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CABLE-STAYED BRIDGES Theory and Design SECOND EDITION M. S. Troitsky, DSc Professor of Engineering Concordia University, Montreal

BSP PROFESSIONAL BOOKS OXFORD LONDON ED INB URGH BOSTON PALO ALTO MELBOURNE

Copyright© M.S. Troitsky 1977, 1988 All rights reserved. No part of this publication may be reproduced, stored in a retrienl system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior permission of the copyright owner. First Edition published by Crosby Lockwood Staples in 1977 Second Edition published by BSP Professional Books 1988 British Librarv Cataloguing i~ Publication Data Troitsky, M. S. Cable-stayed bridges: theory and design.Znd ed. 1. Bridges, Cable-stayed-Design and construction I. Title 624'.55 TG405

BSP Professional Books A division of Blackwell Scientific Publications Ltd Editorial offices: Osney Mead, Oxford OXZ OEL (Orders: Tel. 0865 240201) 8 John Street, London WCIN ZES 23 Ainslie Place, Edinburgh El 13 6AJ 52 Beacon Street, Boston Massachusetts 02108, USA 66 7 L)tton Avenue, Palo Alto California 94301, USA 107 Barry Street, Carlton Victoria 3053, Australia Set by Cambrian Typesetters Printed and bound in Great Britain by Butler & Tanner Ltd, Frome and London

ISBN 0-632-02041-5 Acknowledgements Special acknowledgement is herewith made to the following persons, companies, institutions and organizations for supplying the information and photographs for the many bridges discussed in this book: Alaska Department of I lighways, USA; British Railways Southern Region; Compagnie Fran~aise D'Entreprises Metalliques, France; Compagnie BaudinChateauneuf, France; Dip!. Eng. E. Beyer, Landeshaupstadt Dusseldorf, Germany; Department of Public Works, Hobart, Tasmania; Mr A. F. Gee, Mott, I lay and .'l.nderson, Consulting Engineers, England; Dr 0. A. Kerensky, Freeman, Fox and Partners, Consulting Engineers, England; Dip!. lng. H. Thul, Germany; The Institution of Engineers, Australia; Mr A. Zanden, Rijkswaterstaat Directie Bruggen, Holland; Mr J. \'irola, Consulting Engineer, Finland; lng. J. J. 1\1. Veraart, Holland; Quebec Iron and Titanium Corporation; .\lr Arvid Grant and Associates, Inc., Consulting Engineers, USA; Modjeski and Masters, Consulting Engineers, USA; Dr P.R. Taylor, Buckland and Taylor Ltd, Civil and Structural Engineers, Canada. I am especially grateful to the American Society of Civil Engineers li1r permitting me to use excerpts of the paper 'Tentative Recommendations for Cable-stayed Bridge Structures'.

Contents

Preface to the second edition

Vll

Chapter 1 The Cable-stayed Bridge System 1.1 1.2 1.3 1. 4 1.5 1.6 1. 7 1. 8 1. 9 1.10 1.11 1.12

Introduction Historical review Basic concepts Arrangement of the stay cables Positions of the cables in space Tower types Deck types Main girder and trusses Structural advantages Comparison of cable-stayed and suspension bridges Economics Bridge architecture References

1 2 19 20 21 24 25 26

29 31 34 36 39

Chapter 2 Typical Steel Bridges 2.1 2.2 2.3 2.4 2.5 2.6 2. 7

Two-plane bridges One-plane bridges Inclined tower bridges Railroad bridges Combined railroad-highway bridges Pipeline bridges Pontoon bridges References

42

69 91 95 99 103 105 108

iv

COl\TE:'\TS

Chapter 3 3.1 3.2 3.3

Concrete cable-stayed bridges Railroad concrete bridges Pipeline concrete bridges References

Chapter 4 4.1 4.2

Typical Concrete Bridges

114 139 143 144

Typical Composite Bridges

Introduction Composite cable-stayed bridges References

147 148 154

Chapter 5 Typical Pedestrian Bridges 5.1 5.2

Introduction Cable-stayed pedestrian bridges References

155 155 173

Chapter 6 Structural Details 6.1 6.2 6.3 6.4 6.5 6.6 6. 7 6.8 6. 9 6.10 6.11 6.12 6.13

Stiffening girders and trusses Towers Types of cable Modulus of elasticity of the cable Permissible strength of the cables Fatigue tests and strength of the cables Corrosion protection Behavior of the bent cable Cable supports on the towers Anchoring of the cables at the deck Stiffening girder anchorages Erection methods Adjustment of the cables References

175 176 180 185 191 191 195 195 198 203 211 213 217 221

Chapter 7 Methods of Structural Analysis 7.1 7.2 7.3 7. 4 7.5

Introduction Linear analysis and preliminary design Nonlinear analysis Dynamic analysis Application of computers References

223 223 224 227 229 230

CONTENTS

V

Chapter 8 Approximate Structural Analysis Participation of the stiffening girder in the bridge system Optimum inclination of the cables The height of the tower and length of the panels The relation between the flanking and central spans 8.5 Number and spacing of the cables Multispan bridges 8.6 Multiple cantilever spans 8.7 Inclined cable under its own weight 8.8 Bridge systems 8.9 8.10 The degree of redundancy 8.11 Performance of the cable system 8.12 Linear analysis and preliminary design 8.13 Approximate weight of the bridge system 8.14 Approximate methods of analysis 8.15 Nonlinear analysis References

8.1 8.2 8.3 8.4

231 233 236 237 238 240 240 241 245 247 247 251 261 265 269

272

Chapter 9 Exact Methods of Structural Analysis 9.1 9. 2 9.3 9.4 9.5 9.6 9. 7 9.8 9. 9

Methods of analysis The flexibility method Force-displacement method Reduction method Simulation method Stiffness method Finite element method Torsion of the bridge system Analysis of towers References

273 274 282 297 309 317 323 328

345 361

Chapter 10 Model Analysis and Design 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9

Introduction Basic concepts Planning Static similitude conditions Sectional properties and geometry of the model Design of the model Determination of influence lines Nonlinear behavior Post-tensioning forces in cables References

364

365 366 370 374 375 376 392 397

401

vi

CONTENTS

Chapter 11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11

Introduction Wind forces Static wind action Dynamic wind action Vibrations Vertical flexural vibrations Torsional vibrations Damping Wind tunnel model tests Prevention of aerodynamic instability Conclusions References

Chapter 12 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11 12.12 12.13 12.14 12.15

Wind Action and Aerodynamic Stability 404 407 408 410 413 416 421 428 435 440 446 446

Abbreviated Tentative Recommendations for Design of Cable-stayed Bridges

Introduction Loads and forces Design assumptions Pylons Analysis Cables Saddles and end fittings Protection Camber Temperature Aerodynamics Fatigue Fabrication Erection Inspection References

450 450 451 452 452 453 454 455 455 455 456 456 457 457 458 459

Author Index

460

Subject Index

463

Preface to the second edition

Since the first edition of this book was published a decade ago, there has been considerable development in the state of the art of cable-stayed bridges. In this second edition, the contents have been revised to reflect recent developments in research, analysis, design and construction of new structures. Although much of the data of the first edition has been retained, the arrangement of material has changed, chapters have been expanded and new ones have been added. For the convenience of the users, the following changes and additions were made in the contents of the second edition. The first edition contained seven chapters, while the second edition consists of twelve chapters, as follows: Chapter 1, The Cable-stayed Bridge System has an additional discussion on the problems of economics and aesthetics. Chapter 2, Typical Steel Bridges contains additional data on new steel single and two-plane bridges, as well as pipeline and pontoon bridges. Chapter 3, Typical Concrete Bridges contains additional data on new concrete structures. Chapter 4, Typical Composite Bridges describes new deck types of cablestayed bridges. Chapter 5, Typical Pedestrian Bridges presents additional types of pedestrian bridges. Chapter 6, Structural Details provides additional structural details. Chapter 7, Methods of Structural Analysis presents a discussion on the structural behavior of bridges and methods of analysis. Chapter 8, Approximate Structural Ana(ysis treats methods of preliminary analysis. Chapter 9, Exact Methods of Structural Analysis presents additional methods. Chapter 10, Model Analysis and Design discusses experimental methods of design. Chapter 11, Wind Action and Aerodynamic Stability provides expanded treatment considering aerodynamic action. Chapter 12, Abbreviated Tentative Recommendations for Design of Cablestayed Bridges is a new addition. Every effort was made to correct some errors detected in the first edition.

