Cable stay bridge design

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Cable stayed bridges

D J Farquhar Mott Macdonald

Following a brief history of cable stayed bridges this chapter describes the various

materials and forms of construction that have been adopted for the major structural

components of these bridges, focussing in turn on the cable system, the pylon and

the deck By the use of examples the most appropriate use of these materials and

component forms is discussed A step-by-step approach is given for the preliminary

design of the cable stayed bridge from outline proportions of the structure to the

static and dynamic analysis including requirements for erection calculations and wind

loading on stays The dynamic behaviour for the cable stayed bridge includes the

phenomenon of stay oscillation, which is reviewed in detail including discussion of the

various types of dynamic cable response together with the available preventative

measures

Introduction

The use of inclined stays as a tension support to a bridge

deck was a well-known concept in the nineteenth century

and there are many examples, particularly using the

inclined stay as added stiffness to the primary draped

cables of the suspension bridge Unfortunately, at this

time, the concept was not well understood As it was not

possible to tension the stays they would become slack

under various load conditions The structures often had

inadequate resistance to wind-induced oscillations There

were several notable collapses of such bridges, for example

the bridge over the Tweed River at Dryburgh (Drewry,

1832), built in 1817, collapsed in 1818 during a gale only six

months after construction was completed As a result the use

of the stay concept was abandoned in England

Nevertheless, these ideas were adapted and improved by

the American bridge engineer Roebling who used cable

stays in conjunction with the draped suspension cable for

the design of his bridges The best known of Roebling’s

bridges is the Brooklyn Bridge, completed in 1883

The modern concept of the cable-stayed bridge was first

proposed in postwar Germany, in the early 1950s, for the

reconstruction of a number of bridges over the River

Rhine These bridges proved more economic, for moderate

spans, than either the suspension or arch bridge forms It

proved very difficult and expensive in the prevailing soil

conditions of an alluvial floodplain to provide the gravity

anchorages required for the cables of suspension bridges

Similarly for the arch structure, whether designed with

the arch thrust carried at foundation level or carried as a

tied arch, substantial foundations were required to carry

these large heavy spans By comparison the cable-stayed

alternatives had light decks and the tensile cable forces

were part of a closed force system which balanced these

forces with the compression within the deck and pylon

Thus expensive external gravity anchorages were not

required The construction of the modern multi-staycable-stayed bridge can be seen as an extension, for largerspans, of the prestressed concrete, balanced cantileverform of construction The tension cables in the cable-stayedbridge are located outside the deck section, and the girder is

no longer required to be of variable depth However, theprinciple of the balanced cantilever modular erectionsequence, where each deck unit is a constant length anderected with the supporting stays in each erection cycle, isretained

The first modern cable-stayed bridge was the mund Bridge (Wenk, 1954) in Sweden constructed by thefirm Demag, with the assistance of the German engineerDischinger, in 1955 At the same time Leonhardt designedthe Theodor Heuss Bridge (Beyer and Tussing, 1955)across the Rhine at Dusseldorf but this bridge, alsoknown as the North Bridge, was not constructed until

Stroms-1958 The first modern cable-stayed bridge constructed inthe United Kingdom was the George Street Bridge overthe Usk River (Brown, 1966) at Newport, South Waleswhich was constructed in 1964 These structures weredesigned with twin vertical stay planes The first structurewith twin inclined planes connected from the edge of thedeck to an A-frame pylon was the Severins Bridge (Fischer,1960) crossing the River Rhine at Cologne, Germany Thisbridge was also the first bridge designed as an asymmetricaltwo-span structure

The economic advantages described above are valid tothis day and have established the cable-stayed bridge inits unique position as the preferred bridge concept formajor crossings within a wide range of spans The long-est-span cable-stayed bridge so far completed is theTatara Bridge in Japan with a main span of 890 m At thetime of writing (2008) several other bridges are planned,

or are in construction, with main spans in excess of

1000 m, notably the Sutong Bridge (1088 m) and theStonecutters Bridge (1018 m), which are both in China

CONTENTS Introduction 357 Stay cable arrangement 358 Stay oscillations 364

Stay cable arrangement

Two basic arrangements have been developed for the layout

of the stay cables:

1 the fan stay system (including the modified fan stay

system)

2 the harp stay system

These alternative stay cable arrangements are illustrated in

Figure 1

Fan cable system

The fan system was adopted for several of the early designs

of the modern cable-stay bridge, including the Stromsmund

Bridge (Wenk, 1954) The method of supporting the stays

on top of the pylon was taken from suspension bridge

tech-nology where the cable is laid within a pylon top deviator

saddle The floor of the saddle is machined to a radius so

that each cable stay anchored in the main span can pass

over the pylon and be anchored directly within the back or

anchor span This arrangement is structurally efficient with

all the stays located at their maximum eccentricity from

the deck and a minimum moment is applied to the pylon

The fan arrangement initially proved suitable for the

moderate spans of the early cable-stay designs, with a small

number of stay cables or bundled cables supporting the

deck There were, however, obvious difficulties, with the

corrosion protection of cables at the pylon head, their

sus-ceptibility to fretting fatigue arising from bending and

hori-zontal shear stresses within the cable bundle and with the

replacement of any individual stay in the event of damage

In addition, when this arrangement was adopted for larger

spans the size of the limited number of cables increased,

eventually becoming uneconomically large and difficult to

accommodate within the fan configuration The anchorageswere also heavy and more complicated and the deck needed

