PDF- -BRIDGE BASICS - Loudoun County Public Schools / Overview - Cable-Stayed Bridges, Theory and Design, 2nd ed.pdf

- tayed Bridges,

Theory and Design,

Description

Troitsky,

Copyright© M

Troitsky 1977,

- 1988 All rights reserved

- stored in a retrienl system,
- or transmitted,
- in any form or by any means,
- electronic,
- mechanical,
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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-Design and construction I

Title 624'

55 TG405

BSP Professional Books A division of Blackwell Scientific Publications Ltd Editorial offices: Osney Mead,

- 0865 240201) 8 John Street,

Australia Set by Cambrian Typesetters Printed and bound in Great Britain by Butler & Tanner Ltd,

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,

British Railways Southern Region

Compagnie BaudinChateauneuf,

Hobart,

I lay and

- nderson,

England

Fox and Partners,

Consulting Engineers,

Australia

Zanden,

- \'irola,

\lr Arvid Grant and Associates,

Modjeski and Masters,

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'

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

- 1 2 19 20 21 24 25 26
- 29 31 34 36 39

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

- 69 91 95 99 103 105 108

COl\TE:'\TS

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

- 114 139 143 144

Introduction Composite cable-stayed bridges References

- 147 148 154

Chapter 5 Typical Pedestrian Bridges 5

- 155 155 173

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

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

- 223 223 224 227 229 230

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

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

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

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

- 365 366 370 374 375 376 392 397

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

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

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

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

as well as pipeline and pontoon bridges

Typical Concrete Bridges contains additional data on new concrete structures

Chapter 4,

Typical Composite Bridges describes new deck types of cablestayed bridges

Chapter 7,

Methods of Structural Analysis presents a discussion on the structural behavior of bridges and methods of analysis

Approximate Structural Ana(ysis treats methods of preliminary analysis

Chapter 10,

Chapter 11,

Chapter 1

The Cable-stayed Bridge System

Introduction

- 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,

inclined cables passing over or attached to towers located at the main piers

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

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,

- 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

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

- 2 Historical review

- 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

- such as shown in Figs

2 and 1

- were builrl

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

- 2 Primithe bamboo bridge in Borneo Fig
- 3 (bdow) Primitive bamboo bridge: in Laos
- 4 Bamboo bridge with bamboo stays O\'Cr Serajoc Ri,er 10 Ja,a,

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

- see Fig
- t HI I \
- 5 Bridge stiffened by eye bars,

designed by Faustus Vcrantius,

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

- 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

- rwo British engineers,

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

- 6 All-timber bridge stiffened by inclined timber stays,

designed by Loschcr in Germany,

- ado"s Bridge,

England,

The system of inclined chains was adopted in a bridge built at Dry burgh Abbey across the Tweed River 6 in 1817

3 m) span,

- and was 4ft ( 1
- 2 m) wide (Fig

and the motion of the chains appeared to be easily accelerated

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

all being anchored at the mp of the tower (Fig

- in 1823,

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

- 10 Chain-stiffened bridge systems proposed by 'a vier,

France,

- 12 Ti,•crton Bridge,

England,

Harp type stayed bridge by Hatley,

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

- 1837 Motley 10 built a bridge at Tiverton,

England,

a highly redundant double cantilever with straight stays (Fig

- with parallel stays,
- now called harp-shaped,

was suggested by Hatley 11 in 1840 (Fig

One interesting structure of the inclined-cable type is presented by the bridge over the Manchester Ship Cana

- l 12 in England (Fig

And in 1843,

- shown in Fig
- 14 The Manchester Ship Canal Bridge,

- -------- 106
- ----------)+(,I S
- 9 Vie" ofS1 Florem Bridge,

29'-9" 19'

- 10 S1 Floren
- ! Bridge,
- gener3l arnngemenl

View ofRees Bridge,

- 5 St Floren
- ! Bridge,

- built in 1969,

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

It has two spans each of 340ft (104m)

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

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

- 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

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

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

concrete approach \iaducts with spans of 37 1 f't 7 in (12-+ m) and

- -+79 ft II in (156

and concrete towers 30•31 (Fig

- 26 \ 'icw of the Tjiirn Bridge,

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

while transverse forces arc transmitted to the foundations at the towers and the abutments

