

∂y2+ Fz = 0. (5.9)

Inserting (5.2) into (5.9) and prescribing Hx = Hy = H0 yields [104]:

H0 = − FzR2

2(fx − fy)(5.10)

Equation (5.10) shows that H0 increases as fy increases. For the Scandinavium

Arena Fz =−0.6 kN/m2 (the combined weight of the cladding and cables), fx = 10

m, fy = 4 m, and R = 54 m. These values gives H0 = 145.8 kN/m. The horizontal

component of the force in a single cable is obtained by multiplying H0 with thecable spacing. For non-equidistant cable spacing, H0 is multiplied with the sum ofhalf the distance between adjacent cables, e.g. for a cable i in the y-direction it is(xi+1 − xi−1)/2.

5.1.3 Finite element model

The Scandinavium Arena will in this section be analysed with finite elements, asa comparison to the original calculations. As a demonstration, the comparisonwill be performed for only one load case: uniformly distributed dead load of −0.6kN/m2 and snow load of −0.75 kN/m2 on the whole roof. Due to symmetry in bothstructure and load case only a quarter of the structure had to be modelled for thiscase, Figure 5.7.

xy

Figure 5.7: One quarter of the Scandinavium Arena.

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

130


conditions: θy′ = 0, θz′ = 0, and ux′ = 0. Cable nodes on x = 0 are prevented tomove in the y-direction and vice versa for the other symmetry line. All columns arepin-jointed to the ground and to the ring beam, while the pylon is rigidly connectedat 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 localmechanisms and numerical instability due to slackening cables. The ring beam andthe pylon were modelled by non-linear three-dimensional beam elements based oncubic shape functions. The co-rotational approach is used to compute the tangentstiffness matrix and internal force vector [90]. The Matlab routines for the beamelement have been developed by Costin Pacoste and have been used to analyseother structures [85]. Since the ring beam is relatively stiff, the results using thenon-linear beam elements were compared with results using linear three-dimensionalbeam elements. The differences in the results were very small. Despite the smalldifference, 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 usedfor these elements. A more accurate analysis, which avoids the short elements butkeeps the cable spacing would require the use of solid elements. However, somedifficulties in connecting the solid elements and the cable elements may arise sincethe solids cannot cope with high concentrated load in the same way as the beamelements. The columns were modelled with straight bar elements. The magnitudesof the prestressing forces are computed according to subsection 5.1.2. The radiallyoriented tension rod, shown in Figure 5.5, was not included in the finite elementmodel of the structure.

As mentioned earlier, the main difference in analysis between cable structures andother structures, such as frames and trusses, is that the initial configuration isgenerally unknown for cable structures. According to Møllmann [76], the followingiterative procedure is used for a cable structure with an elastic boundary structure(arches or beams):

131


1. Assuming that the boundary joints are fixed in the positions corresponding to the un-stressed state of the arch, the shape of the cable net is determined corresponding tocables in vertical planes.

2. The cable forces at the boundary joints obtained from the previous stage are now re-garded as external loads acting on the arch. The arch is then analysed separately forthese forces and for the weight of the arch members.

3. Keeping the boundary joints fixed in the positions obtained from stage 2, the shape ofthe 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 sep-arately until the displacement changes of the joint coordinates of the cable net andboundary are sufficiently small. A somewhat similar procedure was used in theanalysis of the Scandinavium Arena. The only difference is that at step 2 the pre-tensioned cable net is numerically attached to the unstressed ring beam. This meansthat the stiffness of the whole structure is used to compute the displacements of thering beam. Some cables will be unloaded during step 2, but since the ring beam isquite stiff only 3–4 iterations are needed to get a deviation in the horizontal com-ponent of the cable forces of a most 0.5 %. What this error corresponds to in theunstrained length of a cable will now be checked. For a bar the following equationholds:

T = AE∆L

L0

= AEL − L0

L0

. (5.11)

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. Thistolerance can not be reached in practice. However, one should bear in mind that asmall error in unstrained length may give large errors in force; for example, an errorin unstrained length of 0.05 % gives an error in force of 30 % with the cable datagiven 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 non-equidistant mesh. Instead, the following equation expressing the vertical equilibriumis used:

Hix

(zj − zi

xj − xi

− zi − zl

xi − xl

)+ Hiy

(zk − zi

yk − yi

− zi − zm

yi − ym

)+ Fiz = 0. (5.13)

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 cablesare given in Figure 5.10. More data, including coordinates for beam elements, nodeloads, prestressing forces, etc., can be found in Appendix A.

132


(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 sug-gested by Møllmann [76]: (a) the shape of the cable net is obtained byassuming fixed nodes, (b) the cable forces are regarded as external loadson the ring beam (step 1 and 2 are repeated until convergence), and (c)the two structures are connected and the whole structure should now bein equilibrium.

133


x′ y′

z′

00

000

3

45

556

Pylon

E = 32 GPaG = 12.8 GPaA = 4.55 m2

Ix′ = 0.566 m4

Iy′ = 4.645 m4

Iz′ = 17.021 m4

x′ y′

z′

00

00

12

35

Ring beam

E = 32 GPaG = 12.8 GPaA = 4.2 m2

Ix′ = 1.581 m4

Iy′ = 0.504 m4

Iz′ = 4.288 m4

008Column

E = 32 GPaG = 12.8 GPaA = 0.503 m2

Ix′ = 0.040 m4

Iy′ = 0.020 m4

Iz′ = 0.020 m4

Cable

E = 162 GPaA = 2.12 · 10−3 m2

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

134


5.1.4 Calculation results

Some results from the present analysis of the Scandinavium Arena are presentedin Figures 5.11–5.16. The maximum deflection of the net due to the snow loadis 54 cm. Figure 5.11 shows that the contour curves of the net displacement arealmost circular. All cable forces increase from the pretension values due to thesnow load. The forces in the bracing cables increase as a result of the outwardsmovement of the valley of the ring beam. The axial force diagram indicates thatthe pylon significantly reduces the axial force in the lower part of the ring beambetween the pylons. Luckily, the bending moments in these parts are not thatlarge. A discontinuity in the bending moment diagram occurs due to effects fromthe pylon. The ‘height’ of this discontinuity depends on the relation between thebending stiffness of the ring beam and the torsional stiffness of the pylon. Thetwisting moments are much lower than the bending moments. As for the bendingmoment diagram, the pylon affects the twisting moment distribution. Some morecomments concerning the results are given in section 5.1.6, where the present resultsare compared with previous results. Note that in all bending moment diagrams inthis chapter, a positive moment corresponds to tension at the outer side of the ringbeam.