to be further strengthened at the termination point

Therefore when a greater number of stays were requiredthe modified fan layout was introduced whereby the staysare individually anchored near the top of the pylon This

is now the more commonly adopted system In order togive sufficient room for anchoring, the cable anchorpoints are spaced vertically at 1.5–2.5 m Providing theanchor zone is maintained close to the pylon top there islittle loss of structural efficiency as the behaviour of thecable system will be dominated by the outermost cablewhich is still attached to the top of the pylon and anchored

at the supported end of the back span The advantages ofthis arrangement are as follows

n The large number of stays distribute the forces with greateruniformity through the deck section, providing a continuouselastic support Hence the deck section can be both lighterand simpler in its construction

n As each stay supports a discrete deck module, each module can

be erected by the progressive cantilever method without resort

to any additional temporary supports Thus increased speedand efficiency of the deck erection is possible

n The concentrated forces at each anchor point are much reduced

n With the modified fan layout it is also possible to completelyencapsulate each stay, thus giving a double protective systemthroughout its length and, should damage occur, replacement

of the stay can be undertaken as a routine maintenance task

n The large number of stays of varying length and naturalfrequency increases the potential damping of the structure

Freyssinet International has recently reintroduced theconcept of a deviator saddle at pylons in conjunction with

a modified strand system, for use in smaller-span stayed bridges and extradosed bridges The modifiedstrand, known as Cohestrand1, is protected by a poly-ethylene sheath but is filled internally with polymer resininstead of petroleum wax The resin compound is hydro-phobic, resistant to water vapour and oxygen and is capable

cable-of transferring both compression and shear forces from thepolyethylene sheath to the steel wires of the strand Thestrand can thus be continuous through the deviatorsaddle without the need to remove the polyethylenesheath This enables more slender pylons to be constructedwithout having to provide a cross-over stay arrangement.The disadvantages of earlier saddle designs have beenaddressed in that the corrosion protection of each strand

is continuous through the saddle, individual strands arenot in contact and thus not subject to fretting corrosionand the system is replaceable strand by strand The deviatorsaddle is made of a bundle of tubes placed within a largersteel saddle tube All voids between the tube bundles arefilled with a high-strength fibre concrete in the factory If

(a)

(b)

(c) Figure 1 Alternative stay cable arrangements: (a) fan stay system:

(b) modified fan stay system; and (c) harp stay system

necessary, the external surface of the polyethylene sheath

can be locally treated, at the saddle location, to ensure

that the friction coefficient between the saddle and the

strand is greater than 0.5 The arrangement of the strands

and saddle is illustrated in Figure 2

This arrangement of multi-tube saddle and Cohestrand1

has been incorporated in a number of bridges worldwide: in

Malaysia, Vietnam, Korea, Lithuania and the Sudan A

typical design is that of the Sungai Muar Bridge, a 132 m

main span cable-stayed bridge in Malaysia

Harp cable system

With the harp system the individual stays are anchored at

equal spacing over the height of the pylon and are placed

parallel to each other This arrangement provides a visual

emphasis of the flow of forces from the back span to the

main span and, in examples that are well proportioned, is

aesthetically pleasing However, the arrangement is not as

structurally efficient as the fan layout and relies on the

bending stiffness of the pylon and/or deck for equilibrium

under non-symmetrical live loading When loading one

end only of the stay system the load may be divided into

symmetrical and antisymmetrical components of loading

The symmetrical loading will be resisted by the triangle of

forces formed by the stays, pylon and deck but the

anti-symmetrical loading can only be resisted by bending of

the deck, the pylon or a combination of both depending

on their relative stiffness This disadvantage can be

over-come by anchoring the back stay cable at approach pier

locations so that any unbalanced load is resisted by the

pier An elegant example of this arrangement is the Knie

Bridge over the River Rhine at Dusseldorf with its single

pylon and 320 m main span

Multiple span bridges

The main concern with multiple-span cable-stayed bridges

is the lack of longitudinal restraint to the top of the inner

pylons, which cannot be directly anchored to an approach

pier Without providing additional longitudinal restraint amultiple-span structure would be subject to large deforma-tions under the action of live load Increasing the stiffness ofeither the pylons or the deck can provide this additionalrestraint However, increasing the deck stiffness will beaccompanied by an unacceptable increase in the deadload and thus, the more practical approach is to stiffenthe pylon A typical example of the stiffened pylon is theA-frame braced pylon shown in Figure 3(a) However,such an arrangement requires a substantial increase in thepylon materials and a much larger foundation An alterna-tive to increasing the bending stiffness of the pylon is theintroduction of an auxiliary cable system to provide theadditional stiffness and stability Two cable systems areillustrated The first system, in Figure 3(b), connects thetops of the pylons and thus directly transfers any out-of-balance forces to the anchor stays in the end spans Thesecond system, in Figure 3(c), connects the top of the inter-nal pylons to the adjacent pylon at deck level so that anyout-of-balance forces are resisted by the stiffness of thepylon below deck level An example of this latter arrange-ment can be seen with the design of the Ting Kau Bridge,Hong Kong, which is a four-span cable-stayed bridge.The disadvantage of such an auxiliary cable system is thatthe individual cables are very long and the large sag will

be visually dominant when compared with the adjacentstay cable plane Special measures are necessary to limitthe propagation of wind-induced oscillations in these verylong stays (see the section onStay oscillations)