To my wife Tania

Chapter 1

The Cable-stayed Bridge System

1.1

Introduction

During the past decade cable-stayed bridges have found wide application, especially in Western Europe, and to a lesser extent in other parts of the world. The renewal of the cable-stayed system in modern bridge engineering was due to the tendency of bridge engineers in Europe, primarily Germany, to obtain optimum structural performance from material which was in short supply during the post-war years. Cable-stayed bridges are constructed along a structural system which comprises an orthotropic deck and continuous girders which are supported by stays, i.e. inclined cables passing over or attached to towers located at the main piers. The idea of using cables to support bridge spans is by no means new, and a number of examples of this type of construction were recorded a long time ago. Unfortunately, the system in general met with little success, due to the fact that the statics were not fully understood and that unsuitable materials such as bars and chains were used to form the inclined supports or stays. Stays made in this manner could not be fully tensioned and in a slack condition allowed large deformations of the deck before they could participate in taking the tensile loads for which they were intended. Wide and successful application of cable-stayed systems was realized only recently, with the introduction of high-strength steels, orthotropic type decks, development of welding techniques and progress in structural analysis. The development and application of electronic computers opened up new and practically unlimited possibilities for the exact solution of these highly statically indeterminate systems and for precise statical analysis of their three-dimensional performance. Existing cable-stayed bridges provide useful data regarding design,

2

CABU:-SfAYED BRIDGES

fabrication, erection and maintenance of the new system. With the construction of these bridges many basic problems encountered in their engineering are shown to have been successfully solved. However, these important data have apparently never before been systematically presented. In summary, the following factors helped make the successful development of cab:e-staycd bridges possible: ( 1) The development of methods of structur al analysis of highly statically indeterminate structures and application of electronic computers. (2) The development of orthotropic steel decks. (3) Experience with previously built bridges containing basic clements of cable-stayed bridges. (4) Application of high-strength steels, new methods of fabrication and erection . (5) The ability to analyse such structures through model studies.

1.2 Historical review

Fig. 1.1

Egyptian sailing

boat with rope-srjlyed

sail beam

The history of stayed beam bridges indicates that the idea of supporting a beam by inclined ropes or chains hanging from a mast or tower has been known since ancient times. The Egyptians 1 applied the idea for their sailing ships as shown in Fig. 1.1. In some tropical regions of the world primitive types of cable-stayed bridge, such as shown in Figs. 1.2 and 1.3, were builrl. Inclined vines attached to the trees on either bank supported a walk which was woven of vines and bamboo sticks.

Fig. 1.2 Primithe bamboo bridge in Borneo Fig. 1.3 (bdow) Primitive bamboo bridge: in Laos

Fig. 1.4 Bamboo bridge with bamboo stays O\'Cr Serajoc Ri,er 10 Ja,a, Indonesia

F igure 1.4 s hows a primitive bridge of bamboo stays interwoven with vi11es with the ends fastened to trees ar each side. This crude structure indicates that its builders had a vague idea of some of the principles of bridge engineering. In 1617, Faustus Verantius proposed a bridge system ha,'ing a timber deck supported by inclined eyebars3 ; see Fig. 1.5. I

t HI I \

Fig. 1.5 Bridge stiffened by eye bars, designed by Faustus Vcrantius, Italy, 1617.

THE CABLE-STAYED BRIDGE SYSTEM

5

Like all bridge designs of this epoch, it exhibits many departures from what .a structural analysis would dictate; nevertheless, it contains the main features and basic principles of a metal suspension bridge stiffened by stays. In 1784, a German carpenter, Immanuel Loscher4 in Fribourg designed a timber bridge of I OS ft (32 m) span consisting of timber stays attached to a timber tower (Fig. 1.6). In 1817, rwo British engineers, Redpath and Brown, built the King's ,\leadows Bridge5 , a footbridge in England which had a span of approximately II 0 ft (33.6 m), using sloping wire stay cable suspension members attached to cast iron towers (Fig. 1.7). Fig. 1.6 All-timber bridge stiffened by inclined timber stays, designed by Loschcr in Germany, 1784.

Fig. l.i

King's i\lc;ado"s Bridge, England, 1817

6

Fig. 1.8

CABLE-STAYED BRIDGES

Dryburgh Bridge, England, 1817

The system of inclined chains was adopted in a bridge built at Dry burgh Abbey across the Tweed River 6 in 1817. It had a 260ft (79.3 m) span, and was 4ft ( 1.2 m) wide (Fig. 1.8). It was observed that the bridge had a \'ery noticeable vibration when crossed by pedestrians, and the motion of the chains appeared to be easily accelerated. In 18 18, six months after the completion of the bridge, it was destroyed by a violent gale. Around 1821, the French architect Poyet7 suggested hanging the beams up to rather high towers with wrought iron bars. Jn this system he proposed using a fan-s haped arrangement of the stays, all being anchored at the mp of the tower (Fig. 1.9). Poyct's idea was further developed by the famous French engineer Navier who, in 1823, stud ied bridge systems stiffened by inclined chains8 (Fig. 1.1 0). By comparing both the weights of the deck and the inclined chains, Navier found that for a given span and height of the towers, the cost of both systems was approximately equal. Fig. 1.9

Fan type stayed bridge proposed by Poyet, France, 1821.

THE CA BLE-STAYED BRIDGE SYSTEM

l'

II

Fig. 1.10 Chain-stiffened bridge systems proposed by 'a vier, France, 1823

7

8

CABL£-STAYED BRIDGES

Fig. Lll

~.. """"~- ---- ...... ·-

"(EJ·

Bridge ~cross the Saale River, Germany, 1824

.

-~ -

fig. 1.12 Ti,•crton Bridge, England, 1837

Fig. 1.13

Harp type stayed bridge by Hatley, England, 1840

·==·"'

THE CABLE-STAYED BRIDGE SYSTEM

9

In 1824, a bridge was erected across the Saale River at Nienburg, Germany, with a 256ft (78.0 m) span and having the main girder stiffened by inclined members 9 . However, this bridge had excessive deflections under loading and the foUowing year it collapsed under a crowd of people because of failure of the chain-stays (Fig. 1. 11). 1837 Motley 10 built a bridge at Tiverton, England, a highly redundant double cantilever with straight stays (Fig. 1.12). The other type of stay arrangement, with parallel stays, now called harp-shaped, was suggested by Hatley 11 in 1840 (Fig. 1.13). He mentioned that this system provided less stiffness than the fan-shaped one. One interesting structure of the inclined-cable type is presented by the bridge over the Manchester Ship Cana.l 12 in England (Fig. 1.14). And in 1843, Clive 13 proposed an original system of a cable-stayed bridge, shown in Fig. 1.15. Fig. 1.14 The Manchester Ship Canal Bridge, England

;,. ___

t

,

- - - ---- -------- 106 -00

,

.

-- ----------)+(,I S

Fig. 2.9 Vie" ofS1 Florem Bridge, Fr3nce

t-

29'-9" 19' -II"

1 I

I -' N

2 1 -0 11

Fig. 2.10 S1 Floren! Bridge, gener3l arnngemenl

49

Fig. 2.11

View ofRees Bridge, Germany

2.1.5 St Floren! Bridge, France This bridge, built in 1969, crosses the south arm of the Loire River at Saint-Florent-le-Vieil, France 12 • 13 (Fig. 2.9). It has two spans each of 340ft (104m). The main support is formed by the river pier, which is surmounted by a portal type tower, supporting the stay cables arranged in a radial pattern on each side. The stay cables are disposed in the vertical plane of each side girder. They are arranged in three radiating bundles which converge at the top of the portal frame. Each bundle comprises two identical cables. The bridge deck is formed by two solid web side girders interconnected by cross-girders (Fig. 2.10). The deck plate is an all-welded structure extending in one continuous length of 682 ft (208 m) between abutments and supports the reinforced concrete deck slab.