- ection 13 X Hft 9in (-+ X 4

connected by cross-beams at two leYels

TYPICAL STEEL BRIDGES

Side spans

- 27 Tjiirn Bridge,
- general arrangement

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

the structural damping of the main span was determined by tests of the erected bridge by sudden release of static loading

- 15 Kessock Bridge,

- finished in 1982,

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

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

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

- 3450' between bearing centres

North E

- only at supports
- 28 Kessock Bridge,
- general arrangement

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

- lvlississippi Rirer Bridge,

- 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

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

- \ 'iew of the Luling 13ridge,

- 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

To eliminate the possibility of vortex shedding and to improve aerodynamic stability in steady wind,

a fairing plate was added to the main span

- 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

- 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

This bridge,

with symmetrical multi-cable fans and displaced end piers,

- was erected in 1985 35

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

TYPIC\L STEEL BRIDGES

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

- 52 ft 6 in (16 m) from the end of the fan

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 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

- 31 Faro- Foister Bridge,
- general arrangement

rigidity of the closed box section,

resulting in cable economy over a dualcable system

- 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

North Elbe Bridge,

Germany

- 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

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

- one on each side,

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

- 80 m) centers
- 2 Julirher Street Bridge,

- having spans of 105,
- 325 and 105ft (32,
- 99 and 32 m) was completed in 1963 (Fig

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

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

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

- !~~· Elbe Bridge,
- 32 Two views of

- :' { e:t I
- 33 North Elbe Bridge,
- general arrangement

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

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

- 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

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 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

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

- 34 (a) View ofjlilicher Street Bridge,

(b) The Jtilicher Street Bridge under construction

- 35 The Jtilicher Street Bridge,
- general arrangement
- 223'-10" 532'-6"

- ! in (63
- 5 mm) in diameter

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

- 2 m) wide and the bottom flange plate is 32ft (9
- 76 m) in width (Fig

- 0 m) central section,
- dual24 ft (7
- 31 m) roadways,

12ft (3

- 66 m) cycle track and 12ft footpath

- 27 m) intervals

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

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

- 76 (a) The principle for a prestressed cable support for a pontoon

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

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

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)

each one 226 ft (69 m) long and with a diameter of 15 ft 6 in (4

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

a pontoon b n'd ge between s·ICI'1y an d'c a1ab na

- loo ' 101 ,
- proJect proposmg Italy (Fig
- 77 A proposed pontoon bridge for crossing the sea strait between Sicilv and Calabria,

- (b) detail of supports
- j~~:h·
- ""· "-

59' 2 13

-ITI 11 I

I S Bundesallee Footbridge,

- general arrangement

Eric Harvie Bridge,

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

- -shaped box girders with a cross-section 10 ft 4 in (3
- 15 m) wide and 3 ft 7 in (1

10 m) deep

- 16 View of the Eric llarvie Bridge,
- w w >Vl

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

which decreases as the length of cable increases

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

- 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)

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

the following permissible cable strengths may be suggested

following practice in North America and Europe

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

65nl +0

- 35n 2 (1 + Jl)
- where n1 = 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

5 ~ 2 1

The calculated strength of the cable may be expressed as R

RauKm 1 m 2

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

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

- we may use m 2 = 0
- 6 Fatigue tests and strength of the cables

FATIGUE TESTS German specifications require that the allowable working load for steelwire ropes shall be taken as 42% of their calculated breaking load

5 respec-

- ~~v- '"""-"'""'"' '""' w
- ~~3 136

- 20 >--

f---·- f

- 15 Fatigue tests of cables
- rr35 IIRJD(,J
- 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

TI-STR \1\1) \"\

Hand /:>

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

By loading the influence lines by these additional vertical forces,

- we obtain the corrections

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

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+

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

Kireenko,

- 196 7 (in Russian)

Anonymous,

'Rheinbrucke Bonn Nord',

Gimsing,

1989-2003,

John Wiley & Sons,

194-210,

Leonhardt,

- and Zellner,

Nordbrucke Diisseldorj

Landeshauptstadt,

Kachurin,

- and Erunov,

Design of Suspension and Cable-Stayed Bridges,

- 1971 (in Russian)

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

183-194,

- and Lazar,

Model Investigation of Cable-Stayed Bridges,

- ickenbau',

Exact Methods of Structural 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

- 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,

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

based on computer applications,

considering linear and nonlinear behavior of cable-stayed bridge systems

- 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

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

Analysis due to the action of dead and live loads

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

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

- 1 Cable-stayed bridge systems analysed by the flexibility method
- 2 Selection of redundants for analysis due to dead and live loads