Number of cable planesThe cable layout may be arranged as either a single-planesystem or as a twin-plane system

Outer steel tubes

The twin-plane system may either be formed as two vertical

planes connected from the edge of the deck to two pylon legs

located outside the bridge cross-section or as twin inclined

planes connected from the edge of the deck to either an

A-frame or inverted Y-A-frame pylon The A-A-frame pylon was

first adopted for the Severins Bridge (Fischer, 1960) The

ver-tical twin-plane system, with its tensioned stay geometry,

pro-vides considerable rigidity between the deck and pylon when

compared with the free-hanging cables of the suspension

bridge Inclined stays further increase the stiffness and

stability of the structure, with the stays and deck forming a

transverse frame Inclined stays are of particular benefit

when adopted for longer spans as they improve the torsion

response of the structure to both eccentric live load and

aero-dynamic effects When comparing the alternative stay

sys-tems, when supporting a deck with low torsional stiffness,

the inclined stay system connected to an A-shaped pylon

will have approximately half the rotation under eccentric

loading when compared with the vertical twin-plane

system The inclined stay system is also aerodynamically

superior, reducing the magnitude of vortex shedding

oscilla-tions and increasing the critical wind speed of the structure

However, the geometry of the inclined stay planes must be

carefully checked in relation to the traffic envelope and the

clearances required may result in an increase in the overall

width of the deck

The single-plane system creates a classic structural form

avoiding the visual interference often associated with

twin-cable planes However, the single plane is not able to

resist torsion loading from eccentric live loading and

there-fore this configuration requires the deck to be in the form of

a strong torsion box A deck section of this form is likely to

have excess resistance to the longitudinal bending of the

deck, particularly when a multi-stay arrangement is used

The single pylon has to be located within the central

median of the carriageway and as such an additional

width of deck is required to provide for the necessary

clearances to traffic

Two outstanding examples of cable-stayed bridges with a

single-stay plane are the Rama IX Bridge (Gregory and

Free-man, 1987) (see Figure 4) and the Sunshine Skyway Bridge

(Figure 5) The Rama IX Bridge crosses the Chao Phraya

River, Bangkok with a 450 m main span and has an

orthotro-pic steel box deck section which is 4 m deep The deck section

carries three lanes of traffic in each direction and is 33 m wide

The Sunshine Skyway Bridge crosses Tampa Bay, Florida

with a 366 m main span and a 4.27 m deep trapezoidal

concrete box deck section The deck section carries two

lanes of traffic in each direction and is 29 m wide

It is possible to combine the use of twin- and single-plane

arrangements in the single structure as incorporated into

the Rama VIII Bridge, Bangkok, completed in 2002

(Figure 6) This bridge has a single inverted Y-pylon with

twin inclined stay planes supporting a main span of

300 m, whereas the back span has a single stay planeanchored directly to a piled abutment

Stay designMany factors must be considered in the design of the staysystem including the characteristic breaking strength andthe effective stay modulus The proportion of the breakingstrength that can be realised depends on the relaxation ofthe stay under permanent loads The irreversible strainarising from relaxation increases rapidly when the perma-nent load in the stay exceeds 50% of the breaking load.The Post-Tensioning Institute (PTI) Recommendations(2001) limit, for normal load combinations, the maximumload in the stay to 45% of the stay breaking load and to50% for exceptional load combinations The French Inter-ministerial Commission on Prestressing Recommendationsfor Cable Stays (2002) limit the maximum load in thestay, at the serviceability limit state, to 50% of the staybreaking load and, at the ultimate limit state, to 70% ofthe stay breaking load This slightly higher loading is

Figure 4 Rama IX Bridge, Bangkok

Figure 5 Sunshine Skyway Bridge, Florida, USA (courtesy Parsons Brinckerhoff )

permitted providing detailing recommendations to limit

bending effects due to eccentricity at the anchorages and

cable stay vibration are incorporated into the design

The permissible load in the stay may also be limited by the

fatigue performance of the stay under repeated live load

cycles However, this will depend on the magnitude of the

live load cycles as a proportion of the permanent loads

Thus it is only the most heavily loaded stays of a highway

structure that are possibly limited by fatigue whereas the

stays for a railway structure where the live load has greater

dominance will be much more fatigue sensitive Fatigue

endurance of a component is usually given in a plot of the

stress range (
) against the number of load cycles (N)

known as the Wo¨hler curve When the Wo¨hler curve is

represented on a log–log scale the plot is represented by a

series of straight lines The PTI Recommendations relate

the design limit to the test acceptance criteria of each type

of stay material, such as strand, bar or wire, and the test

cri-teria of the assembled stay The recommendations for

par-allel wire strand (PWS) and parpar-allel strand are given in

Figure 7 The fatigue endurance of the assembled stay will

not only result from variations in the applied axial

tension but also be influenced by any secondary bending

in the stay, arising from either wind- or structure-induced

vibrations at the anchorage or bending of the stay in a

saddle The response to these factors is extremely important

though complex and varies according to the manufacturing

characteristics of the stay and its anchor Because of this the

PTI Recommendations propose that at least three

represen-tative samples of the stay assembly to be used in a project be

fatigue tested Testing is usually undertaken over two million

cycles The stress range depends on the generic type of stay

being tested but the upper limit of the stress range is always

taken as 45% of the breaking load Acceptance criteria for

the test are based upon a limit to the number of individual

wires in the stay that may break and that a tensile test,

undertaken after the fatigue test, achieves at least 95% ofthe guaranteed breaking load of the stay

