mechanical springs wahl

8/21/2019 Mechanical Springs Wahl

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/ h t t p : / / w w w . h

a t h i t r u s t . o r g / a c c e s s_

u s e # p d - g o o g l e

Mechanical springs, by A.M. Wahl ...

Wahl, A. M. (Arthur M.), 1901-

Cleveland, O., Penton Pub. Co., 1944.

http://hdl.handle.net/2027/uc1.$b76475

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http://slidepdf.com/reader/full/mechanical-springs-wahl 9/462

PUB I H IN GCO .,C E V E L A N D13,O H IO ,U.S .A .

F S T E L . MA C H IN E D S I GN . TH E

W E Q U IPM N T DI G S T • R E V I TA I N D U T R IA L

P u b l i c D o m a i n , G o o g l e - d i g i t i z e d

/ h t t p : / / w w w . h





ManufacturingCompany

H I N G CO M PA N Y

D ,O H I O


/ h t t p : / / w w w . h





PUB I H IN GCO MPA N Y

D ,O H I O

t . , W e st m in s te r , S . W . 1


/ h t t p : / / w w w . h





undamentalprinciplesunderlying

pringsand bringstogetherincon-

erofmachinesthe moreimportant

eoryandtestingwhichhaveta en

lthoughmechanicalspringsoften

ponentsofmodernmachinesandde-

ngstoooftenhavebeendesignedon

ruleof thumb methodsw hichdo

e limitationsofthematerialsused

esset upduringoperation.Thisis

tionswherefatigueorrepeated

conse uence,unsatisfactoryoperation

hasresultedin manycases.

thatthepresentboo maycontribute

idanceofsuchconditionsbyhelping

springsonamorerational basis.

esearchescarriedoutunderthe

esearchCommitteeonMechanical

canS ocietyofMechanica lEngineersand,

heWarE ngineeringBoardSpringC om-

utomotiveEngineers, havebeenf reely

aterialin theboo hasalsobeenbased

heTransactionsof theA . . M. . , the

anics,theS .A . .Journalandtheauthor s

n MachineDesignduringthepast

eofthehelical compressionor

argeamountof spacehasbeende-

nlyhavethetheoreticalaspectsof

pe ofspringbeentreatedin con-

emphasisalsohasbeen laidupon

chsprings,as wellasonthe fatigue

singeneral.This hasbeendone

periencethatthe lim itationsduetothe

aterialsareaptto beoverloo edby


/ h t t p : / / w w w . h





antaspectsofthehelical springdesign

chaptersincludecreep effectsunder

lateralloading,vibrationandsurging.

thefundamentalsofdesignof

esincludingdis ,Belleville,flat,leaf,

ngspringshave beentreated.Be-

ance,thevolute typeofspringhas

abledetail.

dinarilythoughtof asaspring

applicationofrubberspringsandmount-

adetheinclusionofa chapteronthis

ble. S incethesub ectofv ibrationisin-

plicationof rubbermountings,some

rationabsorptionandisolationalso

thatinaboo o f thisnature it

thefactorswhich enterintothe

enapplication.Conse uentlythe

esultsinanyparticulardesign,close

ntainedbetweenthedesignerandthe

dertobenef itf romthelatter se per-

ow ever, ak now ledgeof thefunda-

udgingthefeasibilityofadesign.

wouldnothavebeenpossible

estinghouseE lectricandManu-

ctiontheauthor sthan saredue,

hubb, Director, R esearchL aboratories,

, Manager, MechanicsDepartment. Than s

e llso f theL aboratories, andtoTore

y , B. S t e r n e an d H . O . F u c h s o f th e S . A . .

dS pringCommittee.Theencouragement

or sresearchbyS . T imoshen oand

ppreciated.ToL .E .Jermy,E ditor,

W. Greve, A ssociateE ditor, thew riter

ablesuggestionsconcerningthepres-


/ h t t p : / / w w w . h





pringDesign1

S pringMaterials—TyposofLoading—

S urfaceC ondit ionsandDecarburization—

ariationsinDimensions—F actoro fS afety

ompressionandTensionS prings25

ppro imateTheory—E x actTheory , Ef fect

pringsw ithLargeDef lection50

eetoR otate S tress, Def lection, Unw inding

p r in g s wi t h E n d s F i e d A g a i n st R o t a t io n D e -

tress

stsonHelica lSpringsandSpringMateria ls. . . 69

DeflectionTests—V ariationsinModulus

a ining, SurfaceDecarburization—Determina-

ty DeflectionMethod,DirectMethod,

od—ModulusofR igidityV alues Car-

l loy , Sta inless, MonelandPhosphorB ronze,

atigueTests S mall-sizeS prings,S hot

prings, F ew S tressCycles

rStaticL oading95

glectingC urvature—L oadforC omplete

nofF ormulastoSpringTables C urvature

e la ationTests—A naly tica lMethodsof

Creep, R e la ation, S hearStresses

oadingofHelica lS prings119

S ensitiv ity toStressC oncentration, A p-

tationsofMethods—ComparisonofTheo-

A lternativeMethodofCalculation

DesignofHelica lC ompressionSprings134

sUsedinPrarice—S pringTables—Design

ation


/ h t t p : / / w w w . h





tionsforH elicalCompressionS prings.157

rns—E ccentricityo fLoading, A llow able

ngTolerances,Deflection—E ffectofMod-

ssatS o lidC ompression O ver-stressing,

tresses

ompressionSprings169

o a d , H i n g ed E n d s, F l e u r al R i g i d i ty , S h e a ri n g

ds—Def lectionUnderC ombinedL oading—

imumSpaceE f f iciency183

o l id a n d F r e e -H e i g h t V o l u me , I nf r e u e nt L o a d-

, Ma imumEnergyStorage—S pringNests

taticLoading

gs StressandDeHectioninEndL oops,

fE ndC oils, Wor ingStresses—C om-

ssionS prings

nular-WireC ompressionS prings203

, S mallP itchA ngle MembraneA nalogy

r i n g s of S m a ll I n de S m al l P it c h A n g l es ,

x actTheory—R ectangular-WireS prings

hartsforCalculatingS tress,Calculationof

hA ngles—TestsonSq uare-WireS prings

mulas

gofHelica lS prings222

esonance, Principa lF re uencies, Surge

onforV ibratingSpring A cce leratingF orces,

t u ra l F r e u e nc y S p r in g E n d s F i e d , O n e

n e S p r in g E n d W e i g h te d — S u r g i ng o f E n g i ne

ignE x pedients

leville)S prings238

S implifiedDesignforConstantL oad

heory—W or ingStresses—F atigue


/ h t t p : / / w w w . h





prings263

gs—S pringsofC onstantThic ness—L arge

dCalculation

gs28C

apezo ida lP rof i leSprings—S impleLeaf

oading—PlateSpring—S tressC oncentration

tches, S harpB ends, C lampedE nds

gs314

dtions—B inding—B uc ling—WireS ection—

arTorsionS prings—CircularW ireTor-

i n g S t r e ss e s

sWithoutC ontact C lampedOuterE nd,

pringswithF ew Turns—L argeDef lections,

nerR ing, OuterR ing—Def lection—Design

le B ottomingL oads, Def lection, Stress—V ari-

L oad, Def lection, TypeC urves

Mountings378

impleShearS pring—C y lindrica lS hear

ight, C onstantStress—C y lindrica lTorsion

c ness, ConstantS tress—A llow ableStresses

S teady - tate , Shoc

cityofV ariousS prings399

prings—C antileverSprings R ectangular

eafSpring—H elicalTorsionS pring

ularW ire—S pira lSprings—R oundB ar

lCompressionSprings—TensionS prings

eCapacities

operties—E nduranceL imits—Descrip-

dMateria ls.


/ h t t p : / / w w w . h





BO L S

ec/in.2

rala istocentero fgrav ity inch

nginch

city lb/ s in.

ationcycles/ sec

tyin./(sec)

b/s in.

nch

rea in.

ertiain.4

—

/ in.

ionfactor—

tationfactorduetocurvature—

threductionfactor—

concentrationfactor—

cientofmodulusdegrees '

on sratio—

wistingmomentsperinchin.-lb/in.

mentsperinchin. - lb/ in.

omentin.-lb

mberofleaves,etc —

frubberslab —

oils(he lica lspring)—

ieldinglb


/ h t t p : / / w w w . h





BO L S

—

area lb/ s in.

l loadlb

engthlb/in.

s

omentin.-lb

springperunitvolumein.-lb/in.

runitvolumein.-lb/in.

rdinates—

aminch

matanypoint inch

edegreesorradians

radius. —

ctiondegreesorradians

ardegreesorradians

melb/in.

ngleradians

#/dt)radians/sec

-

n—

in.

riablecomponentsofnormalstresslb/ s in.

t re s s lb / s i n .

ionlb/ s in.

mtinbendinglb/ s in.

asedonstrengththeory . lb/s in.

l st r es s es l b /s i n .

in.

riablecomponentsofshearstresslb/s in.

hearlb/ s in.

on,angulardeflectionradiansordegrees

y , circularf re uency radians/ sec


/ h t t p : / / w w w . h





/ h t t p : / / w w w . h





S P R IN G

N S I D R A T IO N S

I GN

ybedefinedasan elasticbody

odeflector distortunderloadand

shapewhenreleasedafter being

stmaterialbodiesareelasticandwill

not allconsideredassprings.Thus

willdeflectslightlywhenaweightis

tisnotconsideredasa springbecause

todeflectunderloadbut ratherto

elicalspringusedinthe ordinary

esignedso astodeflectby arelatively

d.Conse uently,thedeflectionand

ned.This,therefore,functionsasa

tstressedbeyondthe elastic

ingwillhaveastraight-lineload-

wninF ig.2.Thismeansthatthe de-

e load,i.e.,iftheloadis doubled

ed.Therelationwill holdtrueeven

ueormoment,providedlineardeflec-

rdeflection.

r load-deflectiondiagrams,how-

eflectiondiagramsasshownin F ig.

maybeobtainedwithathin flatcir-

gedeflection.CurveBmayresultfrom


/ h t t p : / / w w w . h





S P R IX G

orBelleville)spring.Thesetwospring

aptersX V andX IV , respectively . B e-

ct betweenturns,theordinarycloc

a lineartor ue-anglecharacteristic,

Y III.

