análisis estructural del puente colgante de clifton situado en bristol

Análisis estructural del Puente Colgante de Clifton situado en Bristol, Reino

Unido.

Tomo1–Memoria

Trabajofinaldegrado

Titulación:GradoenIngenieríaCivil

Curso:2015/16

Autor:Toms,CameronIan

Tutor:PayáZaforteza,IgnacioJavier

Cotutor:—

Valencia,juniode2016

TrabajofinaldeGrado Toms,CameronIan

AnálisisestructuraldelPuenteColgantedeCliftonsituadoenBristol,ReinoUnido 1

Contents

Contents................................................................................................................1Abstract.................................................................................................................21.Introduction.......................................................................................................32.Objectives..........................................................................................................33.Methodology......................................................................................................34.HistoricalContext...............................................................................................4

4.1.Ahistoryofsuspensionbridges...............................................................................44.2.Graphicalmethodsofstructuralanalysis.................................................................6

5.TheCliftonSuspensionBridge............................................................................95.1.Thehistoryofthebridge.........................................................................................95.2.Analysisofexistingdocumentationanddescriptionofbridge...............................105.3Structuralanalysis..................................................................................................12

5.3.1Transversalgirders.................................................................................................125.3.2.Chains....................................................................................................................215.3.3.Othercalculations.................................................................................................27

6.Comparisonofanalyticalmethods...................................................................297.Criticalevaluationofthebridge........................................................................30

7.1.Themodernstructure............................................................................................307.2.Othersubmissionsfromthedesigncompetition....................................................31

7.2.1.‘Giant’sHole’design–Brunel...............................................................................317.2.2.GothicRevival–Telford........................................................................................317.2.3.Stonebeam–Burge..............................................................................................327.2.4.Combinedarch-suspensionbridge–Hill...............................................................32

8.Conclusion........................................................................................................349.References.......................................................................................................3510.Appendices.....................................................................................................37



AbstractThisprojectisanevaluationandanalysisoftheprincipalelementsoftheCliftonSuspension

Bridge inBristol,UK.Theanalysis isdonewithanumericalmethod solvedona computer

andagraphicalmethodcalledgraphicstatics.Themethodsandthedesigns forthebridge

are studied in theirhistorical context.Despite the fact that graphic statics ismore limited

thanthenumericalmethod, itproducesresultsthatagreewiththenumericalanalysisand

physical tests.Theerrorscausedbygraphicstaticsareanalysedanddiscussed.Theresults

showthat thebridgewaswelldesigned for theoriginaldesign loads,andthat thecurrent

restrictions on vehicles well suited. Finally the bridge and several alternative designs are

qualitatively evaluated in terms of the quality of design and their value in a social and

historicalcontext.

Resumen:Esteproyectoesunaevaluaciónyanálisisde loselementosprincipalesdelpuentecolgantedeCliftonenBristol,ReinoUnido.Elanálisissehaceporunmétodonuméricoenordenadoryunmétodográficoquesellamaestáticagráfica.Losmétodosylosdiseñosparaelpuenteseinvestiganensucontextohistórico.Apesardelhechodeque laestáticagráfica tienemáslimitaciones que el método numérico, produce resultados que están de acuerdo con elanálisis numérico y los ensayos físicos. Los errores causados por la estática gráfica seanalizan y se discuten. Los resultadosmuestran que el puente fue bien diseñado para lascargasdediseñooriginales,yque lasrestriccionesactualesestánbien.Alfinal,elpuenteyvariosdiseñosalternativosseevalúancualitativamentedesdeelpuntodevistadelacalidaddediseñoysuvalorenuncontextosocialyhistórico.Resum:EsteprojecteésunaavaluacióianàlisidelselementsprincipalsdelpontpenjolldeCliftonenBristol,RegneUnit. L'anàlisi es faperunmètodenumèricenordinador iunmètodegràficques'anomenaestàticagràfica.Elsmètodesielsdissenysperalponts'investiguenenelseucontext històric.Apesardel fet deque l'estàticagràfica témés limitacionsqueelmètodenumèric, produïx resultats que estan d'acord amb l'anàlisi numèrica i els assajos físics. Elserrors causats per l'estàticagràfica s'analitzen i es discutixen. Els resultatsmostrenque elpont va ser ben dissenyat per a les càrregues de disseny originals, i que les restriccionsactuals estan bé. Al final, el pont i diversos dissenys alternatius s'avaluen qualitativamentdesdelpuntdevistadelaqualitatdedissenyielseuvalorenuncontextsocialihistòric.



1.IntroductionModerncivilengineeringalmostexclusivelyusesnumericalmethodsofanalysisinthedesign

ofbuildingsandcivil structures.Civilengineeringstudentsaretaughthowtodostructural

analysis with hand calculations while computers analyse complex models by solving

thousandsofmatrices.Howeverthemathematicsbehindthesemethodsisarelativelynew

invention. Many of the world’s most famous monuments, palaces and bridges were

designed and constructed before the invention of these numericalmethods. The primary

aimofthisprojectistocompareanalternativemethodofanalysiswithamodernnumerical

method, and explore more deeply the history of structural analysis and its effect on

construction.

Tocomparethemethods,theywillbeusedtoanalysetheCliftonSuspensionBridge

inBristol,UK.Thebridge itself isoneof theoldest survivingexamplesofearly suspension

bridge design and is an illustration of how the evolution of technology and the

understanding of structures affects the way structures are designed. The results of the

analysiswill also be used to evaluate the design of the bridge,whichwas built in a time

whenthebehaviourofsuspensionbridgeswaspoorlyunderstood.

Unlike thesciences,where theaim is investigationandtheemphasis isonwhat is

notunderstood,theemphasisinstructuralengineeringisonwhatwealreadyknow.Itoften

seems that things thatwedonotunderstandorcannotcalculateareavoided in favourof

something that we understand. The history of suspension bridges is one full of

experimentation, mistakes, unknowns and guesses. It is this side of engineering that

intriguesme.

2.ObjectivesThe objectives of thiswork are to analyse part of the structure of the Clifton Suspension

Bridgeinasimplifiedmannerusingagraphicalandnumericalmethodofstructuralanalysis.

Theresultsoftheanalysiswillpermitanevaluationofthedesignofthesuspensionbridge

andacomparisonofthetwoanalyticalmethods.

3.MethodologyTheprojectwillbeput intohistoricalcontextsothatthethinkingbehindthedesignofthe

CliftonSuspensionBridgecanbebetterexplained.Theanalysisofthebridgewillbebroken

downintopartsprimarilyencompassingthetransversaldeckgirders,thesuspensionchains,

and the longitudinal girders. This will provide enough information to make a critical

evaluationofthesuperstructureofthebridge.Theanalysiswillbecarriedoutusingboththe

graphicalandnumericalmethodswherepossibleandwillbebasedonthedesignloadused

duringconstructionand the loads that thebridge is subjected to today.The resultsof the

twomethodswill be compared inorder to identify the causesof anyerrors andpermit a

quantitativeandqualitativeevaluationofthemethods.



4.HistoricalContext

4.1.AhistoryofsuspensionbridgesSuspendedbridgesmadeofropes,bambooandironchainshaveexistedformillennia(Body,

1976), however the firstmodern suspension bridges camewith the advent of the use of

structuraliron.TheCoalbrookdaleBridgeinShropshire,UK,wasbuiltin1779andiscredited

asbeingthefirstcastironbridge.Althoughtheuseofthematerialwasnovel,thestructure

wasnot:thebridgeusesthesamecirculararchdesignthattheRomanshadbeenusing2000

yearsearlier.Thistrendofbuildingironbridgeswithstructuresmoresuitedtobeingmade

ofwoodorstonecontinuedfordecades;in1802theFrenchgovernmentorderedthreenew

‘modern’ bridges to be built – all had structures based on wooden and stone bridges

(Grattesat,1978).

The first stepwas the inventionof the ironeyebar, patentedby SamuelBrown in

England,1817.ThreeyearslaterhebuilttheUnionChainBridgeovertheRiverTweed.With

a spanof137m, itwas the longest suspensionbridge in theworldat the time (Grattesat,

1978). Itwas exceeded by Thomas Telford’s bridge over theMenai Strait in 1826,whose

largest span is 177m. The Menai Suspension Bridge is regarded by many as the first

importantstepinthehistoryofsuspensionbridgedesign(Gimsing,1984).

At the same time in France, iron suspension bridgeswere being constructed. The

TournonBridgeovertheRhône,builtin1825byMarcSeguin,usedinnovativecablesmade

of 3mm iron threads to support the deck (Gimsing, 1984). Although thismethod became

popular in France, it was not widely adopted elsewhere because of the difficulty of

constructionandpoorqualitycontrol(Drewry,1832).

Figure1.TheBrooklynBridge,withverticalhangersandinclinedstaysvisible(User:Postdlf,2005).



As technology became more refined, the size of the bridges increased. In 1849,

Charles Ellet Jr. built theWheelingBridgeover theOhioRiver in theUSA.With a spanof

308m, itwassignificantly larger thananythingbuiltbefore itandwassupportedwith iron

cables.Fiveyearslateritcollapsedduringastorm.Followingthisandthecollapseofseveral

otherbridgesinhighwinds,anemphasiswasplacedonthesafetyandstiffnessofbridges.