( , )( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

x (m)

y(m

)

0

0

0

0

00

00

00

00

00

0

0

0

0

0

0000000

1

1

11

11

1

1

1

11

22

22

2

2

2

2

2

22

3

3

3

3

3

3

3

33

4

4

4

44

4

4

4

4

44

5

5

55

5

55

5

5

5

5

5

5

55555555

6

6

6

7

7

8

−

−

−

−

−

−

−

−−

−

−

−

−

−

−

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

135


Incr

ease

info

rce

(kN

)

y (m)

000

0

00

0

00

0

00

0

0000000

1

1

1

1

11

2

22

2

2

2

22

3

333333333333

3

3

33

4

4

44

55

5

5

5

555555

666 777777

8

888 9

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

Incr

ease

info

rce

(kN

)

x (m)

0

0

0

0

0

0

0

0

0000000

1

11

1

11

2

22

3

3

33

4

4

44

44

44

44

5

5

5

5

55

555555

66

6

7

7

77

8

8

8

9

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

136


... ...

.

. . . . .

.

Axi

alfo

rce

(MN

)

x (m)y (m)

ColumnPylon

0

0

0

0

0

0

0

0

0

0

00

00

00

0

1

11 11

11 1 11 11 11

1

1

1

1 1

2 2

22

3

33

3 3 3

33

4 4

4

4

5

5

5

5

6

6

6

6

777 8 8 8 8

8

9 9

9

− − − − − −

−− −

− − − − − − −

−

−

−

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

Ben

ding

mom

ent

(kN

m)

x (m)y (m)

ColumnPylon

0

00 0

0

0

000

0000

000

0

000

0000

0000

0

0

0

0

0

0

00

00

00

0

1

1 1

1

1

1

1

1

1

1

1 1

22

2

2

2

2

2

2

22

33

3

3

3

33

4

4

4

4

4

55

55

5

5

5

5

5

5

66

66

6

77

7

7

7

7

7

7

8

8

88

88

8 88

99

9

9

9

9

−

−−

−− −

−

−

−

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

137


Tw

isti

ngm

omen

t(k

Nm

)

x (m)y (m)

ColumnPylon

0

0

0

000

00

000

00

0

00

0000

0

0

0

0

0

0

00

00

00

0

11

1

1

1

1

1 1

2

2

2

22

33 3

33

4

4 4

4

4

5

55

55 55

5

5

5

5

5

5

6

66 6

6

6

7

77 7

8

88

8

9

9− − − − − − −−

−

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

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 thecables with straight bar elements. As mentioned above, the ring beam and the cablenet were analysed separately with the stiffness method and connected by a flexibilityapproach. Two load cases were considered: snow load (−0.75 kN/m2) on the wholeroof and wind load (0.4 kN/m2) on the whole roof. The maximum deflection underfull snow load was 64 cm.

The finite element model in [86] differs from the model analysed in the previoussection. The most important differences between the two structural models are:

• Cable spacing. While the cable spacing according to the drawings was usedin the present analysis of the Scandinavium Arena, a non-equidistant meshof cables was used in [86]. The cables are connected to the joints betweenthe 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 numberof unknowns.

• Contour shape. No information on the x- and y-coordinates of the ring couldbe obtained from any of the references 56, 86 and 104. Nonetheless, the ring

138


appears to be circular in the figures in [86]. In the present analysis the co-ordinates of the ring beam were obtained from the construction drawing inFigure 5.4. It is important to know the ‘exact’shape of the ring beam as itconsiderably affects the distribution of the bending moment, see section 5.2.

• Cable forces. As mentioned above, the cable forces in the upper bracing cablesare lower than those in the lower bracing cables. In addition, the five lowestbracing cables were also post-tensioned by introducing a gap between the ringbeam and the cable net. The final cable forces after the post-tensioning werenot given. Therefore, a uniform cable force distribution was used in the presentanalysis.

• Young’s modulus. No information concerning the modulus of elasticity forthe concrete is given in [86]. According to the drawings, concrete K400 (oldnotation) was used for the ring beam. A characteristic value of 32 GPa for themodulus of elasticity for this concrete class is found in the Swedish buildingcodes. This value was used in the present analysis. The shear modulus is0.4E, 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 facilitatecomparison.

( , )( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

( , )

x

y

00

00

00

0000

00

00

00

00

00

00 00

00

1

1

1

1

1

1

1

2

2

2

2

2

2

3

3

3

3

3

3

4

4

4

4

4

5

5

5

5

5

5

66

6

6

7

77

7

7

7

8

8

8

99

9

9

9

−

−

−

−

−

−

−

−−

−−

−

−

−−

−−

−

−

−

− −

−

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

139


Incr

ease

info

rce

(kN

)

y (m)

0

0

00

00

00

00

00

00

00

00

00

00000000

1

1

11

2

2

22

33

3

33

4

4

4

44

5

5

5

5

5

55

5

55555555

6

6

6

6

6

7

77

77

7

88

8

9

9

9

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

Incr

ease

info

rce

(kN

)

x (m)

0

0

00

00

00

00

00

00

00

0

000000000

111

1

1

111

2

2

2

2

2

22

3

33

4

4

44

4

44

5

55

5

55555555

6

66

6

7

7

8

9

9

−−−−−−

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

140


Ben

ding

mom

ent

(kN

m)

x (m)y (m)

ColumnPylon

0 000

0

000

0000

000

0

000

00000

0

0

0

0

0

0

00

00

00

0

1 1

1

1

1

1

1 1

22

2

22

2

22

3

33

3

3 3

33

44

4

44

4

44

4

4

4

5

5

5

5

5

5

5

666

66

6

6

66

6

777

7

77

88

8

8

8

8

99

9

9

9

99

− −

−

−−

−

−

−

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

Tw

isti

ngm

omen

t(k

Nm

)

x (m)y (m)

ColumnPylon

0

00

000

00

0

00

0000

0

0

0

0

0

0

00

00

00

0

11

11

111

1

1

1

1 1

2

2

2 2

22

3

33

33

33

4

4

44

4

4

5

5

5 5

5

5

5

5

5

5

6

6 6 6 6

6

6

788

8

9

9

− − − − − − −−

−

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

141


5.1.6 Comparison of the results

The differences in midpoint deflection and in-plane displacements are probably dueto a higher Young’s modulus of the concrete in the present analysis and differentinitial cable forces in the analyses. For both analyses, both the cable forces in thex- and y-directions increase. This is due to the outwards displacement of the valleyof the ring beam. A detailed comparison of the increase in the cable forces can notbe done since the intial forces differ much in some parts of the roof. No axial forceswere presented in [86]. However, Figure 5.14 clearly shows that the pylon takes alarge horizontal force. Concerning the twisting moments, they agree very well; thedistributions of the twisting moments are similar and the values do not differ verymuch. In the case with the bending moments, their distributions differs qualitativelyvery much. The maximum magnitude of the bending moments is about the samefor the two analyses.

