1913-theory of the non-elastic & elastic catenary as applied to transmission lines by pierce

THEORYOFTHENON-ELASTICANDELASTIC CATENARYASAPPLIEDTO TRANSMISSIONLINES BY C.A.PIERCE,F.J.ADAMSandG.I.GILCHREST Presentedunder theauspices ofthe HighTensionTransmissionCommittee PERCYH.THOMAS,Chairman,2RectorStreet,NewYork. H.E.BUSSEY,Atlanta,Ga.HAROLDPENDER,Boston,Mass. MAXCOLLBOHM,Madison,Wis.NORMANROWE,MexicoCity,Mez. G.FACCIOLI,Pittsfield,Mass.C.S.RUFFNER,St.Louis,Mo. P.T.HANSCOM,SanFrancisco,Cal.DAVIDB.RUSHMORE,Schenectady,N.Y. JOHNHARISBERGER,Seattle,Wash.HARRISJ.RYAN,StanfordUniversity,Cal. R.F.HAYWARD,Vancouver,B.C.P.W.SOTHMAN,Toronto,Ont. 1373 THEORYOFTHENON-ELASTICANDELASTICCATENARYAS APPLIEDTOTRANSMISSIONLINES BYC.A.PIERCE,F.J.ADAMSANDG.I.GILCHREST ABSTRACTOFPAPER Equationsforlengthofconductor,span,tensionandsagarederived onthebasis ofa flexible elasticconductor.These equationscontainfunc-tions of ,theangle ofbending ofthe curve inwhichtheconductorhangs, andaconstant.Theconstantiseliminatedintwowaysleading,(a),to thecharacteristicratiosoftheelasticandnon-elasticcatenaries,(b),to threeequationswhichgivethevaluesoftension,lengthofconductorand sagintermsofeachother.Numericalvaluesofthecharacteristicratios ofthesimplecatenaryaretabulatedforangleslessthansixtydegrees. Bymeansofthistableproblemsbaseduponthetheoryofthenon-elastic catenarymaybesolvedreadily. Thecharacteristicratiosoftheelasticcatenaryarereducedtomore simpleapproximateformsinvolvingthecharacteristicratiosofthenon-elasticcatenary.Theequationswhichgivetheexactvaluesofthe ratiosoftheelasticcatenaryaretoocomplicatedtouse. Theresults oftests on anexperimentalspanapproximatelytwohundred feetlongaregivenintwotablesandthesevaluesarecomparedwiththa theoreticalvaluesbasedonthenon-elasticcatenary. 1374 Apapertobepresentedatthe30*AnnualCon-ventionoftheAmericanInstituteofElectrical Engineers,Cooperstown,N.Y.,June25,1913. Copyright,1913.ByA.I.E.E. (SubjecttofinalrevisionfortheTransactions.) THEORYOFTHENON-ELASTICANDELASTIC CATENARYAS APPLI EDTOTRANSMISSIONLINES CA.PI ERCE,F.J .ADAMSANDG.I .GI LCHREST Thoughmanyengineershavewrittenarticlesonthesubjectof tensionsandsagsinsuspendedwires,fewhavegivenanyatten-tiontothetheoryofthesubject,beingsatisfiedtorefertosome text-book,orotherguide,forauthority.Whenthenoviceturns to these references,he usually finds them insufficientfortheunder-standingofthearticlesinwhichthereferencesoccurandheis forcedtospendmoreorlesstimeinrecreatingthearticles.I t wouldseemthent hatthereis needforanarticledealingwiththe theoryofthecatenaryasappliedtotransmissionlines.Further-more,thereseemstobeneedformoreexperimentaldatatotest theaccuracyoftheequationswiththeactualmeasuredvalues onrealspans.Itisbelievedt hatthesedatacanbeobtainedin thelaboratoryonshortspanswithsmallwiresbetterthanwould bepossibleoutofdoorsonlongspanswithlargerwires,because ofthereadinesswithwhichvariousconditionscanbecontrolled inthelaboratory. Thisarticledealswiththetheoryofthecatenaryasappliedto transmissionlines,andexperimentaldataarecomparedwiththe valuesderivedbyuseofthetheoreticalequations. THEORETICAL Whenaperfectlyflexibleelasticstring'hangsbetweentwo horizontalsupportsandisactedonbygravitationonly,ittakes theformofacurvewhichhasbeencalledtheelasticcatenary. Theequationforthiscurveisdeducedasfollows: ReferringtoFig.1,letthelengthofthearcoftheelastic catenary,P2OPi.bemeasuredfrom0,thelowestpointofthe 1375 1376PIERCE,ADAMSANDGILCHREST:[June25 arc.Consideranelement,dl,ofthearcbetweentwopoints,P andP' .Theelementdl,isundertensionandconsequentlyis stretched.Iftheunstretchedlengthofdlisda,thenby Hooke'slaw, dl=da{I-) , whereistheelasticconstantofthestringandTisthetension whichstretcheslengthdaintolengthdl.Iftheweightofunit lengthoftheunstretchedstringisW,thentheweightofdl, whichisequaltoweightofda,is equaltoWda.Substituting thevalueofdaasgivenintheformulaabove,theweightof elementdl is equaltoWdl-r-(1+). Theverticalcomponent,V, ofthetensionatPdiffersfromt hat atP 'bytheweightoftheelementdl,hence, dV=W dl 1+T FIG.1 ButV=Htan,whereHisthehorizontalcomponentofthe tensionatPand is theanglebetweenthetensionatPandthe horizontalcomponentH.Hence, ()=W dl 1+T or,sinceHisconstantalongthearc, Wdl d(tan)= H1+ LettingW+H=1+Kand\H=N,whereKandNare constants,andsubstitutingT=Hsec, dl=K(1+Nsec) d(tan