S P R I N G

tionsofspringsthe followingare

nt:

andMitigateS hoc : Inordertoabsorb

epea loadsthespringmustdef lectby

fie S ons

calesprings

n e ampleof theuseofaspringtoab-

arspringshowninF ig. 4. A nother

vespringsforindependentsuspension

hichmustbe abletoabsorbtheenergy

esoverabump.Thesealsofunctionas

evehicle.


/ h t t p : / / w w w . h





S I GN CO N S I D R A TI O N S

orceohTor ue: A n automotiveva lve

hichholds thevalvefolloweragainst

gsuppliesthetor uenecessary toover-

vingmechanism.S ometimesspringsare

-def lec-

as etpressureasin thehigh-voltage

etspringsshowninF ig.6.Thefunction

tainanoil-tightgas etsealregardless

emperaturechange. Loc w asherscommonly

headsalsofunctionessentiallyassprings

oad-

f

gardlessofslightchangesinthe bolt

mperaturechanges,etc.Thisforce

romunwindingeventhoughvibration

assesonToIsolateV ibration:The

edto supportmovingorvibratingmasses


/ h t t p : / / w w w . h





S P R IN G

evibrationorimpact.Thuselectric mo-

gsupportedtopreventthetransmission

tontothefoundation. Li ew ise , thesprings

sionsnotonlytendto mitigateshoc

roadsurface(Point1)but alsopre-

rStee lC o.

ew of ringspringindraftgear

ecar bodyofob ectionablevibration

waves(wash-board)intheroadcon-

gsona railwaycar,F ig.7,tendtoprevent

tshoc sf romtruc irregularit ies. A n

heuseof springstosupportavibrating

anautomaticwashingmachine,F ig.

iblyonsprings,transmissionof

dmassesinthe eccentrically-loaded

oadorTor ue: Oneof themostim-

sforindependentsuspensionoffrontwheels


/ h t t p : / / w w w . h






sis thatoffurnishingafle iblemember

siderableamountwhensub ecttoa

eof suitablemechanismsthisdeflection

whichindicatestheamountofloador

e isthesca lespringinF ig. 1.

P ivotorGuide: Sometimesoneormore

ncombinationtofunctionasan elastic

nalfrictionsuch pivotsoftenhavereal

sorantifrictionbearings.Thusthefle ible

untingsfortheMt. Palomartelescope,

formainta ininggas etpressure

agroupofstraightbars radiallydisposed

iscanbe consideredessentiallyasaspring

cpivotis re uired.A napplicationwhere

isinthebalancing machine,F ig.10.

I A L S

allydeflectbya considerablede-

owsthatarelativelylarge amountof

enthespring isinthe deflectedposition.

andload formostspringsarepropor-


/ h t t p : / / w w w . h





S P R IN G

energyisproportionaltodeflectiontimes

eraltheamountofenergywhichmay

othes uareof thestress. H encefor

wor ingstressesmustbeused.This

gmaterialshavehightensile strengths

higherstresses thaninotherfields.

erialforsprings atpresentiscar-

zesofwire,foran .8to.9per cent

gand patenting,ultimatetensile

0,000to400,000poundspers uare

ofthewire)maybeobtained.This

nasmusicwire.In thelargersections,

bonsteelandheattreatingafterform-

mayreach200,000to 240,000pounds

larva luesmaybeobtainedf romchrome-

Propertiesofvariousspringmate-

springsonN ew Y or C entra l" Mercury

erX X III.Inthe largersizes,helical

atedafterforming,the latterbeing

fhelicalsprings, ontheotherhand,are

etemperedmaterialormusicwire.

lievinglow-temperatureheattreat-

ordifferentspringmaterials,includingdataonheat treat-

rX X III.


/ h t t p : / / w w w . h






arepresent,stainlesssteelsprings

8percentnic e lcomposit ionaref re-

rial(whichisalso usedforhigh-tem-

yhaveatensilestrengthvaryingfrom

ortforautomaticw asher

areinchfor3 16-inchwireto280,000

or1/32-inchwire.Phosphorbronzealso

present.Thismaterial,however,has

thanstainlesssteel,valuesofultimate

0,000poundspers uareinchare

A DIN G

anycases, springsaresub ecttoa load

stantor whichisrepeatedbuta few


/ h t t p : / / w w w . h





S P R IN G

espring. Suchspringsarek now nas

x amplesaresafetyvalvesprings

ctedtopopof fbutafew timesduring

ucinggas etpressure , typif iedby thecon-

/ g. 6 springsincircuit-brea ermech-

eroperatesbutafewtimes initslife.

loadedspringsitisfre uentlyim-

ntainits calibrationtoasufficient

faspringcompressedbya given

time goeson,theloadshouldnot drop

mount(usuallyafewpercent). This

sk now nasrela ation. F ore ample,

hereis somelossinloaddue tore-

ftera periodoftime,thevalvewill pop

hat forwhichitwasdesigned.S imilar-

ducegas etpressure , re la ationof the

lementsforgimbalo fMt. Palomartelescope

ofpressure.W hilesomelossinpres-

ted,toomuchrenders thespringin-

asealingdevice.


/ h t t p : / / w w w . h






toaconstantloadratherthanacon-

stressistoohigh,there willbeaslow

susuallyk nownassettageandis due

susedinbalancingmachine

hematerial.Inmanycases thisisun-

spring isusedtosupporta givenload

-actioncar,settageofthespringwill allow

elativeto thecar.S uchadeflection

thesteeringmechanism.A nother

asespringsshow ninF ig. 11. If settageof

,ob ectionablelossinpressurebetween

dwirewouldresult.Thereforethe

othat thissetisk eptsmall.

s, if thepea stressinthespringis

iclimit oryieldpointofthe material,

a tionwillse ldomoccur. H ence indesign,

arenotinvolved,thema imumstress

y ieldpo intdiv idedby thefactoro f safety .

o f safety ista ene ua lto1. 5a lthough

n manycases.S omeofthefactors

ofa factorofsafetyarediscussed

estheef fectsofcreeporrela ationare


/ h t t p : / / w w w . h





S P R IN G

e designstressshouldbebasedon

ontests.Unfortunately,notmuchdata

todesigners,althoughsometests were

P.Z immerli-.Thesetestsindicatedthat

prings,theeffectsoftemperatures

ahr.werenotverypronounced.

ahr.theuseofstainless-steelspringsis in-

furtherdiscussedin ChapterV .

b ecttostaticloadingit issug-

pringmaterialwhichhassomeduc-

tas muchasstructuralmaterials),

tssuchasthosedue tocurvaturemay

mple , inasimpleplatew ithasmallhole

nload,theoreticalanalysisbasedonthe

hat,forelasH cconditions,thepea

oleisaroundthree timesthestress

orfatigueorrepeatedloadingthis pea

ever,forstaticloadstheusualpractice

glectsuchstressconcentrationeffects4,

esarelocalizedandmaybe relievedby

uenceofthematerialductility.A vailable

fact thatsimilarstressconcentration

tocurvatureinhelicalsprings maybe

dsareinvolved.

manyspringapplicationsthe load

butvariesw ithtime. F ore ample , an

initiallycompressedbya givenamount

goperation,itis compressedperiodic-

nt.It may,therefore,beconsidered

mumandama imumloadorstress.

issaidtobesub ecttovariableorfatigue

chstress cyclesforsucha springsub-

c stressbetweentheminimumstress

mvaluea „ ^ . Thisise uiva lenttoastaticor

atureonC oiledS tee lSpringsatV ariousL oadings , Transac-

1 9 41 . P ag e 3 63 . A l n " R e l a a t io n R e s i st a nc e o f N i c e l A l l o v

ta l. T ransactionsA . . M. . , In ly, 1942. Page465.

engthofMateria ls, PartII. SecondE dit ion. 1941. V anN ostrand,

scussionof stressconcentration alsoTheoryofE lasticity.