The French CIP Recommendations for Cable Stays (2002)include the effect of coincident bending through a modifiedfatigue test whereby the cable specimen is made to deviatesinusoidally from the anchorage centreline at the samefrequency as it is being subject to an axial stress variation.Stay types

Problems arose with the stays of early cable stay bridges as

a result of deficiencies with the anchorage design, steelmaterial problems and inadequate corrosion resistance.The development of modern stay systems has largelyovercome these problems providing designs that minimisebending of the stay at the anchorage face and incorporate

a double corrosion protection system throughout able stay systems include:

Avail-n locked coil (prefabricated)

n helical or spiral strand (prefabricated)

Locked coil stays

Locked coilstays have been incorporated into many of theearliest cable-stay bridges The stays are factory produced

on planetary stranding machines, each layer being applied

in a single pass through the machine and contra-laidbetween each layer The core of the stay is composed

of conventional round steel wires while the final layers

Figure 6 Rama VIII Bridge, Bangkok (courtesy A Yee)

comprise Z-shaped steel wires which lock together creating

an extremely compact stay cross-section A typical example

of a locked coil stay is illustrated in Figure 8 Modern

locked coil stays provide all the wires in a finally galvanised

condition and will achieve a tensile strength of up to

1770 N/mm2 The stays are commonly anchored by

zinc-filled sockets although sometimes stays that are sheathed

with a polyethylene protection have their sockets filled

with epoxy resin The largest locked coil stays

manufac-tured to date are the 167 mm diameter stays supplied for

the Rama IX Bridge over the Chao Phraya River, Bangkok

Helical or spiral strand stays

Helical or spiral strandstays, which are illustrated in Figure

9, are also factory fabricated on a planetary stranding

machine similar to the locked coil stay but are entirely

manufactured from finally galvanised round steel wires

The wires are usually of 5 mm diameter with a tensile

strength of either 1570 N/mm2 or 1770 N/mm2 The largestspiral strand stays manufactured to date are 164 mmdiameter, as supplied for the Queen Elizabeth II Bridgeover the River Thames at Dartford

Bar bundlesBar bundles contain up to ten threaded steel bars with atensile strength of 1230 N/mm2 coupled together in 12 mlengths The bars have been conventionally placed within

a steel tube and protected with a cement grout The use ofcouplers connecting the bars will give a much reducedfatigue resistance when compared with the equivalent wire

or strand systems Coupled bar systems are thus rarely usedwhere significant variations in the stay load are likely tooccur Tests have also been undertaken to assess the effective-ness of cement grout as a protective medium These testsconcluded that transverse and longitudinal cracking of thegrout rapidly develops due to temperature effects, live loadstrains and wind vibration It may be assumed that thecement grout provides little protection against corrosion

Parallel wire strand stays

Parallel wire strand (PWS) most commonly comprises

7 mm diameter finally galvanised round steel wires with atensile strength of 1570 N/mm2 PWS stays may either beprefabricated or assembled on site, the wires being installedwithout a lay or helix within a polythene tube and injectedwith cement grout or wax When manufactured without alay the prefabricated stays are difficult to handle and coil

on to the reel and the system also suffers from the doubtsassociated with grouted stays

New parallel wire strand stays

The new PWS system, as illustrated in Figure 10, was oped with a tensile strength up to 1770 N/mm2 The stay isprefabricated and with a long lay helix to improve coiling

devel-on to the reel The largest stays can cdevel-ontain up to 400wires and a coating of high-density polyethylene (HDPE)

is applied in the factory using the continuous extrusionprocess The stays can be socketed using a patented

Figure 8 Locked coil cable (courtesy Bridon International Ltd)

Figure 9 Spiral strand cable (courtesy Bridon International Ltd)

