KUNGL TEKNISKA HÖGSKOLAN
INSTITUTIONEN FÖR BYGGKONSTRUKTION
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Numerical Analyses of Cable
Roof Structures
Gunnar Tibert
TRITA-BKN. Bulletin 46, 1999
ISSN 1103-4270
ISRN KTH/BKN/EX--46--SE
Licentiate Thesis
Numerical Analyses
of
Cable Roof Structures
Gunnar Tibert
Department of Structural Engineering
Royal Institute of Technology
SE-100 44 Stockholm, Sweden
TRITA-BKN. Bulletin 46, 1999 ISSN 1103-4270
ISRN KTH/BKN/B--46--SE Licentiate Thesis
c
Gunnar Tibert 1999
Abstract
This thesis deals with the techniques used in the numerical analysis of cable roof structures. These structures are usually very light and flexible and require analysis methods, which take their non-linear behaviour into account.
An extensive literature survey, concerned with both practical and theoretical aspects of cable roofs, is presented. Some aspects included are: structural systems, roof erection procedures, different cable types and their properties, structural details, roof loads and analysis methods.
As the initial shape of a cable roof depends on the internal force distribution, it cannot be described by simple geometrical models. Special iterative methods, usu-ally not familiar to the structural engineer, have to be utilised in order to find the pretensioned configuration of the roof. The simple force density method is presented in detail and applied to a number of different types of cable roof structures. The method worked well for structures composed of only cables, but not for structures with compression members.
Three analytical finite cable elements are presented. Two elements are mathemat-ically exact and can accurately model both taut and slack cables using only one element per cable. It is shown that the analytical elements are advantageous in modelling cable behaviour.
A static analysis of the Scandinavium Arena in Gothenburg has been performed. The results from this analysis were compared with results from the original design of the same object. It was found that the bending moments in the supporting structure—the concrete ring beam—were very sensitive to its shape. This explained the large discrepancy in the bending moment distribution between the analyses. Results from a simplified method, used for preliminary calculations, agreed well with those of the more accurate finite element calculations, for a studied symmetric load case.
Failure stage analysis of the class of self-stressed cable structures called tensegrity structures has been identified as an area of further research.
Keywords: cable roof structures, loads, form-finding, force density method, finite cable elements, static analysis, the Scandinavium Arena.
Preface
The research work in this thesis was carried out at the Department of Structural Engineering, Structural Mechanics Group, at the Royal Institute of Technology in Stockholm, under the supervision of Professor Anders Eriksson. The work reported in this thesis was financed through a personal grant from KTH.
First of all, I express my gratitude to my supervisor Professor Anders Eriksson for his scientific guidance and valuable advice.
I also thank Docent Costin Pacoste for help with the selection of a suitable beam element for the static analyses.
I would also like to thank Professor Emeritus Alf Samuelsson at Chalmers Uni-versity of Technology in Gothenburg and Mr. Nils Dahlstedt, Technical Manager at the Scandinavium Arena in Gothenburg, for the valuable information about the Scandinavium Arena.
Finally, I am grateful to all people at the Department of Structural Engineering that have helped me in the work with this thesis.
Stockholm, April 1999
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
2.5.2 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
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
4.6.2 Comparison example 2 . . . 114
4.6.3 Comparison example 3 . . . 115
4.6.4 Conclusions from the comparisons . . . 117
5 Static analysis 123 5.1 Static analysis of the Scandinavium Arena . . . 123
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
List of symbols
The following is a list of the most important symbols that appear in the chapters of the thesis. Symbols not included in this list are defined when they first appear. The number refer to the page where the symbol first appear.
A cross-sectional area, 45
A0 cross-sectional area of core wire, 19
Ai cross-sectional area of a wire in layer i, 19
A equilibrium matrix, 52
B compatibility matrix, 54
C length of cable chord, 102
Cp pressure coefficient, 30
C connectivity matrix for free nodes, 56
Cf connectivity matrix for fixed nodes, 56
Cs connectivity matrix for all nodes, 56
d vector of nodal displacements, 54
E Young’s modulus, 45
E0 Young’s modulus of core wire, 19
Ei Young’s modulus of wires in layer i, 19
e vector of bar elongations, 54
F force in global coordinate system, 99
F component of cable force in local coordinate system, 87
f vector of nodal loads, 52
H horizontal component of the cable force T , 88
h projection of cable profile on z-axis, 87
I moment of inertia, 87
Ii moment of inertia of wire in layer (around its own centerline), 92
K tangent stiffness matrix in global coordinate system, 101
K tangent stiffness matrix in local coordinate system, 102
KE elastic stiffness matrix in local coordinate system, 100
KG geometric stiffness matrix in global coordinate system, 50
L length, 43
L0 unstrained length, 45
l projection of cable profile on x-axis, 87
mi number of wires in layer i, 19
n number of wire layers, 19
p wind pressure at time t, 30
q vector of force densities, 58
Ri wire radius in layer i, 19
ri radius of wire centerline helix in layer i, 19
s arc length (elastic cable), 87
s0 arc length (inextensible cable), 88
T cable force, 43
Tb cable force at the base (s0 = 0), 91
T transformation matrix, 100
t vector of bar axial forces, 52
U mean wind velocity, 29
u turbulence component of the wind field in the x-direction, 29
u vector of free x-coordinate differences, 58
V total wind velocity, 30
v turbulence component of the wind field in the y-direction, 29
v vector of free y-coordinate differences, 58
w turbulence component of the wind field in the z-direction, 29
w vector of free z-coordinate differences, 58
xg vector of nodal coordinates, 50
x vector of free x-coordinates, 58
xf vector of fixed x-coordinates, 58
y vector of free y-coordinates, 58
yf vector of fixed y-coordinates, 58
z vector of free z-coordinates, 58
zf vector of fixed z-coordinates, 58
αi angle of wire centerline helix in layer i, 19
β angle between the cable chord and horizontal, 102
ν Poisson’s ratio, 92
φ angle between x- and x-axis, 103
ρ air density, 30
θ angle between tangent to cable profile and x-axis, 87
θm mean wind direction, 30
θv azimuth angle of turbulent wind component v, 30
θw elevation angle of turbulent wind component w, 30
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 memmem-bers is that they can be as light as the tensile strength permits. With new materials, such as high strength steel cables and silicone-coated glass fibre membranes, larger distances can be spanned using the same amount of material as before.