75.

ionarticlebvC . R . S oderhergon" W or ingStresses ,

..1933,A PM55-16isrecommended.A lsoTimoshcn o— S trength


/ h t t p : / / w w w . h






altoha lf thesumofma imumandminimum

mposeda variablestress< t,.Thevari-

a lf thedif ferencebetw eena, „ x andami ,

eingconsidered.Thediscussion con-

thischapteralsoapplies,in general,

oadingconditionismuchmore

icatedinF ig. 12. F ore ample , an

springs

spring. F ig. 5, issub ecttopractica lly

ar istravelingoversmoothpavement.

roads,however,thespringmaybe

oadingconditionwithstresscyclesof


/ h t t p : / / w w w . h





S P R IN G

atedinF ig.13.Thesamethingis also

motivesprings.Insuchcasesthe de-

tressesismoredifficult,particularly

ngresults forcaseswherethevariable

geswithtimearealmost whollylac ing5.

gsaresub ecttoloadsor stresses

ontinuouslybetweenaminimumand

dif ferencebetw eenthema imumand

ow nasthestressrange thisisa lsotw ice

stressar.This stressrangeisof

nfatigueorrepeatedloadingisin-

erialstheendurancerangeis prac-

heyieldpointis note ceeded.

s avwhichthespringwill j ust

emeanstressa onw hicho> issuper-

agramsuchasthatshownin F ig.14

pointP onthiscurvemeansthat,if

variablestresslargerthana, if super-

willeventuallycausefatiguefailure

ana,willnotcause fatiguefailure

twhenthestaticcomponentof

onditionofcompletelyreversedstress

ressac forzeroa shouldtherefore

atigueF a iluref romStressC yclesofV ary ingA mplitude , Jour-

s,December,1937,givesafurtherdiscussionofthis problem.

vcrstressintheF atigueofF errousMeta ls , byR usse lland

. . T . M. , 1936, Part2, page118.


/ h t t p : / / w w w . h






durancelimita> forcompletelyre-

on orbending(dependingonwhether

sareconsidered).O ntheotherhand,

ow, testsshowthatthecurvetends

trengtha . O fcourse, forhighvalues

ecreepmaybe e pectedtooccurso

esofvariableamplitude

oneofavoidinge cessivesettageor

enduranceorfatiguetest results

onsisshowninF ig. 15. Heretw o

ngo- „ uanda, ( respectivelyareplotted

anddash linecwhichrepresentsthe

peri-

C T0

d bcurvesrepresentupperandlower

stre uiredtocausefatiguefailurefor

ntsonthesamevertical line.Itis clear

nateofthemeanstress linecrepre-

omponentofstressa whilethevertical


/ h t t p : / / w w w . h





S P R IN G

nstresslineand eithertheupperor

blestresscomponento> .Theupper

atw hilethe low ercurvebrepresents

mumandminimumvaluesofstresso-ma

toanycombinationofstatic andvari-

maybereaddirectly fromthecurves

fdeterminingw or ingstressistore-

ecurveby astraightlineconnecting

heendurance lim ita , asshow nin

wever,beconsiderablybelowthe

thatwor ingstressesdeterminedin

whattooconservative.H owever,the

manycaseson accountofitssim-

nationofstatic andvariable

nypointPon thelineA Bconnecting

vnanda / nisdef inedashavinga

isthestaticcomponentof the

, thevariablecomponent, f romthege-

ybeshownthatthefactorof safetymay

methodof rep-

results

s

dbyS oderberg—" F actorofS afetyandW or ing

onsA . S . M. E . , 1930. A PM52-2.


/ h t t p : / / w w w . h






owingrelation:

onas wellasforbending,pro-

, areta enasthey ieldpo intand

respectively.Torsionstresses,how-

ytheCree lettert.

ostspring steelswillusuallybe

calrelationshipbetweenthe values

sesnecessaryto causefatiguefailure.

ndurancecurvewill berepresentedby

pse, A C B inF ig. 17andintersecting

Tvandt, . If itbeassumedthatno

eyieldstrength,thedottedportionof

acedby the lineC B e tendingatan

abscissae.Thisisdonesince, when

ponentsofstress representedby

hema imumstressw ill j uste ua l

asisof thisellipticallaw,thefactor

onof

a -


/ h t t p : / / w w w . h





S P R IN G

esamemeaningasthose usedin

tests areplottedinF ig.17to

madebetweenthe ellipticalandthe

nthe caseoftorsionalfatiguestress-

ttheresultsobtainedby Weibel7

eelwiretestedbothin pulsating(0to

edtorsion,thesurfaceof thewires

d condit ion. Whiletheseresultsshow

eamongthe fewavailableforsuch

n F ig.17representtestresults by

sofsilico-manganesespringsteelwithma-

seen thatinthesecasesthe elliptical

test results.Itshouldbenoted,

N S T R E S S L B ./ Q . IN .

T IGU T S T B W E IB L O N T MP R E D

E L W IR E . ( U R F A C -A S R E C IV E D )

A T IGU T S T B H A N K I N S O N S I -MN

. ( U R F A C MA C H IN E D )

verepresentingfatiguefailureandcom-

ancetestresults

inelaw,F ig.16,issaferto usein

ecompletetestdataarelac ing.

meanstressa ,theconsensusof

pringWireB endingandTorsionTests —E . E . Weibe l,

.,N ovember,1935,page501.

e T es t s on S p r i n g S t e e ls — G . A . H a n i n s, D e pt . o f S c i en t if i c &

it ish)Specia lR eportNo. 9.


/ h t t p : / / w w w . h






stress-concentrationeffectsmaybe

rials9.Thisisconsistent withneg-

oneffectswherestaticloadsonlyare

pea sduetocurvature inhelica l-com-

ctinfatigue

n,.7per

lundEisen,

S S

sarelocalized,it isbelievedthatfor

beconsideredasdue tostresscon-

cemaybeneglectedin computingo- .

eeffectofdirectshear inhelical-

dbeconsidered,becausethisis not

g thevariablecomponenta> ,how-

maynotbeneglected.

rtofthis methodliesincertain

rsunder combinedstaticandvariable

ch test1 on.7percent carbonsteel

,thefull linesbeingresultsforspeci-

ntrationandthedashedlines for

ybeseenthat themeanorstatic

efinedbyS oderbergasthosehavingelongationsover5

ostspring materials.

oudremontandB enne , S tahlundEisen, V o l. 52, page660.

. PetersonofR eporto fR esearchC ommitteeonF atigueof

. T . M. , 1937, V o l. 37, Part1, Page162.


/ h t t p : / / w w w . h





S P R IN G

otand dashlineisnot diminished

neffect,whilethevariablestressrep-

tancebetweeneitherthe fulllines

nishedinamoreor lessconstant

trationeffectofthenotch.W hileit

lablefatiguetestdata madeforthe

ss-concentrationeffectsundercom-

ressare rathermeagre,itisbelieved

areathand,stressincreasesdue to

gsmaybe treatedinthismanner.

ethodtothedeterminationofw or ing

giveninChapterV I.

ingtheproblemofcombinedstatic

alculatethestressrangebyta ing

cts(duetocurvature,fore ample)into

wherestressconcentrationispres-

elocationof stressconcentrationwill

ntsince thematerialwillmerelyyield

ema imumpointo f thestressrange

tstressofthematerial.To determine

durancerangeof thematerialiscom-

culatedinthismanner . A further

thatthestressat ma imumload,

ressconcentration,mustnote ceed

erial,sinceotherwisee cessiveyield-

onofthismethodinthe designof

tratedinChapterV I.

T O P R A TIO N

ecttore lative ly few cyclesof load-

ingstressmaybeconsiderablyin-

eforaninfinite number.E x amples

usedincertaincontrol mechanisms.

raphforhelicalcompressionspringsof

mzerotoama imumisshow nin

e factthatthematerialmaynot befullysensitiveto

thattheactualstress rangeasfoundbytestmay begreater

sinutheoreticalstress concentrationfactors),areduction

emade,providedtestdataareavailable.F urtherdiscussion

Two-andThree-DimensionalCasesofS tressConcentration

tigueTests PetersonandWahl, Journa lo fA ppliedMechanics,


/ h t t p : / / w w w . h






orthistypeof stressapplication,the

failurein tenthousandcyclesofstress

wiceasgreat asthatre uiredto

cycles.Providedsomepermanent

ionable,thissuggeststhataconsiderably

ouldbeusedif thespringistobesub ect

C E S T O BM UR E

-cyclecurveforhelicalsprings

es.It mustnotbeinferredhowever,

ewor ingstressisusuallypossible

tivelyfew.Inmost casestheincreased

ablyinterferewiththeoperationof

pringsforman integralpart.

D ITI O N S A N D D CA R B UR I Z A T IO N

surfaceconditioninspring steels

hefatiguestrength ofthematerial.The

emanufactureofsprings,becauseof

eatingandformingoperations,the

edtosomee tent.Thusthereis,in

carbonsteel(whichisrelatively

thespringw hichiscomposedof the

bonoralloysteel.Underfatigueor

nsthewea erlow-carbonsteelonthe

c whichthenspreadsacrossthe


/ h t t p : / / w w w . h





S P R IN G

ceofthehighstress concentrationatthe

tua ltestshaveshow nthatavery thinlayer

alissufficientto greatlywea enthe

ntheef fecto f surfacecondit ionson

ngshas beencarriedoutbythe

oratory inEngland. Theresultso f this

used inleafspringsshowconclusively

ronthe surfacecombinedwiththe

tofsurfaceirregularitiesproducedby

smayreducetheactualendurance

lessof thattobee pectedonthe

dorgroundspecimens. F ore ample ,

ch barsofheattreated.61per cent

steel(asusedin leafsprings)made

howedanendurancerangewiththe

undof0 to128,000poundspers uare

asleftuntouched,the endurancerange

spers uareinch,areductionofabout

ecurvesofF ig.20areplottedfrom

perimentersandshowthetremendous

tionsin thisparticularcase.A lthough

nunusuallygreatreductionin strength,

owimportantarethesurfaceconditions.