Thus, it can be concluded that all the results except the bending moment distri-bution have a satisfactory agreement. The disagreement in the bending momentdistribution 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 thecalculations of the Scandinavium Arena and to verify the finite element programanother structure will now be analysed. The structure shown in Figure 5.22 hasbeen analysed by Møllmann [76] and was chosen because of its similarities with theScandinavium Arena.

5.2.1 Description of the structure

The system used for the calculation consists of nine hanging and nine bracing cablesin the net, and 28 straight space beam elements which form the ring beam. The jointsbetween 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 thatvertical 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 thecolumns, joints 8 and 15 are prevented from moving in the y-direction and joint 15cannot move in the x-direction.

142

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

Dimensions in m

x

x

y

y

z

z1

4

5

8

11

12

15

18

19

22

25

26 8.60

8.60

5.40

5.40

52.0052.00

52.0052.00

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 modulusE = 160 GPa. The concrete ring beam is assumed to be elastic with the materialand cross-sectional properties given in Figure 5.23. The cross-section of the ringbeam twists along the perimeter. The longer side of the rectangle is parallel tothe tangent of the roof surface in the direction normal to the boundary. In theinitial state, the cables are in vertical planes and the cable joints are very nearlylocated on a hyperbolic paraboloid, Figure 5.22. The initial state is the equilibriumconfiguration where the cables are pretensioned and the combined weight of thecables and the cladding acts on the net. The initial equilibrium configuration isdetermined according to the procedure described in section 5.1.3. The horizontalcomponents of the cable forces in the initial state are 2600 kN for both the hangingand the bracing cables.

x′y′

z′

2000

1300

Ring beam

E = 20 GPaG = 10 GPa (ν = 0)A = 2.6 m2

Ix′ = 0.877 m4

Iy′ = 0.366 m4

Iz′ = 0.867 m4

Figure 5.23: Cross-sectional and material properties. Drawn dimensions in millime-tres.

143


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 theroof (x > 0).

3. Uniformly distributed dead load on the whole roof plus snow load on half theroof (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 onthe 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 modelledusing the same program and finite elements as for the analysis of the ScandinaviumArena.

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 arcs26–1–4, 5–8–11, 12–15–18 and 19–22–25 were replaced by parabolas with the samerise as the corresponding circular arcs. This was done in order to make the projectedboundary curve conincide with the line of compression for the projected cable forcesin the initial state. As is seen in Figure 5.22, along these arcs the ring beam is in theplane loaded only in one direction (either the x- or the y-direction). To investigatethe effects of a change of the shape of the ring beam arcs, the structure studied byMøllmann 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 arc26–1–4 (−3R/5 ≤ x ≤ 3R/5):

ycircle = (R2 − x2)1/2, (5.14)

yparabola = −5x2

9R+ R, (5.15)

ycosine = R cos

[5x

3Rcos−1

(4

5

)]. (5.16)

The y-coordinates for the different shapes are shown in Figure 5.24.

144


12

3

4

5

Node x (m)y (m)

z (m)Circle Parabola Cosine

2 10.400 50.949 50.844 50.808 −4.8403 20.800 47.659 47.378 47.288 −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 mostinteresting 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.

DescriptionShape of ring beam arcs

Ref. 76Circle Parabola Cosine

Midpoint displacement (m) −1.174 −1.176 −1.177 −1.171Ring beam displacement in 0.189 0.190 0.190 0.188

y-direction at (x = 0, y = R) (m)Ring beam displacement in −0.222 −0.222 −0.222 −0.221

x-direction at (x = R, y = 0) (m)

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

145


changes, but not as much as the bending moment. Nevertheless, the twisting mo-ment is not so important in this case since its maximum value is less than 10 % ofthe maximum value of the bending moment. The reason for the large difference inthe bending moment distribution will be discussed below.

Structures with circular or parabolic shape are optimised for a certain load case (cf. abicycle wheel or a stone arch). If the load case changes, the structure may undergolarge changes in displacements and force distribution. Anyone who has broken aspoke in a well-built bicycle wheel will agree that the structure adjust itself to thenew load case and for some structures the adjustment might be large. The reversemust also hold: change the shape of the structure but keep the load distributionand 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 momentcompared to the normal force. Remember, in the present example, the axial forceN changed very little when the shape of the ring beam was varied. This means thatif the shape of the ring beam is changed, the eccentricities of the axial forces arechanged.

M1

N1

N1

M2

N2

N2

e1

e2

(a)

(b)

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

For the parabolic arcs, the eccentricities in the initial state for nodes 2 and 3 aree2 = 227/12431 = 0.018 m and e3 = 189/13700 = 0.014 m. The coordinate changesfor these nodes are ∆y2 = 0.105 m and ∆y3 = 0.281 m. Hence, if the coordinatechanges are larger than the eccentricities one can expect significant changes of thebending 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


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

σ = −N

bh+

6M

bh2= −N

bh+

6Ne0

bh2= 0 (5.17)

Equation 5.17 yields e0 = h/6. If the eccentricity e is smaller than e0 no tensilestresses will occur. However, the compressive stresses on the other side of the cross-section can be very high and may crush the concrete. It is therefore desirable tohave a small eccentricity in the whole ring beam.

It is concluded that the bending moment distribution is strongly dependent on theshape of the ring beam. The shape of the ring beam in the present analysis of theScandinavium Arena and in the previous analysis was different. Thus, the bendingmoment distributions from the two studies cannot be compared. Also for the exam-ple by Møllmann, the present analysis gave bending moments values that differedquite much from those given in [76], see Figures 5.31 and 5.34. As no coordinatevalues for the ring beam were given in reference [76], a detailed comparison of thebending moments is not possible.

( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( )[ ]

[ ] [ ] [ ] [ ] [ ] [ ] [ ]

[ ]

Cab

lefo

rce

(kN

)

y (m)

0 0

00

00

000

00

000

00

000

00

000000000000

111

1

11111

1

1

11

22222

22

2

2

22

33 33 3

33333333333333

33

333333333333333

33

33333333333333333

3

3

3

33

444444444444

44

55

55

5

5

5

5

55

6666

7 7

7777

77

788

8888

9999

9

−−−−−

Figure 5.26: Forces in the hanging cables under full snow load. The following nota-tion is used: ·=Circle, (·)=Parabola, [·]=Cosine, ·=Møllmann [76].

147


( )

( )( ) ( ) ( ) ( ) ( ) ( )

( )

[ ]

[ ] [ ] [ ] [ ] [ ] [ ] [ ]

[ ]

Cab

lefo

rce

(kN

)

x (m)

0 0

0000

00

0

00

000

00

000

00

000

00

000000000000

11

11

11

11

11

1

1

11

22

222222

2

2

22

33 3333

33333333333333

33

3

3333333333333 33

3

3

33333333333333

3

3

3

33

4 4

44444444444

44

44

55555 5

5

5

5

5

55

66

6

77

8

88

8

99

99

−−−−−

Figure 5.27: Forces in the bracing cables under full snow load. The following nota-tion is used: ·=Circle, (·)=Parabola, [·]=Cosine, ·=Møllmann [76].

( ) ( ) ( ) ( ) ( ) ( )

( )[ ] [ ] [ ] [ ] [ ] [ ]

[ ]

Axi

alfo

rce

(kN

)

x (m)y (m)

00

000

000

0

0

000

000

000

000

0000

000

0000

0

0

0

0

0

0

00

00

00

0

111

11111111111

111

1111111111 1

1 11 1 11

111

1

1

1 1

2222 2222

22 222

2

2

2

22

3

3

33333333

3 3 3

33

44

444

444

4 44 4

4

4

4

4

5

5

5

6

6666

6 6

6

6

6

77

7

77777

7777

7 7 7

7

8

8

88

99

9

9

−−

−−−−−−− −−−−−−−

− − − − − −−

−

−

−

−

−

−

−

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

148


( )( ) ( ) ( ) ( ) ( )

( )[ ] [ ] [ ] [ ] [ ] [ ]

[ ]

Axi

alfo

rce

(kN

)

x (m)y (m)

00

0000

0000

00

0

000

000

000

000

0000

000

000

000

0000

0

0

0

0

0

0

00

00

00

0

11

1

111111111

111111111

11 1 11 1 1

1

1

1

1

1

1

1 1

2

22 2

2

2

22

3

3

333

33

4

44

44 44

44

4

4

4

4

4

4

5

5555555555

55

5 5 5 5

5

5

666

6 6

6

6

6

6

7

7777

77777

777 77 7

88

888

8

8

8

99

9

9 99

99

99

−−

−−−−−−− −−−−−−−

− − − − − −−

−−−−−−−−−

Figure 5.29: Axial force in beam elements under full snow load. The following no-tation is used: ·=Circle, (·)=Parabola, [·]=Cosine, ·=Møllmann [76].

Ben

ding

mom

ent

(kN

m)

x (m)y (m)

0

0

00

000

00

000

00

0

00

000

00

0000

0

0

0

0

0

0

00

00

00

0

11

1

11

11

1

1

1

1

1

1

1 1

22

22

2

2

2

22

3

3

3

33

4

4

4

5

5

5

5

5

5

5

5

66

6

7

7

88

9

9

9

9

9

−−

−

−

−−−−

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

149


Ben

ding

mom

ent

(kN

m)

x (m)y (m)

0

0

0

0

00

00

00

0

00

00

00

000

0

0

0

0

0

0

00

00

00

0

11

1

1

11

1

1

1

1 1

2 22

2

2

2

22

3

3

3

3

33

4

4

4

4

4

5

5

5

5

6

6

777 8

8

8

9

9

9

−

−

−

−−

−

−

−

−

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

Ben

ding

mom

ent

(kN

m)

x (m)y (m)

0

0

0

00

00

00

00

0

00

00

00

00

0000

0

0

0

0

0

0

00

00

00

0

1

1

1 1

2

2 2

2

2

22

3

3

3

3

33

4

4

4

4

4

4

4

4

4

5

5

66

6

6

6

7

7

7

7

8

8

8

9 9

−

−

−

−−−−−

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

150


Ben

ding

mom

ent

(kN

m)

x (m)y (m)

0

0000

000

000

0

000

000

000

0000

0

0

0

0

0

0

00

00

00

0

1

11

1 1

2

2

2

2

22

3

33

33

4

4

4

44

4

4

4

4

4

5

55

5

6

6

6

6

6

6

6

6

7

8

8

8

8

8

8

9

−−

−

−

−

−

−

−

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

Ben

ding

mom

ent

(kN

m)

x (m)y (m)

0

0000

0

000

000

000

0

000

000

000

000

0000

0

0

0

0

0

0

00

00

00

0

1

1

1

1

1 1

222 22 22

2

2

2

2

22

33

3

3

3

33

4

4

4

44

4

4

4

55

5555

5

5

5

5

6

6

6

77

8 8

8

9

−

−

−

−

−−−−−

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

151


Ben

ding

mom

ent

(kN

m)

x (m)y (m)

00

0

0

000

000

000

000

0

000

000

000

000

0000

0

0

0

0

0

0

00

00

00

0

1

1

1

1

1 1

22

2

2

2

22

3

3

33

3

3

3

33

444

4

4

44

4

4

4

4

5

5

5

5

6

66

6

7

8

8

9 99

9

−

−−

−−−−−

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

Tw

isti

ngm

omen

t(k

Nm

)

x (m)y (m)

0

0

00

00

0

00

00

000

0

0

0

0

0

0

00

00

00

0

1

1

1

1

1

1

1

1 1

2

2

2

2

22

3

33

3

3

33

4

4

5

5

5

5

6

6

6

8

9

−

−

−

−

−

−

−

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

152


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

Tw

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

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5.3 Comparison with a simplified method

In reference 116, a simplified method to analyse a prestressed cable net anchoredin an elliptic contour beam is presented. The simplified method is well suited forpreliminary design, where the dimensions of the structure and its structural membersare to be decided. However, in further design work a more accurate finite elementmethod is often needed. In this section the reliability and limitations of the simplifiedmethod will be investigated.