Tensile structures have always fascinated architects and engineers, mainly because of the aesthetic shapes they produce. Despite this, very few tensile structures have been built. Why are they not more common, if they are both economic and beauti-ful? One answers might be that tent-like structures have always been thought of as temporary. Although, a probably more correct answer is that they are more difficult to analyse and construct than traditional buildings. From a structural viewpoint, tension structures have several special features, such as light weight and flexibility. These features require special care in the design; for example, an error in the distri-bution of the pretensioning forces may lead to damage of the cladding under large loads.
If the numerical analysis of building structures is concerned, the finite element method is the dominating tool. In this method, the structural characteristics and external loads are described by matrices and vectors. The sought parameters, e.g. displacements and internal forces, are found by matrix operations.
The first step in the analysis process is the definition of the geometry of the struc-ture, which generally is known a priori. However, this is not the case for tensile structures. Due to the negligible flexural stiffness of cables and membranes, the initial configuration of these structures must be stressed, even if the self-weight is disregarded. Thus, before the analysis of the behaviour of the structure to external loads can be performed, the initial equilibrium configuration must be found. The shape of a tensile structure, which very much depends on the internal forces, also governs the load-bearing capacity of the structure. Therefore, the process of deter-mining the initial equilibrium configuration calls for the designer’s ability to find an optimum compromise between shape, load capacity and constructional require-ments. Several numerical methods, applicable to the initial equilibrium problem,
CHAPTER 1. INTRODUCTION
can be found in literature. Most of these methods are not included in general finite
element programs, e.g. ABAQUS1 and are not familiar to the practising structural
engineer.
After the initial reference configuration has been determined, the structural members have to be described by stiffness matrices and force vectors. Special elements for cables or chains are often not available in commercial finite element programs. The single cable is instead modelled by one or several other elements depending on the sag-to-span ratio. Nevertheless, this approach has problems such as numerical instability of the solution algorithms. To avoid these problems it is desirable to have at hand a robust element which accurately describes the behaviour of both taut and slack cables. Since cable structures in general are very flexible, a geometrically non-linear solution method has to be used. The most common is the Newton-Raphson algorithm, embedded in more or less sophisticated load incrementation techniques. The final step in the analysis process is to define the external loads on the structure. For civil engineering structures there are a number of loads that must be considered: self-weight, vehicles, wind, rain, snow, ice, earthquakes, temperature, etc. The magnitude and distribution of these loads is a constant source of research. The present knowledge in the area is found in the national building codes, which aid the engineers in their decisions. Tensile structures often have irregular shapes and low self-weights which may give rise to unforeseen effects such as very high snow loads and flutter instability due to wind. To ensure the safety of the structure, experimental tests have to be undertaken together with statistical analyses to find the magnitudes of the snow and wind loads.
Even with the right tools, the design of tensile structures will not be straightforward. Each new roof type has its own features. It is no surprise that experience and good engineering judgement are frequent characteristics among famous designers
of tensile structures: Fritz Leonhardt, J¨org Schlaich, Frei Otto, Horst Berger and
David Geiger, to mention a few.
1.1
Aims and scope
The aim of this work is to study the mechanical aspects of cable supported shell type structures (roofs, cooling towers, etc.). The first part of the work is concerned with the basic aspects of these specific types of structures. These include: the principal arrangements of the cables, pretensioning schemes necessary to obtain a prescribed shape, and the practical aspects of connections between and supports for the cables. Further, basic computational models are to be studied. These include, but are not limited to, the methods for analysis and force distribution. The second part is concerned with the formulation of suitable finite elements, which take into account the non-linear behaviour of a cable. Basic analyses are performed, and verified. Theoretical and numerical studies are included in this thesis, but no experimental
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.
1.2. GENERAL STRUCTURE OF THESIS
work is conducted. In the discussed methods, only elastic structures and static loads are considered.
All of the numerical calculations in this thesis has been coded in the Matlab2
lan-guage. Some expressions have been derived using the computer algebra package
Maple3
1.2
General structure of thesis
To get an overview of the structure of this thesis, the contents of the chapters are presented below.
In Chapter 2, an extensive literature study on cable roof structures is presented. The study includes both practical and theoretical aspects of cable roofs. Among the practical aspects are: different structural systems, roof erection processes, 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 equilibrium configuration of cable structures and structures of mixed type (cables and stiff struc-tural members) are presented. One of the methods—the force density method—is further described in detail and coded. A variety of examples are analysed to 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 based on the conventional approach, i.e. using shape functions, are discussed. Further, four finite cable elements are presented: the straight bar, the parabolic cable, the elastic catenary and the associate catenary. The internal force vectors and tangent stiffness matrices are presented and the elements are compared by some simple examples. In Chapter 5, an existing cable roof structure is analysed by a finite element program written by the author. The structure is the Scandinavium Arena in Gothenburg, which consists of a pretensioned cable net anchored in a nearly circular concrete ring beam. The results of the calculations are compared to the results from the initial design process and the reasons for discrepancies in the results are discussed. In addition, results from a simplified method, mainly used in preliminary design, are compared to the results from the finite element calculations.
In Chapter 6, the conclusions of this study are stated and directions for further research are suggested.
In Appendix A, data used in the analysis of the Scandinavium Arena are presented.
2Matlab is a registered trademark of The MathWorks Inc., 24 Prime Park Way, Natick, MA
01760-1500, U.S.A. Internet: http://www.mathworks.com.