ta inedbyH an insandF ord tw ho

nesesteela± 60,000poundspers uare

ersedbendingon specimenswhich

rgrindingto size.Inthiscase there

elayerleftthere bytheheat-treating

weremadeonspecimensof thesame

eat treatmentbuthavingathinlayer

offafterheat treatment,theendur-

103, 000poundspers uare inch. F ur-

ecimenswhich hadbeenheattreated

such awayastopreventthe forma-

r inthiscasethe endurancelimitwas

gthofC arbonandA lloyS tee lP latesasUsedforL aminated

g InstituteofMechanica lEngineers, 1931, Page301.

tallurgica lP ropertiesofS pringStee lsasR evea ledbyLab-

insandF ord- JournalIronandStee lInstitute , 1929, N o. 1,


/ h t t p : / / w w w . h





S 1 GN C O N S ID R A T IO N S 21

uare inchorpractica lly thesameasfor

rheat-treatment.Thisindicatesthat

bythe usualheattreatmentwasto

bleforthelowerendurancelimits found

otbeenmachinedafter heattreatment.

elyinterestinginthattheyaffordani n-

neby meansofspecialheattreat-

tiguestrengthof actualsprings.

estrengthbecauseof surface

rvedinreversedtorsion fatiguetests

4onchrome-vanadiumspringstee lw ires.

that,wherethewireshadbeenma-

C E S O F S T R E S S

I CA T IO N S F R O M Z E R O T O MA X I MU M)

ecursesfor. 61percentcarboncom-

romtestsbyBatsonand

.,1931,Page301)

rsionalendurancelimitswerein-

evalueobtainedfrom specimensin

n. Sw an, Sutton, andDouglas1 1a lso

umsteelunderpulsatingtorsional

o m eS t e el W i r e s U nd e r R e p ea t ed T o rs i on a l S t r es s es — L e a

ngsInstituteofMechanicalE ngineers,1927,Page403.

nofStee lsforA ircra ftE ngineV alveS prings , P roceedingsInstitute

s,1931,V ol.120,Page261.


/ h t t p : / / w w w . h





S P R IN G

imumtothema imum), anincreaseof

urancerangewherethespecimens

ed.Theresultsofthese varioustests

esurfaceconditionsof springsteels

estressing.O ntheotherhand,it

orsionfatiguetestson S wedishvalve

showedpracticallynodifferenceinthe

etweenspecimenswiththesurface

ngthesurfacelayergroundoff. Prob-

nedby thefactthatthismateria lhasa

nsothatthe effectofdecarburization

t-blastinghelicalsprings hasbeen

stheendurancerangeby50per cent

rings16.Thisprocessconsistsofpro-

highvelocityagainstthespringsur-

centrifugaltypeof machine.A ppar-

steel shotpropelledagainstthesur-

oco ldwor andthusincreasethe

decarburizedsurfacelayer.Thismethod

calwayofobtainingsatisfactory

utthe e penseofgrindingthesur-

owever,shotblastingorshotpeen-

nnotbee pectedtogivesatisfactory

decarburizationorsurfacedefectsare

ussionof thisisgiveninC hapterIV .

E F F E C T

sub ecttoevenmildlycorrosive

stressing,theendurancelimit for

reducedgreatly.In suchcases,fa-

outformanymorethanthe usual

argenumberofcorrosionfatiguetests

doutby McA dam18,showthetre-

H ow ShotB lastingIncreasesF atigueL ife , MachineDesign,

lsoL essellsandMurray—" TheEf fecto fS hotB lastingandIts

P roceedingsA . . T . M. V o l. 41, 1941, Page659.

ueofMeta ls —H . J. C ough, E ngineer, V o l. 154, 1932, Page284.

sionF atigueofSpringMateria ls . —D. J. McA dam, Jr. ,

.,1929,A PM51-5.


/ h t t p : / / w w w . h






ndurancelimitfor springmaterials

orsa lt-watercorrosionfatigue. F orspring

watercorrosionfatigue,thevalueof en-

sbutone-fourthtoone-ninththat ob-

nsin air.H ighervaluesofendurance

tionswereobtainedoncorrosion-

mium-platedspringsshowedmuch

dersuchconditions,i.e.,abouttwice

a springsteelwithaplating than

sshowthatthespringdesignermust

omcorrosiveaction, orelseusee

sses.E venthen,ifcorrosionis present,

eventualfatiguefailurewillnotoccur,

erof stressrepetitionsofstressta e

I N D IM N S I O N S

espringdesignershouldk eepin

sanunavoidablevariationinthesize

a ingsprings. Theef fecto f these

ge,especiallywhenitcomes toob-

tioncharacteristics.F ore ample,in

,acumulativevariationinbothcoil

percentwillresult ina7 percent

oncharacteristicofthe spring.Thus

entvariationwouldcorrespondto

nly.001-inch.S uchvariationsareeasily

ctice.H ence,itmaybenecessary

cturersomeleewayinchoosing the

compensateforunavoidablevaria-

ia lw irestoc . F ore ample, if thew ire

shappensto beslightlyundersize,the

beabletocompensateforthis by

ameter.In mostcases,thisslightre-

uldnot bedetrimentaltotheopera-

mercialsprings,actualstressesand

sticsmayeasilydeviateby5to10 per

luesandthis mustbeconsideredin

onsaretobeta enby thespring


/ h t t p : / / w w w . h





S P R IN G

maybereduced, butatincreasede -

cussionof thisisgiveninC hapterV III.

A F E T

safety , n(asdef inedinE q uation1)

dby manyconsiderations.Ifthe

areserious,thena higherfactormust

enspringcausesbutlitt le inconvenience,

designertolowerthe factorofsafety.

ofuniformandhigh-gradematerial

he manufacturingprocessismain-

etymaybeused.If, inaddition,

particularspringmaterialsemployed

rethetest conditionsappro imate

designfactorof safetymayagainbe

and, ignoranceof thepea loadsacting

wnfactorssuchascorrosionor tempera-

increasein thisfactor.

o f thisboo toac ua intthede-

alsunderlyingspringdesigninorder

nintelligentselectionofsprings for

heless,itisadvisable,incaseswhere

onsareconcernedtohavethe design

withthespringmanufacturer,inorder

se perience.


/ h t t p : / / w w w . h





U N D -W I R E CO M PR E S S I O N

S P R IN G

nceinmachinedesign arehelical

rtensiontypes.Theyaremadein

dusedintremendousq uantities.

ideacceptanceandgeneraluse are

ringsarerelativelycheapto manu-

eenoughq uantitiesarere uired

maticspring-windingmachinery.

relativelycompact,aconsiderable

ueezedintoasmallspace.

erialisstressedfairlyefficientlyunless

ofcoildiametertowirediameter)

discussedinChapterX X II.

e helicalspringisas broad

self. S everalofthemoreimportant

licalspringshavealreadybeen men-

omotivefieldthese includein-

rontwheels,F ig.5,suspensionofrear

R ailroadsarelargeusersof helical

ghtandpassengercar suspensions.

ricale uipment,springsareused

sinswitchgearandcontrole uipment,

sms,etc.Innumerableotherapplica-

ned.A typicalapplicationofaheli-

ermechanismisshow ninF ig. 21.

portanceofthis typeofspring,

o f space isdevotedto it inthisboo .

ssesthetheoryfor stressanddeflec-

rings, theapplicationofthistheory

coveredinsubse uentchapters. The


/ h t t p : / / w w w . h





S P R IN G

pterwillbelimitedto springswhere

nottoolarge(notmore thanhalf

of largepitchangles,however,

desmostpracticalsprings.

ulatinghelicaltensionsprings

hat usedforcompressionsprings

eeffectsoftheend loopswhichare

ngs,additionalconcentrationsof

. F orthisreasona low erw or ingstress

aspecial typeofendfasteningis

r,effectsdue toendturnsboth

nspringsare e cludedfromthe

theseeffectsareconsideredlaterin

Y TH E O R Y - L A R G IN D X A N D S MA L L

E

rings,theelementarytheoryas

boo sonstrengthofmateria lsorma-

heassumptionthatthespringmay

asastraight barundertorsion.This

ate ly truewherethespringinde is

angle issmall. Sincethee lementary

oaccountthedifferencein fiberlength

tsideofthe coilwhicharisesbecause

ingbarorwire,considerableerror

oryisusedforsprings withsmall

eflytheelementarytheoryis as

inde underana ia lloadPas

ressedbetweentwoparallelplates as

esultantload ingeneralwillbeslight-

sshown.Thiseccentricityis neglected,

scussion.R eferringtoF ig.22,each

pringcoil maybeconsideredto

emomentPrw herer= meancoilradius.

e lementsof lengthtl iscutf romthe

pitch anglescombinedwithlargedeflectionsareconsidered


/ h t t p : / / w w w . h





IN G 27

diculartothebara is. A ssumingthat

r distortduringdeformation,itfol-

rmationsandhencethe shearing

stributionalongaradiusas shown.

tressdistributioninastraight bar

tadistancep fromthecenterO

nofhelica lspringinacircuitbrea er

t— 2ptm/d(fromsimilartriangles)

umshearingstressatthesurfaceof thebar

hemomentta enupby theshaded

uspw illbedM= 2nrp-dp 2ptm/ d)

mentPrw illbe

- - C 3,

mumstressT ,


/ h t t p : / / w w w . h





S P R IN G

for calculatingstressinhelical

nte tboo sorhandboo s. A sstated

erableerrorforspringswithsmall

(1)The effectofdirectshearstress

sneglected and(2)The increase in

ceinfiberlengthbetweentheinside

n g o l a rg e i nd e ,

producedbywirecurvatureis not

willbemorefullydiscussedlater.