Ferdinand Arnodin improved deck design by using the parapet to improve stiffness and

Seguin invented cable-winding machines to so that cables could be constructed on-site

(Grattesat,1978).

TheBrooklynBridge,NewYork(seefig.1),wascompletedin1883byJohnRoebling

andisagoodillustrationoftheprogressionofsuspensionbridgedesigninthatperiod.The

onlymathematical theories to describe themechanics of suspension bridges at that time

were 1st order theories such as Rankine’s, published in 1869 (Buonopane and Billington,1993),whichdescribethebridgeaccordingtoitsundeflectedshape.Thismeantitwasnot

possibletocalculatehowthe loads inthestructurechangedas itdeflectedunder load.To

makeupforthis,theBrooklynBridgeissupportedbyacombinationofcable-staysaswellas

suspended cables and has a deep ‘stiffness truss’. The entire structure is highly

indeterminate andheavily over-engineered; Roebling said that if all the suspended cables

whereremoved,thebridgewouldstillnotcollapse(Gimsing,1984).

Shortlyafterwards in1888, JosephMelanpublishedhis2ndorderdeflection theory

(Gimsing, 1984). Although itwas difficult to implement by hand, it took into account the

displacementsofthesuspensioncablesunderload,sotheresultsweremoreaccuratethan

using the 1st order geometry. Gimsing believes that thismarked the beginning of amove

towardsmathematically simplerdesigns, suchas theWilliamsburgBridge,built in1903. It

has a similar span to thatof theBrooklynBridge (490m)butno stays, as their effectwas

impossible to calculate. Unlike earlier designs, bridges from this period have cables the

attachtothebottomofthestiffnesstruss,asitismoreeconomical.ThelateralforcetheorydevelopedbyMoissiffandHudsonin1932meantthattheeffectsofhorizontalwindloads

could be calculated, making the ‘wind girder’ in the decks of older bridges unnecessary

(Gimsing,1984).Theseadvancesinmathematicalability,writesGimsinginCableSupportedBridges (1984),meantthatengineersbecameincreasinglyreliantoncalculationsandwere

‘blindly trusting of results’. Although the behaviour of the bridge could still not be

completely calculated, there started a trend towards extreme slenderness. Bridge decks

were builtwith evermore slender and flexible decks because theywere calculated to be

adequate.AgoodexampleistheGoldenGateBridge,SanFrancisco(seefig.2),whichhada

span todepth ratioof 168whenbuilt. Theproblemof torsional stiffness anddynamic air

pressurescametolightafterthespectacularcollapseoftheTacomaNarrowsbridgeinlight

winds.TheGoldenGateBridgewassubsequentlymodifiedandinvestigationwasmadeinto

theeffectsofaerodynamics(Gimsing,1984).Newanalyticalmethodsweredeveloped.

Figure2.TheGoldenGateBridge,SanFrancisco(WPPilot,2015).



Figure3.PhotographoftheSevernBridge,showinginclinedhangersandaerodynamicdeckprofile

(Edwards,2007).

After theSecondWorldWartherewasan increase inbuildingactivityanddesigns

begantotendintheoppositedirectiontothoseofthe1930s.Gimsing(1984)notesthatthe

MackinacbridgeinMichigan,US,isdesignedforamaximumwindspeedof995km/h–farin

excessofwhatisrequired.

Further advances in structural analysis lead to the construction of the first cable-

stayedbridgein1956.Ithasbecomethepreferredformoflong-spanbridgeasitallowsfora

cantilevermethodoferection.

In1959, theTancarvilleBridge,France,becamethefirstsuspensionbridgeoutside

of America to have a span ofmore than 500m.Unlike American-style bridges,which had

steel towers, the Tancarville Bridge had concrete towers and a continuous deck that ran

through the centre of the towers. A culmination of modern analytical and construction

techniquespermittedthebuildingoftheSevernRoadBridge,UKin1966(seefig.3).Itsvery

slenderdeck (withaspan:depthratioof324)thathadbeenunthinkable just twentyyears

earlier was made possible by an aerodynamic box girder deck section which gave high

torsionalstiffnessandcostsavingsovertraditionalmethods.Anintersectingarrangementof

inclinedhangersprovidedsufficientverticaldamping(Gimsing,1984).

Suspensionbridgescontinue tobeused in the largest spans today,as theycanbe

mademorecheaplyandwithlessmaterialthanotherbridges.Thelongestsuspendedspan

to date is the Akashi-Kaikyo Bridge, in Japan andmeasures almost 2000m (Miyata et al.,

2002).

4.2.GraphicalmethodsofstructuralanalysisAlthough Newton published his theories on mechanical mathematics in 1687, the use of

mathematicsinstructuralengineeringanddesignisstillrelativelymodern.Computershave

only been used in structural design and analysis for the last half a century, yet it is now

unthinkabletheitcouldexistwithoutthem.Oneoftheearliestrecordsofstructuralanalysis

isGiovanniPoleni’sMemorie istorichedellagrancupoladel tempioVaticano,published in1748.Init,PoleniexaminesthecupolaofSt.Peter’sBasilicaintheVaticanandformulatesa

plantohaveitrepaired.



Figure4.Poleni'sdiagramofthecupola(Poleni,1748).Compressiveloadsinthedomeaboveare

comparedtothetensileloadsthatoccurinacatenary.

The first theories regarding suspension bridge design were published by Claude

Navierin1823,havingstudiedearlysuspensionbridgesinEngland.Howeveritonlyprovided

basic calculations about the effect ofwind loading,withNavier noting that ‘the accidents

thatwouldresult fromthisactioncanbeappreciatedandpreventedonlyfromknowledge

providedbyobservationandexperience,’(BuonopaneandBillington,1993).Thechangesto

the design of bridges over the last 200 years illustrate howmuch our understanding has

advanced since then. However, as the use of computers for calculation only became

commonplace inthe1950s,manyfamous landmarks–suchastheBrooklynBridge–were

builtwithoutthisanalyticalpower.

Graphicalmethodsofanalysisareanalternativetomodernnumericalmethods.The

architectAntoniGaudífamouslybuiltmodelsofstringsandweights(seefig.5),which,when

viewed upside down, represented the force paths in a building. The graphical method

addressed in thisproject is calledgraphic statics,whichcanbedoneusingsimpledrafting

tools. It has its origins in the works of the Dutchman Simon Stevin and was further

developed by JamesMaxwell and Luigi Cremona in the nineteenth century (Baker et al.,

2013). The graphic statics method introduced by Cremona for solving trusses became so

popularthatisoftencalledtheCremonamethod.



Figure5.Gaudí'sforcemodeloftheColòniaGüellintheSagradaFamíliaMuseum(Canaan,2009).

Themethodusestwodiagramstosolveaxialloadsinelementsofastructure.Using

amethodsimilartothetriangleofforcesdrawnfortheobjectinequilibriumshowninfig.6,

theaxialforcesintheelementsofatrusscanbecalculated.

Figure6.Anobjectinequilibrium(left)andthecorrespondingtriangleofforces(right)

Thefirstdiagram–theformdiagram–showsthepositionanddirectionofthelines

ofactionofthestructuralmembers,thesecond–theforcediagram–showsthedirectionandmagnitudeoftheforcescarriedbytheelementsoftheformdiagram.Thedoesnottake

intoaccountmomentsordeflection,andcannotcalculateshearforcesorbendingmoments

developed in theelements.Typically it isused to solveproblems in2Dstructuresbut it is

possible,althoughdifficult, tocalculate in3D (VanMeleetal.,2012).The fullmethodwill

not be described here, but various books are available on the subject, such as Allen and

Zalewski’sFormandForces:DesigningEfficient,ExpressiveStructures(2009).



5.TheCliftonSuspensionBridge

5.1.ThehistoryofthebridgeThestoryoftheCliftonSuspensionBridgestartedmorethan100yearsbeforetheopening

of the bridge, when in 1754 William Vick, a local merchant, bequeathed £1,000 for the

building of a stone bridge over the Avon Gorge between Clifton Down in Gloucester and

LeighWoods in Somerset (McIlwain, 1996).By1829 the legacywasworth£8,000but the

estimated cost of a stone bridge was around £90,000, so a competition to design a

suspension bridge was announced (McIlwain, 1996). Twenty-two entries were received,

includingonefromBrown,thedesigneroftheUnionChainBridge,andfourfromengineer

Isambard Kingdom Brunel. Thomas Telford was appointed to select a final design, but

rejectedallofthemandsubmittedhisowndesign(Body,1976).

Telford’s design had three spans and was supported by two towers in a gothic

revival style. He maintained that 183m was the greatest admissible span. However his

designwasheavilycriticisedbythepublic,andsoasecondcompetitionwasheld in1830.