3Maple is registered trademark of Waterloo Maple Inc., 57 Erb Street W., Waterloo, Ontario,
Chapter 2
Literature review
2.1
Historical review
The first structures regarded as cable roofs are four pavilions with hanging roofs built by the Russian engineer V. G. Shookhov at an exhibition in Nizjny-Novgorod in 1896. During the 1930’s a small number of roof structures of moderate sizes were built in the U.S.A. and Europe, but none of major importance [88].
A big step in the development of suspended roofs came in 1950 when Matthew Nowicki designed the State Fair Arena, Figure 2.1, at Raleigh, North Carolina, USA. Sadly, Nowicki died that same year in a plane crash, but his work continued through the architect William Henry Deitrick and civil engineer Fred Severud and in 1953 the arena was completed [88].
(a) (b)
Figure 2.1: The State Fair Arena at Raleigh, North Carolina, U.S.A., (a) Repro-duced from [10], (b) Structural system, reproRepro-duced from [16].
On an exchange visit to the U.S.A. in 1950 a German student in architecture, named Frei Otto, previewed the drawings for the Raleigh Arena in the New York office of Fred Severud. Otto saw that the project embodied many of his own ideas about how
CHAPTER 2. LITERATURE REVIEW
to construct with minimal amount of material. After graduation in 1952 Otto began a systematic investigation of suspended roofs. The investigation was presented in
the doctoral thesis Das H¨angende Dach (The Suspended Roof), which was the first
comprehensive documentation on the subject [31].
The thesis caught the attention of Peter Stromeyer of Stromeyer Company, one of the largest tent manufacturers in the world. Stromeyer contacted Otto and they began a fruitful cooperation. In 1957 Otto formed the Development Centre for Lightweight Construction in Berlin in order to further increase the research about tensile architecture. In 1964 he incorporated the centre into the Institute of Light Surface Structures at the University of Stuttgart. A massive research work was 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 modern tensile architecture. He was involved in the construction of many of the large tensile structures during the mid 1960’s to early 1970’s. Among these was the first large cable net structure with fabric cladding, the German pavilion at the World’s fair in Montreal 1967 [10], Figure 2.2.
Figure 2.2: The German pavilion at the World’s fair in Montreal 1967. Reproduced from [10].
Another pioneering structure at this time was the large low-profile super elliptic air-supported roof, Figure 2.3, with a membrane attached to a diagonal cable net. This structure was designed by David Geiger for the United States pavilion at the World’s fair in Osaka 1970 [10].
2.1. HISTORICAL REVIEW
Figure 2.3: David Geiger’s air-supported roof at the World’s fair in Osaka 1970. Reproduced from [10].
Following the success of the cable net in Montreal, Frei Otto produced a very elegant development of the Montreal design for the Olympic Stadium in Munich 1972 [87], Figure 2.4.
Figure 2.4: The Olympic Stadium in Munich. Reproduced from [31].
After the Osaka dome, several air-supported domes were built around the world, because they provided the economically best alternative to span large distances. However, several of them deflated due to heavy snow loads or compressor failure. To overcome the deflation problems, David Geiger invented another structure 1986— the cable dome. The cable dome concept was inspired by the tensegrity principle by
CHAPTER 2. LITERATURE REVIEW
Kenneth Snelson and Richard Buckminster Fuller. The first two domes were built for the 1988 Seoul Olympics. The latest and biggest, the Georgia Dome, was built in Atlanta 1994, Figure 2.5.
Figure 2.5: The Georgia Dome in Atlanta, U.S.A., during construction. Reproduced from [10].
In the year 2000, the Millennium Experience will be held in Greenwich, London, close to the Greenwich meridian. This exhibition will be held inside the largest dome ever. The diameter of the dome is 364 m and the height is 50 m [70].
Figure 2.6: The Millennium Dome in London, during construction. Reproduced from the cover of Bautechnik, Vol. 75, No. 11, 1998.
2.2. STRUCTURAL SYSTEMS
2.2
Structural systems
In this section, the traditional cable roof systems are presented together with a fairly new one. Each roof type is presented very briefly, but references to more information are given.
Cable roofs can be divided into different categories depending upon the criterion used for classification. In accordance with how the cables are used, they can be classified as [57]:
1. cable supported roofs, and 2. cable suspended roofs.
Cable supported roofs are, in principle, similar to cable-stayed bridges. In these roofs, the cables only provide additional support for elements which themselves carry a major part of the load. In cable suspended roofs the load is carried directly by the cable system [57]. The cable supported roofs, for which the cables only have an auxiliary function, will not be considered in this thesis.
The cable suspended roofs may be divided into the following categories [16]: 1. simply suspended cables,
2. pretensioned cable trusses, and 3. pretensioned cable nets.
Further, the pretensioned cable structures may be either self-balancing or non-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 increase the stiffness of a roof. In the following, the traditional types of cable suspended roofs will be described. In each category, the structural systems are illustrated by a limited number of figures. More examples can be found in the references 16 and 57.
CHAPTER 2. LITERATURE REVIEW
2.2.1
Simply suspended cable structures
The first type is the simply suspended roof. These roofs have a single curvature or a positive double curvature (like a bowl). Systems of this type have no stiffness. To reduce the displacements caused by any form of applied loading, the roof cladding must either be very heavy or stiff. Concrete is perhaps, therefore, the most suitable roofing material; both prefabricated slabs and in situ cast concrete are used [16]. One can compare this roof type to a suspension bridge which is stiffened by the bridge deck.
Figure 2.7: Simply suspended roof.
The simply suspended roofs, which are stiffened by the cladding material, will not be considered in this thesis; only systems, which can be pretensioned before the cladding is applied, will be analysed.
2.2.2
Pretensioned cable trusses
Lighter and stiffer systems than the simply suspended systems can be achieved if a second set of cables with reverse curvature is connected to the hanging cables. A ca-ble truss is quite stiff if it is tensioned to a level which ensures that both the hanging and the bracing cables remain in tension under any load case. The basic cable truss configurations with vertical connecting elements are shown in Figure 2.8. Another system is the cable truss with diagonal ties, Figure 2.9, developed by the Swedish engineer David Jawerth. Generally, the cable trusses with the vertical connecting elements are structural mechanisms if they are considered as pin-jointed trusses. However, the cable truss with diagonal connecting elements is statically indetermi-nate [76]. Therefore, the Jawerth truss is stiffer than the other trusses [51]. The cable trusses may be arranged in parallel planes, Figure 2.8, or radially, Figure 2.10. A parallel Jawerth system was used in the Johanneshov Ice Stadium in Stockholm, Sweden. An extensive study of cable trusses is presented in reference 76.