In the simplified method, a number of assumptions concerning the behaviour of thestructure and load cases are imposed to obtain closed-form solutions. The mostimportant assumptions are:

• The cable net is substituted by a continuous shear-free membrane. This canbe considered to be valid for a net with a dense mesh.

• The projection of the ring beam in the horizontal plane is an ellipse.

• The ring beam is assumed to deform like a linear plane beam. Thus, thespace-curved shape of the ring beam is not taken into consideration.

• The ring beam is supported by a continuous wall which permits the ring tomove freely in the horizontal plane.

• In the initial state, the roof is uncladded. This is in contrast to the initialstates of the Scandinavium Arena and Møllmann’s structure.

The complete list, which includes 16 assumptions is given on the pages 34–36 inreference 116.

First, the example by Møllmann will be investigated. The geometry of the roof willdiffer from that shown in Figure 5.22 due to the last assumption given above. Forthe same prestressing intensity H0 in both directions and Fz = 0, equation (5.9) canbe written as:

H0∂2z

∂x2+ H0

∂2z

∂y2= 0. (5.18)

Solving (5.18) yields fx = fy as expected. The heights fx and fy will be different alsofor the Scandinavium Arena as the self-weight is zero in the initial configuration.

5.3.1 Results and discussion

The results for Møllmann’s example and the Scandinavium Arena are presented inTables 5.2 and 5.3. Møllmann’s example were calculated with both circular andparabolic shapes of the contour arcs.

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5.3. COMPARISON WITH A SIMPLIFIED METHOD

Table 5.2: Comparison between the simplified method and the finite element methodfor 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

Description Finite elementSimplified

Circular Parabolic

fx (m) 7.002 7.002 7.000fy (m) 6.998 6.998 7.000Midpoint displacement (m) −1.689 −1.692 −1.634Force in the mid cable 3009 2996 3196

in the x-direction (kN)Force in the mid cable 2974 2991 3107

in the y-direction (kN)Axial force at (x = 0, y = R) (kN) −13538 −13549 −15430Axial force at (x = R, y = 0) (kN) −13464 −13471 −15001Bending moment at (x = 0, y = R) (kN) 2877 4238 5577Bending moment at (x = R, y = 0) (kN) −7154 −5237 −5577

For Møllmann’s example, the midpoint displacement agree very well for both the cir-cular 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. Itshould be mentioned that axial forces given for the finite element calculation are notthe maximum axial forces, cf. Figure 5.29. The maximum axial force for the circularand parabolic shapes are −15785 kN and −15816 kN respectively. Nevertheless, themaximum positive and negative moments occur at the bottom and the top of thering beam. The bending moments for the ring beam with a parabolic shape agreebest with the simplified method. The difference in the positive bending momentmay 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 thebending moments since the structure must be in equilibrium.

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

DescriptionMethod

Finite element Simplified

fx (m) 6.611 6.703fy (m) 6.795 6.703Midpoint displacement (m) −1.999 −1.821Force in the mid cable 891 932

in the x-direction (kN)Force in the mid cable 720 707

in the y-direction (kN)Axial force at (x = 0, y = R) (kN) −10423 −12197Axial force at (x = R, y = 0) (kN) −8797 −9250Bending moment at (x = 0, y = R) (kN) 24985 39729Bending moment at (x = R, y = 0) (kN) −40350 −39729

All comments on Møllmann’s example also hold for the Scandinavium Arena: mid-point deflection and cable forces show good agreement but the differences in axialforces and bending moments are larger.

It should be pointed out that the simplified method assumes an elliptic contourbeam. As the cable net is assumed to behave like a continuous membrane, theforces 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 suchcases these sections should have a parabolic shape otherwise the bending momentdistribution will not be smooth as assumed in the simplified method. Nevertheless,it can be concluded that the simplified method is well suited for preliminary designworks. 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.

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Chapter 6

Conclusions and further research

6.1 Conclusions

The conclusions from the study are divided into three subsections, each having thesame 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 acombined approach. A simple method is first used to obtain a starting shapethat is close to the final shape. Then, a more accurate, iterative method isused to obtain the final shape.

• The linear version of the force density method is very simple to use, but it isnot simple to specify force densities that give the desired force distribution orshape. Therefore, the linear force density method is used to obtain a startingshape for the non-linear force density method.

• The linear force density method is not suitable for a cable net with an orthog-onal projection in the horizontal plane. A better method for this case is thegrid method.

• For a structure composed of only cables, the force density stiffness matrix D ispositive definite and, thus, the solution is unique. However, this matrix can besingular for structures composed of both tension and compression members.Therefore, the force density method, as presented in this thesis, is not suitablefor such structures.

• No suitable method to find the initial equilibrium configuration of pure tenseg-rity structures has been found. For a tensegrity structure with known geome-try, the independent states of prestress and the number of inextensional mech-

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CHAPTER 6. CONCLUSIONS AND FURTHER RESEARCH

anisms are given by the dimensions of the subspaces of the equilibrium matrixA.

6.1.2 Finite cable elements


• The elastic and associate catenary elements are mathematically exact underthe assumption that the cable is perfectly flexible. Only one element percable is necessary to model the static behaviour of both slack and taut cablessubjected to uniformly distributed loads.

• In the case when the self-weight of the cable approaches zero the tangentstiffness matrix of the elastic cable element approaches that of the straightbar element. Hence, both light and heavy cables can be modelled using theelastic catenary element.

• The parabolic element is not mathematically exact but yields extremely accu-rate results for cables with small sag-to-span ratios.

• The tangent stiffness matrices of the analytical cable elements are functions ofthe horizontal force and have to be obtained by iteration. Good starting valuesand a modified Newton-Raphson method ensure that the correct horizontalforce is found with only 3–4 iterations.

• Comparisons between the finite cable elements show, as expected, similar re-sults for taut elements. For slack elements, however, the discrepancies arelarger and are not negligible.

• A real cable has a bending stiffness, but this can be neglected for very tautcables. However, for span-to-length ratios less than 0.8 the bending stiffnesscannot be disregarded.

6.1.3 Static analysis


• In the comparison between the new and old results for the ScandinaviumArena, the maximum vertical net displacement, in-plane ring beam displace-ments, cable forces and twisting moments show good agreement. The differ-ences can be explained by differences in cable spacing, initial cable forces andmodulus of elasticity for the concrete. However, the large discrepancies in thebending moment distribution cannot be explained in this way.