system such as BBR’s DINA or HiAm anchorages The

individual wires within these anchorages incorporate

button heads transferring the full load to the anchor The

socket is then filled with a proprietary epoxy compound

that is claimed to enhance the fatigue resistance of the

stay Other manufacturers provide these stays with

con-ventional socketed anchorages filled with zinc or epoxy resin

Parallel strand stays

Parallel strand stays are usually manufactured from

15.7 mm (or 15.2 mm) diameter seven-wire strands, which

are usually galvanised and have a tensile strength of

1770 N/mm2, to give a characteristic breaking load per

strand of 265 kN Some manufacturers also supply strand

with a tensile strength of 1860 N/mm2 The strand bundle

can typically comprise up to 110 strands and anchoring is

by means of a pre-stressing anchor head with individual

strands gripped by wedges In order to provide adequate

fatigue resistance it is essential that rotation of the strands

at the face of the wedge grips, due to changes in stay load or

wind oscillation, is minimised Guides or dampers located

some distance in front of the anchor face provide the

necessary restraint (see the section on Stay flexure at the

anchorage) An outer polyethylene tube covers the strand

bundle, providing protection against impact damage and

prevents dynamic oscillation or rattling of the individual

strands Early designs filled the tube with cement grout or

wax as corrosion protection but in later designs each

strand is manufactured with a continuously extruded

HDPE coating over a corrosion-inhibiting petroleum

wax With this level of protection further cement or wax

injection is unnecessary European and Japanese practice

has been to use galvanised strands but some American

bridges have been constructed using epoxy-coated strand

The risk with epoxy-coated strand is that small pinholes

or minor damage can propagate corrosion, forming a

notch in the strand, which will create a local stress

concen-tration The use of Galfan, a licensed zinc and aluminium

mixture, gives two to three times the protection for the

equivalent weight of zinc coating, but the fatigue properties

of the strand are reduced

The outer covering to the stay has conventionally been

manufactured from steel or polyethylene pipe Where

polyethylene pipe is used, UV resistance was achieved by

the use of carbon black pigment in the material The

design temperature differential between the stays and the

deck or pylon can vary considerably The PTI

Recom-mendations (2001) note that values of 98C and 228C have

been used for white painted or taped stays and black

stays respectively The use of black stays is not preferred,

particularly in tropical zones where there is a high solar

gain Early attempts to wrap the stays, as in the case of

the Pasco-Kennewick Bridge over the Columbia River

USA, where a white plastic wrapping was used, were

unsuccessful as the coating deteriorated within a fewyears Later coverings using Tedlar, as in the case of theSecond Severn Bridge (Mizon et al., 1997), have beenmore successful Subsequently a range of light-colouredpolyethylene pipe has been developed with a high UV resis-tance The pipe is manufactured in a bi-extrusion processwhere a thin coating of light-coloured polyethylene isextruded over a black pipe core

Advanced composite stays

Advanced compositestays are manufactured from aromaticpolymide fibres, abbreviated to arimid, developed byDupont in the 1970s under the trade name Kevlar Kevlar

is manufactured in three grades and has an exceptionallygood strength-to-weight ratio The structural grade

‘Kevlar 49’ has a tensile strength in the range 3600–

4100 N/mm2 and a density of 14.4 kN/m3 Due to its goodresistance to corrosion the material has found favour inthe manufacture of rope for use in a marine/offshoreenvironment An experimental cable-stayed footbridgehas also been constructed in Aberfeldy, Scotland Thestructure, which has a main span of 63 m, utilises Kevlararamid stays, protected with a low-density polyethylenecoating For further information on the use of non-metallicstay systems reference should be made to the chaptertitled Advanced fibre polymer composite structural systemsused in bridge engineering

Stay behaviourThe behaviour of the stay under load must be represented inthe analysis of the structure The modulus of the stay underload is a characteristic of the stay manufacture and a non-linear variation with respect to both stay length and axialtension When comparing the modulus of various types ofstay that are manufactured, parallel wire strand (PWS)achieves the highest modulus at 205 kN/mm2 This isclose to the modulus of the steel wire itself Seven-wirestrand achieves a modulus of some 195 kN/mm2whilelocked coil will be approximately 155 kN/mm2 Themodulus of helical strands will be within the range 155–

175 kN/mm2 The modulus of both locked coil and helicalstrands is variable depending on the lay angle, the galvanis-ing and the stay diameter Factory-produced locked coiland helical strand cable, which are pre-stressed as part ofthe manufacturing process, will give the stays a predictableelongation in service Pre-stressing, where the cable issupported in a bed and preloaded, is the method used toremove the non-elastic stretch, resulting from the initialcompaction of the strand Pre-stressing is not required forPWS or parallel strand stays The non-linear behaviour ofthe stay may be represented by an equivalent modulustaking into account the sag or catenary effect in theloaded stay The variation in the equivalent modulus ofelasticity of the stay (Eeq) is given in Figure 11 and may

where E is the guaranteed modulus of the straight stay, L is

the horizontal length of the stay, is the specific weight of

the stay and is the tensile stress in the stay

Allowance must be made during the erection of the deck

for the dead load extension of the stay In the case of a

prefabricated stay this is achieved in manufacture, by

reducing the stay length to compensate for this extension

In the case of the stay fabricated on site, using a strand

system, the calculated extension of the stay is taken up

within the anchorage The length of the stays will also vary

with changes in temperature and it is therefore necessary to

measure the temperature of the stay and deck at the time of

stay installation and calibrate the stay load accordingly

Stay flexure at the anchorage

Some stay systems are designed with rotational adjustment,

through a pin and clevis arrangement, which assists with the

proper alignment of the stay during its erection However,

when the cable is subject to small angular deviations in

ser-vice the inertia of the anchorage components will inhibit the

cable end from rotating and thus the anchorage connection,

whatever arrangement is adopted, should be considered

fixed ended Angular deviations at the anchorage may

arise through a number of cumulative effects as follows:

n installation error when the anchorage is built into the structure

n vibration of the cable stay due to wind and other effects causing

an oscillating rotation at the anchorage

n structural displacements causing a varying rotation of the

anchorage with respect to the stay cable

n varying load in the stay cable due to the effect of imposedloading on the structure causing varying rotation of the staycable at the anchorage due to the catenary effect