Tocalculatedeflectionofthe

duremaybeemployed.Consider-

urfaceofthebar andparallelto

se lement, a fterdeformation, w illro tate

thepositionac.F romelastictheory

a ltot, div idedby theshearingmodulus

Eq uation4.

7d , forsmallanglessuchasare

mentaryangledctthroughwhichone

respecttotheotherw illbee ua lto


/ h t t p : / / w w w . h





IN G

umingthatthespringmaybeconsideredas

2irnrwheren— numberofactivecoils,

tingthe angulardeflectionofoneendof

eotherw illbe , usingEq uation5,

mentarmof the loadP ise ua ltor,

willbe

formulafor springdeflections.In

stressformula,whichmaybe incon-

c-

tionformulaisq uiteaccurateeven

esandforlargeheli angles. Tests


/ h t t p : / / w w w . h





S P R IN G

eaccuracyof thise uationarediscussed

TH E O R Y — S MA L L O R M O D R A T IN D X C O N -

eofaheavyhelical springwhich

gisshownin F ig.25.Itwill benoted

afatiguecrac neartheinsideofthe

angleofabout45degreestothea is

ailuresare typicalofheavyhelical

erathersmallinde es, itmaybee -

umstressoccursatthe insideof thecoil

6a. Thereasonsforthee istenceof thema i-

re:F irst,thefiberlengthalong the

sswhich meansthatahighershear-

ivenangularrotationof ad acent

g. 26b, if theradia lsectionsbb and

allanglewith respecttoeachotherand

inside(andshorter)f ibera b w illbe

ershearingstressbecauseofits short

erah whichislonger.S econd,the

a b isincreasedbecausetheshear

ia lloadP isaddedtothatduetothe

ispo int. Intheoutsidef iberabthisstress

tothetor uemoment.Theresult

nsideofthe coilreachvaluesaround

outsideforspringsof inde 3, asmaybe

perIV ) andtheory forlargerinde es

A T UR E E F F E C T

nalele-

n

e fracturedsurfacema inganangleof 45degrees

fatiguefractureofastraightcylindricalbar underalter-


/ h t t p : / / w w w . h





IN G

,not sopronounced.

ee actso lutionof theproblemof

gsofsmall inde iscomplicated

ro imatesolutionwhichissufficiently

withinabout 2percentfor practical

follows3:

isassumedsincethisassumptionisvalid

gs. Consideringanelementofan

thmeanradiusofcurv aturercutby two

ectionsaa andbb asshowninF ig.

efailure

hiselementareresolvedintoa twisting

aradia lplaneandadirecta ia lshear-

etupbythis twistingmomentare

differsinseveralrespects fromanappro imatesolu-

cr, " B eanspruchungZ y lindrischeSchraubenfedemmitK re is-

913.Page1907,butthefinal numericalresultsareonlyslightly

hor spaper" S tressesinH eavvClose lyC o iledH elicalS prings ,

.,1929paperA .P.M.,51-17.


/ h t t p : / / w w w . h





S P R IN G

resuperimposedonthe stressesdue

omentM= Prthetwocross-

w illro tatew ithrespecttoeachotherthrough

ntionedbefore,thiswillresult inmuch

e fiber

springsof

tress^ act-

nF ig. 27f

edinto

, para lle lto

ndatrans-

ndicular

o

saa and

achotherand

rpendicularto

through

butionof

thestress

erper-

a isw ill

the

uchadis-

toa

siblesince

center

othe le f tand

ecouldbe

brium.If,however,rotationoccursabout

29, whichisdisplacedtow ardthea iso f the

ntO ,adistributionofstress isobtained

actionofapuremomentM. F rom

etransversestresscomponentst will

mwhenrotationoccursaboutanypoint

. 27b. Po intO' maybefoundasfollows:

rotationaboutO' , thestressract-

withcoordinates.vandy maybefound.

andbb haverotatedthroughasmallangle

calspring


/ h t t p : / / w w w . h





IN G

her,therelativemovementoftheends

e sp o nd i ng t o d A w i ll b e d /8 ( 2 + { / 2 ) ' 4

co ilo fhe lica lspring

d is(r-y - )dtl, theshearingstress7

be

ofthisstresswill be,F ig.27,

ade,thisdistributionofstress is

vedbar4andthedistributionof the

cn o, S (rmcf/ infMateria ls, V anNostrand, Part2, S econd

en o, Mrenf if /infMateria ls, va

discussionof curvedbartheory


/ h t t p : / / w w w . h





S P R IN G

cinform, F ig. 29. Thisdistancey isde-

onthatthe integra lo f r dA (w here

ea)mustbezerow henta enoverthe

edbartheorythedistance * maybe

at e ly a s r :

tedin thedenominatorsincein

mgreaterthan1/3 andhencet/2/16r-

y. PuttingE q uation10in9,

at theordinaryformulaforangleof

uation6)w illapplyw ithsuf f icient

ressdistributiona longa

tionaboutthecenter

nofdfi/dd (borneoutbyactualtests,

).Thus

hisinE q uation11,

ge7-1.


/ h t t p : / / w w w . h





IN G

sclearthatthema imumvalueof t

2—d2/ 16r, i. e ., a tpo inta inF ig. 27b.

butionalongatransversediameter,

hepointO '

uation13andalsoputtingthespring

essata inF ig. 29becomes

tpo inta( ig. 29)ontheoutsideof

d/ 2—d2/ 16rinE q uation13anddropping

a loadPactsa longthea is, as


/ h t t p : / / w w w . h





S P R IN G

ee ternalforcesactingoverthecross

toa twistingmomentM= Pranda

,assumingthat thepitchangleis small.

ng momentmaybefoundbysubstitut-

ons14and15. Onthesestressesthe

( ig. 27b)duetothedirecta ia lload

tappearsreasonableto ta eforthis

eoryofelasticityatthe outeredges

cantileverofcircularcross section

heory6givesavalueof stresse ual

isstressto thatduetothemomentPr

m a i m um s t re s s t, u . a t a m ay b e e p r es s ed

pressionforthema imumstress

g,a iallyloaded.Comparingthe

his formulawiththoseofamore

Goehner(seePage42),itmaybe

pringswherethe inde cise ua lto3

ithin2percentof themoree act

esarenegligiblefromapractical stand-

oagreesw ellw ithe perimenta lresults

mentsonactualsprings(ChapterIV ).

fthecoilat a,( ig.28b),may

ation15andsubtractingthestressdue

actsintheoppositedirection,giving

imenta lresultsindicatesthatthise ua-

eycorrect.

thema imumshearingstressinahelica l

oryofE lasticity , McGraw-Hill , Page290.


/ h t t p : / / w w w . h





IN G

nfactorK is

a imumstressissimply thestress

mulaofEq uation4multipliedbya

thanunity andwhichdependsonthe

nvenience inca lculationva luesofK are

springinde cinF ig. 30. It isseen

3thefactorK = 1. 58, w hichmeansthat

X

ectionfactorforhe lica lround

nsprings

dinaryformulamustbemultipliedby

imumstress.

ethat thestressesderivedhold

ditionsprevail,i.e.,aslongasthe yield

ematerialisnote ceeded. If thisisnot

stressmaybelessthan thatcalculated.

caseswhereyieldingoccurs,the formula


/ h t t p : / / w w w . h





S P R 1N G

tresswhichis ofmostimportance

urtherdiscussionoftheuse of

e loadingisgiveninC hapterV I.

Y

developedintheprecedingsec-

helicalspringsofsmallor moderate

reviously,sufficientlyaccurateformost

beingaccurateto within2percent

erthanthree).W heregreateraccuracy

thodofcalculationdevelopedby

ismethodwillbe brieflyoutlined8.

eferringtoF ig. 31, if t9-andrroare

gstressactingon theelementA ofa

springasshown,thecoordinatesofA

ch angleissmallsothat theelements

sideredunderpuretorsion,allshear

eptt : andrromaybeassumedzero.

mthetheoryofelasticity,the condi-

indricalcoordinatesgivethe follow-

uation

ore uires• thatthefollowing

w hatarek now nasthe" compatibil ity

satisf ied:

0( 21 )

spannungsverte ilungimQuerschnitte inerSchraubenfeder ,

0, Page619 " S chubspamiinigsverteilungimQuerschnitte ines

A nw endungaufS chraubenfedern , Ing. -A rch. V o l. 2, 1931,

gsverte ilungine inemandenE nd uerschnittenbelastetenR ingstab-

o l. 2, 1931, Page381 and" DieB erechnungZ y lindrischeS chraub-

rch12, 1932, Page269.

S. T imoshen o, McGraw -Hill , Page355, givesamore

method.


/ h t t p : / / w w w . h





IN G 39

,21and22 astressfunction< /> is in-

I d< h \

d. B ysubstitutionofEq uation23in21

uationsareobtained

= 0 ( 24 )

d t > \

pressioninparenthesismustbea

enotedby—2c . Thus,

0 ( 26 )

eoustointroducenewcoordinates

omes

0 ( 27 )

beconsideredsmall,

)

solveE q uation27bymeansof a

imations.F romtheconditionthat

ssatthe boundaryofthecrosssection


/ h t t p : / / w w w . h





S P R IN G

undary,thefunction< f> maybeshown

boundary . Withthiscondit ionthee -

s:

( 29 )

0

, 3 d < t > o

3 $ d $ o

0 e t c.

