An Introduction to Cable Roof Structures

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An introduction to cable roof structuresSecond edition


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An introduction to cable roof structuresSecond edition

H. A. Buchholdt

'"[I Thomas Telford


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Published by Thomas Telford Ltd, 1 Heron Quay, London E14 4JD.URL: http://www.t-telford.co.uk

Distributors for Thomas Telford books areUSA: ASCE Press, 1801 Alexander Bell Drive, Reston, VA 20191-4400Japan: Maruzen Co. Ltd, Book Department, 3—10 Nihonbashi 2-chome, Chuo-ku, Tokyo103Australia: DA Books and Journals, 648 Whitehorse Road, Mitcham 3132, Victoria

First published 1985 by the Press Syndicate of the University of CambridgeThis edition 1999

A catalogue record for this book is available from the British Library

ISBN: 978-0-7277-2624-7

© H.A. Buchholdt, 1999

All rights, including translation reserved. Except for fair copying, no part of this publicationmay be reproduced, stored in a retrieval system or transmitted in any form or by any means,electronic, mechanical, photocopying or otherwise, without the prior written permission ofthe Books Publisher, Thomas Telford Publications Ltd, 1 Heron Quay, London E14 4JD.

This book is published on the understanding that the author is solely responsible for thestatements made and opinions expressed in it and that its publication does not necessarilyimply that such statements and/or opinions are or reflect the views or opinions of thepublishers.


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Contents

Preface to the second edition xiPreface to the first edition xiii

1. Structural systems 1Introduction 1Simply suspended cable structures 2Pretensioned cable beam structures 4Pretensioned cable net structures 7Pretensioned cable grid structures 8General structural characteristics 11Bibliography 13

2. The nature and statistical properties of wind 14Introduction 14The nature of wind 15Mean wind speed and variation of mean velocity withheight 17Statistical properties of the fluctuating velocity componentof wind 19

Variance and standard deviation 20Auto-correlation and auto-covariance functions 21Spectral density functions of longitudinal velocityfluctuations 21Cross-correlation and cross-covariance functions 24Cross-spectral density and coherence functions forlongitudinal velocity fluctuations 25

The probability density function and peak factor for thefluctuating component of wind 26The cumulative distribution function 28Pressure, drag and lift forces 28References 32

3. The nature and statistical properties of earthquakes 33Introduction 33Types and propagation of seismic waves 33Recording of earthquakes 35Magnitude and intensity of earthquakes 35Influence of magnitude and surface geology on thecharacteristics of earthquakes 35Representation of ground motion 37


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Construction and representation of response spectra 39Power spectral density functions for earthquakes 42Soil-structure interaction 43References 44

4. Generation of wind and earthquake histories 45Introduction 45Generation of single wind histories by a Fourier series 45Generation of wind histories by the autoregressive method 46Generation of spatially correlated wind histories 50Numerical illustration 52Generation of earthquake histories 52Cross-correlation of earthquake histories 58Design earthquakes 59References 61

5. Freely hanging cables 63Introduction 63The governing equation for freely hanging cables 63Cables with assumed distributed load wx along the span 65

Uniformly distributed load 65Uniformly tapering distributed load 68Two symmetrically placed triangularly distributed loads 70

References 71

6. Static analysis of cable structures 72Introduction 72Structures subjected to point loads only 75

The total potential energy at x in displacement space 75The gradient vector of the total potential energy at x indisplacement space 78The method of steepest descent 79The method of conjugate gradients 80The Newton-Raphson method 80The total potential energy at x^+x in displacement space 82Determination of S from the steplength polynomial 83Member forces and displacements at Xk+i in displacementspace 84

Laterally loaded cable elements 84Slackening cable elements 85Cable rupture 85Cable elements with nonlinear stress-strain relationships 85Buckling of strut elements 87Change of temperature 87Numerical ill-conditioning and scaling 87Convergency criteria 88Summary of the iterative procedures 88


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Numerical example 89First iteration 91Second iteration 94

References 97

7. Dynamic analysis of weakly nonlinear cable roof structures:frequency domain analysis 99The importance and extent of dynamic analysis 99Aeroelasticity and dynamic response 99Dynamic response analysis of aeroelastically stable cableroofs 100Dynamic response of one-DOF systems to turbulent wind 101Relationships of response, drag force and wind velocityspectra for one-DOF systems 102Dynamic response of multi-DOF structures to turbulentwind 106Summary of expressions used in the frequency domainmethod for multi-DOF systems 109Modal force spectra for wind for three-DOF systems 109Aerodynamic damping of multi-DOF systems 110Dynamic wind response analysis of weakly dampedstructures 112Dynamic response of cable structures to earthquakesstudied using frequency domain analysis 113Dynamic response of one-DOF systems to earthquakesstudied using power spectra 113Influence of the dominant frequency of the ground on themagnitude of structural response 114Dynamic response of multi-DOF structures to earthquakesstudied using power spectra 115Conclusion 116References 117