Brunel’s plans were rejected again, however he arranged ameeting with the judges and

convincedthemtoaccepthisdesignwiththetowersdecoratedinanEgyptianstyle.Having

raisedover£32,000of the£57,000totalestimatedcost,workbegan in1831 (Body,1976;

McIlwain,1996).Theprojectwasplaguedbydelays,disputesand financialdifficultiesand

by1843onlytheabutmentsandtowershadbeenbuilt,atatotalcostof£45,000.Themain

contractorwentbankruptandtheprojectwasabandoned(McIlwain,1996).

Brunel’s death in 1859 brought about renewed interest in the project. Engineers

JohnHawkshawandWilliamBarlowformedtheCliftonSuspensionBridgeCompanyin1861

and work was resumed in 1862 (McIlwain, 1996). It was decided to use the chains from

Brunel’s Hungerford Bridge, London, which had recently been demolished. Various

modificationsweremadetotheoriginaldesign;increasingthenumberofsuspensionchains

fromtwotothree;thedesignofthetowerswassimplifiedandtheirheightincreasedby5m;

the chain-anchorages were brought nearer; the wooden deck girders were replacedwith

iron and, using Arnodin’s recent improvements to deck design, the parapet was used to

stiffenthelongitudinalgirder(BarlowandBen,2003).Workwascompletedin1864andto

test the bridge, 500 tons (508023kg) of stone was evenly distributed across the deck,

producingadeflectionofonly180mm(BarlowandBen,2003).

Since then the opening of the bridge, little about the design has changed. The

woodendeckwasfirstasphaltedin1897(Cullimore,1986).In1925extensivereinforcement

ofthechainanchoragewascarriedout(Body,1976)andin1953amaintenanceframewas

addedandthemodernvehicleweight limitof4tons(4064kg)was introduced.Thetimber

deckhasbeenreplacedonseveraloccasionsandthemetalworkhasbeengrit-blastedand

zinc-sprayedtoreducecorrosion(Cullimore,1986).

Morerecently,in2003,thebridgebecameoverloadedwithcrowdsfortheAshton

Court Festival and the Bristol International Balloon Fiesta. Since then, the bridge has

remainedclosedonthesedaystopreventdamagetothestructure(“Suspensionbridgeshut

forevents,”2005).In2014,thebridgewasclosedforthefirsttimeduetohighwinds(“High

windsforceclosureofBristol’sCliftonSuspensionBridge,”2014).

Althoughnotcompleteduntilthe1860s,theCliftonSuspensionBridgeisavaluable

example of early British suspension bridge design. Its iron chains, wooden deck and

longitudinal plate girder make it one of a few remaining examples, including the Menai

bridgeandtheUnionChainbridge.



5.2.Analysisofexistingdocumentationanddescriptionofbridge

The information in thispapercomes fromseveraldistinctsources;Barlow’s1867paper,ADescription of the Clifton Suspension Bridge; various plans held by the Clifton SuspensionBridge Trust and The Clifton Suspension Bridge: preservation for utilisation, by Cullimore,

1986. Brunel’s original sketches and calculations are held in the Brunel Institute, but the

modificationsmadebyBarlowandHawkshawmakethemoflittlevalueforthepurposesof

calculations.

A Description of Clifton Suspension Bridge contains the dimensions of the bridge;

diagramsofthebridgeprofileandasectionofthedeck;notesontheconstructionmethod

andcalculationsforthemaximumstressdevelopedinthechainsandhangers.Itprovidesan

almostcompletedescriptionofthebridge.Havingbeenpublishedbeforetheintroductionof

SIunitsandstandardnomenclature,allmeasurementsareinimperialunits.Morenotableis

theBarlow’sdiscrepantterminology;hewritesthat‘Thestrain….atthecentreofthechains

is597tonsapproximately,’butgoesontowrite inthenextparagraphthat ‘themaximum

strainupontheironis….4.76tonspersquareinch’.Itcanbeassumedinthiscasethat‘the

strain’referstoaxialloadandstressrespectively.However,atothertimestheinformationis

meaningless: ‘thesuspension-rodsareeach rathermore than2 inches in section.’For this

reason, information fromBarlow’spaper isonlyusedwhen it agreeswith thatofanother

source.

InTheCliftonSuspensionBridge:preservationforutilisation,Cullimoredescribesthe

results of fatigue and fracture testing in the deck and hangers as well as details of

operationalpracticeandmaintenance.

Figure7.ThesouthernsideoftheCliftonSuspensionBridgewithAvonGorgeinthebackground.The

LeighWoodstower(left)hasadifferentdesigntothatoftheCliftontower(Gothick,2009).

AccordingtothedescriptiongivenbyBarlowandBen(2003),theCliftonBridgeisa

single-span, eyebar-chain suspension bridge. It has three chains on either side of the

roadwaywhicheachchainsupportsaplategirderlongitudinallyalongthebridgebymeans

of81verticalhangers.Directlybeloweachhangerthelongitudinalgirdersareconnectedby



transversal open lattice girders that are riveted to the lower flange of the longitudinal

girders,anddiagonalcrossbracingthatisboltedtotheundersideofthedecktimbers.The

areabetweenthetwolongitudinalgirdersisthecarriagewayandthecantileversectionsof

the transversal girders support thepedestrianwalkway and theparapet. Theparapet is a

lattice girder and works to increase the longitudinal stiffness of the deck. The chains,

hangers,girdersandparapetareallmadeofwroughtiron.Ratherthanpassingcontinuously

overthetowersaddles–asiscommoninmoremodernsuspensionbridges–thesuspension

chainsandlandchainsareattacheddirectlytoaframeoncastironrollers.

Figure8.Sectionthroughthedeck.Nottoscale.

The 26m (85ft) towers are made of local sandstone (Cullimore, 1986) and are

positioned214m(702ft)apart,whilethesuspendedsectionoftheroadwayisonly193.9m

(636ft)long(BarlowandBen,2003).BarlowandBen(2003)writethatalongthesuspended

section, there is a transversal girderandhangersevery2.4m (7.95ft). In the centreof the

bridge,thechainsare21.3m(70ft)lowerthanatthetowers.Thedistancebetweenthetwo

setsofchainswidthofthecarriagewayis6.1m(20ft)–one3mwidelaneoftrafficineach

direction.Eachwalkwayis1.5m(5ft)wide.TheCliftonsideofthebridgeis0.9m(3ft)higher

thantheLeighWoodsside,somethingthatBrunelbelievedwouldmakethebridgeappear

absolutelylevelagainstthebackdropofthegorge.Thedeckisslightlycambered,sothatthe

centreis0.61m(2ft)higherthanattheends.

Figure9.Diagramofthesouthernviewofthebridge.Nottoscale.



The deck is made of 127mm deep timber planking laid longitudinally across the

transversaltrussesandcoveredwithalayerof50mmthicktransversalplanking.Thesurface

isfinishedwith32mmnominalthicknessmasticasphalt(Cullimore,1986).

Figure10.PhotographoftheLeighWoodstowerwiththelandchainsintheforeground(D,2007).

Figure11.Detailofthesuspensionchains,showingthearrangementofthehangersandeyebars.

5.3StructuralanalysisIn the following section, the chainand the transversal girdershavebeenanalysedusinga

numericalmethod (a computermodel analysed in the program SAP2000) and a graphical

method(graphicstatics).Thispermitsacriticalevaluationofthedesignofthestructureand

acomparisonbetweensaidanalyticalmethodswith regards to theiraccuracy,easeofuse

andrelativepower.

5.3.1Transversalgirders

5.3.1.1.ModelAsimplifiedmodelofthetransversalgirdershasbeenusedfortheanalysis. Itcomprisesa

top and bottom boom, eachmade of two 100×100×16 angles back-to-back with flat bardiagonalbracingsandwichedbetweenthetwo(seefigure12).Itissimplysupportedatthe

twonodesthatcorrespondtotheconnectionswiththe longitudinalbeamsandallapplied

loadsactwithintheplaneofthegirder.Thematerialisassumedtobeuniformwroughtiron



with a density,ρ of 7750kg/m3 (Shackelford et al., 2016). Details such as bolts and rivets

havenotbeentakenintoaccount.

Scaleelevationofhalfofthegirder. Crosssection.Nottoscale.

Figure12.Approximationofthetransversalgirderasusedintheanalysis.

5.3.1.2.LoadingTheloadsappliedtothemodelaredividedintothecategories.Theself-weightofthegirder,

thesuperimposeddeadload–theweightofthedeckandtheparapet–andtheliveload.In

thenumericalmethod,theselfweightoftheelementsiscalculatedfromtheirvolumesand

the specificweight of thematerial. In the graphicalmethod the selfweight has not been

takenintoaccount.

The superimposed dead loads have been calculated as follows assuming the deck

compositionshowninfigure13andthatthegirdersupportsasectionofdeck7.95ft(2.4m)

long(seefigure14).Thedensitiesofthetimberandtheasphaltaretakenas530kg/m3and

2300kg/m3respectively(Richards,2010).

Figure13.Sectionthroughtheroadway.Thewalkwayis

assumedtobethesamebutwithoutthebottomlayer.

Figure14.Thetributaryareaofeach

hangeris7.95ft(2.4m)long.