When no guide is installed to limit the rotation of the staythe maximum bending is at the face of the anchorage anddecreases exponentially over a characteristic length, whichdepends upon the bending stiffness of the cable Thebending stresses at the face of the anchorage are high, arelocated where they are most damaging, and are typically

of the same order as those for live loading However, theyare identical whether the stay is monolithic, such as with

a spiral or locked coil strand, or is composed of separatetensile components that can slide over each other, such asparallel strand or parallel wire stays Monolithic staysvary from separate tensile components in that the charac-teristic length is much longer and this has to be consideredwhen designing any guide system

To limit these harmful bending effects a guide systemshould be provided at a distance from the anchor face Theguide, acting as a simple support, is usually located at theend of the anchor pipe that is an integral part of the bridgepylon or girder, so that the stay is subject to continuous bend-ing over the guide When adopting a rigid guide, subject to itbeing located a sufficient distance from the face of the anchor,the moment will be reduced by half the original moment atthe face of the anchor However, by manufacturing theguide from an elastic material, typically poly-butadiene,with an optimum spring stiffness designed for each individualcable arrangement, the bending stress in the cable can befurther reduced to a third of the original moment Thiseffect is illustrated in Figure 12

Stay oscillations

A phenomenon peculiar to cable-stay bridge construction isthe effect of stay oscillation During cantilever erection aslender deck may be particularly prone to wind-inducedmovement This can in turn excite the stays, producingviolent oscillations which, in some projects, have had to

Figure 12 Effect of guides on stay anchor bending

be restrained with temporary straps In service the

wind-induced vibration of stay cables has also occurred on a

range of cable-stayed bridges under a variety of wind and

traffic conditions Both standing waves and travelling

waves have been observed and these oscillations can reach

amplitudes of more than a metre This behaviour was

reported to have occurred with the Helgeland Bridge

(Svensson and Jordet, 1996)

A cable’s natural frequency for the fundamental mode

may be calculated from the following:

is the cable tension and m is the cable mass per unit length

(Note: It is normal to provide an anchor guide system

which gives an apparent fixity to the cable In this case

the chord length of the cable may be assumed to be the

length between guides at the top and bottom of the cable.)

Cable vibrations can occur as a result of a number of effects

which can be categorised under the following headings:

appropriate measures need to be planned to suppress cable

vibration due to these phenomena

Vortex shedding

This effect is due to the alternate shedding of vortices from

opposite sides of the strand, inducing a periodic load in the

cable As steady wind is required for this effect to occur, the

most damaging vibrations generally occur at low wind

velocities They can be mitigated by reducing or eliminating

the underlying excitation along the lengths of the stay

cables that drives stay oscillation by introducing

projec-tions, or texturing, of the stay pipe This roughens the

surface of the cable and presents an irregular surface to

the wind flow

Vortex shedding is expected to occur when its frequency

becomes approximately the same as the natural frequency

of a stay, that is to say, a wind velocity approaching the

reso-nance-inducing velocity In order to avoid this phenomenon

the French CIP Recommendations for Cable Stays (2002)

require that the natural frequency of stays should not be

the same as the vortex-shedding frequency described below:

where U is the wind velocity, D is the outer diameter ofthe stay and Stis the Strouhal number 0.20 for a circularcable

Wake-induced vibrationsWake-induced vibrations occur when the leeward stay lies

in the wake of a windward obstacle, such as another stay.This effect has most often occurred in moderate winds,which are not turbulent, and the effect is sometimes related

to rain or ice accretion The effect can be mitigated byimproving the cable system stiffness through tying thecables together using cross-cables These couple themodes of the different stays and thus stiffen the combinedcable system so that any excitation causes less oscillationand self-excited modes are avoided This method of mitigat-ing cable oscillation was employed on both the Normandyand Helgeland Bridges

Rain–wind instabilityRain–wind instability results from perturbing the smoothsurface of a stay and has occurred when water rivuletsform on the top and bottom of a stay, with wind in thedirection of the span, and for stays that slope downwards

in the direction of the wind, as illustrated in Figure 13.The French CIP Recommendations for Cable Stays (2002)note the results of research which show there is a possibility

of rain–wind excitation if the steady wind velocity is in therange 8–15 m/s The wind must be from an oblique direc-tion, between 308 and 808 from the perpendicular to thecable, so that the wind will tend to lift the cable Thefrequency of the cable oscillation is typically 1–3 Hz.Below the stated wind speed range, the top rivulet doesnot form because the wind is insufficient to prevent itfrom running down the side of the stay Above the range,the wind forces tend to blow the rivulets off the stay.Wind turbulence also prevents the rivulets from forming