, us i ng P ~ r - - i ,

actingoverthespring crosssec-

f r(3D

t/ r)- inEq uations30maybee -

ollows:

3 2 )

ndi, bymeansofsuccessive

ncarriedoutin thisgeneralmannerby

ngresultsforround-wire,helical

ssforacircular ringsectorwithzero


/ h t t p : / / w w w . h





IN G

1 6 )

withinonepercentevenfor

scomponentsincross-

e acttheory )

esnot applyaccurately,however,

ppreciable.

rethe pitchangleais notzero,

a forma imumshearstressbecomes

actualradiusofcurvatureof theheli ta -

count.Thefirsttermof thise uation


/ h t t p : / / w w w . h





S P R IN G

on33wheretheactualradius ofcurvature

econdtermof thise uationarisesbe-

he shearloadPcosa. Thetermin

econdparto f therightsideofEq ua-

heseries whicharisesinthecalcula-

ation34isconsideredveryaccurate , it is

calculations forsuchcasesthefollow-

with anaccuracywithinonepercent:

reaterthan3andforpitchanglesa

a imumshearingstressis

/ j _ y v

2 P / J

rthan4anda< 20degrees.

s(whichincludesmostpractical

ula fort, „ . , e pressedinterms

onvenientandisto bepreferred:

c T + - cd ( 37 i

osa= l. E q uation37dif fersby

q uation16derivedbyappro imate

esof threeormore.

vetheshearingstressduetothetwist-

andthedirectshearPcosa. How ever, there

mu presentduetothebending moment

tensionorcompressionstressduetothe

tthema imume uiva lentstressinthe

o- , „ . rmustbecombinedw iththetor-

basisofa strengththeory(Page44).

umbendingstress(T, „ x theresults


/ h t t p : / / w w w . h





IN G

ybeused.A somewhatmoreac-

he generale uationsofthetheory

Goehner .Thisinvolvesessentially

uil ibriume uationsandthecompatibil ity

nding<pin

ulafor

and

A N D IA M T R

T R

oordinatesandtheirsolution bya

ro imations,similartothatpreviously

torsionstresst, , , . F ina lresultsare:

stress:

m + 4 / d

" 2p

onstant= reciproca lo fPo isson sratio .

. 3thisformulasimplifiesto

Thelastterminthebrac etsd/ 8ry ie ldsthe

ace381. A lsoTheoryofElasticity—Timoshen o, Page361.


/ h t t p : / / w w w . h





S P R I N G

nsionorcompressionwhichis

nderyieldsthestressdueto thebending

thedenominator1—d/ 2p represents

swhicharisesfromthe methodofsuc-

s.

needonlybeusedforre latively large

de es. F ortheusua lcasethefo llowing

sufficientaccuracy:

4

m 8

rrespondingtoPo isson sratio= . 3

tressinthespring,the shearing

dingstresso- , u . w hichactatagiven

dbefore,becombinedaccordingtoa

strengththeorywhichiswidelyused at

m-sheartheory,whichstatesthatfailure

imumshearingstressatanypointin a

shownfromelastic theorythatifa

dashearing stresstm< uactata givenpoint,

resst, basedonthema imumshear

mu f 1 +

whichiscomingmoreand moreinto

heory1:(alsok nownasthevonMises-

heorystatesthatfa ilurew illoccurw hen

yof distortion)ofthehigheststressed

oshen o—S trengthufMateria ls, Purt2, SecondE dit ion, Page

ussionol strengththrones.

, Part1, Page122.

art 2,Page479givesafurther discussionofthistheory.

. X uda i, Eng. S oc. Monographs, McGraw -H ill , 1931.


/ h t t p : / / w w w . h





IN G

shearenergyofane lementinana ia lly

eldpoint (orattheendurancelimit

atiguefailure).Ifo- a2,ando-3are

ure,a mathematicalstatementofthe

r ) + ( < r l — < r 3 )- = 2 oc = C on st an t (4 3)

thertotheyield pointortheendur-

ftside ofthise uationcanbe

othe shear-energyorenergyofdis-

al.This energyise ualtothetotal

nergyofthree-dimensionaltensionor

nabendingstresscrl nranda

simultaneouslyasinthiscase, thee uiva-

gto theshear-energytheoryis

bydeterminingtheprincipal

clefor acaseofcombinedtensionand

seprincipalstressesare

( r , ) ( 4 5)

t ^ Y ( 46 )

ncenonormalstressactsonthesurface

ingE q uations45and46inEq uation42

a lentshearstressiy= av / 1. 73(w hichis

lenttoasimpletensionstressa- , ) , Eq ua-

nusuallylargeamax willbesmall

formulasforprincipalstress forcombinedtensionand

ngthofMaterials,loc.cit.,Part1, Page48.


/ h t t p : / / w w w . h





S P R IN G

sothatingenera lformostpractica lsprings

romt , x .

eformulasgiveninthischapter

asthedeflectionsperturn aresmall

er)so thatthecoilradiusand pitch

onstant.Theseconditionsusually

acyinpractical springswherethe

eforsmallormoderate inde ese cessive

hedeflectionsperturnapproachthe

ampleof theapplicationof themore

hischapter,assuminga springwithan

anglea= 12degrees, sothatEq uation

s< x = . 978andtana= . 2126. Using

a

tion41

44 4 s in a

imum-sheartheoryapplies,bysub-

2,

beta enasabasis, bysubstitu-

mthe valueobtainedbyusingthe


/ h t t p : / / w w w . h





IN G

.Italsodiffersbyonlyabout1^ percent

viouslybyappro imatemethodsand

( q uation16). Thisindicatesthat

disaccurateenoughformostpurposes.

ssumingthatthespringdeflec-

1relativetothecoil radiusandthatthe

redverysmall,the springdeflec-

. It isseenthatthisissimply theor-

q uation7multipliedbva termin

sonthespringinde . The largerthe

rw illthistermapproachunity . How -

ptiona lly smallinde o f3, theterminthe

on48ise ua lto . 977w hichindicatesthat

2.3 percentsmallerthanthat cal-

mula.H owever,forbestaccuracy

considered.

ingis considered,theprocedure

tictheory1 itmaybeshow nthatthe

elicalspringis

naland initialpitchangles,respectively.

ectionis smallsothatthecoil radius

ant.Multiplyingthis bythetorsional

ngmomentM, = Prcosoc. Li ew ise

notw illbee ua ltothef le ura lrigidity

acte pressionforthechangeincurva-

tions(whichmay occurwithoute cessivestressonlyfor

dinC hapterIII.

2, Page272.

e—TheoryofE lasticity , third. edit ion, CambridgeUniv . P ress,


/ h t t p : / / w w w . h





S P R I N G

ngwireorbar willbe

sandtheresultsof elastictheory,amore

thedef lectionofhelica lspringshasbeen

rateformula,whichassumessmall

n

a t a n a (5 2 J

figuredbyusualformula,E q uation7,

dingonthespringinde c= 2r/ dandon

. 32va luesof theconstant<phavebeen

pring inde cforvariouspitch angles

ulations, aratioG/ - . 38w asta en(cor-

tely toPo isson sratio= . 3), w hichho lds

rmostspringmaterials.H owever,

nthisratioG/ w il lhavebuta

ueof i^ .Thedashedcurveforzero

zedin practicalspringsbecauseof

coils alsoapartof thecurvefor5

wndashedsince itcannotberealized.

cticalspringswherethepitch angle

greesandforthesmallerinde esthe

y,whichmeansthatthe actualdeflec-

thenominaldeflection,E q uation7.

st sinceonewoulde pectthatthe

ncreasethedeflectionoverthatgiven

tobedescribedlater(C hapterIV ), how -

conclusion.Itshouldbe notedthat

tisusua llyverysmall thusforinde es

15degrees,thedeviationsbetween


/ h t t p : / / w w w . h





IN G

aandtheordinary formulaEq uation7

nt. F ormostspringsw herea<10de-

enceisunder onepercent,a figurewhich

tice sinceotherfactorssuchas the

andwiresize,shapeofend turns,

ordinarilybe greaterthanthis.In

sfore ampleincertaininstrument

mumaccuracy,itmaybedesirableto

uation52inma ingdeflectionca lculations.

gastee lspringof thefo llowingdimen-

in.,meancoilradiusr= .286in.,wire

tveturns, springinde (2r/ d)=3. 28,

grees,wor ingload140lb,the deflec-

nalformulaforthese dimensionsis

tchangleof7% degreesandinde c= 3. 2

985. Thedef lectionca lculatedby themore

= . / , „ = . 985(. 0745)= -0733in.


/ h t t p : / / w w w . h





H E L I CA L S PR I N G

W ITH L A R G D F L E CTIO N S -

y isre uiredthetheorydeveloped

edhelicalspringsissatisfactoryfor

spring deflectionsandstresseswhere

der10degreesand thedeflectionper

coilradius.H owever,forcases

e islargeorwherethe deflectionper

ntheuseof theusualformula , Eq ua-

g deflectionswillresult.Thiserroi

initialpitchanglesaround20 de-

urne ualtotheinitial coilradius.The

usualformulaispartly thatthe

eroin thepreviousderivationand

us changeswithdeflection.Thus

giscompressedfromthe initialposi-

thatshowninF ig. 33b, themeancoil

tor. S incethespringdeflection, other

oportionaltothecube ofthecoilradius,

becomesmoref le ibleasit iscom-

ct,ofcourse,occursin tensionsprings.

he effectofpitchanglemaybe

reaccurateformula,E q uation51,

gle intoaccount.Ifthespring deflec-

owever,thisformulawillalsobesome-

deflectionssincethechangein coil

neglectedinthe derivation.

pterwillbelimitedto springs

effectsdueto changesinthepitch

mostpronouncedinsuchsprings.

moderate inde highstressesaresetup

meslargeenoughsothatchanges in

areof importance. H enceEq uation51

s.