EXACT METHODS OF STRUCTURAL ANALYSIS

- ,,:r 19
- 3 Flow chart continued on next page

CABLE-STAYED BRIDGES

- 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 16,

20 and 23,

consists of influence coefficients for bending moments,

- axial and shear forces,
- reactions and displacements

axial and shear forces for the most critical combination of dead and live loads

the uniform distributed dead load and LL,

the uniform distributed live load

AMA1,6s

AMA 2 ,

1 AMA 93 ,

AMA93,6s

- 1) contain bending moments (lines 3,
- axial forces (lines 1,
- 91) and shear forces (lines 2,
- 92) at member ends for 67 locations of the unit load along the stiffening girder

- axial and shear

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 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

- bending moments,

interaction of axial forces and shortening of members due to bowing

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

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

- 4 Displacements

uses data determined in the previous step,

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

- 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

- 5 Selection of cables as redundants
- 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

- 1 is represented in Fig

- -=-==···

3X3+···+a2

12X12 = A2

- 2 X2+ a12,3X3 +a12,
- 12 X12 = A11,
- 2X1 Equation (5
- 4) may be written in matrix form as [a]*{X} =

{X} = {X2,

From (9

- 2) may be rewritten as

- 11) and taking account of (9
- 12) becomes

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

- -1)-(M,)*[ar 1 *{A 1 } 1
- +(M,)*[arl*{A2}

and X may be calculated from expression (9

With X 1 and X known,

- 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

The first

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

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

- 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

- 1) This time,
- however,

the cables had to be chosen as redundants (Fig

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,

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

- (3) The matrices AMQ,

(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

- 6 is obtained

the displacements indicated in Fig

- 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 The flexibility matrix F 1 of the substructure shown in Fig
- 6 is obtained by removing the row and column corresponding to cable 1 from matrix F
- 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
- 4 The vector of displacements associated with the released sub2

EXACT METHODS OF STRUCTURAL ANALYSIS

- structure I,

DQL 1 ,

- is calculated:

AMQi*FM 1 *AML

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

Steps 2

- 6 are repeated for alll2 substructures

The matrix of displacements indicated in Fig

- 4 due to unit loads acting along cables 1-12 is computed:

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

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

- axial and shear forces,

reactions and displacements due to post-tensioning are calculated from

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,

The procedure developed above allows the determination of posttensioning forces to be applied in cables to reduce displacements due to dead load

- 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'

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

Vaha+ Vb

- /i,b+ Vehc

V Jea+ VJeb+ V Jec = Ac

where Va and Ve are the vertical components of the cable forces at D'and E,

/i,a is a flexibility coefficient for the deflection at B due to unit load at D,

- for the simple beam AC

or the vertical flexibilities of the cables,

- at D'and E,
- are respectively
- -A,-----E---,--,--

2:-ll-=-

- e_ Le fe- A e E Sln
- a s1n ua
- where A,

- modulus of elasticity,

length and slope to the horizontal,

- of the cables,
- respectively

EXACT METHODS OF STRUCTURAL ANALYSIS

INFLEXIBLE LEVER

- 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)

where His the height of the tower and AT its cross-sectional area

-BE x ¢,

- respectively

- 18a) and (9

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

due to the flexibility of the springs

- 18a) and (9
- the coefficient

faa must be modified to faa+ fa and lee to lee+J~

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
- 18a) and (9

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

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

- 6 1 1 + 6EI' (6e 2 L'cos {3 + 3eL 2 sin {3- L'3 tan2 {3- 3eL2 tan {3
- + 6L cos {3)Nn

- f3 )Mn wn+L\LYSIS
- where = = = = w

,A1 lh li S

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

40 (b)):

For a roller saddle (Fig

40 (c)):

76)- (9

- 78) can be written as: (9
- 79) where C 1 and C2 are matrices which are related to the cable characteristics

75) and (9

the redundant forces can be computed through the following equation:

CJiz + Cz(iz' hz- hz

- 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

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

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

- 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

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

- 41 Examples of cable-stayed bridges
- ~:::::::::::::::4i t/m
- [email protected] lttlllll

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

- f (a) Axial
- force in cable
- !i#llif
- force in cable 4
- 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

classical computation methods,

is difficult to take into account,

- as the force,

displacement or energy equations become very complex

if a computer program is employed,

t

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