Four combinations of live loads have been used in order to find the most

unfavourablecombinationofloads.OneisbasedonBarlow’sestimatedmaximumliveload

of70lbspersquarefoot(3.35kN/m2)distributeduniformlyoverthewholedeck.Theother

loadcasesaretoreflectthemaximumloadingthatoccurstoday.Themaximumliveloadon

thepedestrianwalkwaysisassumedtobe5kN/m2,inaccordancewithEN1991-1-1(British



Standards Institution, 2002) for areas susceptible to crowds and IAP-11 (Ministerio de

Fomento,2011)forpedestrianloadsonroadbridges.Europeanstandardsarenotvalidfor

the loading on the carriageway as a 4 ton vehicle weight limit is enforced. Therefore a

uniformly distributed load of 2.65kN/m2 has been calculated assuming a 4 ton vehicle

situatedevery5malongthedeckinbothlanes.Theautomatictollbarriersregulatetheflow

ofvehiclesacrossthebridge(Body,1976),sothisvalueisconservative.

LC1:Barlow’smaximumload LC2:Modernmaximumload

LC3:Loadtoproducemaximumsaggingofgirder LC4:Loadtoproducemaximumhoggingofgirder

Figure15.Thefourloadcombinations(LC)usedintheanalysis.

Thedead loadsaremultipliedbya coefficientof1.35and the live loadsby1.5 to

account forunderestimatesof themaximumloads–providingaso-called factorofsafety.The coefficientsareapplied inall casesexcept LC1, soas to compare the resultswith the

calculationsmadebyinADescriptionoftheCliftonSuspensionBridge.

5.3.1.3.ResultsofnumericalmethodForallofthe loadcases,theaxial load ineachelementwasrecorded.Forthepurposesof

this paper, an element refers to linear member between two nodes in the model. For

example,thetopboomisasingleobjectcomposedof28elementsofdifferentlengthsanda

piece of bracing is a single element. Figures 16—19 show the axial load envelopes of the

boomsandbracingofthegirder. Itshouldbenotedthatwhilethediagramsofthebooms

showtheloadinasingleobject,thediagramsofthecrossbracingshowtheaxialloadmany

individualobjects.

Figure16.Axialforceenvelopeforthetopboom.Thegreatesttensileload(74.0kN)occursduringLC4

andthegreatestcompressiveload(162.0kN)duringLC3.

-200

-100

0

100

Axialload[kN]

LC1:500t LC2:Maximumload LC3:Sagging LC4:Hogging



Figure17.Axialforceenvelopeforthebottomboom.Thegreatesttensileload(162.4kN)occurs

duringLC3andthegreatestcompressiveload(88.3kN)duringLC4.

Figure18.AxialforceenvelopeforthetensionbracingasseeninFigure12.Approximationofthe

transversalgirderasusedintheanalysis.

Figure19.Axialforceenvelopeforcompressionbracingasseeninfigure12.

Thegreatestloadintheboomsisfoundinthecentreofthespan,wherethegirder

isdeepest.Axialforceintheboomsisprimarilycausedbybending,soitisgoodtonotethat

theenvelopesinfigures16and17closelyresemblethebendingmomentgraphsofasimply

supportedbeamsubjectedtoauniformlydistributedload–thegreatestbendingmomentis

found in themiddleof thespan.Thesameobservationcanbemade in figures18and19,

wherethegreatestaxialloadsinthebracingoccurattheconnectionswiththelongitudinal

girders; when girder is seen as a simply supported beam, the shear force diagram peaks

wherethereactionforcesareappliedtothebeam.

-100

0

100

200

Axialload[kN]


0

20

40

60

80

Axialload[kN]


-60

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Figure20.Thedistributionofbendingmoments(left)andshearforces(right)inasimply-supported

beamwithauniformlydistributedload.

Withtheseresults,thecriticalelementandofthegirdercanbecalculated–thatis

theelementthatwillfailfirstinthecasethatthebeamisoverloaded.Therearenomodern

guidelinesfortheuseofwroughtironinstructuraldesign,sothecalculationswereusingthe

method in EN 1993-1-1 (British Standards Institution, 2005) and material properties of

wrought iron as listed in ASTM A207 (ASTM International, 1939); the ultimate tensile

strength (UTS) is taken to be 330kN and the yield strength,!!as 190kN. To account forimperfectionsinthematerial,theUTSand!!aredividedbyareductioncoefficientof1.15.

The critical tensile load,!!" and the flexional buckling load!!,!" were calculatedfor the top and bottom booms and the critical bracing elements. There are two sizes of

bracing,asshowninfigure21,soloadshavebeencalculatedforboth.

Figure21.Thegirderbracingismadeoftwosizesofflatsection.Thelightelementshavesectionsof

2”×3/8”(50×10mm)andthedarkelements2”×0.5”(50×13mm)sections.

Table1.Calculationofthecriticaltensileloadforeachelementtype.

Object Topboom Bottomboom 2”×0.5”bracing 2”×3/8”bracingSectionarea,A[mm] 5888 5888 806 605

Tensileload,T[kN] 73.9 162.4 58.3 20.3

Stress,σ[MPa] 12.6 27.6 72.3 33.6

Criticalload,Ncr[kN] 1689.6 1689.6 231.4 173.6

Factorofsafety,F 22.8 10.4 4.0 8.5

Thefactorofsafety,Fistheratio!!" !. If!!" ≫ !,thevalueofFishighandsuggestsaninefficient use ofmaterial. If! < 1, the element is not strong enough andwill fail during

normaluse.Theresultsoftheanalysisshowthatallthesectionshaveasufficientfactorof

safetyandthatthegirderappearstobeinefficientlydesignedfortensileloads.The2”×0.5”sectionhasthelowestfactorofsafety,whichsuggeststhatitwouldbethefirsttoundergo

tensilefailureiftheloadswerefurtherincreased.

Tocalculate thecompressivecapacityofeachsection, thebuckling load!!,!" wascalculated.Alltheelementsareslender(! > 0.2)sowillfailinbucklingandnotcrushing.Tomake the calculation, itwas assumed that all the elementswere fixed at both ends. This

means the effective length,!! = 0.7×!, where!is the actual length of the compressed

section. The topboom is bolted to the longitudinal deck timbers, effectively reducing the

unconstrainedlength,howeverthisisnottakenintoaccount.



Table2.Calculationoftheflexionalbucklingloadforeachelementtype.

Object Topboom Bottomboom 2”×0.5”bracing 2”×3 8”bracingCompressiveload,C[kN]

162.0 88.2 52.5 27.2

Stress,σ[MPa]

27.5 15.0 65.1 44.9

Momentofinertia[mm4] 2834350.3 2834350.3 10839.4 4572.9

Effectivelength[mm] 4267 4267 186 206

Non-dimensional

slenderness,λ2.0 2.0 0.7 1.1

Flexionalbucklingload,

Nb,Rd[kN]233.2 233.2 116.5 59.9

Factorofsafety,F 1.4 2.6 2.2 2.2

Theresultsshowthatthesectionsaremuchmoreefficientincompressionthanintension.

The critical section is the top boom, which agrees with the findings of the test to

destruction, where failure occurred due to buckling of the top boom. However the

calculated buckling load is likely to be conservative, as Cullimore (1986) notes that

longitudinaldecktimbershaveaconsiderablestiffeningeffectonthegirder.Healsowrites

thattherewasvisiblebowingofthebracingandbottomboom,suggestingthattheyhavea

similarfactorofsafety.Thisagreeswiththeresultsofthecalculationsasseenintable2.

5.3.1.4.CalculationwithgraphicstaticsAsthegirderandtheloadcasesarebothsymmetrical,andthegraphicstaticsmethoddoes

nottakeintoaccountmomentsofanykind,itisonlynecessarytoanalysehalfthetruss,as

shown in Fig. 22. To emulate the graphic statics as it would have been done in the 19th

century,allthegraphicanalysiswasdrawnbyhandonA0paper.Thispermittedprecisionof

up to0.5kN,whenusingmeasurements inmillimetresanda scalewhere10kN isequal to

1cm. The analysiswas repeated using computer-aided design (CAD) software, so that the

errorresultingfromthehanddrawingscouldbecalculated.

Figure22.Halfof thegirdermodel,asanalysedwiththegraphicstatics.Thesectionsof thebooms

underthewalkwayareblueandthoseunderthecarriagewayarered.Thecrossbracingisingrey.



Figure23.CADforcediagramofthesaggingloadcase.Theverticalblacklinerepresentsthetotalload

uponthegirder.Theforces inthetopandbottomboomwheretheypassunderthewalkway(blue

lines)areinsignificantincomparisontowheretheypassunderthecarriageway(redlines).

Figure24. Forcediagramof thehogging load case. Theverticalblack line represents the total load

uponthegirder.Themaximumaxialforcesofthesectionsbelowthewalkwayandcarriagewayareof

similarmagnitudesandlessthanhalfthatofthemaximumforceinfig.23.