2.1.6

Rees Bridge, Germany

This bridge, built near Rees in I 967, has a multi-cable system supporting the main span of 837 ft (255 m) and side spans each 341 ft (104 m) long14. 15. 16 (Fig. 2.11). The cables are formed in two planes outside the roadway and are supported from single towers built into the piers. The cable stay system is

·n PICAI. STEI-.1. IIRID(il·. of diameter in (7 mm)supporting a railway and dual lane road.

TI

2. 1.I-+ 1}ijm Bridge, Stveden This bridge across Askerofjord has a cable-stayed steel main span of 1200 ft (366 m), concrete approach \iaducts with spans of 37 1 f't 7 in (12-+ m) and -+79 ft II in (156.6 m), and concrete towers 30•31 (Fig. 2.26).

Fig. 2.26 \ 'icw of the Tjiirn Bridge, S"cdcn

The deck has a clear width of -+9ft (15.75 m) comprising a roadwa) of three lanes and a combined sidewalk and cycle track. The orrhotropic deck surfaced with asphalt is supported b) a \\elded rectangular bo' girder of width 27ft II in (8.5m) and height 9ft lOin (3m) and with f1oor beams at 13 ft (4 m) centres. The in (12 mm) deck plate of the box and its cantilever extensions as well as the bottom plate are stiffened by trapezoidal ribs and the webs by angles. The supcrstructme is longitudinally anchored at the abutment of the eastern viaduct, while transverse forces arc transmitted to the foundations at the towers and the abutments. The concrete towers ha,·e two parallel legs of constant cross-c;ection 13 X Hft 9in (-+ X 4.5 m), connected by cross-beams at two leYels. The two legs were cast simultaneously by slip-forming in two stages. Steel anchorages for the cables are attached in recesses at four le,·els of each tower leg (Fig. 2.27).

H

TYPICAL STEEL BRIDGES

51.7'

I

65

•

!, ..~.y

1

1·~-.1 ,

11,9'

27.9

II. 9

Main span 51.7'

Fru=TI9,,.j

[(

"'

.I

Side spans

Section B. B

Fig. 2.27 Tjiirn Bridge, general arrangement

*

The stay cables, located outside the superstructure, each consist of two strands of the locked coil type, of diameter 3 - 4 in (77 - 108 mm), with hot galvanized (1600 N/mm 2) wires. The strands are anchored individually to facilitate their replacement. Apart from the wind-tunnel tests carried out in connection with the design, the structural damping of the main span was determined by tests of the erected bridge by sudden release of static loading.

2.1.15 Kessock Bridge, Scotland The highway Kessock Bridge, finished in 1982, spans the navigation channel between Inverness Harbour and the Caledonian Canal 32 •33 . The I

66

CABLE-STAYED BRIDGES

navigation span is 782 ft (240 m) with a clear headroom of 95 ft (29 m). There are seven spans of 275-262 ft (84-80 m) on the south side and five spans of 197-262 ft (60-80 m) on the north side. The overall length of the bridge is 3450 ft (1052 m) (Fig. 2.28). In cross-section the deck is of the orthotropic type consisting of steel plate with longitudinal trapezoidal stiffeners and supported by two lOft 10 in (3.3 m) deep plate girders at the outer edges, with cross-girders at 13 ft (4 m) centers. The completed deck was surfaced with mastic asphalt. The total width of 71 ft 11 in (21. 92 m) between the parapets consists of a 6ft 5 in (1.95 m) central reservation, two 23 ft 11 in (7.30 m) wide roadways, two 2 ft 11 in (0.885 m) outer guidance strips and two 5 ft 11 in (1.80 m) footpaths. The two towers have a modified A-shape and extend 351 ft (107m) above the top of the concrete pier. Each leg of the tower is a nine-cell steel box tapered in both the longitudinal and transverse directions. High strength weathering steel has been used in the fabrication. 3450' between bearing centres

North E.

Navigational clearance

only at supports

77.15'

Fig. 2.28 Kessock Bridge, general arrangement

I

TYPICAL STEEL BRIDGES

67

Cable stays are in two planes in line with the tower legs. Three groups of stay cables arc attached to the top of each tower in a fan arrangement. The cables are rigidly connected to the cross-girders and to the tower tops. Each stay consists of two or four cables arranged about 2 ft (0.6 m) apart. Each cable is a bundle of ~ in (6 mm) diameter steel wires placed inside a polyethylene tube to protect against corrosion during slipping and erection. Once the full dead load was in place, the tubes were filled with a special cement grout as further protection.

2.1.16

lvlississippi Rirer Bridge, Luling, USA

This cable-stayed steel bridge spanning the M ississippi River in the delta area near New Orleans, at Luling, was opened for traffic in 1983 3·u s. This stretch of the river is navigable by ocean-going vessels requiring a horizontal clearance of 1200 ft (366 m) and a vertical clearance of 133 ft (40.5 m). The main span of the bridge was set as 1220 ft (372 m) and side spans of 508 ft (155 m) and 495 ft (151 m) together with adjacent spans of 259 ft (79 m) (fig. 2.29).

\ 'iew of the Luling 13ridge, USA

rig. 2.29

The deck carries four traffic lanes and a 2 ft 6 in (0. 7m) median barrier for a total width gutter to gutter of 79 ft (24 m). A steel orthotropic deck is supported by two longitudinal trapezodial box girders spaced 39 ft (I 1. 9 m) apart. The metalwork is high-strength weathering steel. The wearing surface is 2 in (57 mm) of epoxy asphalt. To eliminate the possibility of vortex shedding and to improve aerodynamic stability in steady wind, a fairing plate was added to the main span. The towers are internally stiffened box members with dimensions of 5 ft 3 in (1.60 m) in the transverse direction and 7 ft 3 in (2.20 m) in the longitudina l direction of the bridge, rising 138 ft (42 m) above the bridge deck (Fig. 2.30).

+

68

CABLE-STAYED BRIDGES

85'

Fig. 2.30 Luling Bridge, general arrangement

The cables are galvanized spiral bridge strands with maximum cable forces from 738 k (3280 kN) to 497 k (2210 kN), requiring a diameter up to 3.96 in (101 mm). At lower cable anchorages the vertical cable components are carried by cantilevers of the cross-girder. The horizontal cable components are carried by horizontal cantilevers at the level of the deck plate.

2.1.17

Faro-Folster Bridge, Denmark

This bridge, with symmetrical multi-cable fans and displaced end piers, was erected in 1985 35 - 38 , with a central span of951 ft (289m) and two

i. TYPIC\L STEEL BRIDGES

69

side spans each of 394ft (120m) (Fig. 2.31). The end pier is positioned under the anchor point of the second cable, 52 ft 6 in (16 m) from the end of the fan. Thus, the three outer cables of the side span can be regarded as forming the anchor cable. The steel deck has a box-type cross-section of trapezoidal form 73 ft: 6 in (22.4 m) wide at the top and of height 11.50 ft (3.5 m) reinforced by longitudinal box-type stiffeners. Both pylons are made of reinforced concrete, romboidal in shape. The pylon legs have a hollow cross-section and a hexagonal cross-section at the pier junction.