2.2. STRUCTURAL SYSTEMS
(a) Convex cable truss structure with corrugated metal roof decking
(b) Concave cable truss structure with corrugated metal roof decking
(c) Convex-concave cable truss structure with corrugated metal roof decking
CHAPTER 2. LITERATURE REVIEW
Figure 2.9: Cable truss system developed by the Swedish engineer David Jawerth. Redrawn from [16].
Figure 2.10: Radial cable truss structure—Lev Zetlin’s cable roof over the 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, this surface must be anticlastic (saddle-shaped) at every point [16].
The stiffness of a cable net depends mainly on: the curvature of the net surface and the level of pretension. In order to minimise the material in both the net and the supporting structure it is advantageous to have a surface with a relatively small
2.2. STRUCTURAL SYSTEMS
radius [31]. The prestressing force must not be exceeded by any type of loading, or the cables become slack. Areas of slack cables may damage the cladding or give rise to the destructive phenomenon of flutter [16].
Cable nets can be designed with masts and edge cables or with stiff boundaries such as beams, arches and rings, Figure 2.11. The first type is generally less stiff and more complicated to construct than the latter ones. The cladding is often placed directly on the cable network [16, 57].
Figure 2.11: A cable net structure—the Scandinavium Arena in Gothenburg, Swe-den.
More information on different types of cable nets and their properties can be found in references 16 and 76.
2.2.4
Tensegrity systems
A pure tensegrity structure is a structure composed of a relatively few non-touching, straight compression members which are suspended in a net of tension members. The key feature of such structures is that they are self-stressed; no external devices to equilibrate the cable forces are needed. Tensegrity structures can be said to have been invented by Kenneth Snelson and Richard Buckminster Fuller [99].
Several new systems, based on the tensegrity principle, have been developed in recent years. The most well-known of these new systems is the cable dome concept by David Geiger. The cable dome, Figure 2.12, is not a pure tensegrity structure since a curved ring beam is used to balance the cable forces. The cable dome concept was developed as an economically equal alternative to air-supported structures, which several times have deflated due to mechanical failure or excessive snow loads. Today, at least eight cable domes exist, but more will surely be build.
CHAPTER 2. LITERATURE REVIEW
Figure 2.12: The cable dome by David Geiger. Redrawn from [99].
More about tensegrity structures and some other fairly new structural concepts can be found in reference 99. Tensegrity structures are further discussed in Chapter 3.
2.3
Roof erection
Theoretically, cable and membrane structures can be given infinitely many different shapes. In practice, the number of configurations is restricted, as shown in the previous section. Of course, with the use of scaffolding, more shapes would be possible, but this eliminates some of the benefits with cable structures. Generally, cable roof construction has two advantages over other forms of roof construction: very little or no scaffolding is required, and quite rapid erection process. However, these advantages do not indicate that the erection of a cable structure is an easy task. Every step of the erection process must be computer controlled to avoid over-stressing of the supporting structures. It is important that the contractor responsible for the roof erection fully understands and exactly follows the erection plan specified by the designer [57].
Cable trusses have the easiest erection process among cable roof structures and may be assembled in the air or on the ground, of which the latter is to prefer. After being assembled on the ground the truss is hoisted into position and prestressed by applying tension at both ends simultaneously. Depending upon whether the centre of the truss needs to be lifted or lowered, tension is applied to the suspension or prestressing cable. Since the cable trusses usually do not interact with each other before the cladding is applied several trusses can be erected and prestressed simultaneously to reduce overall construction time [16]. Double layer grids with radial symmetry can be erected in the same fashion as trusses, but care has to be taken to not over-stress the compression ring as bending moments are introduced when just a few trusses are tensioned. An erection scheme for radial double layer grids is given in [57].
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 methods can be used [16]. With use of computational methods (see Chapter 3) the shape of a net and the corresponding cable forces can be very accurately determined. To obtain the computed shape of the real net a high dimensional accuracy in fabrication is required. Small errors in unstrained length may cause large errors in force. One method to achieve a high accuracy at a minimal cost is to specify a net with a square unstrained mesh and uniform cable stresses. In this way, the same cable dimension can be used for the whole net (not the edges) and the equidistant cable-to-cable connections can be factory-assembled. But, even with a high accuracy some ad-justment can be necessary after the net has been lifted into its final position. This adjustment is possible if tensioning devices (turnbuckles) are incorporated at the ends of the cables [63].
For tensegrity structures, suitable methods for prestressing large tensegrity frame-works have not yet been developed. This is probably the main reason for the very few large tensegrity structures today. Nonetheless, one exception is the cable domes by David Geiger. These domes were developed as an economically equal alternative to air-supported structures, but without the risk for deflation. From the economic point of view, it was necessary that the domes could be constructed without any scaffolding. Figure 2.13 shows the steps of erection of a cable dome [99]. For a complicated structure, the best way to plan the erection steps is to build a physical model of the structure [99].
2.4
Cables
The main load carrying element in the structures considered in this thesis is the cable. In structural applications, the term ‘cable’ means a flexible tension member. However, a cable can have different configurations. In this section, the different types of cables and their characteristics will be examined.
2.4.1
Products
The smallest single tension element in a cable is the steel wire. It is usually circular in cross section, with a diameter between 3 and 8 mm, but may be non-circular in locked coil strands. The wire has a high tensile strength that is obtained by cold drawing or cold rolling [35].