In order to avoid the onset of rain–wind oscillation it hasbeen suggested that the Scruton number (Sc) for the stayshould be at least 10 according to the following formula

specified in the PTI Guide Specifications (2001):

m

where m is the mass per unit length of the stay,  is the

damping ratio (typically in the range 0.5–1%),  is the

density of air and D is the outer diameter of the stay

Rain–wind excitation is therefore unlikely to occur if the

damping ratio satisfies the above formula Where stay cable

pipes have effective surface texturing it has been suggested

that the above requirement may be relaxed such that

Sc55 However, this conclusion is based on limited testing

of regularly spaced stay arrangements and a study by

Jones et al (2003) recommends that a careful case-by-case

evaluation of the above limits be undertaken

Cable galloping

The possibility of galloping oscillations of either single

cables or groups of cables should also be investigated

The PTI Recommendations (2001) propose the following

equation to check if cable galloping occurs:

where c¼ 40 (PTI) and 35 (CIP) for circular cables

The French CIP Recommendations for Cable Stays (2002)

raise doubts as to the validity of the above formula;

how-ever, no other advice is given Clearly, as the wind speed

increases there could be instances of cable galloping

occur-ring, but usually increasing wind speed is accompanied by

the wind becoming more turbulent and in this state the

likelihood of galloping will be reduced

Parametric excitation

Parametric excitation can occur when the frequency of an

applied load, derived from either vehicular or wind

excita-tion of the girder/pylon, causes small vibraexcita-tions of the deck

or pylon cable anchorages which match a stay frequency or

any multiple (harmonic) thereof Stays with an oscillation

amplitude of several metres can occur although these

more pronounced effects are usually when the deck is

poorly streamlined, such as when twin I girders are

adopted The deck anchorages are usually the main driver

for such oscillations but it is possible for an unbraced

pylon to vibrate under wind loading The Øresund Bridge

for example, is reported to have experienced stay

oscilla-tions due to this phenomenon The new Stonecutters

Bridge in Hong Kong has 300 m high pylons which were

originally conceived as tapered tubular steel members

above the deck level However, the transverse frequency

of the pylons was found to be close to the cable frequencies

and consequently vulnerable to parametric oscillation

under longitudinal wind conditions The pylons were

modified to concrete construction, with a composite steelskin, in order to increase the pylon mass and hence their fre-quency It is therefore possible to set limits on the wind-induced motions and frequencies of the girder and pylon

so as to limit or preclude objectionable parametric tion of the stays, by ensuring separation between thedeck/pylon frequencies and the stay frequencies

excita-Rattling

A stay that is made up of a bundle of sheathed strandsexperiences aerodynamic interaction whereby the outerstrands will move in and out of the bundle and slap againstthe inner strands, eventually initiating a general motion ofthe whole cable The solution is to encase the strandbundle within an HDPE pipe and this forms an integralcomponent within all modern stay cable designs

Methods of damping stay oscillationsThere are three methods of damping stay oscillations:

1 incorporating internal and external damping mechanisms

2 texturing the external surface of the cable cover

3 installing stabilising cables

Dampers

The damping ratio of a cable stay is the sum of its intrinsicdamping, which will vary according to the type of stay andthe aerodynamic damping which increases with increasingwind speed The intrinsic damping of a cable stay is lowwith a typical range of 0.1–0.3% (logarithmic decrement

of 0.6–1.8%) Wind–rain instability can be avoided if thetotal stay damping exceeds 0.5% (logarithmic decrement

of 3.0%)

Various devices are available to supplement the cabledamping External dampers are hydraulic devices whichapply a transverse damping force to the stay and aremounted on structures which are fixed to the deck close

to the stay anchorages Internal dampers are ring shapedand placed between the stay and the steel anchorage tubewhich is built into the structure Internal dampers use thedistorsion of a dissipating material (specially formulatedneoprene) or viscous friction or dry friction The object of

a damper is to minimise the amplitude of any cable stayvibration However, unlike a guide, which would com-pletely inhibit cable displacement, a damper must permitsome displacement if it is to effectively dissipate energy

Texture of the stay pipe

Applying a texture to the external surface of the stay isalso an effective method of preventing wind–rain instability

as it helps prevent the rivulets forming long continuouslengths Initially on Japanese bridges longitudinal finswere adopted but when this type of profile is adopted thedrag coefficient increases dramatically to 1.35 Later helical

ribs, as illustrated in Figure 14, or a series of random

dimples were used and tests have confirmed that the

recommended drag coefficient of 0.7 is still appropriate

Tests have shown that the helical ribs are more effective

in preventing wind–rain instability than the random

dimples

Stabilising cables

Stabilising or secondary cables stiffen the stay cable array

and can also provide additional damping They are

how-ever of limited effectiveness in preventing wind–rain

instability but can successfully inhibit vertical vibration

modes of the stays arising from parametric excitation

They are less effective in reducing transverse vibration

modes Stabilising cables must be sufficiently pre-stressed

such that they do not de-tension under extreme variations

in structure loading Failures occurred in early designs,

with insufficient pre-stress, where they were subject to

repeated shock following unloading The trajectory of the

stay cables will be modified by the stabilising cables so

kinking of the stay cable at the anchorage will occur The

setting of the rigid anchor tubes at deck and pylon needs

to be corrected so that they follow the modified cable

geometry

Stabilising cables cannot therefore be easily used as a

remedial measure when cable vibration arises, as only

lim-ited pre-stressing of the stabilising cables is possible without

unacceptable modification of the stay geometry of the main

cables leading to anchorage kinks

Summary of preventive measures

Wind and rain instability should be avoided by the

inclu-sion of suitable texturing of the stay cable pipe together

with a minimum Scruton number between 5 and 10,achieved by the installation of suitable damping devices.Long stays (more than 80 m) should have dampersinstalled such that the stay has at least 0.5% damping.Parametric resonance should be assessed at the designstage, by a study of the eigenmodes of both the cablesand the structure When the cable and structure frequencyare close (within 20% of each other) the use of stabilisingcables should be considered