• The bending moment distribution is found to be very sensitive to the shape ofthe elastic ring beam. A circular ring beam yields an irregular bending moment

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6.2. FURTHER RESEARCH

distribution in both the initial and loaded states. A ring beam consisting ofparabolic in-plane arcs is shown to provide a smoother moment distributionwith lower values. This can be explained in that the parabolic shape is closerto 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 momentsand do not seem to be a problem in design.

• The pylons stiffen the ring beam significantly; much larger bending momentsand net displacements are obtained if the pylons are not present. The pylonsalso considerably reduce the axial compressive force in the lower parts of thering beam. With a low compressive force in the concrete ring beam, theimportance of the bending moment increases as large tension stresses mayarise.

• Most of the results from a simplified method, which assumes that the cable netbehaves like a shear-free membrane, agree well with the results from the finiteelement calculations. However, due to the many assumptions, the simplifiedmethod can only be recommended for preliminary calculations. Since thebending moments of the accurate and simplified methods differ more thanthe other quantities, the most important of the assumptions in the simplifiedmethod 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 literaturewhich are shown by the published monographs, e.g. [16, 48, 57, 61, 113], related tothis topic. The author, therefore, does not consider it necessary to do more workon elastic static analyses of these two types. Also their dynamic behaviour in theelastic 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 ratioof structures, and cable structures are very much involved in this trend. Structuraloptimisation leads to non-linear phenomena and parameter sensitivity. For safetyreasons, 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, andmaterial failure at ultimate strength. The first failure type—the elastic instabilityphenomenon—has been extensively analysed for bar, beam and shell structures atthe Department of Structural Engineering at the Royal Institute of Technology,Stockholm. An analysis tool has been developed for these structures, and this toolcan also be used to analyse the stability of cable structures. In the following sectionsthe other aspects related to failure analysis of cable structures are reviewed. In thelast section, a structural concept is discussed.

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Plasticity

The high strength steel used in modern cables exhibits linear stress-strain character-istics over only a portion of its usable strength. Therefore, in ultimate load analysesof general cable structures, the resulting formulations must consider material non-linearity. Under large external loads, some cables will go into the plastic range whilesome will stay in the elastic zone and other will lose their pretension. Hence, thecable elements must handle all three cases. In their present form, the elements inthis thesis are only valid for the elastic and the slack cases and therefore have tobe extended to consider plasticity. In reference 1 the general cable element wasextended to include material non-linearity. Since the general cable element formsthe basis in the derivation of the analytical cable elements, it is anticipated that itis possible to include material non-linearity also in the analytical elements.

Some works have been concerned with the plastic analysis of cable structures. Forexample, Ma et al. [73] used four-node isoparametric cable elements in the plasticdynamic analysis of a saddle-shaped cable net. More recently, Atai and Miodu-chowski [6] derived conditions for the stability of cable structures in the plasticstate and addressed important issues, such as load history, path-dependency of so-lution and unloading of a cable from a plastic to a slack state. From these workssome important conclusions have been drawn and they ought to provide a goodstarting 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 itis necessary to determine the sensitivity of the structure to changes in some of theseparameters. Also different types of imperfections, e.g. non-straight compressionelements or misplaced cables, may have a serious impact on the maximum allowableloads on the structure. Two works that can be referred to in this context arepresented 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 forratios found in practice the cladding significantly contributes to the net stiffness.As a result of the cladding, the prestressing forces could be lowered. However, inthis case the composite action between the cladding and cable net must be assuredthroughout the lifetime of the structure. The connection between the cladding andnet must be able to transmit large shear forces, which may give rise to higher costand thereby decreases the benefits of the cladding-net interaction.

In general, space trusses are regarded as highly redundant structures with the abil-ity to survive the loss of several members without losing overall stability. Thesestructures also have the property to be very sensitive to imperfections. Wada andWang [123] have investigated the effect that different types of imperfections have onthe load-bearing capacity of a double layer space truss. Their investigation included

160


variation of member strength, initial imperfection of member length, and errors inthe assembly process. The conclusions were that the fabrication errors have minorinfluence 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 spacetruss analysed, if two or more members out of 288 members had errors the structurehad a large probability of collapsing. A reduction of the load-bearing capacity mayalso occur in cable structures if cable connections are not assembled in their rightpositions.

Analysis of tensegrity structures

During the last decade new structural principles have been developed, the struc-tural behaviour of which are not yet fully understood. The most interesting of thenew structural principles is that of self-stressed systems, also known as tensegritysystems. Self-stressing is interesting as it allows cable-strut structures to be builtwithout the need for supporting structures to equilibrate the stresses in the initialconfiguration. In addition to this advantage, tensegrity structures have other bene-fits that make them interesting for research; for example, they are lightweight andearthquake resistant. There is also a strong desire from architects to implement thetensegrity principle in buildings because of the pure shapes it produces. Althoughthe 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 applica-tion can be a reality. In this section, analysis aspects of tensegrity structures willbe discussed.

Morphology studies. From the invention of the tensegrity concept in the late 1940’suntil the beginning of the 1990’s most research projects have dealt with the geo-metrical shapes of tensegrity networks. According to Hanaor [46] “It appears thatthe morphological study of tensegrity networks has reached a degree of saturation,whereby the range of conceivable patterns exceeds by far the likely range of appli-cations.” Nevertheless, double-layer tensegrity grids have been developed by joiningtensegrity simplexes [79]. Wang [126] concludes that “the future work will be con-centrated on applying other simplexes in space structures.”

Initial equilibrium configurations. Several different shapes of tensegrity structuresare available from the morphology studies. But, form-finding with geometrical meth-ods does not guarantee mechanical equilibrium and solutions must be checked witha numerical method [81]. Motro et al. [82] applied both the dynamic relaxationmethod and the force density method to solve the initial equilibrium problem. Theyanticipated that the latter method is more suitable for large systems, but that morework has to be done to check its efficiency. One possible way to modify the force den-sity method to apply to tensegrity systems might be to adopt Mollaert’s approachgiven in reference 75. In that approach, the compression and tension membersare separated to ensure a solution out of the plane. Recently, Bruno [15] used amethod based on the minimisation of the potential energy to find the equilibriumconfiguration of simple two-dimensional tensegrity systems. A more mathematical

161


approach was developed by Roth and Whiteley [101] and extended by Connelly andWhiteley [25]. This approach has been used by Burkhardt [17] to analyse quitecomplicated tensegrity domes.