A spiral strand, Figure 2.15(a), is an assembly of wires laid helically around a central straight wire. An assembly of a small number of wires is called a spiral strand and if there are more than three layers it is called a spiral bridge strand. The successive layers are usually wound in opposite directions to get equal torsional stiffness in both directions [35].
CHAPTER 2. LITERATURE REVIEW (a) (b) (c) (d) (e)
Figure 2.13: Cable dome erection steps: (a) The upper cables are hung, then (b) a hoop and struts are hung, raising the inverted ridge cables. More hoops and struts, (c)–(e), further raise and tension the ridge cables. Drawn from data given in [39].
2.4. CABLES
Figure 2.14: A wire rope and its parts. Reproduced from [29].
Locked coil strands, Figure 2.15(b), are similar to spiral strands but are composed of two types of helically laid wires: the core is a spiral strand with helically laid circular wires, and at least the two outer layers have wires with a special Z-shape that interlock with each other. The special shaped wires together with the self-compacting effect of the helical arrangement result in a tight surface and a low void ratio in the outer layers [42].
A wire rope, Figure 2.15(c), is an assembly of spiral strands that are laid helically around a central core that can be a strand or another independent wire rope. The spiral strands are usually laid in the opposite direction to the wires in the spiral strands (ordinary lay) but can be laid in the other direction (Lang’s lay) [35]. The helical lay of wires increases the flexibility of the cable, but reduces the strength and stiffness. In some applications, particularly suspended bridges, a high strength and stiffness are more important than flexibility and therefore products with parallel wires and strands have become popular. Other benefits with parallel strands and wires are easier handling and transportation. In the last decade, parallel strands have also found use in roof construction. Parallel strand systems were used as the hoop and ridge cables in the cable domes by David Geiger [100]. The development of parallel products over recent years is reviewed by Walton [125].
CHAPTER 2. LITERATURE REVIEW
(a) Bridge strand (b) Locked coil bridge
strand
(c) Wire rope
Figure 2.15: Cable cross sections. Reproduced from [16].
2.4.2
Strength
For the wires commonly used in cables the guaranteed minimum tensile strength is 1570 MPa and the guaranteed 0.2 % proof stress is 1180 MPa. The limit of proportionality (0.01 % proof stress), which is the absolute upper limit for the stresses in the service condition, has a value of 65–70 % of the tensile strength. When deciding the allowable stress level, the effect of relaxation must also be taken into account. Tests on steel wires show that the relaxation accelerates when the wire is held under a permanent stress larger than 50 % of the tensile strength. Therefore, the stresses from permanent loads should not exceed 45 % of the tensile strength [42].
2.4.3
Axial stiffness
For structural applications, the perhaps most important property of the cable, be-sides the tensile strength, is the axial stiffness. As mentioned above, a cable with helical wires has a lower stiffness than a cable with straight wires. In the design of cable structures, it is of cardinal importance to know the axial stiffness of the cables since the force distribution in, for example, a cable net is very sensitive to small errors in the cable properties (modulus and length). Several methods have been developed to calculate the axial stiffness of a helically wound cable, see for example reference 22. Most of these methods are based on contact theories and are, thus, very complex. Nevertheless, two simple and accurate methods have been found and will be presented in this section. For explanation of the notations see Figure 2.16.
2.4. CABLES
Figure 2.16: Geometry of a helically wound cable. Reproduced from [58].
In reference 58, Kumar and Cochran linearised the equations from Costello [29] and arrived at the following closed-form expression for the axial stiffness:
(AE)eq = A0E0+ n i=1 miAiEisin αi 1− (1 + ν)picos2αi , (2.1) where Ai = πR2i, (2.2) and pi = 1− νRi ri cos2αi 1− R 2 i 4r2 i 1− ν 1 + ν cos 2αi cos2αi . (2.3)
Kumar and Cochran [58] also provide an even simpler expression for the equivalent axial stiffness (AE)eq = A0E0+ n i=1 miAiEisin3αi 1− ν cot2αi . (2.4)
Another method, in which the wire layers are modelled as orthotropic sheets, has been developed by Raoof [98]. The method is quite cumbersome and not suitable for practical design work. Therefore, Raoof derived a simplified procedure, by
para-metric studies of different cable dimensions. In that, Hruska’s1 parameter is first
computed as: κ = n i=1 miAi AT cos4αi, (2.5) in which AT = n i=1 miAi. (2.6)
CHAPTER 2. LITERATURE REVIEW
The relation between the full-slip modulus (no friction between wires) and steel modulus is computed as:
Efull-slip
Es
=−0.26442 − 2.004046κ + 6.5735κ2− 3.3068κ3, (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 and
Eno-slip, respectively, with AT. The method by Raoof, equations (2.5)–(2.8), are included in Eurocode 3 [35]. Raoof’s method has been checked against experimental results in [47]. It was found that the experimental moduli of newly manufactured cables agreed well with the theoretical full-slip modulus.
The methods presented above have also been compared to other analytical methods and it is concluded that the overall elastic behaviour of helical cables under axial loading is well represented by the available mechanical models. Which model one should use is dependent on the size of the cable [22]. The expressions by Kumar and Cochran is expected to yield higher accuracy for cables with few layers of wires, while the opposite can be said about the method by Raoof, [22].
Although any of the methods presented above gives an accurate value for the axial stiffness, a newly assembled cable does not have a linear stress-strain relationship. The reason is that a cable consists of moving parts which need a run-in period. In order to obtain a more linear behaviour the cable is, after the assembly, loaded repetitively to a load well within the elastic limit of the wire material. The purpose of this procedure is to remove the constructional stress and, thereby, obtain an almost linear stress-strain curve [16]. However, despite this linearising process, the cable stiffness will vary; it is lower when the cable is new and becomes higher during the useful life of cable [97].
2.4.4
Corrosion protection
Cables made of high strength steel wires are extremely vulnerable to phenomena such as stress and fretting corrosion. Add to this that most of the wires will be inaccessible for inspection and maintenance in the completed cable and that numerous of cavities are present between wires, and one understands that it is essential to ensure that the corrosion protection is of highest quality, particularly in the regions of end or intermediate fittings [35, 42].