In order to avoid the visual concern of the user it isrecommended in the French CIP Recommendations forCable Stays (2002) that the amplitude of the stays shouldalso be limited Ratios of amplitude to stay length (L) inthe range L/1000 to L/1600 have previously been adopted.Pylons

The pylon is the main feature that expresses the visual form

of any cable-stayed bridge, giving an opportunity to impart

a distinctive style to the design The design of the pylonmust also adapt to the various stay cable layouts, accom-modate the topography and geology of the bridge site andcarry the forces economically

The primary function of the pylon is to transmit theforces arising from anchoring the stays and these forceswill dominate the design of the pylon The pylon shouldideally carry these forces by axial compression wherepossible, such that any eccentricity of loading is minimised.Steel pylons

Early cable-stay pylon designs were predominantly structed as steel boxes, and bridges such as the StromsmundBridge (Wenk, 1954) took the form of a steel portal frame,which was intended to provide transverse restraint to thestay system However, this restraint is largely unnecessary

con-as sufficient transverse restraint can be provided withinthe stay system itself When a single mast supports eachstay plane, any lateral displacement at the top of the mast

is accompanied by a rotation of the stay plane This tion of the stay plane ensures, for the simple fan layout ofstays attached to the top of the mast, that the resultantreaction from the main span and back span stays cableswill pass through the pylon foot The weight of the pylonwill remain vertical but the reaction from the stays willnevertheless be dominant Thus the effective length of themast in buckling will not be that of a simple cantilever,twice times the height (2H), but equal to the height (H).This effect is illustrated in Figure 15 With the harplayout of stays, the loads will be applied at various levelsalong the pylon but similar principles apply An example

rota-of the harp layout is the Theodore Heuss Bridge (Beyerand Tussing, 1955) where one slender strut supports eachcable plane When considering the buckling behaviour ofsuch a pylon, allowance must be made for constructional

Figure 14 Outer sheath for parallel strand stay with helical ribs (courtesy

BBRV Systems Ltd)

inaccuracies; typically an eccentricity of 100 mm of the stay

reaction is adopted

In the longitudinal direction the main and back stay

cables will restrain the pylon against buckling providing

the deck, to which the stays are anchored, is adequately

restrained against longitudinal movement The pylon

beha-viour when the deck is allowed to float is illustrated in

Figure 16 When the deck is unrestrained any disturbing

forces can only be resisted by the pylon acting as a

canti-lever with maximum bending at the base Thus the effective

length of the mast in buckling will be twice the height (2H)

When the deck is effectively restrained at either an

abut-ment or at one of the pylons the top of the mast is held in

position by the stays and the effective length of the mast

in buckling will be approximately 0.7 times the height

Con-necting the back stays to an independent gravity anchorage

is an equally effective solution Early cable-stay pylon

designs, such as for the Stromsmund Bridge (Wenk,

1954), incorporated a pin at the pylon foot so as to

ensure that the mast did not have to be designed for large

bending moments Later designs have adopted a fixed end

cantilever mast, which is simpler and also more stable

during erection Nevertheless, the effect of a frictionlessbearing can still be achieved with a fixed end mast providingthe member is sufficiently slender, such that the maximumaxial load approaches the buckling capacity of the mast

in free cantilever As the axial load increases to oneeighth of the buckling capacity, the location of the maxi-mum moment will move up from the base and the basemoment will tend to zero In this condition the mast willoffer no resistance against a longitudinal displacement atthe top In practice the adoption of a particular slendernessmay be limited by the need to maintain stability of the pylonduring construction when restraint from the stays is notavailable

For the single mast pylon supporting a single plane ofstays two methods have been used to connect the mast atits base

1 It can be constructed encastre into a transverse girderforming part of the deck In this case a bearing isrequired on top of the pylon foundation immediatelybeneath the mast and two further bearings at each end

of the transverse girder so as to provide the necessarytorsional restraint to the deck and pylon forces

2 Alternatively the mast can pass directly through thedeck to sit upon the pylon foundation In this caseonly bearings at each end of the transverse girder arerequired and these only have to provide the resistance

to the deck forces

Method 2 is the more efficient of the two and has beenuniversally adopted in more recent designs An example

of the use of this method of support is in the design of theRama IX Bridge (Gregory and Freeman, 1987)

Concrete pylonsConcrete is very efficient when supporting loads in axialcompression Advances in concrete construction andmodern formwork technology have made the use ofconcrete increasingly competitive for pylon construction,despite the much greater self-weight when compared with

a steel alternative Concrete has proved particularly ble to the more complex forms of pylon Many varied types

adapta-of pylon have been developed to support both the verticaland inclined stay layouts These include H-frame, A-frameand inverted Y-frame pylons as illustrated in Figure 17.With the H-frame pylon the stay anchors are normallylocated above the level of a crossbeam and with themodified fan arrangement of stays this crossbeam locationwould be between mid-height and two-thirds of the pylonheight above the deck When the harp arrangement ofstays is adopted the anchors are distributed over the fullheight of the pylon above the deck Therefore, as in thecase of the Øresund Bridge between Denmark andSweden, a crossbeam can only be provided, in practice,below the deck level

H

Figure 15 Transverse deflection of a simple pylon

Figure 16 Longitudinal restraint of the pylon by the anchor stays