Mechanism elimination. Finding a configuration which satisfies equilibrium is notenough to solve the initial equilibrium problem for a tensegrity system; the internalmechanisms for that configuration must be identified, classified and, if possible, elim-inated by prestressing [80]. The method by Calladine and Pellegrino [20,93] can beused to find the mechanisms and determine the stability of the initial configuration.Recently, Tomka [120] introduced a technique called the method of stabilising forceto analyse the stability of cable structures. The advantage of this method is thatin addition to the qualitative result (stable or unstable), obtained by the methodby Calladine and Pellegrino, quantitative conclusions can also be drawn concerningthe measure of stability.

After an acceptable solution has been found, the behaviour of the tensegrity systemhas to be studied under the effect of external loads. In particular, the stability ofthe self-stressing configurations should be studied [80]. It is of cardinal importanceto know if mechanisms can reappear as a result of external loadings. According toMotro [80] this subject “still remains a fairly open matter.”

Construction. Besides the theoretical aspects mentioned above, the analysed struc-tures must be possible to build. In the present state, there has not been muchapplication of the tensegrity principle in the construction field. The reason for thisis that several fundamental technical problems still need to be solved [46, 80]. Themain problems are:

• suitable prestressing procedures,

• efficient node systems, and

• incorporation of cladding.

Finding a suitable construction and prestressing procedure is quite difficult. Theseprocedures tend to be cumbersome and uneconomic because general tensegrity sys-tems are geometrically complex and lack rigidity prior to prestressing [80]. Theprestressing methods must be reliable and assure the level and permanence of thetension that has been put in. The efficiency of the node systems is very much relatedto the construction procedure and the prime objective is to have compact connec-tions. Concerning the roof covering, a flexible membrane is preferable because of theflexibility of the tensegrity frameworks. It is important that the membrane formsan integral part of the design, as it is not a trivial matter to obtain a correct stressdistribution in the membrane [46]. According to Hanaor [46] “none of the studiescarried 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 lossof the prestress are more severe in these structures than in other cable structures.Cable structures relying on supporting structures or foundations to equilibrate theunbalanced loads have generally a higher safety to failure (if the supporting struc-tures themselves are stable). Tensegrity structures are in many aspects very inter-esting, but also very complex. From the review in the previous section, the followingresearch directions are suggested:

• Analysis of the elastic stability of tensegrity structures under external loads.

Most tensegrity structures are kinematically indeterminate, but in many casesit 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 canreappear under external loads due to the loss of the prestress. First, thestability of an ideal configuration composed of only cables and struts shouldbe analysed. Then, it would be of interest to study what additional effects themembrane 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 boundof loading. Real structures have imperfections of different kinds and in manycases, these greatly affect the stability of the structures. A sensitivity anal-ysis identifies those parameters that significantly affect the behaviour of thestructure. More or less automatic procedures for these analyses are highlydesirable.

• 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 andpath-dependency of the solution. To simplify the analyses certain assumptionsmay be introduced to avoid some of these problems, for example that onlylinearly increasing loads are considered.

163

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174

Appendix A

Numerical data for theScandinavium Arena

Zoom

0

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00

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11

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1

111

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2

2

2

2

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2

2

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3

3

3

3

3

3

3

3

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4

4

4

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5

5

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5555

6

6 6

6

6

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7

7

7

7

7

7

7

7

8

8

8

8

8

8

9

9

9

9

9

Figure A.1: Computational model for the Scandinavium Arena. A circle indicatesa node. Unfilled circles in the cable net are nodes which are loaded.

175

APPENDIX A. NUMERICAL DATA FOR THE SCANDINAVIUM ARENA

0

0

00

0

1

1

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11

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1

1

1

11

1

1

1

11

1

1

1

1

1

1

1

1

1

1

1

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1

1

1

2

2

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3

33

333

3

3

3

3

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3

3

4

4

4

4

5

5

5

5

55

5

6

66

6

6

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7777

7

7

7

8

8

8

8

8888

8

8

8

9

9

9

99

99

9

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

x

y

z

x′

y′

z′

1

2

3

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

176

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

Node Node coordinates (m)number x y z

150 0.000 53.927 −3.948151 2.000 53.927 −3.948152 3.550 53.927 −3.948153 6.000 53.629 −3.785154 10.000 53.143 −3.520155 10.671 53.062 −3.475156 14.000 52.237 −3.031157 18.000 51.229 −2.490158 21.300 49.944 −1.827159 25.202 48.424 −1.044160 26.000 47.942 −0.826161 29.180 46.000 0.048162 29.500 45.811 0.133163 32.436 44.017 0.941164 33.900 42.570 1.452165 34.462 42.000 1.650166 37.194 39.294 2.605167 38.504 38.000 3.064168 41.585 34.000 4.257169 42.500 32.822 4.610170 43.222 31.883 4.892171 44.376 30.000 5.367172 46.600 26.377 6.284173 46.834 26.000 6.379174 47.437 25.015 6.629175 48.817 22.000 7.225176 50.612 18.000 8.001177 51.835 14.000 8.547178 52.733 11.058 8.947179 52.894 10.000 9.020180 53.506 6.000 9.299181 53.855 3.710 9.458182 53.855 2.000 9.458183 53.855 0.000 9.458184 3.550 53.927 −19.500185 10.671 53.062 −19.500186 18.000 51.229 −19.500187 25.202 48.424 −19.500188 32.436 44.017 −19.500189 38.504 38.000 −19.500190 43.222 31.883 −19.500191 47.437 25.015 −19.500192 50.612 18.000 −19.500193 52.733 11.058 −19.500194 53.855 3.710 −19.500

177

APPENDIX A. NUMERICAL DATA FOR THE SCANDINAVIUM ARENA

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

ElementNode number Coordinates for third point (m)