It is nowadays normal practice to protect the wires in a cable by galvanization. Both electrolytic and hot-dip techniques can be used, although the hot-dip technique has become the preferred method. There are different classes of coating thickness dependent on the severity of the exposure conditions. The coating is usually of pure zinc but zinc-aluminium alloys are also used. Hydrogen embrittlement of galvanized steel is not recognised as a real problem with wire ropes and strands [125].
2.5. CLADDING
It is today generally agreed that cables should have two barriers against corrosion. For spiral strands, wire ropes and locked coil strands the second barrier consists of filling the interstices between the wires with a blocking material and coating the outer surface. The primary purpose of the blocking material is to prevent ingress of moisture [42]. Suitable blocking materials are synthetic waxes and compounds based on petrolatum (petroleum jelly), which are hydrophobic and have good adherence. The final coating can be ordinary paint or, if necessary, a more displacement resistant compound [125].
If the cable is exposed to an aggressive environment it is normally sheathed with a tube made of steel or polyethylene. The space between the tube and the cable is filled with a suitable compound such as polymer cement grout or petroleum wax [42]. Sheathing is the most effective method for corrosion protection and it is considered as impermeable. Materials used for sheathing must be ductile and if polyethylene is used it must be resistant to ultraviolet radiation. An alternative sheathing method is to extrude polyethylene directly onto the cables (no filling) [35].
2.5
Cladding
In analysis of a prestressed cable structure the cladding is usually assumed not to add any contribution to the structural stiffness. Some contribution will in any case be added to the performance of the building, which cannot be neglected. Especially the damping properties of the roof will be enhanced, which have significant importance for the dynamic behaviour of the structure.
There are two main categories of cladding: continuous membranes and unit cov-erings. Membranes can be made of fabric, foil or metal sheet. Unit coverings are panels of metal, wood or plastic [23]. The choice of cladding material depends on the type of structure (e.g. its shape), the expected lifetime, static and dynamic be-haviour, security and maintenance. What type of cladding to be used should be decided upon at an early stage in the design process in order to avoid large changes, which might effect the cable spacing and the design of structural details [16].
2.5.1
Fabrics and foils
Fabric is today the most common cladding material used for lightweight tension structures. As a structural element, the fabric must have the strength to span between supporting elements, carry wind and snow loads, and be safe to walk on. To comply with these requirements, the fabric must be prestressed, since it has a negligible bending stiffness. The amount of prestress and the patterning of the membrane, i.e. how the membrane should be cut and assembled, is given by the structural analysis of the roof. Besides the structural requirements, the fabric must meet the requirements which affect the environment inside the building; these are air tightness, water protection, fire resistance, heat insulation, light transmission, acoustic properties, maintenance and durability [10].
CHAPTER 2. LITERATURE REVIEW
Fabric membranes are composite materials. Inside the membrane there are filament fibre yarn, designed to resist tensile forces, woven in different directions forming an anisotropic surface. For permanent buildings with expected long lifetimes only
two types of fibres can be used: glass and aramid (Kevlar2) fibres, of which glass
fibre is the most common. The mechanical properties of these fibres compared to the properties of a steel wire are shown in Table 2.1. To protect the fibres from
environmental degradation, they are coated with some resin. Resin used are PTFE3
(Teflon2), silicone and PVC4 [99].
Table 2.1: Comparison of filament yarn characteristics [42, 130].
Property Glass Aramid Steel
(E-HTS glass) (Kevlar 49) (Cold drawn wire)
Density (g/cm3) 2.55 1.44 7.86
Young’s modulus (GPa) 69 124 205
Tensile strength (MPa) 2410 2760 1570
Max. elongation (%) 3.5 2.5 4.0
Temp. 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 without damaging the coating. It is also resistant to abrasion and highly reflective, absorbing little light as well as heat. The fact that Teflon comes in two forms, PTFE and
FEP5, with different melting points makes it possible to heat weld seams, which
enables a fast installation of the roof cladding. In addition to its high initial cost, PTFE-coated fibreglass has two disadvantages: the material is brittle and requires considerable care in the packing, shipping and installation of panels, and it has little elastic forgiveness and must therefore be accurately patterned [99].
Fibreglass coated with silicone is more flexible than PTFE-coated fibreglass, so it is less likely to be damaged during shipment and installation. With a silicone coating, the fabric can be made more translucent than with PTFE and the need for artificial lightning during daytime can be almost eliminated. Fabric joints are chemically bonded or glued. The self-cleaning properties of silicone rubber are not yet as good as those of PTFE; it is recommended to clean the membrane once a year [99]. Fabrics of Kevlar have high tensile strength, high stiffness and very low weight. These properties make it possible to span large distances with Kevlar fabrics without a supporting cable net. One major disadvantage with fibres of Kevlar is that they are highly susceptible to ultraviolet radiation and cannot be coated with translucent resin. The fibres must be shielded with an opaque carbon black coat. Due to the sensitivity to ultraviolet radiation the joints of Kevlar fabrics cannot be heat welded with clear Teflon. The seams must instead be sewed, but it is impossible to develop
2Kevlar and Teflon are registered trademarks of E. I. du Pont de Nemours and Company
3Abbreviation for Polytetrafluoroethylene
4Abbreviation for Polyvinyl Chloride
5Abbreviation for Fluorinated Ethylene Propylene
2.5. CLADDING
the full strength of the fabric through the joints due to the high strength of the base material [40].
The newest membrane material is EFTE6 foil, which is not a woven fabric but a
polymer film sheet. From a structural viewpoint, EFTE foil is interesting because of its high tear resistance. In addition to the structural properties, the foil has many properties that make it work well as enclosure material. For example, it can be considered as incombustible, impervious to ultraviolet radiation and most chemicals, and it can be manufactured with a translucency of over 90 % [99].