8. Dynamic analysis of nonlinear cable structures:time domain analysis 118Basic concepts 118The force components of dynamic time-dependent loads 119

Force components due to wind 119Force components due to support movements such as thosecaused by earthquakes and explosions 120Force components due to other forms of dynamic loading 120

Dynamic analysis in the time domain 120Assumptions 121Total potential work at time {% + At) 122Total potential energy 124Total potential structural energy dissipation 125Total potential work of inertia forces 126


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Total potential work of wind 127Total potential work of inertia forces due to supportmovements 128Total potential work of independent dynamic forces 129Convergency and scaling 129

Stability and accuracy 130Numerical illustration 130References 136

9. Damping ratios and damping matrices 138Introduction 138Measurement and evaluation of damping and damping ratios 138The influence of air at resonance 139Damping matrices 142Modelling of structural damping by orthogonal dampingmatrices 142

First method 142Second method 143

References 145

10. Cables and terminals 146Wire strand rope 146Steel 148Manufacture of cables 148Environmental factors affecting steel cables 150

Moisture 151Water in mass (sea, river, lake or pond water) 151Water as discrete droplets (rain or driven spray) 151Water vapour 151Heat and cold 152Solar radiation 152

Solid particles 152Protective coatings 153Cable properties 153Cable terminations 154Linearization of cables—prestressing 157Creep 160Fatigue 161Flexibility of cables 162References 163

11. Tension anchors 164Introduction 164Types and suitability of tension anchors 165Gravity anchors 167Plate, mushroom and other anchors 167

Theory 167Strip anchors 168Circular anchors 169


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Rectangular anchors 170Group action 170Pull-out tests in sand 171Pull-out tests in clay 171Long-term uplift capacity 171Factor of safety 172

Tension piles 172Tension piles in granular material 172Tension piles in clay 173Factor of safety 174

Ground anchors 175Sand and gravel 175Medium-to-fine sand (k < 103 m/s) 176Clay 111Soft rock 177Other points 178Caution 178

Rock anchors 178Mechanical anchors 178Bonded anchors 178Rock sockets 180Design considerations 180

Concluding remarks 180References 181

12. Cable beams and cable grids 182Introduction 182Structural characteristics 182Preliminary design analysis of cable beams and grids 188Design and construction 195Cladding and cladding materials 214Erection of cable beams 221Influence of boundary geometry on the forces at theboundary 221Preliminary design—example 224References 230

13. Cable net roofs 231Introduction 231Shape finding 231Static and dynamic characteristics 239Loading 261Preliminary design analysis 261Static and dynamic modelling of cable net structures 261Design details 268Methods of erection 272Cladding and cladding materials 273References 274


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14. Design considerations 276Introduction 276Architectural requirements 276Site location and geological information 277Shape finding 277Roof cladding and cladding materials 278Wind and snow loading 278Computer analysis and the use of models 279Corrosion protection 280Fire rating 280Choice of contractor 281Design and construction costs 281Conclusion 282

15. Index 283


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Preface to the second edition

In the second edition of this book a better and more efficient method forgenerating single wind histories and spatially correlated wind fields, and abetter method for generating earthquakes and families of earthquakeswith the correct dominant ground frequencies, are presented. As manycable roofs are only weakly nonlinear, the frequency domain method ofdynamic analysis using spectral density functions for wind and earth-quakes has also been incorporated.

Thus, the chapter on wind and earthquake loading in the first editionhas been enlarged and divided into three chapters: one on wind,Chapter 2; one on earthquakes, Chapter 3; and one on the generationof spatially correlated wind fields and the generation and families ofearthquakes with predetermined spectral density functions, Chapter 4.Similarly the chapter on dynamic analysis has been expanded. In thesecond edition there is one chapter dealing with the frequency analysis ofweakly nonlinear structures, Chapter 7; one chapter on time domainanalysis, Chapter 8; and one on damping, Chapter 9. Chapter 9 includescomments on the measurements and use of damping ratios, aerodynamicdamping, and the numerical modelling of damping by damping matrices,the modelling of which is required in time integration methods ofnonlinear structures. At the end of Chapter 8 some results are givenfrom a numerical investigation into the dynamic response of a largecircular saddle-shaped roof in which the magnitude of responses of theroof when calculated by quasi-static frequency domain and time domainmethods are compared.

Of the redrafting undertaken the most significant part is undoubtedlythe new method presented for generating single wind histories, spatiallycorrelated wind fields, earthquakes and families of earthquakes.