2.2 One-plane bridges In one-plane bridges, unsymmetrical loading on the roadway does not affect the cable load, but is resisted by the great torsional strength and

Fig. 2.31 Faro- Foister Bridge, general arrangement

70

CABLE-STAYED BRIDGES

rigidity of the closed box section, resulting in cable economy over a dualcable system. The chief structural element of the bridge structure, apart from its cables, is the central box girder, chosen for its great torsional strength and rigidity. It should be noted that the idea of applying a middle type main carrying system to bridges originated with Haupt, a German engineer. He proposed similar systems in 194839-42 .

2.2.1

North Elbe Bridge, Germany

In 1962, the North Elbe Bridge, the first bridge with single-plane cables, was built over the Elbe River in Hamburg. The central span is 565 ft (172 m) long and the flanking spans each measure 210 ft (64 m) 43 - 51 (Fig. 2.32). The central towers standing 174 ft' (53 m) above the deck support a star-shaped configuration of cables which gives the bridge an interesting appearance. Although the configuration cannot be justified from a purely economical viewpoint, a visually satisfying solution has been achieved, which is complemented by the increased height of the tower above the cable saddles. The bridge has a cross-section with a central box girder and two single web girders, one on each side, which are joined at about 72 ft (22 m) centers by transverse beams (Fig. 2.33). In the side spans, the bottom flange of the central box is replaced by diagonal bracing. The central box and side girders are about 10ft (3 m) deep, and the central box web and plate girder webs are equally spaced at 25.6 ft (7.80 m) centers.

2.2.2 Julirher Street Bridge, Dusseldorf; Germany The Jiilicher Street crossing, having spans of 105, 325 and 105ft (32, 99 and 32 m) was completed in 1963 (Fig. 2.34). The bridge consists of a 52 53 single box girder supported by a single-plane cable-stay system · . The towers, rising 53 ft (16.2 m) above the deck, are of a rectangular box section and are both longitudinally and transversely restrained between the two inner webs of the deck box girder. In cross-section, the main girder consists of a shallow 4ft 11 in ( 1.50 m) rectangular box supporting the cantilevers and incorporating a roadway, cyclist path and sidewalks (Fig. 2.35). The top and bottom plates of the main girder are stiffened by open type longitudinal ribs.

~!~~· Elbe Bridge, .

32 Two views of

Germany

72

CABLE-STAYED BRIDGES

37.8'

Ill

37.8'

I I

5. 8'

''['!! :y];:' { e:t I

Fig. 2.33 North Elbe Bridge, general arrangement

I· 2.2.3

25.6'

I

I· 25.6'

25.6' ... I

Leverkusen Bridge, Germany

This bridge, erected over the Rhine at Leverkusen in 1965, has a single twin cellular box girder from which sloping struts, which support the deck, cantilever out 54-56 . The river part of this structure comprises a continuous suspended box girder over three spans of 347, 920 and 347ft (106, 281 and 106m) (Fig. 2.36). Support is provided to the deck by a single-plane system of two sets of parallel cables dividing each of the side spans into two equal lengths and the center span into five parts. The towers rise 147ft (44.8 m) above the bridge deck, and are of rather unusual design in that they taper towards the base which is built into the deck structure and is supported on a hinged bearing. By tapering the tower, a reduction in the required width of the median strip is obtained, thus providing a substantial saving in cost. The steel roadway deck is stiffened with triangular shape box ribs, and the walkway is of reinforced concrete (Fig. 2.37). 2.2.4

The Wye River Bridge, England

The cable-stayed portion of the Wye River Bridge, built in 1966, consists of a 770 ft (235 m) central span and two 285 ft (87 m) side spans 57 •58 (Fig. 2.38). At each end of the central span a single box-section tower, 96 ft (29.3 m) high and pivoted at road level, supports a staying cable, anchored to the box girder 255 ft (78 m) either side of the tower. The cable passes into the box girder through a 13ft (3.96 m) wide median strip between the two lanes and is anchored into a system of diaphragms.

TYPIC \I . STI~F I . BRIDGES

Fit,!. 2.34 (a) View ofjlilicher Street Bridge, DUsseldorf, Germany

(b) The Jtilicher Street Bridge under construction

73

74

CABLE-STAYED BRIDGES

Fig. 2.35 The Jtilicher Street Bridge, general arrangement

223'-10" 532'-6"

The cable itself consists of 20 galvanized wire spiral strands each approximately 2! in (63.5 mm) in diameter. The strands are arranged in the form of a truncated equilateral triangle with a horizontal base, permitting the use of a simple saddle at the tower top and simplifying erection. The box girder section is 10ft 6 in (3.20 m) deep at the centerline with a slight fall to either side for run-off. The top of the trapezoidal section is 43ft (13.2 m) wide and the bottom flange plate is 32ft (9.76 m) in width (Fig. 2.39). The deck carries a 13ft (4.0 m) central section, dual24 ft (7.31 m) roadways, 12ft (3.66 m) cycle track and 12ft footpath. The box girder consists of stiffened steel plates with transverse diaphragms at 14ft (4.27 m) intervals. The top flange, which also forms the roadway decking, consists of steel plate stiffened by the trapezoidal longitudinal ribs. The bottom flange and web plates are stiffened by single-sided bulb flats.

2.2.5

Bonn-Nord Bridge, Germany

This bridge over the Rhine at Bonn, completed in 1967, has a stay system in the form of a multi-stringed harp comprising twenty cables strung one above the other Sm''' J!I!1Jn; >nn;,-;'}Phv;;;;;~J;;;;;

Fig. 2.76 (a) The principle for a prestressed cable support for a pontoon; (b) prestressed cable supports for a bridge over the Ganges River, India

·~

I .

_.,>,

(b)

When one pontoon is under a load ~ which is greater than the actual load for the bridge P, and at this position the bridge is stiffened by a prestressed anchor cable, then after deloading the cable is prestressed in tension. Under this loading the pontoon is effectively supported by a prestressed cable acting as stiff support. This principle was applied in 1912, during the construction of a pontoon bridge supporting a deck of rigid girders over the Ganges River, in Calcutta, lndia99 (Fig. 2.76 (b)). The bridge has three spans with a total length of 1407 ft (4 29 m) and a width of 98 ft (30 m). The bridge supports were built of eight cylindrically shaped pontoons, each one 226 ft (69 m) long and with a diameter of 15 ft 6 in (4.7 m). The pontoons were anchored to the bottom of the river by prestressed cables. It should be noted that the same principle of using post-stressed anchor cables may be applied to achieve stiffness during construction of oil rigs at sea. 2. 7.2

Proposed pontoon bridge, !ta(y

The principle discussed in the previous section was also applied in a . . a pontoon b n'd ge between s·ICI'1y an d c a1ab na . loo ' 101 , proJect proposmg Italy (Fig. 2.77).

(a

Fig. 2.77 A proposed pontoon bridge for crossing the sea strait between Sicilv and Calabria, Italy: (a) .view, (b) detail of supports

l

(b)

j~~:h·

fl/.? / I. I

""· "-..\~ \

,, 29' -5"

59"

I

65' -6"

59' 2 13. - 2"

11' -S"

0.81'

I

-ITI 11 I

r

5.2. 10

3. -3"

I I

0

I

II 3 '- 3"

Fig. 5 .I S Bundesallee Footbridge, general arrangement

Eric Harvie Bridge, Canada

T his cable-stayed precast prestressed concrete pedestrian bridge 18 over the Bow River in Calgary was completed in 1982 (Fig. 5.16) . The deck is composed of five precast pretensioned T -shaped box girders with a cross-section 10 ft 4 in (3.15 m) wide and 3 ft 7 in (1.10 m) deep. The structure is symmetrical with a central span of 262 ft 6 in (80 m) and side

Fig. 5. 16 View of the Eric llarvie Bridge, Canada

168

-

Vl

-'

w w >Vl

2.8

1.4

2000

Assuming numerical values for the span Land stresses CJ, it is possible to determine from formula (6. 9) different values of the ideal modulus E;, shown in the diagram (Fig. 6.14). Static calculations for the live load are based on an idealized modulus of elasticity E; which decreases as the length of cable increases. If the load on the sloping cable is increased, its sag is reduced and its ends move away from each other. Solely from this elongation of the chord, an apparent Young's modulus can be derived which increases with increasing load. This effect, together with the elastic deformation of the cable, can be used to calculate an idealized modulus of elasticity which is then introduced into the static calculations. In Fig. 6.14, this modulus is diagrammatically shown on the ordinate as a function of the cable stress, and the horizontal distance between the tower and the anchor of the stay cable is shown on the abscissa. For very long bridges the loss of E; can be as large as 40%. The economical limit for cable lengths for inclined cable systems is therefore between 658 and 987ft (200m and 301m). Nonetheless, longer lengths of cables could be subdivided by intermediate supports to avoid this disadvantageous effect, but it is debatable how far such a design could be made to look attractive. It is certain, however, that even with longer cables, the inclined cable bridge could still successfully compete with the conventional suspension bridge.

