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Theory and Design,

Description

CABLE-STAYED BRIDGES Theory and Design SECOND EDITION M

Troitsky,

DSc Professor of Engineering Concordia University,

Montreal

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

Copyright© M

Troitsky 1977,

No part of this publication may be reproduced,

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,

Cable-stayed bridges: theory and design

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

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,

institutions and organizations for supplying the information and photographs for the many bridges discussed in this book: Alaska Department of I lighways,

British Railways Southern Region

Compagnie Fran~aise D'Entreprises Metalliques,

Compagnie BaudinChateauneuf,

Landeshaupstadt Dusseldorf,

Germany

Department of Public Works,

Hobart,

Tasmania

I lay and

Consulting Engineers,

England

Kerensky,

Freeman,

Fox and Partners,

Consulting Engineers,

England

Germany

The Institution of Engineers,

Australia

Zanden,

Rijkswaterstaat Directie Bruggen,

Holland

Consulting Engineer,

Finland

Veraart,

Holland

Quebec Iron and Titanium Corporation

\lr Arvid Grant and Associates,

Consulting Engineers,

Modjeski and Masters,

Consulting Engineers,

Taylor,

Buckland and Taylor Ltd,

Civil and Structural Engineers,

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

Chapter 1 The Cable-stayed Bridge System 1

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

Chapter 2 Typical Steel Bridges 2

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

COl\TE:'\TS

Chapter 3 3

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

Chapter 4 4

Typical Concrete Bridges

Typical Composite Bridges

Introduction Composite cable-stayed bridges References

Chapter 5 Typical Pedestrian Bridges 5

Introduction Cable-stayed pedestrian bridges References

Chapter 6 Structural Details 6

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

Chapter 7 Methods of Structural Analysis 7

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

CONTENTS

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

Chapter 9 Exact Methods of Structural Analysis 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

345 361

Chapter 10 Model Analysis and Design 10

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

CONTENTS

Chapter 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

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

Author Index

Subject Index

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,

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,

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

Introduction

During the past decade cable-stayed bridges have found wide application,

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,

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,

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,

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,

CABU:-SfAYED BRIDGES

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

Egyptian sailing

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

In some tropical regions of the world primitive types of cable-stayed bridge,

2 and 1

Inclined vines attached to the trees on either bank supported a walk which was woven of vines and bamboo sticks

Indonesia

F igure 1

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

designed by Faustus Vcrantius,

THE CABLE-STAYED BRIDGE SYSTEM

Like all bridge designs of this epoch,

it exhibits many departures from what

a structural analysis would dictate

it contains the main features and basic principles of a metal suspension bridge stiffened by stays

In 1784,

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

In 1817,

Redpath and Brown,

built the King's ,\leadows Bridge5 ,

a footbridge in England which had a span of approximately II 0 ft (33

using sloping wire stay cable suspension members attached to cast iron towers (Fig

designed by Loschcr in Germany,

King's i\lc

England,

CABLE-STAYED BRIDGES

Dryburgh Bridge,

England,

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,

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

Poyct's idea was further developed by the famous French engineer Navier who,

stud ied bridge systems stiffened by inclined chains8 (Fig

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

Fan type stayed bridge proposed by Poyet,

France,

THE CA BLE-STAYED BRIDGE SYSTEM

France,

CABL£-STAYED BRIDGES

Bridge ~cross the Saale River,

Germany,

England,

Harp type stayed bridge by Hatley,

England,

·==·"'

THE CABLE-STAYED BRIDGE SYSTEM

In 1824,

a bridge was erected across the Saale River at Nienburg,

Germany,

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

England,

a highly redundant double cantilever with straight stays (Fig

The other type of stay arrangement,

was suggested by Hatley 11 in 1840 (Fig

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

And in 1843,

Clive 13 proposed an original system of a cable-stayed bridge,

England

29'-9" 19'

View ofRees Bridge,

Germany

France This bridge,

crosses the south arm of the Loire River at Saint-Florent-le-Vieil,

France 12 • 13 (Fig

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

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

Rees Bridge,

Germany

This bridge,

has a multi-cable system supporting the main span of 837 ft (255 m) and side spans each 341 ft (104 m) long14

16 (Fig

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

IIRID(il·

of diameter in (7 mm)supporting a railway and dual lane road

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

and concrete towers 30•31 (Fig

The deck has a clear width of

-+9ft (15

The orrhotropic deck surfaced with asphalt is supported b) a \\elded rectangular bo' girder of width 27ft II in (8

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

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

TYPICAL STEEL BRIDGES

Main span 51

Fru=TI9,,

Side spans

Section B

The stay cables,

located outside the superstructure,

each consist of two strands of the locked coil type,

- 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

Scotland The highway Kessock Bridge,

spans the navigation channel between Inverness Harbour and the Caledonian Canal 32 •33