/ h t t p : / / w w w . h






eica lspringissub ecttoana ia l

ction,thereis atendencyforthe

rwords,oneendof thespringtendsto

eotheraboutthespringa is. If this

reelyandwithoutrestraint,we have

lly- loadedspringasindicatedinF ig. 34.

econditionintensionspringswith

ehoo sarenotrigidlyhe ldbuthavesome

espringa is.If,however,theends

nbyfriction(as isusuallythecase

by clamping,endmomentsacting

ila resetupw hichtendtopreventthis

itis necessarytodistinguishtwocases:

ringa ia lly loadedandw ithendsf reeto

isand

aendsaref i edsothatrotationaboutthe

d.

N D F R E E TO R O T A T

nopen-coiledhelicalspringof large

ig. 34issub ecttoatensionloadP ,

f reetorotateabouttheco ila is. If

ngleandrtheactua lco ilradius, the

gontheelementA oflengthdswill

momentPrsin a,atwistingmoment

sa andatensionforceP sina.The

w istingmomentPrcosaw illbee ua l

he torsionalsectionmodulusfora

Thus

stressaduetothebendingmoment

nde c

isassumedlarge, stressesduetothe


/ h t t p : / / w w w . h





S P R IN G

ndthe tensionPsina willbeneglected

elementofthesurfaceofthe coil

reactingasindicatedinF ig. 34c. A s

hapterIIthesestressesmaybe com-

imum-sheartheoryand,therefore

used. Hencethee uiva lentshearstress

tgivenbyEq uationsS 3and54

ee pressionfore uiva lentshear

a = — — — ( 56 )

under theradicalisunity.

heoryapplies, thise uationshow s

uiva lentshearstressise ua ltothatgiven

Pr/W ' regardlessofthepitchangle

m- hearandS hear- nergyTheories

/ t c

nddirect shearbeingneglected),pro-

a enasthatactua llye ist ingwhenthe

w ninF ig. 33, thisw illbedif ferentf rom

atzero load.

rgytheoryofstrength(asdiscussed

andtgivenbyE q uationsS 3and54

ation43. If thisisdoneane uiva lent

ua lto

n -a ( 5 7)


/ h t t p : / / w w w . h






enE q uations56and57show sthat,

spring,thedifferencebetweenthe

ories (ma imum-shearandshear-

tforpitchanglesunder 20degrees

nglesbelow30degrees,( TableI).

encebetweenthetwotheories,

springwithlargedef lection

uation56willbeused inthefollowing:

simplerforthedesigner touse

o loadasabasisforca lculationsince it

sured.Itisshown,Page57,that foran

eactua lco ilradiusisgivenby r= K 2r

f theratio8, , / nr betw eennomina lde-

coilradius,andofthe initialpitch

eatzero load). Thenomina ldeflectionS „

ydeflectionformulausing theinitial

, is


/ h t t p : / / w w w . h





S P R IN G

essedasafunction8 /nr andofa and

comes, f romE q uation56,

)

ssedsimplyastheordinaryformula

multipliedbyafactorK 2w hichmaybe

s of F i g . 35 i f < x a n d $ / n r a r e k n o wn .

aybeca lculatedf romEq uation58foragiven

thatnegativeva luesof8 / nr cor-


/ h t t p : / / w w w . h





H E L I CA L S P R I N G 55

positivevaluestoe tensions.The

unityforopen-coiledcompressionsprings

asesasthespringis compressed the

nsionsprings.

. 35itmaybeseenthat, if the init ia l

greesandthecalculateddeflection

rethanthe init ia lcoilradiusr , the

aduetopitchanglechanges areunder

reach15per centiftheangle a

edeflectionper turne ceedsthe

t actualapplicationswherex ,<


/ h t t p : / / w w w . h





S P R IN G

< . 5, theerrorisunder3percentand,

edformostpracticalpurposes.

der todeterminetheeffectof

reasein stressduetobar curvature

enneglected. A sanappro imationto

ess,thestressesfiguredinthis way

ythe curvaturecorrectionfactorK

9. E venforspringinde esbetween10

yfrom1.07to1.14andis thusofim-

ate,butmorecomplicated,methodis

, niandama givenbyE q uations34and38,

s A ssumingahelica lspringa ia l-

eetorotateasthe springdeflects,F ig.

ryofelasticity1that thechangeinbar

pringdeflectsfromaninitial pitch

hangleais

hecoilradiicorrespondingtothe init ia land

eThise uationissim ilartoE q uation49

nco ilradiusisconsidered. F romF ig.

usingthischange incurvatureis

stbee ua ltothef le ura lrigidityEI

incurvatureA k . Thus

felasticity,/= momentofinertiaofcross-

t ion60:

mayalsobeshow nthatthetw istA f

asthespring deflectsfromapitch

ea is

lasticity , C ambridgeUniversityP ress, ThirdEdit ion, Page421.


/ h t t p : / / w w w . h






.C O S a c o

etorsionalrigidity GI (for

wistingmomentmt.Thislatteris,

cosa . Thus

o s a \

dityandIp= polarmomentofinertia

nd63,

- a

-

elengthI ofthespringremainscon-

pringlengtliwith

forsprings oflargeinde )thespring

ontheheli cylinderasindicatedin

ometryof thisf igureastheanglechanges


/ h t t p : / / w w w . h





S P R I N G

ringdef lectionS becomes

rnr , w herenisthetota lnumberofactiveco ils,

ritten:

* „ ) ( 66 )

nd66thetota ldeflection8of thespring

enomina ldeflection8 (ca lculatedf rom

edbyafactorThus

D F L E C TI O N P R t UR N

R A D IU

ndingdeflectioncorrectionfactorV ' , ,


/ h t t p : / / w w w . h






( 6 7)

ntheinitial pitchanglex .,theratio E /G

armoduli, andontheratio8 / nr be-

perturnandinitial coilradius.V alues

ensiondiagramsforopen-co iledhelicaltension

rotate

tedforaratioE/ G— 2. 6(whichapplies

springsteels)and theresultsplottedin

esofaandSc/ nr . A lthoughava lueof

gtoPo isson sratio= . 3hasbeenassumed,

erablechangemaybehadin thisratio

hangein thefinalresults.

o fF ig. 37show sthatforpitchangles

ectionsperturn lessthanhalfthecoil


/ h t t p : / / w w w . h





S P R IN G

theerrorintheusua lspringdef lectionformula

ent,i.e.,doesnotdifferfrom unity

percent.F orpitchanglesaround20 de-

urne ualtothecoil radius,however,

rcent.

alcalculationsof open-coiledhelical

sarytodeterminethedeflection8 for

ordinaryspringformula,E q uation58,

esorcharts. F romthisva lueof8 the

T N S I O N P R T UR N

D IU

culatingtwistofspringends.

reeto rotate

ound. Thenk now ingthisandthe init ialpitch

/ < , maybereadf romF ig. 37. Thema imum

Pw illthenbee ualto^ 18 .

ectiondiagramsdeviatefroma

al pitchanglesa andforvarious

curvesofF ig.38havebeenplotted,

tensionspringswherethe endsarefastened


/ h t t p : / / w w w . h






ationaboutthe springa isissmall.

representvalues ofPX 64r,r/Gd and

theload,while theabscissasrepresent

rndividedbyinitial coilradius.These

d,whichmeansanincreasein spring

eflection)withload.Thiswouldbe

ngssincethe coilradiusrdecreaseswith

hlinerepresentsthedeflectionas fig-

la.In thiscaseitmaybe seenthat

tchangles thereisaconsiderablede-

erepresentingthe ordinaryformula.

nds—Whenatensionspringise -

n,thecoilstendto unwind,atleastat

ountofthisunwindingmaybe cal-

mF ig.36,theanglein radianssubtended

totalspring lengthintheunloadedposi-

ulartothea isw illbe

anglewillchangeto

rtherelativerotationofone endwith

ethedifferenceinthesevalues.

2 w nr / c os ( „ ,

ssedinradians.

nd67thisanglemaybee pressedin

r asbefore . Ex pressing$indegrees


/ h t t p : / / w w w . h





S P R IN G

rturnin degreesandmaybereadfrom

/ nr arek now n. Itw il lbenotedf romthisf ig-

tlynegativefor smallvaluesof8 /nr,

greaterthanzero.Thismeansthat

ringhas aslighttendencytowind up.

thedistortionsof theelementsofthe

insuchadirectionastocausethiswind-

ngdeflectionincreases,this tendencyis

geinpitchangle whichcausesanun-

N D F I X E D A G A I N S T R O TA T IO N

Wherethespringendsaref i ed,

ngaboutthea isofthespring during

realizedin manycompressionsprings

he licalspring

ntsare

ntheendsand thesupportingplate

,asimilaranalysismaybemadeto that

occurs withoutrestraint.Inthiscase,

a e intoaccountthemomentactingat

entsthe coilsfromunwinding.


/ h t t p : / / w w w . h






famomentM anda loadPareacting

slyasindicated,thebending andtwist-

ting onthewirewillbe

(72)

(73)

saandM siniofthe a ialmoment

ctorsinF ig. 40b.

the wireA k duetothemoment

entdiv idedby thef le ura lrigidityEI.