STRUCTURAL DETAILS

6.5

191

Permissible strength of the cables

Based on existing practice, the following permissible cable strengths may be suggested. The general safety factor of the cable may be taken asK= 2.5, following practice in North America and Europe. This coefficient represents the reserve of the strength of the cable with respect to the loading. Assuming that the dead load of the superstructure constitutes 60-70% of the total load, and the corresponding values of the live load are 30-40%, then the resulting coefficient of the reserve strength of the cable is

K Kres = - - - - - - - 0.65nl +0.35n 2 (1 + Jl)

(6.10)

where n1 = I. I = the coefficient of the overloading for the dead load n2 = I. 4 = the coefficient of the overloading for the live load 1+ J1 = 1.1 = the dynamic coefficient. Substitution of the above values into (3.10) yields

Kres =

0.65 X

2.5 ~ 2 1.1 +0.35 X 1.4 X 1.1

(6.11)

The calculated strength of the cable may be expressed as R

=

RauKm 1 m 2

(6.12)

where m 1 = 0. 8 is the coefficient of the performance of material in structure and m2 =

Kml Kres

=

1 = 0.78 0.8 X 0.8 X 2

However, some authors recommend decreasing the calculated strength of the cable by 5% to take into account the decrease of its bearing capacity under the transverse compression at supports. Assuming such a decrease in the allowable stress, we may use m 2 = 0.95 x 0.78 = 0.74

6.6 Fatigue tests and strength of the cables

l. FATIGUE TESTS German specifications require that the allowable working load for steelwire ropes shall be taken as 42% of their calculated breaking load. The effective safety factor against fracture or yielding is then 2. 4 or 1. 5 respec-

192

CABLE-STA YEO BRIDGES

~

''.l

~~v- '"""-"'""'"' '""' w --'

;;; 40

~~3 136

z w f-

w

I' I I I

0

f-

•'

«

:0::

- 20 >--

ALLOVo'0 =4~%. f B

0

f---·- f - - - ·

~ f---.

35,

I

I I

I

.

I I I I

I

--'

J14

:

I I

:0

0

0

'

' I

.._1 0

Fig. 6.15 Fatigue tests of cables

rr35 IIRJD(,J.'>

Fig. 6.29

frcyssinc1 anchor

dead load the anchor tube is filled with epo.Ky resin. Under live load the additional cable force will be trans form ed by shear from the cable strand to the tube.

2.

\\LI.TI-STR \1\1) \"\. V, /:>.Hand /:>.S may have positive or negative values. The first step in the design is the determination of the influence on the bridge system of the additional components /:>. V and !l.H, and also of the additional bending moments due to the forces V and H which result from the deviation of the girder axis from the initial position. The sum of the additional vertical forces should satisfy the condition 2: V = 0. By loading the influence lines by these additional vertical forces, we obtain the corrections !l.M, /:>.N and /:>.S to the first approximation for the bending moments and normal forces in the stiffening girder and stresses in the stays, and are able to determine the resultant additional deformations of the system.

272

CABLE-STAYED BRIDGES

In the same manner, we repeat the calculation in the second approximation, using the forces and deformations found in the first approximation. The final values of the forces in the system after n approximations are expressed as

M 9 = M+~M1+~M2+ .. ·+~Mn N 9 = N+~Nl +~N2+ .. ·+~Nn S 9 = S+~S1 +~S 2 + .. ·+~Sn

(8.86)

where

M,N,S

are the bending moments and normal forces in the stiffening girder and forces in the cables according to the preliminary design

~M, ~N, ~S

are the corresponding additional values, obtained after each approximation.

The series converge very fast, and generally it is enough to use only the first approximation.

References 1. Kireenko, B. 1., Cable-Stayed Bridges, Kiev, 196 7 (in Russian). 2. Anonymous, 'Rheinbrucke Bonn Nord', Tiefbau 29-40, 1968. 3. Gimsing, N.]., 'Multispan Stayed Girder Bridges,' Proc. ASCE Struct. Div., 1989-2003, October, 1976. 4. Gimsing, N.]., Cable Supported Bridges, Concept and Design, John Wiley & Sons, New York, pp. 194-210, 1983. 5. Leonhardt, F. and Zellner, W., 'Cable-Stayed Bridges-Report on Latest Developments', Canadian Structural Engineering Conference, Toronto, 1970. 6. Beyer, E., Nordbrucke Diisseldorj; Landeshauptstadt, Dusseldorf, 1958. 7. Kachurin, V. K., Bragin, A. V. and Erunov, B. G., Design of Suspension and Cable-Stayed Bridges, Moscow, 1971 (in Russian). 8. Smith, B. S., 'The Single Plane Cable-Stayed Girder Bridge: A Method of Analysis Suitable for Computer Use', Proc. Ins/. civ. Engrs, 37, 183-194, July, 1967. 9. Troitsky, M. S. and Lazar, B., Model Investigation of Cable-Stayed Bridges, Report No. 1, Sir George Williams University, October, 1969, pp. 22-26. 10. Feige, A., 'Stahlbri.ickenbau', Stahlbau, Handbuch for Studium und Praxis, Band 2, Stahlbau-Verlags-GMBH, Koln, 1964.

Chapter 9

Exact Methods of Structural Analysis

9.1

Methods of analysis

A cable-stayed bridge is a highly statically indeterminate structure in which the stiffening girder behaves as a continuous beam supported elastically at the points of cable attachments. Except in the case of a very simple cable-stayed bridge, a computer is necessary for the solution of this type of structure, its use being primarily in analysis rather than in design application. Computer programs are necessary to generate the influence diagrams for cable forces, stiffening girder, bending moments and shears, and tower and pier reactions. The computer is also required for the rapid solution of various parametric efforts and loadings that have to be taken into account in achieving a reasonably efficient design. Probably the most important problems are the determination of the optimum section of the stiffening girder section, and cable configuration and size. In a simplified approach to the solution, the structure is assumed to be a linear elastic system which may be analysed using the standard stiffness or flexibility method. Several general computer programs are available which use this approach, e.g. FRAN, STRESS, STRUDL. The nonlinear behavior of cables, whose sag varies with changing axial load, presents problems in the solution of the bridge system more complex than those of a structure oflinear behavior. A convenient method of accounting for the nonlinear behavior of the stay cable bridge system is to introduce the concept of a straight line chord member with a modified or ideal modulus of elasticity substituted for the actual cable member. The use of this concept allows the application of a plane frame computer program properly adapted to account for the nonlinearity by an iteration procedure. In the following sections are discussed different methods of analysis, based on computer applications, considering linear and nonlinear behavior of cable-stayed bridge systems.

274

CABLE-STAYED BRIDGES

9.2 The flexibility method To develop a computer program for a cable-stayed bridge, either the stiffness or the flexibility method, or both, may be applied. If the flexibility method is employed, bending moments at fixed and flexible supports should be chosen as redundants in order to obtain a well-conditioned, banded, flexibility matrix. In the following sections, the computer methods are discussed first for dead and live load and then for post-tensioning forces 1 .