2.5.2
Metal sheets
Instead of a fabric membrane a metal membrane can be chosen. Sheets of aluminium or steel sheets with thicknesses of 1 to 5 mm are found to be suitable for this appli-cation. Due to the low bending stiffness of the sheets, it is necessary to prestress the membrane to prevent buckling. Prestressing is achieved by applying the membrane before the roof is fully erected. When the roof is raised to the final position the membrane is pretensioned. The metal membrane is composed of small accurately cut sections jointed by welding, gluing or bolting. Metal sheet membrane is a fea-sible choice for long-life structures and can be designed with openings covered with glass to provide natural lightning. Heat loss is prevented by attaching insulation material internally [23]. In [131], Yeremeyv and Kiselev describe the manufacturing and erection of a number of large projects in Russia where metal sheets are used as covering.
2.5.3
Panels
A cable net with cable spacing of around half a meter is ideal for small elements (panels). The elements are either shape-cutted or jointed in such a way so that they will conform to the shape of the structure. The panel system is most economical if it is made of light material, not to impose extra weight on the cable structure. Panels of fibreboard, aluminium and plastic are appropriate to use for covering roofs [23]. For the Olympic Stadium in Munich, a system with translucent plastic panels
(Plex-iglas7) with thickness of 4 mm and size of 2.90 m × 2.90 m was used. The panels
were fastened to the supporting cable net with shock absorbing flexible connections to prevent cracking of the panels under roof movements. The joints between the panels were sealed with continuous neoprene profiles, as seen in Figure 2.17 [63]. However, it should be mentioned that many architects, e.g. Philip Drew [31], find the Plexiglas cladding of the Olympic Stadium ugly. Therefore, it will probably not be used again.
6Abbreviation for Tetrafluoroethylene
CHAPTER 2. LITERATURE REVIEW
Figure 2.17: Acrylic panels for the Olympic stadium in Munich. Reproduced
from [57].
2.6
Structural details
Already at early stages of the design process the designer has to pay attention to the design of the structural details. Structural details are fittings, saddles and anchorages. Fittings are attachments used to grip the cable at the ends or along its length. They can be classified, in accordance with the type of application, as the friction or clamp type, the pressed or swaged type, and the socketed type. Saddles are used when the cable has to run continuously over masts and other supports. In self-supporting systems, cables are anchored into structural members, such as a concrete ring or an arch. In other systems the cable forces are resisted by anchors in the ground [57]. A comprehensive survey of structural details is given by Chaplin
et al. [23].
2.6.1
End fittings
An end fitting (terminal) is an attachment, which transmits the cable force to the supporting system. To be totally effective, the end fitting must withstand the full breaking force of the cable without significant yielding, endure dynamic loading without risk of fatigue failure and not induce fatigue failure of the cable. For ap-plications where large forces are to be transmitted to the supporting structure two different end fittings are accepted [125]: the socketed type and the swaged type, Figure 2.18.
2.6. STRUCTURAL DETAILS
(a) Socketed type (b) Swaged type
Figure 2.18: Cable end fittings with pin connectors. Reproduced from [16].
The most reliable, but also the most expensive, of the end fittings is the socketed type. It is manufactured by splaying the end of the cable a prescribed length and cleaning the individual wires. When the wires are cleaned and dried the conical socket of machined or casted steel is positioned on the splayed cable section. Then molten socketing material is poured into the socket, hardens and forms a cone, Fig-ure 2.18(a). As tension is applied to the cable the cone is drawn into the socket and wedging forces are developed which grip the wires. As socketing material either of zinc or resin is used. Pure zinc has been used for over a century and it offers a cathodic protection for the cable, but it is sometimes criticised for impairing the fatigue resistance of the cable in this region. Another, more important, disadvan-tage with sockets filled with pure zinc is that they are prone to creep effects under high stresses. Therefore zinc alloy, with improved creep resistance, is often used. Polyester or epoxy resin has better creep resistance. As the resin is casted at low temperature the fatigue resistance of the cable will not be impaired. Socketed end fittings can be used for all cable sizes but cables of smaller diameter, approximately less than 38 mm, can be terminated by means of hydraulically compacted fittings called swaged end fittings. Swaged end fittings are cheaper than socketed types but they are only guaranteed to resist 95 % of minimum breaking load of the ca-ble. All end fittings are manufactured, installed and rigorously tested by the cable manufacturer [16, 125].
2.6.2
Intermediate fittings
Intermediate fittings are used to connect cables to other cables. These fittings are usually not standard appliances and their behaviour depend on the frictional force between the cable and the clamp. To prevent sliding of the clamp, the clamping force must be large and thereby high radial stresses are induced. Cables are more prone to fatigue when the pressure between adjacent wires is high and it is, therefore, important to use fittings where the clamping force is evenly distributed over the cable. The resistance of a spiral strand and a locked coil strand to clamping forces, where the latter has the higher resistance, can be found in Eurocode 3 [35]. When the cable is tensioned the diameter will decrease and consequently the clamping force. It can therefore be necessary to retension the clamp bolts to prevent sliding.
CHAPTER 2. LITERATURE REVIEW
To avoid abrasion between the clamp and cable under cable movements, which can result in fatigue failure, the ends of the fittings must be radiused. Different types of intermediate fittings are shown in Figures 2.19–2.20.
(a) Clamp connection (b) Swaged clamp connection
Figure 2.19: Cable connections for dual-strand cable nets. Reproduced from [16].
(a) Single U bolt connection (b) Double U bolt connection
Figure 2.20: Cable connections for two-way cable nets. Reproduced from [16].
In the search for the best economical solution one key is to use few types of structural details, as the number of fittings in, for example, a cable net can be quite large. A way to achieve this is to use a fitting which can be adjusted for different angles between cables. The fitting shown in Figure 2.19(b) can be mounted in a factory and thereby it is possible to reach a high accuracy. As mentioned above, accurate assembly of the fittings is necessary in order to obtain the desired internal force distribution in a cable net.
2.6.3
Saddles
When the cables have to run continuously over supports like columns and masts, they have to be supported by saddles, Figure 2.21. When designing a saddle one has to take the bending stiffness of the cable into account. Two factors have to be checked:
• the tensile stress in the outer wires, and
• the pressure between the cable and the saddle.