1 2 x y z

275 150 151 0.000 55.659 −4.202276 151 152 3.653 55.659 −4.202277 152 153 6.223 55.347 −4.026278 153 154 10.223 54.863 −3.753279 154 155 11.006 54.769 −3.699280 155 156 14.440 53.920 −3.228281 156 157 18.527 52.893 −2.660282 157 158 21.951 51.562 −1.951283 158 159 25.947 50.010 −1.126284 159 160 26.908 49.425 −0.854285 160 161 30.085 47.491 0.045286 161 161 30.403 47.297 0.135287 162 163 33.473 45.429 1.003288 163 164 35.095 43.819 1.584289 164 165 35.662 43.257 1.787290 165 166 38.390 40.548 2.764291 166 167 39.826 39.124 3.279292 167 168 42.918 35.106 4.510293 168 169 43.828 33.924 4.872294 169 170 44.642 32.868 5.196295 170 171 45.807 30.961 5.689296 171 172 48.024 27.334 6.628297 172 173 48.255 26.955 6.725298 173 174 48.949 25.820 7.019299 174 175 50.334 22.769 7.634300 175 176 52.196 18.601 8.462301 176 177 53.425 14.547 9.026302 177 178 54.370 11.431 9.460303 178 179 54.544 10.286 9.541304 179 180 55.150 6.285 9.825305 180 181 55.521 3.837 9.999306 181 182 55.521 2.000 9.999307 182 183 55.521 0.000 9.999308 152 184 4.550 53.927 0.000309 155 185 11.671 53.062 0.000310 157 186 19.000 51.229 0.000311 159 187 26.202 48.424 0.000312 163 188 30.851 44.235 0.941313 167 189 39.504 38.000 0.000314 170 190 44.222 31.883 0.000315 174 191 48.437 25.015 0.000316 176 192 51.612 18.000 0.000317 178 193 53.733 11.058 0.000318 181 194 54.855 3.710 0.000

178

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

Nodes Load area (m2)1–48 16.049–60 14.661–71 16.072–82 16.483–92 15.893–102 15.4103–111 17.2112–119 18.8120–125 16.2

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

Cable Force (kN)

number x-dir. y-dir.

1 583.20 583.202 583.20 583.203 583.20 583.204 583.20 583.205 583.20 532.176 583.20 583.207 583.20 597.788 583.20 575.919 583.20 561.3310 583.20 626.9411 583.20 685.2612 583.20 590.49

179

List of Bulletins from the Department of Structural Engineering, The Royal Institute ofTechnology, 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., Dampning av 13 m lang stalbalk—“Ullevibalken”. Utprovning av dampmassoroch fastsattning av motbalk samt experimentell bestamning av modformer och forlustfaktorer.Vibration tests of full-scale steel girder to determine optimum passive control. Licentiatavhandling,1993. Bulletin 2.

Silfwerbrand, J., Renovering av asfaltgolv med cementbundna plastmodifierade avjamningsmassor.1993. Bulletin 3.

Norlin, B., Two-Layered Composite Beams with Nonlinear Connectors and Geometry—Tests andTheory. Doctoral Thesis, 1993. Bulletin 4.

Habtezion, T., On the Behaviour of Equilibrium Near Critical States. Licentiate Thesis, 1993.Bulletin 5.

Krus, J., Hallfasthet hos frostnedbruten betong. Licentiatavhandling, 1993. Bulletin 6.

Wiberg, U., Material Characterization and Defect Detection by Quantitative Ultrasonics. DoctoralThesis, 1993. Bulletin 7.

Lidstrom, 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 withoutStirrups. Licentiate Thesis, 1994. Bulletin 9.

Krus, J., Betongbalkars lastkapacitet efter miljobelastning. 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 Pro-cess. Licentiate Thesis, 1994. Bulletin 12.

Zhitao, H., Influence of Web Buckling on Fatigue Life of Thin-Walled Columns. Doctoral Thesis,1994. Bulletin 13.

Kjorling, M., Dynamic response of railway track components. Measurements during train passageand dynamic laboratory loading. Licentiate Thesis, 1995. Bulletin 14.

Yang, L., On Analysis Methods for Reinforced Concrete Structures. Doctoral Thesis, 1995.Bulletin 15.

Petersson, O., Svensk metod for dimensionering av betongvagar. Licentiatavhandling, 1996.Bulletin 16.

Lidstrom, T., Computational Methods for Finite Element Instability Analyses. Doctoral Thesis,1996. Bulletin 17.

Krus, J., Environment- and Function-induced Degradation of Concrete Structures. DoctoralThesis, 1996. Bulletin 18.

Editor, Silfwerbrand, J., Structural Loadings in the 21st Century. Sven Sahlin Workshop, June1996. Proceedings. Bulletin 19.

Ansell, A., Frequency Dependent Matrices for Dynamic Analysis of Frame Type Structures.Licentiate Thesis, 1996. Bulletin 20.

Troive, S., Optimering av atgarder for okad livslangd hos infrastrukturkonstruktioner.Licentiatavhandling, 1996. Bulletin 21.

Karoumi, R., Dynamic Response of Cable-Stayed Bridges Subjected to Moving Vehicles. LicentiateThesis, 1996. Bulletin 22.

Hallgren, M., Punching Shear Capacity of Reinforced High Strength Concrete Slabs. DoctoralThesis, 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—proceed-ings 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 pa bestandigheten. LicentiateThesis, 1997. Bulletin 30.

Karawajczyk, E., Finite Element Approach to the Mechanics of Track-Deck Systems. LicentiateThesis, 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. Bulletin34.

Ikaheimonen, J., Construction Loads on Shores and Stability of Horizontal Formworks. DoctoralThesis, 1997. Bulletin 35.

Racutanu, G., Konstbyggnaders reella livslangd. Licentiatavhandling, 1997. Bulletin 36.

Appelqvist, I., Sammanbyggnad. Datastrukturer och utveckling av ett IT-stod for byggprocessen.Licentiatavhandling, 1997. Bulletin 37.

Alavizadeh-Farhang, A., Plain and Steel Fibre Reinforced Concrete Beams Subjected to CombinedMechanical and Thermal Loading. Licentiate Thesis, 1998. Bulletin 38.

Eriksson, A. and Pacoste, C., (editors), Proceedings of the NSCM-11: Nordic Seminar on Compu-tational 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.

Tarno, I., Effects of Contour Ellipticity upon Structural Behaviour of Hyparform Suspended Roofs.Licentiate Thesis, 1998. Bulletin 42.

Hassanzadeh, G., Betongplattor pa pelare. Forstarkningsmetoder och dimensioneringsmetoder forplattor med icke vidhaftande spannarmering. Licentiatavhandling, 1998. Bulletin 43.

Karoumi, R., Response of Cable-Stayed and Suspension Bridges to Moving Vehicles. Analysismethods and practical modeling techniques. Doctoral Thesis, 1998. Bulletin 44.

Johnson, R., Progression of the Dynamic Properties of Large Suspension Bridges during Construc-tion—A Case Study of the Hoga 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 bepurchased 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 homepagehttp://www.struct.kth.se

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