9.2.1

Analysis due to the action of dead and live loads

(A) Linear analysis. Based on the flexibility method, a computer program for analysis of a cable-stayed bridge has been developed. The program reads input data regarding the geometry and sectional properties of the system and calculates the following: (a) Influence lines for bending moments, axial and shear forces, displacements and reactions. (b) Envelopes of maximum bending moments, axial and shear forces for the most critical combination of dead and live loads. The computer program applies to a bridge system having an overall geometry and supports as represented in Fig. 9 .1. The connections between towers and the stiffening girders are fixed and the cable-tower and cable-girder connections are hinged. For the system considered, the redundants have been chosen as shown in Fig. 9.2. The developed flow chart is shown in Fig. 9.3. In the flow chart, Steps 2 and 3 represent the statements required to read and store the geometrical and sectional properties of the system to be analysed. This data is employed in Step 4 to determine sin, cos, tan, and cot functions of the angles between the cables and the stiffening girder. Fig. 9.1 Cable-stayed bridge systems analysed by the flexibility method

Fig. 9.2 Selection of redundants for analysis due to dead and live loads

EXACT METHODS OF STRUCTURAL ANALYSIS

275

11

12

13

14

:;

15

! ,,

,,'!'I

·~I

16

i:

.;I,,

,, 17

18

,,:r 19

20

Fig. 9.3 Flow chart continued on next page

276

CABLE-STAYED BRIDGES

21

24

22

25

26

23

Fig. 9.3 continued

Steps 6 to 23 were developed on the basis of computer methods described by Gere and Weaver 2 • 3 • To determine influence-line ordinates for 67locations of the unit load (the intervals taken along the girder were one-fifth of the length of one member), Steps 6 to 23 are repeated in a DO loop 67 times. The total computer time required is 3 min 8 s. The output, Steps 16, 20 and 23, consists of influence coefficients for bending moments, axial and shear forces, reactions and displacements. The displacements calculated are shown in Fig. 9.4. Steps 24-26 determine envelopes of bending moments, axial and shear forces for the most critical combination of dead and live loads. Step 24 reads DL, the uniform distributed dead load and LL, the uniform distributed live load. Step 25 scans the matrix AMA of axial and shear forces and bending moments at member ends. The general form of AMA is

AMA =

AMA 1 , 1

AMA1,2

AMA1,6s

AMA 2 , 1

AMA 2 , 2

AMA 2 , 68

AMA 93 , 1 AMA 93 , 2

(9.1)

AMA93,6s

Columns 2-68 in (9.1) contain bending moments (lines 3, 6, ... , 93), axial forces (lines 1, 4, ... , 91) and shear forces (lines 2, 5, ... , 92) at member ends for 67 locations of the unit load along the stiffening girder. Column l contains member ends bending moments, axial and shear

EXACT METHODS OF STRUCTURAL ANALYSIS

277

~ 18

2Q

;

forces due to a uniform distributed load of 1 kips per linear foot along the stiffening girder. To obtain moment envelopes, each third line (columns 2-67) is scanned and all positive terms are accumulated successively in a column vector AMAP. The same is done for the negative terms which are added and stored in AMAN. The next operation is to multiply AMAP by (L x LL )/5 and the first column of AMA by DL/1000 and to add the results. This gives the final AMAP, that is the ordinates of the bending moments at bar ends due to the most critical combination of dead and live loads. The same procedure is employed for AMAN and also for axial and shear forces. The total computer time required to calculate and print the envelopes is 15 s. The computer program has been written in USASI FORTRAN language for the Control Data Corporation (CDC) 3300 computer. This machine has 80k words of core storage (one word is equal to twenty-four bits) which represents a memory roughly equivalent to 320k bits on the IBM 360 series. The computer has full floating points and character hardware, eight disk drives with a total capacity of about 65 million characters, 5 tape units, 1 printer, 2 terminals, l card reader, one punch, one plotter and a multiplexor connected to the TWX network. (B) Nonlinear analysis. The nonlinearity of a cable-stayed bridge system is caused by large displacements, bending moments, interaction of axial forces and shortening of members due to bowing. Relations between stresses and strains at any cross-section are assumed to be linear. Analysis of plane frames which display the above type of nonlinearity has been studied extensively in the past decade 4 • 5 • 6 • 7 . Saafan 4 has developed a physical concept which allows nonlinear analysis by successive iterations of linear subroutines. The first step of the analysis determines a vector of displacements based on the initial geometry of the system and on the external loads. In the second step, an additional displacement vector, due to the difference between the joint loads and the resultants of internal bending moments and axial and shear forces at each joint, is determined. In performing the second step, the stiffness matrix of the system is assembled on the basis of the deformed geometry and of the axial loads determined in Step 1.

Fig. 9.4 Displacements

278

CABLE-STAYED BRIDGES

Each subsequent step, i, uses data determined in the previous step, i-1. The iteration stops when the last displacement vector obtained is a negligible fraction of the total displacement.

9.2.2 Calculation of post-tensioning forces After erection, the cable-stayed bridge is under the action of dead load only. The bending moments and deflections of the stiffening girder may be reduced by post-tensioning the cables. A procedure which permits the reduction of the maximum bending moment due to dead load may be programmed on a digital computer. The released structure will be chosen as shown in Fig. 9.5.

Fig. 9.5 Selection of cables as redundants

Fig. 9.6 Substructure for calculation of posttensioning forces

To determine unit displacements and bending moments due to unit loads applied along the cables, twelve substructures are considered. Each substructure consists of the original structure with one cable removed. Substructure No. 1 is represented in Fig. 9.6. The basic equations for this case are

(9.2) N{

N~

A1

A2

(9.3)

-=-==···

and a2.2X2+a2.3X3+···+a2.12X12 = A2.1+A2.2X1 a3,2X2+a3,3X3+ .. ·+a3,12X12

=

A3,1+A3,2X1

(9.4)

a12, 2 X2+ a12,3X3 +a12, 12 X12 = A11, 1+A12. 2X1 Equation (5.4) may be written in matrix form as [a]*{X} =

{A}

(9.5)

EXACT METHODS OF STRUCTURAL ANALYSIS

279

where

(9.6)

[a]=

{X} = {X2, X3, ... , X12} {A}= {A 1 }+{A2}X 1

(9.7) (9.8)

In (9.8)

(9.9) and

From (9.5) (9.11)

il',,

Relation (9.2) may be rewritten as

M,(C 0 -1) =

M; X 1 +(M,)*{X}

(9.12)

!'

(9.13)

i

where

Substituting {X} from (9.11) and taking account of (9.8), (9.12) becomes

xl may now be obtained from X _ M,(C 0 -1)-(M,)*[ar 1 *{A 1 } 1 -

M;+(M,)*[arl*{A2}

(9.15)

and X may be calculated from expression (9.11). With X 1 and X known, N{ may be determined from

N{ = N;+X 1 Nf+···+X 12 Nf 2

(9.16)

where i varies from 1 to 12. The above procedure makes it possible to obtain identical unit stresses in all cables. Another approach to the problem of post-tensioning the cables is to reduce all displacements due to dead load to a specified value. A procedure has been developed to achieve this reduction. The first