The method developed for the creation of wind fields is the result of anumber of research projects at the University of Westminster, where thedynamic response of guyed masts to wind and earthquakes was investi-gated. In the case of wind the calculated dynamic responses of a 45 mexperimental guyed mast at the University of Florence, Italy, and a 275 mIB A mast, UK, were compared with recorded values. It was found thatthe response was similar in the two cases, and the magnitude of responseagreed with the measured ones. Characteristics of the generated earth-quake histories have been compared with those of real earthquakes, andthe responses of a cable-stayed bridge and a guyed mast to real andartificial quakes were compared, with satisfactory results.

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An introduction to cable roof structures

Errors discovered in the first edition have been corrected, and thescaling used in conjunction with the method of conjugate gradients,which is used as an alternative to the Newton-Raphson method tominimize the total potential energy and total potential work, has beenupdated.

H. A. Buchholdt1999

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Preface to the first edition

During the sixties and early seventies it was thought by many that thearchitectural and economic potential of cable roof structures would leadto an increasing demand for this type of building. Consequently a greatdeal of work was carried out world-wide to study the behaviour ofdifferent types of structural system. The interest in cable structures alsostimulated an increasing interest in the use and development of numericalmethods for solving large systems of nonlinear equations, made possibleby the emergence of the high-speed electronic computer.

The early expectations for the demand for cable roofs have, however,so far not been realized, and to date the total number of cable roofs inexistence is a relatively modest one. There may be several reasons for this.The need for larger clear spans has been less than expected. Architectsand engineers are not in general familiar with their design and havetended to consider them only for the less usual structures such as sportsstadia, ice rinks and exhibition halls. Their industrial potential and theiruse in earthquake areas have never been properly exploited. The cost oftension anchors where these have been required has been relatively highand is thought to be partly due to lack of constructional expertise andlack of research.

This book is written to encourage the use of cable roof designs andshow that a large variety of practical structural forms can be simply andcheaply constructed by the use of such common and well-known materi-als as prestressing strand and metal decking, timber or concrete. Thebook is not intended as an exhaustive study of the architectural,structural and technological aspects of cable roofs but rather, as the titleimplies, as an introduction to this type of structure. It is, however, hopedthat it contains sufficient information to enable interested engineers inpossession of a small computer to carry out their own designs withoutany outside assistance.

Those who are particularly interested in the architectural aspects oflightweight structures should consult the work of Professor Frei Otto,University of Stuttgart, whose endeavours in this field span severaldecades. For those whose interest is more in the analytical field, thework of Dr H. Mellmann and Dr H. Irvine is recommended, togetherwith one of the many books on numerical mathematical methods. (Seereferences 3.1, 4.15 and 4.19.)

In writing the chapter on wind and earthquake loading, the mainobject has been to present the reader with methods for generating wind

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An introduction to cable roof structures

and earthquake histories that may be used in conjunction with thenonlinear theory given in Chapter 5. The generated histories are strictlyonly applicable to linear structures, but can also be used for nonlinearstructures if the approach suggested is adopted. More research, however,is required in order to develop theories for generating histories which areimmediately suitable for nonlinear structures.

When writing the chapters on cable beams and grids it was originallyintended to include a number of nondimensional graphs for the purposeof design. It very soon became clear, however, that the number of graphsrequired was so large as to be impracticable. In their place a number oftables with nondimensional values of forces and displacements forstructures the author has analysed have been inserted, and it is hopedthat these will help the reader to obtain some feeling for the influence ofthe variation in design parameters on the structural characteristics ofcable roofs.

In writing the book the author has been helped and encouraged bymany people. In particular, he is grateful to M. J. Tawse of British RopesLtd for writing the chapter dealing with the manufacture and propertiesof cables, to Dr ing. P. Spinelli and Mr D. Kay for supplying the originaldrafts on wind and earthquake loading respectively, and to Mr R. Dixonfor contributing to the chapter on design considerations. The author alsowishes to thank Mr J. Armishaw for helping him to update a paper ontension anchors which the author had previously written together withMr N. Vadgama, whose contents now constitute the chapter on tensionanchors.

The author also wishes to express his gratitude to Dr P. Regan,Professor P. Krishna and Professor H. Tottenham for their many usefulsuggestions and for checking the manuscript, to Dr S. Moosavinejad andDr H. Tabar-Heydar for their help with the computer analysis, to Mr D.Mutlow for preparing most of the illustrations, and to his wife Mrs R.Buchholdt for preparing the tables, graphs and diagrams and for typingthe manuscript.

Finally the writer would like to express his gratitude to the Science andEngineering Research Council and to White Young and Partners, both ofwhich have supported the work on cable roofs for a number of years, andto those of his colleagues who have facilitated the author's research andthe writing of this book by undertaking a larger share of the under-graduate teaching than they otherwise would have had to do.

H. A. Buchholdt1984

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An Introduction to Cable Roof Structures