Theresultsofthehand-drawngraphicstaticswerecomparedtothosefoundusing

theCADdrawings.Themeanerrorforthefourloadcaseswas0.3kNandthemaximumerror

wasfoundtobe3.1kN.Asthetheoreticalprecisionofthehand-drawingis0.05kNatascale

of 1kN:1cm, this method was not accurate. However for the purposes of structural

engineering,whereloadsareoftencalculatedtothenearest1kN,itcouldbeconsideredto

beaccurate.Howeveritisworthnotingthataveragepercentageerrorofthedrawingswas

8%. Small values, especially those less than 1kN aremore sensitive to the effects of the

error.



Figure25.Graphshowingtheabsolutemanualdrawingerroragainsttheaxialloadsineachmember.

Asseeninfigure25,themajorityoferror is lessthan1kN,howevertheresultsfor

LC1 display some unusual characteristics. For axial loads greater than 60kN, the error

increasesrapidlywithaxial load,withamaximumof3.1kN.Thisisfundamentallyaneffect

ofthegraphicstaticsmethod,wherethepositionsofjointsarefoundfromtheintersection

oflinesofforce.Anyerrorinfindingtheintersectionwillbecarriedoverintothecalculation

ofthelengthofthenextforcelinetobecalculated.Thelongertheline,themoretheerroris

amplified;sosmallerrorsatthebeginningoftheprocesscan leadtoachain-reactionthat

causeslargererrorslater.Itispossiblethattheeffectsofthisphenomenoncanbereduced

bycalculatingthelargestvaluesinthestructurefirst,howeverithasnotbeeninvestigated

forthisproject.

5.3.1.5.Comparisonofgraphicstaticsandnumericalmethod

LC1:Barlow’sload.r

2=0.999.Gradientoftrend

line=0.9571

LC2:Maximumload.r2=0.997.Gradientoftrend

line=0.94.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 20 40 60 80 100 120

Manualdrawingerror[kN]

Axialload[kN]

LC1:500tstone LC2:Maximumload LC3:Sagging LC4:Hogging

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50

100

150

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Graphical[kN]

Numerical[kN]

-100

-50

0

50

100

-100 -50 0 50 100

Graphical[kN]

Numerical[kN]



LC3:Saggingload.r

2=1.000.Gradientoftrend

line=0.9623.

LC4:Hoggingload.r2=0.982.Gradientoftrend

line=0.9867.

Figure26.LoadsobtainedfromtheCADdrawingmethodagainstthoseobtainedfromthenumerical

method,foreachloadcombination.Ther2valueshowsthecorrelationbetweentheresults.

Acomparisonoftheresultsfromthenumericalandgraphicalanalysisshowsavery

stronglinearcorrelation(seefig.26).Itisnotcompletelycertainwhatcausestheslighterror

betweentheresults.ThereisassumedtobenoerrorintheCADdrawingnorintheresults

of the numerical analysis, and the data was recorded to the nearest newton. It is partly

causedbythedifferenceinthemodels;graphicstaticsassumesthatalljointsarepins,sono

momentsaretransferredbetweenelements.Duringthenumericalanalysistheelementsin

theboomswerecalculatedwithfixedends,aseachboomisasingleobject.Tomeasurethe

effect of this difference, the numerical analysis was repeated with a completely pinned

structure.

Figure27.GraphofaxialloadscalculatedwithCADdrawingandthoseofthenumericalanalysisofthe

completelypinnedgirderforLC1.r2=0.99997.

-150

-100

-50

0

50

100

150

-150 -100 -50 0 50 100 150

Graphical[kN]

Numerical[kN]

-80

-40

0

40

80

-80 -40 0 40 80

Graphical[kN]

Numerical[kN]

y=0.9372x-0.0228

R²=0.99997

-150

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-50

0

50

100

150

-150 -100 -50 0 50 100 150

Graphical[kN]

Numerical[kN]



Fig.27showsthatthereisanearlyperfect linearrelationshipbetweentheresults,

whichsuggests that thedisparitybetweenthemodels is theprimarycauseof theerror in

the results. It isunknownwhether thegeometryand typeof loadinghasaneffecton the

fixed–pinneddifference,butitisnotinvestigatedinthepaper.

Inallcases,theresultsofthegraphicalanalysisarelessthanthatofthenumerical

one.Althoughtheresultsareproportional,theyarenotequal;thetrendlinesshowthatthe

loads found using graphic statics are 94-99% of those of the numerical analysis. The

differenceisthatoftheself-weightofthegirder,whichhasnotbeentakenintoaccountin

thegraphicalanalysis.

Basedontheseobservations,thegraphicalmethodofanalysiscanbeseenasavalid

andusefulalternativetonumericalanalysis. Ithas limiteduses(beingonlyabletoanalyse

axial loads on statically determinate structures with pinned joints) and is inaccurate and

highly time consuming when done by hand. However, in this case where a high level of

precisionisnotrequired,itprovidessufficientdatatobeabletocalculatethebehaviourof

the structure. The drawings themselves may be seen as a useful way to visualise the

behaviourofthestructure,andcouldprovidestudentsofengineeringanalternativewayto

learnaboutstructuralmechanicsandanalysis. It is interestingtonotethatthesegraphical

methodswentoutofuseasmathematicalmodellingbecamemoresophisticated,atatime

whenengineerswere‘blindlytrustingofresults’(Gimsing,1984).Itshouldbeaskedwhether

graphical methods, although antiquated, help give an innate understanding of structures

thatmodernengineersnolongerhave.

5.3.2.Chains

Knowingthegeometryofandtheloadsappliedtoastructureallowstheforcesinsidetobe

solvedwithgraphicstatics,aspreviouslydescribed.Thenatureofgraphicstaticsmeansthat

thegeometryofstructurescanbeascertainediftheforceswithinthestructureareknown.

This canbeused to calculate the shapeofahanging cable,or– in thecaseof theClifton

bridge–theoptimalshapeofthesuspensionchain.Theoptimalshapeofacableisthatin

which the loads in the cableareminimised. In caseswhere the self-weightof the cable is

insignificant, that shape is a parabola. Although the chains of the suspension bridge are

madeofdiscretelinksandhavesignificantself-weight–approximately1/3 ofthetotalload(BarlowandBen,2003)–theshapecanstillbeapproximatedtoaparaboliccurve.

Figure28.Theshapeofthechainifitweretohangbetweentowersofequalheightsisaparabola

(left).Thesameshapeadjustedbyincreasingthelevelslinearlyalongitslength–nolongeraparabola

(right).Nottoscale.

ThecalculationoftheshapeoftheCliftonbridgeiscomplicatedbythefactthatthe

LeighWoodstoweris3ft(0.9m)lowerthantheCliftontower,givingthebridgeanaverage



gradientof1:233(BarlowandBen,2003).ItappearsthattheshapeoftheCliftonchainswas

calculatedbytakingtheshapeofaparabolawith levelendsandthenincreasingthelevels

linearlyalongthelengthofthechain,asillustratedinfig.28.McIlwain(1996)writesthatthe

designwas‘closetotheideal’,suggestingthatthismethodisnotaccuratebutmakesagood

approximation to the optimal shape. Using graphic statics and the structural analysis

programSAP2000,itwaspossibletoanalysetheshapeofthechain.

BarlowandBen(2003)calculatethemaximumload inthechainstobe2094tons-

force (21,276kN), approximately 10,638kN in each set of chains. Using the detailed

information provided by Cullimore (1986), it is possible tomake our own estimate to the

maximumloadinthechainsbetweenthetowers.

Table3.Weightsandloadsforhalfofthedeck,tocalculatetheloadinonesetofchains.Thechain

weight is that of a parabolic shape chain. Other ironwork refers to longitudinal and transversal

girders,hangersandparapet.

Load Estimatedvalue

Self-weightof1setofchains 2761kN

Self-weightofotherironwork 903kN

Super-imposeddeadload 1290kN

Liveload(LC1:70lbs/sq.inch) 3070kN

Total 8024kN

Themaximumaxial load of a uniformly loaded light cable,!!"# can be calculatedusingequation1,where!istheuniformlydistributedload,!isthelengthofthechordandℎtheheightofthecurve.

!!"# =!!!!4 + !!!

8ℎ! (1)

The calculated!!"#is 11,100kN,which is approximately4%greater thanBarlow’s

estimatedmaximum.Thisissufficientlyaccuratetobeusedinthecalculationsoftheshape

ofthechain.

Figure 29. Force diagram of one set of chains (supporting half the deck)with uniform live loading

acrossthedeck.



Having verified the Barlow and Ben’s figures, the force diagram in fig. 29 was

created. The lengthof the vertical line represents the totalmagnitudeof the loadon the

chains;eachsmallverticalline,theloadtransferredtothechainsbyonehanger.Eachofthe

convergentlinesistheloadinonesectionofthechains.Theaveragegradientoftheselines

is1:233.Thismeansthatthediagramisnotsymmetricalandthebottomlineislongerthan

thetop.Itrepresentstheforceinthesectionofthechainwiththegreatestload11,100kN.

Knowingthisvalue,itispossibletocreatetheentireforcediagram.

Theshapeofthechaincannowbecalculated,asthegradientofeachchainsection

isknown.