'I (.I,,

280

CABLE-STAYED BRIDGES

step is to determine the displacements, bending moments, axial and shear forces and reactions due to a unit force applied successively along each cable of the bridge. Then, a system of equations is written to express the condition that the sum of the displacements due to the unknown posttensioning forces in cables shall be opposite in sign to the displacements due to dead load and equal in absolute value to a fraction of these displacements. By solving this system of equations, the unknown post-tensioning forces are determined. Finally, the bending moments, axial and shear forces, displacements and reactions due to post-tensioning are determined from the information obtained initially by applying unit forces along each cable. A computer program has been written in the FORTRAN language based on the above principles. The structure considered is the same as for the analysis for dead and live loads (Fig. 9.1) This time, however, the cables had to be chosen as redundants (Fig. 9.5). The program consists of two parts. The first part contains the following steps: ( 1) The sectional properties and joint coordinates of the structure are read into the computer memory. (2) Sin, cos, tan and cot functions of the angles between the main girder and cables are calculated. (3) The matrices AMQ, BRQ, BMJ are calculated and stored in the computer. (4) Matrix FM is calculated and stored. The second part of the computer program contains the following basic steps: (1) Matrix F is computed. (2) Cable 1 is removed from the structure and the substructure shown in Fig. 9.6 is obtained. In this substructure, the displacements indicated in Fig. 9.4, the bending moments, axial and shear forces at all member ends, and the reactions due to a unit load applied along cable 1 are determined. The procedure of Step 2 in detail is as follows: The column corresponding to cable 1 in matrix AMQ is stored in AML. 2.2 The flexibility matrix F 1 of the substructure shown in Fig. 9.6 is obtained by removing the row and column corresponding to cable 1 from matrix F. 2.3 The matrix F 1 is modified by increasing all terms on the main diagonal of the first 12 rows of F 1 by LjEA. 2.4 The vector of displacements associated with the released sub2.1

EXACT METHODS OF STRUCTURAL ANALYSIS

281

structure I, DQL 1 , is calculated:

DQL 1

=

AMQi*FM 1 *AML

2.5

The vector Q 1 of the unknown redundants of the released substructure I is calculated :

2.6

The bending moments, axial and shear forces due to a unit load applied along cable I are calculated:

2.7

Steps 2.1 to 2.6 are repeated for alll2 substructures. Vectors AM 1AM 12 are stored in matrix AM. The matrix of displacements indicated in Fig. 9.4 due to unit loads acting along cables 1-12 is computed:

DJ= BMJT*FM"AM 2.8

The reactions due to unit loads acting along cables 1-12 are calculated:

(3) The post-tensioning forces in cables are determined so that deflections 1-10 and displacements 11-17 are reduced by a factor C 0 < 1. 3.1

3.2 3.3

Displacements due to the action of dead load are read into the computer. Vertical deflections 1-10 and horizontal displacements 11 and 17 are then multiplied by C0 and stored in vector AJ. Matrix JD is assembled from the first eleven rows and row 17 of matrix DJ. The post-tensioning forces in cables are determined from

~I I

H :',

:I ,,

\,I

j

1

X= JD- *AJ (4) Final bending moments, axial and shear forces, reactions and displacements due to post-tensioning are calculated from

AMP= AM 1 X 1 +···+AM 12 X12 DJF = D}1 X 1 +···+D}12 X 12 ARF

=

(9.17)

AR 1 X 1 +···+AR 12 X12

In (9.17), AM 1-AM 12 , DJ1-D}12 and ARcAR 12 are the corresponding column vectors in matrices AM, DJ and AR. The procedure developed above allows the determination of posttensioning forces to be applied in cables to reduce displacements due to dead load.

il

!Il I

282

CABLE-STAYED BRIDGES

9.3 Force-displacement method A method of linear analysis of cable-stayed bridges suitable for a com-

puter was proposed by Smith 8 • 9 in 1967. The analysis is essentially linear and assumes that the deflection is proportional to the load for all parts, as well as for the whole structure. In the following pages, the behavior under a load of single and doubleplane cable-stayed bridges is analysed. It is proposed that the behavior may better be appreciated by isolating and considering separately the rotation and shortening of the tower. Since this method includes displacements, as well as forces among the unknowns, it may be classified as a 'mixed force-displacement method'.

9.3.1

Analysis of the single-plane cable-stayed system

Let us consider a single-plane two-span cable-stayed bridge system (Fig. 9. 7). In the first stage of the analysis, the arbitrary load is applied to span AB, a hinge is introduced at the base of the tower and the elastic stretching of the cables taken into consideration. Considering the case when the cables and tower are fully rigid, the equations of compatibility are then

Vd:Ja+ VJab+ Vefae =Ad

(9.18a)

Vaha+ Vb./i,b+ Vehc

Ab

(9.18b)

V Jea+ VJeb+ V Jec = Ac

(9.18c)

=

where Va and Ve are the vertical components of the cable forces at D and E, Vb is the support reaction at B and./i,a is a flexibility coefficient for the deflection at B due to unit load at D, for the simple beam AC.

The flexibilities of the springs, or the vertical flexibilities of the cables, at D and E, are respectively

La

~~ = -A,-----E---,--,--;;2:-ll-=-

(9.19a)

e_ Le fe- A e E Sln . 2 8e

(9.19b)

a s1n ua

where A, E, L and 8 represent the cross-sectional area, modulus of elasticity, length and slope to the horizontal, of the cables, respectively. The tower flexibility is

H

IT= A E T

(9.20)

283

EXACT METHODS OF STRUCTURAL ANALYSIS

(a)

INFLEXIBLE LEVER

(b)

(d)

(f)

(c)

Fig. 9.7 Single plane cable-stayed bridge system: (a) Simple, stayed continuous girder; (b) Load applied: effect of tower rotation, tower and cables axially rigid; (c) Analogous structure for (b); (d) Effect of cable elasticity, tower axially rigid; (e) Analogous structure for (d); (f) Effect of tower elasticity; (g) Analogous structure for (f)

(e)

(q)

where His the height of the tower and AT its cross-sectional area. The rotation of the tower lowers the connections D and E by BD x ¢ and -BE x ¢, respectively. These additional deflection terms must be added to eqs. (9.18a) and (9.18c). Also, there are the increases Vaf'd and Vef~ in the deflections at D and E respectively, due to the flexibility of the springs. Therefore, in eqs. (9.18a) and (9.18c), the coefficient .faa must be modified to faa+ fa and lee to lee+J~. As the tower shortens, the anchorage of the cables connected to D and E drops by a further amount of ( Va+ Ve)fT. Therefore, the coefficients of Vd and Ve must be further modified by the addition of jT, the tower flexibility, to each of eqs. (9.18a) and (9.18c). Due to the angular rotation ¢ of the tower, the equation obtained by taking moments for the tower about its hinge is (9.21)

284

CABLE-STAYED BRIDGES

Therefore, the equations required for a solution are Vii:Ja+ fd+ fr)+ Vbfab+ Volfae+ fy)+ n+ZEI'(2eL cos {3-L 2 tan {3)Mn 1 . E ,(3eL2 cos {3-L 3 tan {3-6aL sm {3)Q, 6 1 1 + 6EI' (6e 2 L cos {3 + 3eL 2 sin {3- L 3 tan2 {3- 3eL2 tan {3

+

+ 6L cos {3)Nn

(9.40a)

EXACT METHODS OF STRUCTURAL ANALYSIS

wn+l

=

299

1 . f3 )Mn wn+L\LYSIS

325

where = = = = w;, w1, w, = a;, a 1 =

A;,A1 lh li S;, 1 u;, u1, u,

s

sectional areas of cable lengths of cable cable forces horizontal displacement vertical displacements slope angles of cable

For a pinned saddle (Fig. 9.40 (b)):

(9.77)

For a roller saddle (Fig. 9.40 (c)):

S,

(9.78)

S=.1

cos a; --:....5; cosa 1

In general, eqs. (9.76)- (9.78) can be written as: (9.79) where C 1 and C2 are matrices which are related to the cable characteristics. By combining eqs. (9.73), (9.74), (9.75) and (9.79), the redundant forces can be computed through the following equation:

CJiz + Cz(iz' hz- hz

-1

J

{CtYc} y,

(9.80)

where I is the unit matrix. Once the redundant forces have been solved, all the other displacements and internal forces of the structures can be determined.

326

CABLE-STAYED BRIDGES

The computer program developed can be used for solving plane frame cable-stayed structures, since it is only necessary to use the appropriate flexibility coefficients (for the frame member) in eq. (9.73).