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

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

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

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

North E

Navigational clearance

TYPICAL STEEL BRIDGES

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

Luling,

This cable-stayed steel bridge spanning the M ississippi River in the delta area near New Orleans,

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

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

The deck carries four traffic lanes and a 2 ft 6 in (0

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

rising 138 ft (42 m) above the bridge deck (Fig

CABLE-STAYED BRIDGES

The cables are galvanized spiral bridge strands with maximum cable forces from 738 k (3280 kN) to 497 k (2210 kN),

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

Faro-Folster Bridge,

Denmark

This bridge,

with symmetrical multi-cable fans and displaced end piers,

with a central span of951 ft (289m) and two

TYPIC\L STEEL BRIDGES

side spans each of 394ft (120m) (Fig

The end pier is positioned under the anchor point of the second cable,

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

50 ft (3

Both pylons are made of reinforced concrete,

The pylon legs have a hollow cross-section and a hexagonal cross-section at the pier junction

unsymmetrical loading on the roadway does not affect the cable load,

but is resisted by the great torsional strength and

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,

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,

He proposed similar systems in 194839-42

North Elbe Bridge,

Germany

In 1962,

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

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,

which are joined at about 72 ft (22 m) centers by transverse beams (Fig

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

Dusseldorf

Germany The Jiilicher Street crossing,

The bridge consists of a 52 53 single box girder supported by a single-plane cable-stay system ·

The towers,

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

cyclist path and sidewalks (Fig

The top and bottom plates of the main girder are stiffened by open type longitudinal ribs

Germany

CABLE-STAYED BRIDGES

Leverkusen Bridge,

Germany

This bridge,

erected over the Rhine at Leverkusen in 1965,

has a single twin cellular box girder from which sloping struts,

The river part of this structure comprises a continuous suspended box girder over three spans of 347,

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

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

The Wye River Bridge,

England

The cable-stayed portion of the Wye River Bridge,

consists of a 770 ft (235 m) central span and two 285 ft (87 m) side spans 57 •58 (Fig

At each end of the central span a single box-section tower,

96 ft (29

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

TYPIC \I

STI~F I

BRIDGES

DUsseldorf,

Germany

(b) The Jtilicher Street Bridge under construction

CABLE-STAYED BRIDGES

The cable itself consists of 20 galvanized wire spiral strands each approximately 2

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

The top of the trapezoidal section is 43ft (13

The deck carries a 13ft (4

12ft (3

The box girder consists of stiffened steel plates with transverse diaphragms at 14ft (4

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

Bonn-Nord Bridge,

Germany

This bridge over the Rhine at Bonn,

has a stay system in the form of a multi-stringed harp comprising twenty cables strung one above the other Sm''' J

(b) prestressed cable supports for a bridge over the Ganges River,

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,

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

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

Proposed pontoon bridge,

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

Italy: (a)

59' 2 13

-ITI 11 I

II 3 '- 3"

I S Bundesallee Footbridge,

Eric Harvie Bridge,

T his cable-stayed precast prestressed concrete pedestrian bridge 18 over the Bow River in Calgary was completed in 1982 (Fig

The deck is composed of five precast pretensioned T

10 m) deep

The structure is symmetrical with a central span of 262 ft 6 in (80 m) and side

Assuming numerical values for the span Land stresses CJ,

it is possible to determine from formula (6

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

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

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,

the inclined cable bridge could still successfully compete with the conventional suspension bridge

STRUCTURAL DETAILS

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

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 =

65nl +0

I = the coefficient of the overloading for the dead load n2 = I

Substitution of the above values into (3

10) yields

5 ~ 2 1

The calculated strength of the cable may be expressed as R

RauKm 1 m 2

Kml Kres

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,

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

5 respec-

CABLE-STA YEO BRIDGES

I' I I I

- 20 >--

ALLOVo'0 =4~%

f---·- f

I I I I

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

TI-STR \1\1) \"\

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 /:>

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,

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

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 +

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

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,

Cable-Stayed Bridges,

Anonymous,

'Rheinbrucke Bonn Nord',

Tiefbau 29-40,

Gimsing,

'Multispan Stayed Girder Bridges,' Proc

ASCE Struct

1989-2003,

October,

Gimsing,

Cable Supported Bridges,

Concept and Design,

John Wiley & Sons,

New York,

194-210,

Leonhardt,

'Cable-Stayed Bridges-Report on Latest Developments',

Canadian Structural Engineering Conference,

Toronto,

Nordbrucke Diisseldorj

Landeshauptstadt,

Dusseldorf,

Kachurin,

Bragin,

Design of Suspension and Cable-Stayed Bridges,

Moscow,

'The Single Plane Cable-Stayed Girder Bridge: A Method of Analysis Suitable for Computer Use',