5.3.2.1.ChainShapeAsshowninfig.30,theoptimalshapeascalculatedwithgraphicstaticsdoesnotagreewith

the real shapeof the chain. There is amaximumdifferenceof 2.6ft (0.79m)between the

graphicstaticscurveandtherealshapeofthechain.Thedisparityiscausedbythefactthat

thetheoreticalchainshapeshavebeencalculatedforachaininwhichthemaximumloadis

that11,100kN,asmentionedabove.Tocreateashallowercurve,themaximumloadinthe

chain would have to be increased. It may be that the chain shape was designed with a

slightlyhighermaximumloadasaprecaution.

Figure30.Theshapeofthechaincalculatedwithgraphicstatics(blue),theadjustedparabolashape

shown in fig.28. (red), theactual chain shape (black)and the shapecalculated inSAP (green). The

chainshapeisknownfromthemeasuredlengthsofthehangers,soisonlyshownforthesuspended

sectionofthebridge.Heightsarerelativetotheoriginalparabolasimilartothatinfig.28.

Infigure31,theresultsarecomparedtoaparabola702ftwideand70ftdeep,which

aretheapproximatedimensionsusedinequation1.Itiseasytoseethatthechainas-builtis

verysimilartothe‘adjustedparabola’infig.28.Theslightdeviationisprobablyduetothe

-10

0

10

20

30

40

50

60

70

80

0 351 702

Height[y]

Distance[y]



wearingof theeyebarsanddeflectionof thechains.Themethodsusedherearebasedon

first-order static theories and do not calculate the deflected shape as in second-order

theories.Thereisconsiderableerrorinthehand-drawngraphicstaticmethod–itincreases

towardsthecentreofthespantheuntilitis0.9ft(0.27m).Howeveritthendecreasesagain

ontheotherside,sothatthemagnitudeoftheerrorissymmetricalacrossthecentreofthe

span.Thissuggeststhatthecauseisnotrandommeasuringerror,butanerrorintheforce

diagram. The curves of the CAD graphic statics and the cable produced by the numerical

methodareverysimilar,whichsuggeststhatthemethodshavebeencarriedoutcorrectly.

TheeffectoftheUDLincomparisontotheloadscalculatedinthegraphicstaticsissmall,the

chainshapeundertheUDLisslightlyshallower,howeverbyonly0.2ft(6cm).

Theresultsfromthenumericalandgraphicalanalysiscreatecurvesthathavetheir

minimumsclosertothelowertowerthantheupper.Unliketheshapeinfig.28thatclosely

resembles the actual chain shape, the curves have been effectively moved horizontally

towardsthelowertower.Thisiswhatisexpectedandcloselyresemblestheoptimalshape

forthechain.

Figure31.Deviationofcurvesfromoriginalparabola,asshowninfig.28.Theseriesofblackpoints

are the levels of the upper hanger connections,which are attached to each of the three levels of

chainsalternatelyalongthelengthofthechain(seefig.11).

The analysis of the chain shape was done using the loads calculated for LC1 –

Barlow’s original design loads. However it should remain valid for the other load

combinationsbecausetheshapefoundisdependantonthedistributionoftheloadsalong

thelengthofthestructure,notthetotal loaditself.Forthisreasonitcanbeassumedthat

theresultofthesamecalculationwithdifferentloadswouldbesimilar–theoptimalchain

shapetodayshouldnotbedifferenttothatofthe1864.

However it is important to note that thismaynot be themost unfavourable load

combinationforachainofthisshape.Loadsthataredistributedunequallyalongthelength

of the bridgemight cause greater forces in the chain, because the chain shapewould no

longerbeoptimumforthisloadingscenario.

Figure32.Longitudinalmodelofthebridge,asconstructedinSAP2000.

-2

-1

0

1

2

3

4

Deviazon[y]

'AdjustedParabola' Realchainshape GS–CAD



Usingthedimensionsdetailedearlier,asimplified2Dmodelof thebridge (seefig.

32)wasanalysedinSAP2000toseewhattheeffectofnon-uniformloadingalongthelength

ofthebridgewouldhave.Themodelcomprisesonesetofchainswithacross-sectionalarea

of220squareinches(141935mm2)–theminimumarea(BarlowandBen,2003).Thesetof

chainsisconnectedtosinglelongitudinalI-beam3ft(0.91m)indepthbycylindricalhangers

withdiametersof47mm(Cullimore,1986).Thechainsareconnectedtofixedrestraintsat

the land anchors and rollers at the towers. The longitudinal beam is simply supported at

eachandallthehangersarepinned.Thesamewroughtironmaterialasdescribedaboveis

used throughout. During the analysis, point loads were applied at each of the nodes

connecting thehangers and the longitudinal beam to simulate the live load and thedead

loadcontributedbythedeck.Theresultsoftheanalysisareshownintable4.

Table4.ResultsoftheSAPanalysis.

Liveload Fulllength Halflength

Maximumreactionattowersaddles[kN] 8262 7931

Maximumloadinchain[kN] 11013 10550

Minimumloadinchain[kN] 10345 9910

Max.bendingmomentinlongitudinalgirder[kNm] 300 2208

Maximumverticaldeflectionofdeck[m] 0.35 4.05

Thereislittledifferencebetweenthetwocombinationsofloadsintermsoftheload

transmittedthroughthechainsandtowers.Theloadsinthechainandthereactionsatthe

towersforthehalf-lengthloadareapproximately5%lessthanthoseofthefull-lengthload.

This suggests that while the half-length load is transmitted less efficiently through the

structure, the full-length load is still themost unfavourable in terms of axial force in the

chain.

Howeverthemaximumbendingmomentdevelopedinthelongitudinaldeckgirder

forthehalf-lengthloadisseventimesthatofthefull-lengthload(seetable4).Thismeans

that thedeck and chaindeflections for thehalf-length load aremuch greater. Thismodel

does not take into account the stiffening effect of the parapet and the longitudinal deck

timbers so the real deflection is likely to be significantly less than in table 4. Barlow’s

uniform loadingtestcausedthedecktosinkby7 inches (18cm) (BarlowandBen,2003)–

approximately half the deflection calculated in SAP, so the maximum deck deflection is

estimatedtobe2m.

Figure33.Deflectionofthemodelwithmaximumliveloadallthewayalongthedeck.

Figure34.Deflectionofthemodelwithmaximumliveloadonthelefthalfofthedeck.

To estimate the bending moment capacity,!!,!" of the longitudinal beam, the

calculationmethodfromEN1993-1-1wasused.ThebeamhasaClass3cross-sectionandan

elasticsectionmodulus,!!" = 7.542662×10!!!!.Itwasassumedthat!!! = 1.0.

!!,!" = !!",!" = !!",!"#

!!!!!

= 7542662× 3301.0 = 2489!"# (2)



Althoughthebendingcapacitycalculatedinequation2ishigherthanthemaximum

bendingmoment in themodel (see table 4), wrought iron ismore brittle than structural

steel. It is likely that the longitudinal beam would fail before this moment was reached.

Brunel’soriginaldesign,without thestiffeningeffectof theparapet,wouldnothavebeen

strongenoughtosupporttheBarlow’sdesignload.Theadditionofthestiffeningparapetis

keytothestrengthofthebridge.

5.3.2.2.CalculationofloadsTheaxialforceinthechainwasfoundbymeasuringthelengthsoftheforcediagraminfig.

29.,andhasbeenplottedagainstthepositionalongthechaininfig.35.Theminimumload

occurs slightly to the left of the centre of the span, which correlates with the results of

shapeanalysis,becausetheminimumloadshouldbefoundinthelowestlink.

To make the design of the chains more efficient, the cross sectional area of the

chainsisgreateratthetowersthaninthecentre.Theareaofonesetofchainsatthetowers

is 240 square inches (155,161mm2) and the area in the centre of the span is 220 square

inches (141,935mm2) (BarlowandBen,2003). The critical load,!!" hasbeencalculatedat

bothlocationsandforbothLC1andLC2.

Figure35.Distributionofaxial loadsthroughouttheascalculatedfromtheCADgraphicstatics.The

maximumloadattheCliftontowerisalreadyknowntobe11,100kN(calculatedbeforehand)andthe

load at the LeighWoods tower is 10,964kN (calculatedwith graphic statics). Theminimum load is

10,317kNandoccurstwolinkstotheleftofthecentre.

Table5.Calculationofthecriticalload,!!"andthefactorofsafety,!duringLC1.TheUTSistakentobe330MPa(ASTMInternational,1939)andthematerialstrengthcoefficient,γis1.15.

Location Centreofspan Tower

Cross-sectionalarea[mm2] 141935 155161

Tensileload,!!"#[kN] 10315 11100

Stress,σ[MPa] 72.7 71.5

UTS÷ ![MPa] 287 287

Criticalload,!!" [kN] 40729 44525

Factorofsafety,! 3.9 4.0

Table6.Approximatecalculationofthecriticalload,!!"andthefactorofsafety!duringLC2.