9.7.3

Numerical example

Figure 9.41 (a) shows a plane frame cable-stayed bridge. Figure 9.41 (b) shows a three-dimensional bridge with a bridge deck width b = 40 m, such that the overall moments of inertia for all transverse sections are the same. Figure 9.41 (c) shows two different loading cases to be considered: a full uniformly distributed load and a partial uniformly distributed load over half of the deck. These loads are equivalent to the line load for the girder of the plane frame structure. Due to symmetry, only half of the bridge is analyzed, and the deck is divided into 56 shell elements. The cable forces and the deflections and bending moments of the mid-span are compared with the results of the plane frame analysis.

i if t II I J I I I J I I I I J II l 1!! t/m

Icl

Load

1

Load 2

Fig. 9.41 Examples of cable-stayed bridges

~:::::::::::::::4i t/m

2

Jl@llt lttlllll !IIIIBY'-=t-'0/b t/m I

200m

2

~

The results for the full uniformly distributed load (load 1) are symmetrical with respect to the longitudinal center line of the bridge, and the transverse variations of deflections and moments are quite small. Therefore, the bridge can be safely analyzed as a plane frame structure under such loads. On the other hand, the partial uniformly distributed load (load 2) is not symmetrical with respect to the center line of the bridge and the forces and displacements at symmetrical points are now different, with the discrepancies becoming larger with increasing bridge width. Figure 9.42 shows the influence surfaces (b = 40 m) which show that eccentric loads will induce significantly different responses at sym-

EXACT METHODS OF STRliCTliRAL Al'\ALYSIS

I

I

f (a) Axial

force in cable

327

-0.3 0.0 0.3

1

!i#llif!/llll..

1 ob .3o

(b)

Axial

force in cable 4

/~-'im;

(10.5) For the case of a nonlinear elastic cable-stayed bridge system, condition (6.5) may be disregarded, as it applies only when the nonlinearity is due to the behavior of the material itself, and not to that of the system. (2) Poisson's ratio for model and prototype must be equal, therefore the scale reduction factor for Poisson's ratio is (10.6) Equation (10.6) represents the condition that both the model and the prototype should be made of the same material. To neglect this condition is equivalent to neglecting the contribution of shear strain when we compute the magnitude of the elastic displacements. It may be pointed out here that the contribution of shear strain to the magnitude of elastic displacements in a mathematical model, when using

371

MODEL ANALYSIS AND DESIGN

classical computation methods, is difficult to take into account, as the force, displacement or energy equations become very complex. Also, if a computer program is employed, the stiffness matrix of the members contain more terms and hence more computing time is required to perform the calculations. However, if on small-scale structural models that satisfy similitude condition (10.6), moments and axial and shear forces are determined, the influence of shear strain is included in the value of elastic displacements without additional effort. (3) The model and prototype must be geometrically similar; therefore the scale reduction factor for length (10.7) is constant in all directions. Equation (10.7) represents the condition that the longitudinal and cross-sectional dimensions of the prototype and of the model have to be related to the same scale factor KL. This condition is extremely difficult to satisfy in practice. If the assumption is made that sectional properties are represented by I and A only, condition (10. 7) may be substituted on the basis of Buckingham's Pi theorem by and where A = area of cross-section I = moment of inertia of cross-section K 1 = scale reduction factor for moment of inertia of cross-section K A = scale reduction factor for area of cross-section

(4) If the temperatures of the model and prototype, as well as their coefficients of expansion are equal, the scale reduction factor is

(1 0.8) where

Ka = scale reduction factor for linear expansion K, = temperature of prototype divided by temperature model Equation (10.8) represents the conditions to be satisfied by the temperature and linear expansion scale factors. As such investigations are performed and data readings are recorded at constant temperature, and since the bridge system has no constraints which produce temperature stresses, this condition therefore may be disregarded.

372

CABLE-STAYED BRIDGES

(5) Deflection scale factor is equal to the length scale factor, therefore

up Urn

Lv Lm

(10.9)

where

KY = scale reduction factor for deflection. Condition (10.9) is valid for all cases of nonlinear elastic similitude. (6) The materials of model and prototype can be different, therefore

(JP EP -=E-= K(f= KE (Jm m

(10.10)

where K" = scale reduction factor for unit stress KE = scale reduction factor for moduli of elasticity Expression (10.10) indicates that the scale reduction factor for unit stress is equal to the scale reduction factor for modulus of elasticity. (7) The ratio of forces in model and prototype depends on both the length scale factor and the stress scale factor

EPL; _ EmL; pp pm or

EP (Lv)z = pv = K Kz = K

E

m

L

m

p

(I

m

L

p

(10.11)

where

KP

= scale reduction factor for concentrated load.

Equation (10.11) may be rearranged more conveniently, by using (10.10), as

KEKi

=

Kp

(8) Considering condition (10.11) and replacing with Pp and P111 as distributed load acting on a surface, we obtain

or

(1 0.12)

where

Kq = scale reduction factor for distributed load acting on a surface K" = scale reduction factor for unit stress Condition (1 0.12) may be disregarded as the distributed load applied on the bridge system is given in lb/in and not in lb/in 2 . It may be shown

MODEL ANALYSIS AND DESIGN

373

that if the distributed load is given in lb/in, the equivalent condition is

Kw

KEKL

=

where Kw is the scale reduction factor for the distributed load acting on a bar. (9) The ratio of densities of the prototype and model materials is given by the values of the length and stress scale factors

PP LP

aP

(10.13)

or

Equation (10.13) does not apply in the case of statical loads and may be disregarded. Consequently, for a nonlinear elastic cable-stayed bridge system subjected to vertical statical loading, the similitude conditions are

Ky

=

KL

(10.14)

KA

=

KI

(10.15)

KEL 2

=

Kp

(10.16) (1 0.17)

Kv

= 1

KI

=

Ki

(10.18)

KEKL

=

Kw

(10.19)

Employing the same principles that produced the above conditions, similitude requirements between bending moments, axial and shear forces acting on the prototype and bending moments, axial and shear forces acting on the model may be established as follows:

KM = KpKL

(10.20)

KP

(10.21)

KN = Kp

(10.22)

K0

=

Conditions (10.14)-(10.22) represent all similitude requirements to be employed for the model under statical forces. If eqs. (10.14)-(1 0.22) are satisfied, the model is a true model subject to the limitations stated earlier when discussing condition (10.7). If any one of eqs. (1 0.14-10.22) is not satisfied, the model is distorted. In practice, it is very hard to satisfy simultaneously conditions (10.15) and (1 0.18) because of the limited range of small sections commercially available. If condition (10.14) is not satisfied, the model ceases to display a nonlinear behavior. It may be shown, however, that for linear similitude,

374

CABLE-STAYED BRIDGES

this condition is not essential. If only information for influence lines is to be determined, condition (1 0.14) may be disregarded. Equations (10-14)-(10.19) inclusive are used for the design of the model. Equations (10.20)-(10.22) inclusive may be used to compare axial forces and bending moments determined on the model with axial forces and bending moments calculated on the prototype.

10.5

Sectional properties and geometry of the model

The following procedure represents a method to calculate the initial sectional properties for the design of a bridge system with a given geometry, if the sectional properties of a geometrically similar bridge system are known. The method is based on principles of similitude and is restricted to the case when both systems are geometrically similar and of the same material. It may be extended to systems made of different materials. The basic assumptions are

Ks = KAKL KI

=

KsKL

(10.23)

f,=fe where P1 is the uniform distributed dead load given by the nonstructural elements of the bridge system and fis the unit bending stress at extreme fibers. Suffix e applies to the parameters of the bridge system with known sectional properties and suffix n to those with sectional properties to be determined. Fromf,=fe, we obtain (10.24)

where M is the maximum bending moment in stiffened girder and Sis the corresponding section modulus. Substituting M = K 1rf} into (10.24), we have

Kt

PnL~ = PeL~

sn

(10.25)

se

In eq. (10.25), K 1 is a dimensionless constant depending on the location of loads and the type and location of supports, and L is the length of the main span. Equation (10.25) may be rearranged as

K = K I