183-194,

Troitsky,

Model Investigation of Cable-Stayed Bridges,

Report No

Sir George Williams University,

October,

'Stahlbri

Stahlbau,

Handbuch for Studium und Praxis,

Band 2,

Stahlbau-Verlags-GMBH,

Chapter 9

Exact Methods of Structural Analysis

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,

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,

STRESS,

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

CABLE-STAYED BRIDGES

either the stiffness or the flexibility method,

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,

In the following sections,

the computer methods are discussed first for dead and live load and then for post-tensioning forces 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,

(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

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

The developed flow chart is shown in Fig

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,

and cot functions of the angles between the cables and the stiffening girder

EXACT METHODS OF STRUCTURAL ANALYSIS

CABLE-STAYED BRIDGES

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,

The displacements calculated are shown in Fig

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 1 ,

AMA1,6s

AMA 2 ,

AMA 2 ,

AMA 2 ,

AMA 93 ,

1 AMA 93 ,

AMA93,6s

Columns 2-68 in (9

Column l'contains member ends bending moments,

EXACT METHODS OF STRUCTURAL ANALYSIS

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,

1 printer,

one plotter and a multiplexor connected to the TWX network

The nonlinearity of a cable-stayed bridge system is caused by large displacements,

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,

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

CABLE-STAYED BRIDGES

Each subsequent step,

uses data determined in the previous step,

The iteration stops when the last displacement vector obtained is a negligible fraction of the total displacement

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

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

The basic equations for this case are

3X3+···+a2

12X12 = A2

2X1 a3,2X2+a3,3X3+

·+a3,12X12

A3,1+A3,2X1

EXACT METHODS OF STRUCTURAL ANALYSIS

{X} = {X2,

X12} {A}= {A 1 }+{A2}X 1

From (9

Relation (9

X 1 +(M,)*{X}

Substituting {X} from (9

xl may now be obtained from X _ M,(C 0

and X may be calculated from expression (9

With X 1 and X known,

N{ may be determined from

+X 1 Nf+···+X 12 Nf 2

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

CABLE-STAYED BRIDGES

step is to determine the displacements,

axial and shear forces and reactions due to a unit force applied successively along each cable of the bridge

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,

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

the cables had to be chosen as redundants (Fig

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

tan and cot functions of the angles between the main girder and cables are calculated

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

In this substructure,

the displacements indicated in Fig

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

EXACT METHODS OF STRUCTURAL ANALYSIS

DQL 1 ,

AMQi*FM 1 *AML

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

The bending moments,

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

Steps 2

Vectors AM 1AM 12 are stored in matrix AM

The matrix of displacements indicated in Fig

DJ= BMJT*FM"AM 2

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

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

X= JD- *AJ (4) Final bending moments,

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

AR 1 X 1 +···+AR 12 X12

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

CABLE-STAYED BRIDGES

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'

Analysis of the single-plane cable-stayed system

Let us consider a single-plane two-span cable-stayed bridge system (Fig

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

Vaha+ Vb

V Jea+ VJeb+ V Jec = Ac

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,

The flexibilities of the springs,

or the vertical flexibilities of the cables,

2:-ll-=-

L and 8 represent the cross-sectional area,

length and slope to the horizontal,

The tower flexibility is

IT= A E T

EXACT METHODS OF STRUCTURAL ANALYSIS

INFLEXIBLE LEVER

(b) Load applied: effect of tower rotation,

tower and cables axially rigid

(c) Analogous structure for (b)

(d) Effect of cable elasticity,

(e) Analogous structure for (d)

(f) Effect of tower elasticity

(g) Analogous structure for (f)

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 ¢,

These additional deflection terms must be added to eqs

there are the increases Vaf'd and Vef~ in the deflections at D'and E respectively,

due to the flexibility of the springs

Therefore,

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,

Due to the angular rotation ¢ of the tower,

the equation obtained by taking moments for the tower about its hinge is (9

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,

EXACT METHODS OF STRUCTURAL ANALYSIS

,A1 lh li S

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

For a pinned saddle (Fig

40 (b)):

For a roller saddle (Fig

40 (c)):

In general,

76)- (9

By combining eqs

75) and (9

the redundant forces can be computed through the following equation:

CJiz + Cz(iz' hz- hz

{CtYc} y,

Once the redundant forces have been solved,

all the other displacements and internal forces of the structures can be determined

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

Numerical example

Figure 9

Figure 9

such that the overall moments of inertia for all transverse sections are the same

Figure 9

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

!IIIIBY'-=t-'0/b t/m I

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

EXACT METHODS OF STRliCTliRAL Al'\ALYSIS

as it applies only when the nonlinearity is due to the behavior of the material itself,

(2) Poisson's ratio for model and prototype must be equal,

therefore the scale reduction factor for Poisson's ratio is (10

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,

MODEL ANALYSIS AND DESIGN

classical computation methods,

is difficult to take into account,

displacement or energy equations become very complex

if a computer program is employed,

t