Location Centreofspan Tower

Cross-sectionalarea[mm2] 141935 155161

Tensileload,!!"#[kN] 11560 12440

10200

10400

10600

10800

11000

11200

Tensileforce[kN]



Stress,σ[MPa] 81.4 80.2

UTS÷ ![MPa] 287 287

Criticalload,!!" [kN] 40729 44525

Factorofsafety,! 3.5 3.6

ThedesignfactorofsafetyforLC1couldbeseenasexcessive,howeveritshouldbe

takenintoaccountthateyebarchainsarelikelytofailattheconnections.Thevaluesshown

representthestrengthoftheeyebaritself,andnotthestrengthoftheconnectionsbetween

them.TheWheelingBridgecollapsewasduetoonebrokenconnection(Gimsing,1984),and

Cullimore(1986)notesthat thegreatestsignsofwear in thebridgeareat theeyebarand

hanger connection holes. These points act as stress concentrators and can cause the

elementtofailatalowerloadexpected.Thisisarguablyoneofthemainreasonsthatchains

havebeenreplacedbycablesinallsuspensionbridgedesignssincethe19thcentury.Theuse

ofmany threads,means that they have amuch higher level of redundancy– a cable can

continuetofunctionwellevenifafewofthethreadshavebroken.Itislikelythattheactual

maximum load of the chains is significantly lower than the values calculated, which goes

someway to explaining the high values of!. Body (1976)writes that all the chains havebeentestedwithtwicethedesignload,sothelowerboundfor!is2.

5.3.3.Othercalculations

5.3.3.1.Cross-bracingThe cross-bracing under the deck provides stiffness against horizontal wind loading.

AlthoughBarlowandBen (2003)notes that thedeck canmove ‘up to6” [15cm] inheavy

wind’,littlewasdonetocalculateitseffects.Thecross-bracinghasnotbeenanalysedinthis

paperforthefollowingreasons:

1. Nodocumentationhasbeensourcedtocompareresultsofcalculations.

2. Itisunknownwhateffectsthetopographyofthegorgewouldhaveonthewind,so

calculationsarelikelytobeinaccurate.

3. Theactionofwindloadingonsuspensionbridgesisrelatedtotheresonanceand

aerodynamicpropertiesofthebridge,thecalculationofwhichisoutsidethescope

ofthemethodsofanalysisemployedinthispaper.

Howeverwindloadingremainsanimportantfactorinbridgeengineeringandinthe

caseoftheCliftonSuspensionBridge;forthefirsttimeinlivingmemoryin2014thebridge

wasclosedtotrafficduringhighwinds.Thereforetheanalysisofwindloadingonthebridge

wouldbeaninterestingextensiontothisproject.

3.3.3.2.HangersThehighestaxial loadwilloccur inthesecond longesthangers,as the longesthangersare

attached to the end of the suspended section and so have half the tributary area of the

others (see fig. 14). Although the highest loads occur in these hangers, the difference is

causedbythehangers’self-weightandisinsignificant.Thereforeallthehangerswillhavea

similarvalueforF(seetable7).

Table7.Calculationofthecriticalload,NcrandthefactorofsafetyFofthehanger.

Estimatedvalue

Cross-sectionalarea[mm2] 1735

Tensileload,!!"#[kN] 86.6

Stress,!!"#[MPa] 49.9

UTS÷ ![MPa] 287

Criticalload,!!" [kN] 497



Factorofsafety,F 5.8

Thecalculatedcriticalloadisalmost6timesgreaterthan!!"#,whichsuggeststhatthehangersare inefficientlydesigned.Howeverthecalculationdoesnottake intoaccount

theconnectionsateitherend.Likeinthechains,thesearelikelytobethecauseoffailure.

Several hangers have broken, the most recently in 2009 (“Suspension bridge closed by

fault”,2009)andCullimore(1986)notesthatmanyshowsignsofwearattheconnections.It

is likely that the hangers will fail at the bolt-holes or welds, so in this case!!" is notrepresentativeofthefailureload.

5.3.3.3.TowersAccordingtoplansavailableonthewebsiteof theCliftonSuspensionBridge ("YourBridge

Projects|TheCliftonSuspensionBridge"),themasonrytowersarefacedwithlocalPennant

sandstone and the cavities are filled with loose rubble. Richards says that the towers

themselveswerenotsubjecttomuchformalanalysis,butifitisassumedthatrubbleisnot

load-bearingandtheloadbearingwall is0.3mthick,themaximumstressdevelopedinthe

base is insignificant (see table 8), considering sandstone typically has a high compressive

strength–Pennantstonehasacompressivecapacityof168MPa(Bell,2013).

Table8.Calculationofthemaximumstressdevelopedinthetower.Theloadatthebaseofthetower

isthesumoftheloadatthetowersaddleandtheself-weightofthetower.Theload-bearingpartof

thetowerisassumedtohaveadensityof2.5kg/m3.

Location Towersaddle Baseoftower

Totalload[kN] 19,180 27,680

Area[m2] 10.3 16.3

Stress,![MPa] 1.2 2.7



6.Comparisonofanalyticalmethods

As mentioned earlier, graphic statics is a limited method of analysis in comparison with

modern numericalmethods. Numericalmethods aremore precise, faster and are able to

computemuchmorethangraphicstatics.Theprecisionandaccuracyofgraphicstaticswere

primarilyexploredinthisproject,howeverotherqualitativeobservationscanbemadetoo.

The graphic statics method was significantly slower than the numerical method.

Althoughthecomputermodelrequiredsometimetosetup, itwasstillcompletedquicker

than themanual drawings,which inmost cases took 2-3 hours to prepare, complete and

takeresults.Onceacomputermodelwassetup,thecalculationcouldberepeatedquickly

fordifferentloadcombination,whilethegraphicstaticsmethodrequiresthatyoustartfrom

scratchwitheach loadcombination.The fastermethod ispreferable inalmostall cases in

engineering. It shouldbenoted that thegraphic statics for the transversalgirdercouldbe

carried outwithin 20minutes using a CAD program, however this is still slower than the

numericalmethod.

The computational limitationsof the graphicalmethodweremadeobviousduring

this project as several parts of the bridge could not be analysed with it at all. The

longitudinalgirderisasingleobjectandprimarilyactsinbending.Asgraphicstaticscanonly

beusetofindtheaxialloadonanobject,itwasuselessinthiscase.

The precision of the graphic statics was determined by the scale of the force

diagramdrawn.ThedrawingsweredoneonA0paperatscalesof1cm:1kNor1cm:2kNfor

the transversal girders. This allowed for a theoretical precision of 0.05kN an 0.1kN

respectively,howeverthemeanaccuracywastypically0.5to1kN.Thetheoreticalprecision

couldbeincreasedbyusingalargerscale,butitwasnoticedthatinsomecasesthelongest

lineshadthelowestaccuracy.Itislikelythatthelargerthediagram,themoredifficultitisto

draw,sotheaccuracy isreduced.Drawingatthesescaleswasdifficultbecausethe140cm

rulewasunwieldyandtheA0sheetsweretoolargetousecomfortably.Iwouldarguethat

there is a limited scale atwhich useful drawing is possible, as percentage error increases

with scale. Although the method is inaccurate when done by hand, they would be

sufficientlyaccurateandpreciseformanycivilengineeringapplications.

Basedontheseobservations, it isobviousthatgraphicstatics isnotareplacement

for numerical methods. However it does have value as a visual aid. Mathematics is an

integralpartofengineering,however it can causeproblemswhen themethod isnot fully

understood or the results are incorrectly interpreted. Graphic statics could be useful in

teachingorpresentations,wherethespecificresultsarenotasimportantasunderstanding

thebehaviourofthestructure.Acombinationoftheformandforcediagramsisavisualway

to present the information, which would be useful for those who do not have an

understandingofthemathematicsordonotknowthestructure.



7.Criticalevaluationofthebridge

7.1.Themodernstructure

According to David Billington’s The tower and the bridge (1985), a work of structuralengineeringisaworkofstructuralartifitfulfilsthethreecriteria:economy,efficiencyand

elegance.Economyisameasureofthecostofthestructure–duringbothconstructionand

operation – in comparison to it’s social worth. Structures thatmeet this ideal have good

value for money. Efficiency is a quantitative evaluation of the design – how well the

structureissuitedtoitspurpose–andeleganceistheevaluationfromaqualitativepointof

view.

It can be argued that the economy of the Clifton Suspension Bridge has changed

duringits200yearhistory.AccordingtoBody(1976),whenVickdiedinthe18thcentury,the

area around the bridgewas largely uninhabited. Clifton only became a popular suburb of

Bristol inthe19thcenturyandLeighWoodsremainsasmallvillage. It isnotclearwhyVick

wantedabridgebuilt in that location, as therewas seemingly littleneed for it– abridge

there would be of little use to anyone. During the construction, the cost of the bridge

increased enormously; the final cost – approximately £75,000 – was more than double

Brunel’sestimateof1830(Body,1976).Theventureisarguablywastefulanduneconomical

whenthehighcostandthelowpracticalityareconsidered.