2.7. ROOF LOADS
If the pressure between the cable and the saddles is too high the fatigue resistance of the cable will be affected. The common rule is that the diameter of the saddle should not be less than 30d, where d is the diameter of the cable [16, 35].
Figure 2.21: Saddle. Reproduced from [16].
2.6.4
Anchorages
In self-supporting systems, the cables are anchored into the boundary structures, which resist the cable forces due to either geometry or self-weight. These structures are usually rings, arches and masts made of concrete or steel. In open systems the cable forces are resisted by tension anchors in the ground. A survey of exist-ing tension anchors and methods for estimatexist-ing their capacities for various ground conditions can be found in [16]. Which of the two anchorage alternatives that will be most economical, if both are architecturally accepted, depends upon the ground conditions, cost of material, and availability of expertise and labour skill.
2.7
Roof loads
Today, structural analyses are performed using commercial finite element programs, which contain elements for almost every application. New elements are constantly being developed and older refined in an attempt to obtain more accurate results. Nevertheless, the accuracy of the results will mainly depend on the errors in the prescribed loads acting on the structure. Since most loads are environmental loads with random distributions, durations and magnitudes, the ‘exact’ values will never be known. In an attempt to achieve higher accuracy in the results from a structural analysis more reliable data on the extreme loads acting on buildings are needed. Apart from the prestress, the loads acting on cable roofs are the same as any other type of loads acting on more conventional buildings. However, it is well known that non-uniformly distributed loads are more dangerous to cable structures than uniform loads. Therefore, it is important to determine the ‘true’ load distribution on the structure. Nonetheless, the unusual shape of these structures, together with their low weight and large scale, make this a difficult task. A further complication is that practically no guidance is available from codes of practice. This implies additional
CHAPTER 2. LITERATURE REVIEW
costs to the project, because of the need for expertise. The latest methods for determining the loads on roofs of general shapes involve very sophisticated physical and computational modelling techniques, which require expensive equipment and powerful computers. In this section, these methods are reviewed. The loads are viewed in order of their importance on the structural behaviour of tension structures.
2.7.1
Wind load
Due to the low weight of cable roofs with membrane cladding, wind pressure is one of the most important forms of loading. The variability and large number parameters involved in the determination of wind effects on structures make it a very complex problem. Some undesirable effects and partial collapses have been caused by wind on tension structures [16]. Among these can be mentioned the vibrations due to wind on the roof of the Raleigh Arena, U.S.A., which made it necessary to insert supplementary internal cables.
The nature of wind
Wind is initiated by pressure differences between points of equal elevation, caused by variable solar heating of the atmosphere of the earth. The motion of the air mass is modified by the rotation of the earth and close to the ground the velocity of the moving air is reduced due to friction. At a certain height above the surface of the earth the effect of the surface friction becomes negligible. Above this boundary layer a frictionless wind balance is established, and the wind flows with the gradient speed along lines of equal barometric pressure. The height of the atmospheric boundary layer normally ranges from a few hundred meters to several kilometres, depending upon wind intensity, roughness of terrain, and angle of latitude [32].
Physically, the wind is composed of two different velocity components [16]. The first component is the velocity of a steady flow determined by the long-term pressure variations (approximately four day periods). This component is called the mean wind velocity. The second velocity component, which is superimposed on the steady flow, is due to a turbulent fluctuating system with high frequency components, which is caused by the friction between the air and the surface of the earth. The two velocity components are clearly seen when the wind velocity is plotted in a van der Hoven power spectrum, Figure 2.22. This spectrum shows the variations of the mean square of the amplitudes of the fluctuating components against the frequencies of these components.
Hence, the analysis of linear structures can be divided into to two parts: the cal-culation of the quasi-static response due to the steady velocity component and the response caused by the turbulence components. As cable structures have a non-linear behaviour this division is generally not valid. Instead, the total wind load must be used in the dynamic analysis of cable structures. In the sequel to this section the common expressions for description of the wind load on buildings and
2.7. ROOF LOADS
ways to obtain the pressure distribution will be described in brief.
Figure 2.22: Spectrum of horizontal wind speed after van der Hoven. Reproduced from [26].
Mathematical description of natural wind
To describe the wind velocity mathematically a Cartesian coordinate system is 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, are turbulence components of the wind field in the x, y and z directions, respectively. It can be noted that the mean wind velocity U(z) only depends on the height above the ground. The turbulence components are treated mathematically as stationary, stochastic processes with a zero mean value. The mean wind velocity U(z) and the turbulence component u in the wind direction are often most important, as they usually give the main contributions to the wind forces on a structure [32].
Three laws have been proposed to describe the way in which the mean velocity U varies with height [32]. The first law is the power law, which has been adopted in many codes. The second law is the logarithmic law, which is derived not only from empirical data, but also from theoretical considerations. The Deaves and Harris model, which is the third law, is the most exact one since it is fitted to experimental data [16,26]. In urban areas, where stadiums and other large roofs usually are built, the terrain roughness might change if buildings are erected or demolished [32]. This directly affects the mean wind velocity and has to be considered at the design stage. The wind in the boundary layer is always turbulent, which means that the flow is chaotic, with random periods varying from fractions of a second to several minutes, Figure 2.22. In order to describe a turbulent flow, statistical methods must be applied [32].
CHAPTER 2. LITERATURE REVIEW
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 the pressure fluctuations correspond exactly with the variations of the wind velocity. Other methods may be found in [27], but wind loads on buildings are determined using the quasi-steady model in many codes [64]. Therefore, a detailed description of the quasi-steady theory will be given here. For a point on a surface, (x,y,z), the instantaneous pressure, p, is given by [27, 64]
p = 1
2ρV
2
Cp(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 mean
wind direction θm. The magnitude of the wind velocity is given by
V2 = (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 and the velocity magnitude reduces to
V2 ≈ 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 ∂C p(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 pressure coefficient 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