However themodern situation of the bridge is quite different. Although access is

limited and the capacity of the bridge is small in comparison to other bridges across the

Avon,itisusedbymorethan10,000vehiclesaday(“Suspensionbridgetollmaydoubleto

£1,”2010)andprovidesanalternativerouteovertherivertothoseinthecentreofBristol

and in Avonmouth. It is of great historical and structural interest and is a popular tourist

attraction. The toll leveed on motorised vehicles provides money for maintenance and

operation. Instead of becoming less useful and more expensive to run, the bridge has

becomeveryeconomical.

Theefficiencyofthebridge ismoreeasilyanalysed.Suspensionbridgesareoneof

themostefficientbridgedesigns; thisexplains theiruse in the largest spans in theworld.

The cables and hangers act solely in tension and the towers in compression. Thismeans

bendingmomentsandshearforcesinthestructureareverylowandthebridgecanbebuilt

usingminimalmaterial. This project has shown that thedesignof the cable and thedeck

girders isefficient–usingthe leastmaterialpossiblewhileretaininganadequatefactorof

safety–especially considering theanalyticalmethodsavailableat the time.Other factors,

suchasthestiffeningeffectofthedecktimbersandintegratedparapetmeanthatthesize

ofthegirderscanbereduced.

FinallytheeleganceoftheCliftonbridgeisdiscussed.AlthoughnotBrunel’s

first choice, the design is laudedbymany and it is clear that he took great care over the

appearance of the bridge. The slope of the bridgewas designed tomake the bridge look

horizontalwhenviewedfromupstream.Paintingthemetalworkwhitemakesthestructure

less obtrusivewhen viewed fromalong the river and the simple towerdesign, using local

sandstone,isinkeepingwiththearchitecturalstyleofthebuildingsinBristol.Theinscription

ontheLeighWoodstower;‘SUSPENSAVIXVIAFIT’(theroadbecomesbarelysuspended)isa

tributetotheamazementofearlyvisitorstothebridgeandillustratestheeffectivenessof

thedesign– impressivebutnotdominating, so that theAvonGorgeremains theprincipal

subjectofinterest.

Fromthisevaluation,theCliftonSuspensionBridgeisfoundtobeofgreatsocialand

historical value, well designed and with pleasing aesthetic qualities; fulfilling Billington’s

criteriatobeconsideredanexampleofstructuralart.



7.2.Othersubmissionsfromthedesigncompetition

7.2.1.‘Giant’sHole’design–Brunel

Figure36.Brunel's'Giant'sHole'design.FromCliftonSuspensionBridge,GeoffreyBody,1976..

Brunel’sfavouritedesignforthebridgeovertheAvonGorgewassubmittedforthe

firstcompetition.Thebridgewouldbeaccessibleviaanundergroundpassagethat passed

into a cave in rock face. The suspension chains would be anchored in sides of the gorge

itself, removingtheneedforpiers, towersand landchains. It iseasytoseewhy itwashis

favourite,beingfarsimplerandlessobtrusivethantheotherdesigns(McIlwain,1996).The

primary criticism of the design was its ambitious span, which would have been several

hundred feet longer than any other bridge at that time. When evaluated in terms of

Billington’sstructuralartcriteria,thedesignisextremelygood.Byanchoringthesuspension

chains in the cliff-faces, the structure is made very efficient and economical. The

construction of the piers and towers in the Clifton Suspension Bridge accounted for the

majority of the cost of construction (Body, 1976), so this designwould constitute a great

improvementinthepriceofthework.Howevertheviabilityofsuchaspanwasaprinciple

concerntothejudgingpanel,soitisunsurprisingthatthedesignwasnotawardedtheprize.

7.2.2.GothicRevival–Telford

Figure37.Telford'sgothicdesign.FromCliftonSuspensionBridge,GeoffreyBody,1976.



Telford said that thewidthof the gorgewas toogreat tobe coveredbya single span, so

suggestedathree-spandesignwithagothictoweroneithersideoftheriver(Body,1976).

The twin-tower design (see fig. 37), won the first competition but attracted great public

criticismfor itsappearance(McIlwain,1996).Thedesignwouldhavealsobeenparticularly

expensive. The savings made by building a suspension bridge instead of a stone bridge

would have been all but annulled by the cost of building two 80m towers. Although a

suspensionbridgeisaparticularlyefficientbridgedesign,thesheersizeofthetowersinthis

casemakeitfartooexpensivetocomplywithBillington’seconomycriterion.

7.2.3.Stonebeam–Burge

Figure38.Burge'sstonebeambridgedesign.FromCliftonSuspensionBridge,GeoffreyBody,1976.

This stonebeambridgedesignedbyWilliamBurgewas immediately rejectedona

basisofcost (Body,1976),however it is interestingtoexamine it inrelationtoBillington’s

efficiency criterion.Unlike stone archbridges,where the arch acts only in compression, a

hugebendingmomentwoulddevelopinthedeckjustduetotheweightofthestoneitself.

This would cause a tensile load in the bottom of the deck. Stone has good compressive

strengthbutapoortensilecapacity,sothisextremedesignwouldbeimpossibletoconstruct

as thebridgewouldcollapseunder itsownweight. Itmightbepossible toconstructusing

pre-stressed and reinforced concrete, but stone does not have the tensile strength to be

used here. This is a good example of how the choice of material affects the efficiency

criterion–tobeefficientasstructuremustmakeuseofsuitablematerials.

7.2.4.Combinedarch-suspensionbridge–Hill

Figure39.Hill'sironarch-suspensionbridge.FromCliftonSuspensionBridge,GeoffreyBody,1976.



William Hill submitted this rather unorthodox design, where the deck is not only

suspendedbysuspensionchainsbutalsoanironarchandastiffeningtruss.Likemanyother

designs itwasrejectedowingto theestimatedcost (Body,1976). It isanotherexampleof

inefficient design, but because of the structure itself, rather than the material. The

combination of three different elements – the truss, suspension cables and arch –would

havemade an incredibly stiff bridge,more robust than it needed to be. This unnecessary

designwouldjusthavecontributedtothealreadylargecostsofthebridge.



8.Conclusion

Thedevelopmentofstructuralanalysismethodsoverthelast200yearshasgreatlyaffected

the evolution of bridge design. The Brooklyn Bridge is an example of extreme over-

engineering to compensate for a lack of certainty, while the Severn Bridge has an

economical and efficient design that could only be achievedwith the latestmathematical

analysis.Thereisanassumptioninengineeringtodaythat ifthelatestmethodsofanalysis

havenotbeenused,theresultisneitherreliablenorengineering.Theaimofthispaperwas

to explore to what extent non-numerical methods of analysis are useful, and to use a

qualitativemethodtocomparedifferentbridgedesigns.

TheCliftonSuspensionBridgewasanalysedusingamodernnumericalmethodanda

graphical method more representative of the analytical capabilities of the 19th century.

Although the graphicalmethodwas less precise,more time consuming and less powerful

than thenumericalmethod, itwas still sufficientlyaccurate tomodel the forcesand ideal

shapesofsomekeyelementsofthebridge.Thiscomparisonofthetwomethodspermitsa

greater understanding of early suspension bridge designs and an gives an insight into the

roleofcivilengineersinthe19thcentury.

ThedifferencebetweenBrunel’soriginaldesignandtheoneconstructedbyBrunel

andHawkshawwasdiscussedandsupportsthetheorythatitisasmuchtheirbridgeashis;

it iswas suggested that Brunel’s designwould not have been strong enough to last until

today.

Finally, various submissions from the design competition were evaluated using

Billington’sthreecriteriaforstructuralart,inasimilarwaytowhichthecompetitionmight

havebeenjudged.Itprovidedanintegralandcompleteapproachtoqualitativeanalysis;an

evaluationthattakesintoaccountnotjustthemechanicalperformanceofthestructure,but

towhatextentitfulfilsitsroleasapublicfacility.



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10.Appendices

Theappendicesarefoundinvolume2.AppendicesA—Earethehand-drawnforcediagrams

usedforgraphicalanalysis.Thosewhichspanmorethanonepagehavepagesnumbered

fromlefttoright.AppendicesF–JaretheforcediagramsmadewithCAD.AppendicesK—M

arespreadsheets.

Hand-drawnforcediagrams:AppendixA–AnalysisoftransversalgirderunderLC1.(Twopages).

AppendixB–AnalysisoftransversalgirderunderLC2.(Twopages).

AppendixC–AnalysisoftransversalgirderunderLC3.

AppendixD–AnalysisoftransversalgirderunderLC4.

AppendixE–Analysisofsuspensionchain.

CADforcediagrams:AppendixF–AnalysisoftransversalgirderunderLC1.

AppendixG–AnalysisoftransversalgirderunderLC2.

AppendixH–AnalysisoftransversalgirderunderLC3.

AppendixI–AnalysisoftransversalgirderunderLC4.

AppendixJ–Analysisofsuspensionchain.

Spreadsheets:AppendixK–Appliedloadstotransversalgirder.

AppendixL–Resultsoftransversalgirdernumericalanalysis.

AppendixM–Analysisofchainshapes.

análisis estructural del puente colgante de clifton situado en bristol

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