0becomes, f romE q uation62,

o s a .

ons72, and73inEq uations74and75

btained,fromwhichthefollowing

maybefound:

i n a, C O S a '

CO S a \

\

aocosa. \

ofthespringarepreventedfrom

f> 2asgivenbyEq uations68and69aree ua l


/ h t t p : / / w w w . h





S P R IN G

D F L E C TI O N P R TU R N

R A D IU

ngdeflectioncorrectionfactor

aga instrotation

q uations76and77andsimplify ing,

i n a H — 7 T 7~ ' a n a( co s a ~ c os a )

espondingto< / < forthecasew hereno


/ h t t p : / / w w w . h






, Eq uation67. B yusingE q uations66

maybee pressedintermsof init ialpitch

r asbefore , andtheresultsaregivenonthe

arisonofF igs. 37and41indicatesthat

etwocases,i.e.,endsfi edorfree,is not

esof8 / niv• A tlargervaluesthereare

msasdeterminedforcompression

sbyusingEq uation79aregiveninF ig. 42.

pitch anglesanddeflectionsthereis

MP R E S S D N P R T UR N

I R A D U

pressiondiagramsforopen-coiled

dsf i edaga instrotation

mthestraightlinecalculatedfromthe

bnotedthatthequestionofbuc ling

notconsideredhere1 . Wherethebuc -

hecurvesofF igs. 41and42maystil lbe

to preventlateralmovement.

smethodsofdeterminingbuc lingloadsincompressionsprings.


/ h t t p : / / w w w . h





S P R I N G

urvesfortensionspringswithends

boutthespringa isaregiveninF ig. 43.

pro imatelyfortensionspringshaving

o le inaplate , sothat, w henthespringis

cannotrotateappreciably. Ita lsoho ldsw here

T N S IO N P R TUR N

R A D IU

ensiondiagramsforopen-co iled

ndsf i edaga instrotation

ringendswhicharein turnfastenedin

nyrotation.

entStress—F ora ia lly- loadedhelica l

thestressis modifiedbythepresenceof

attheendsof thespring. F romE q uations

dtwistingmomentsactingonthewire

ated.A ssuming,asbefore,thatthe

ofstrengthisva lid, thee uivalentshear

55,foracircular wirecrosssectionbecomes


/ h t t p : / / w w w . h






& andm, givenbyE q uations72and73in

andPrgivenbyEq uations76and

ereducedto

w hichtheusua lformulat= 16Pr / w d

aintheactua lstress. V a luesofK . , a re

L E C tI O N P R t UR N

D IU

ectionfactorK / , endsf i ed

„ andofS „ / nr inF ig. 44. Itshouldbe

reviouslyadditionalstressesdueto

ndthesemaybe ta enintoaccount

apterII.

thischapterindicatesthatwhere


/ h t t p : / / w w w . h





S P R IN G

nt,theerrorin theusualdeflection

houldbeconsidered.Thiserror mayap-

alpitchanglesnear 20degreesandde-

tothecoilradius. F orusua lapplications

e isunder10degreesand thedeflec-

thecoil radius,theresultsindicate

ulaoflessthan3% percent.H ence,

acyisdesired,theseeffectsdueto pitch

ncoil diametermayusuallybe


/ h t t p : / / w w w . h





T IG U T S T

S P R I N G A N D S P R I N G MA T R I A L S

validityofthe formulasderivedin

ndwirehelicalsprings,a seriesof

gH uggenbergere tensometerswas

springsofthetypeused inrailway

oilscutfrom thesesprings .Thesprings

nd3withan outsidediameterof6

ro f IV 2inches. Thislow inde w as

sesvaluesas lowasthisare usedin

more, theuseofa low inde springinthe

encebetweentheresults calculated

ndthosecalculatedby moree act

ble.H ence,abettere perimental

d.

U R E M N T

hefull-sizedsprings weresuchas

oplaceane tensometeronthe insideof

mumstressoccurs),semicoilswere

loadedinsucha wayastosimulate

springunderana ia lload. Todothis,

dtothe semicoilasshowninF ig.45.

edbyspecial eyeboltshavingspheri-

a ia lloads. A photographof thesemi-

ngmachinewiththee tensometer

co ila tthepo into fma imumstress

easurethetorsion stressinthecoil, the

replacedatobanda tob at45degrees

thebar2. F romthesestra inmeasure-

ressesinHeavyC lose lyC o iledH elicalS prings, T ransactions

P . M. 51-17givesfurtherdeta ils.

sistsessentiallyofa tensionstresscombinedwithan

ssatrightanglesthereto bothofthesestressesbeingat 45degrees

trainmeasurementsta enat45degreesto theshearstress

nofthelatter.


/ h t t p : / / w w w . h





S P R IN G

basedonelastictheory,it ispossible

s,providedthe modulusofelasticity

ek now n . B yusingarelative ly shortgage

hepea stresscanbefound withsuffi-

gementofF ig. 46thusma espos-

pea stressontheinside ofthecoil

ico iltestarrangement

a ial-loadingconditioninacomplete

sF ormulas—L oad-stresscurvesob-

surementsontheoutsideand thein-

w nby thefull l inesinF ig. 47. S im ilar

bytests onadifferentsemicoil.F or

sultsthe dashedlinesrepresenting

valuesta enf romEq uation18forthe

moshcn o— StrengthofMaterials, SecondE dit ion, part1,


/ h t t p : / / w w w . h





L ICA L S PR IN G 71

co ilandf romEq uation17forthe

lsoshown.Itwillbenoted thatthe

dfromE q uations17and18agree

thee perimentalresults.Thedashed

helica lspring

tensometer

stresson

scalculatedfromthe ordinary

eeffectsofcurvatureand directshear

thetwoe perimentalcurvesandis

asthema imumstressisconcerned.

that themeasuredstressontheinside

imesthatontheoutside . F orspringsof

nce,ofcourse,wouldbeconsiderably

semicoilswererepresentative

sa iallyloaded,acompletespring

nasshow ninF ig. 48. A sbefore , e -

centimetergagelengthwereapplied

omeasurestrainsat 45degreestothe

d-stresscurvesobta inedinthismanneron

esofthespringare showninF ig.49.

stresson oneside,thefullcircles

ppositesideof thespring.Itmaybe


/ h t t p : / / w w w . h





S P R IN G

esideisabout10 percenthigherthan

atthehigherloads.Thereason forthis

at, becauseofthepresenceofthe end

tlyeccentrictothespring a is(further

PU T D B

E L I CA L

U A

0

T R E S S - B ./ Q . IN .

scurvesforsemicoil

I).Thedashedlineon thisfigurerepre-

tsideofthespringas calculatedfrom

st curve(whichgivesthestressduo

showntogetherwiththecalculated

ula.Itis ofinteresttonotethat this

deswiththat obtainedontheout-

.47,upto aloadof3000pounds,thus

testsdo simulatetheloadingofa

mparisonthestresscurvecomputed

rmula.E q uation4,whichneglects

reffectsisalsoshown.

atfor smallinde esthesimple


/ h t t p : / / w w w . h





L ICA L S PR IN G

pringsmaybe inconsiderableerror.

appro imateformulaofE q uation18

alculatingstressesin helicalsprings,

sumed.

T S T

flectionformula,E q uation7,for

ual springswerecarriedoutsome

r sdirection1.Thesetestsalsoserve

e actdef lectionformulaofE q uation51

nteffectsduetospringinde andpitch

odwastowind threetension

9. 5, 4. 7and2. 7fromasinglebarofcar-

hdiameter.A totalofninespringscut

eretested.Themethodoftesting

eloadbeingapplied throughaneye-

meterusedtomeasurestress

nsweremeasuredbetweenthe

db-b inthebodyof thespringtoe lim inate

eend turns.Byta inganaverage

onHelica lS pringsofR oundandS q uareWire , Transactions

e 2 17 .


/ h t t p : / / w w w . h





S P R IN G

voidableeffectofslight eccentricities

.Itwasfoundthat thetestload-

moste actlystraightlines.A typical

arge inde isshowninF ig. 52, the

own.O nthiscurvethe circlesrepre-

eofthespring,the crossesrepresent

oppositeside.Becauseofthe un-

tyofloading,thesedonot coincide.

arge inde madef romagiven

almodulusof rigiditycouldbede-

on7forthatparticularbar. A veragew ire

bymeasuringdt, d2, d3andd F ig.

ch coilafterthetest.To dothisit

igidity

hespring.Coildiameterswerefound

,andD.,,correctionbeingmadefor

thedeterminationofmodulusof

testedareshowninTableII.

themodulusvaried bylessthan

ifferentbarsand, hence,thatthemate-

25000

T R E S S — L B ./ Q . IN .

scurvesoncompletespring


/ h t t p : / / w w w . h





L ICA L S PR IN G

uniform.

pared—F orthespringsof smaller

s indicatedbythetheoreticalcurves

ectionwas inmostcasesslightlyless

rdinarydeflectionformula,i.e.,^ was

ltestcurvefora springofsmallinde

0 2500030000

T R E S S — L B ./ Q . IN .

scurvesforcompletespringundera ia lloading

ig.53,thiscurverepresentingthe

dondiametricallyoppositesides.The

narydeflectionformula,E q uation7,

ltsobta inedonsi springshav ing

iveninTable III, w hichshowstheper-

nthetestcurvesandthecurves cal-

ormula,E q uation7,forthevarious

vedeviationmeansthatthedeflection

calculatedbymeansoftheordinary

t theaveragetestdeviation(for

ars1, 2,and3)was— 1.7percent

and—1. 0percentforspringsof inde

tedbyusingthe factoripofF ig.32

leandspringinde w ere—2. 4percent

husseenthat theaveragetestvalues

ecorrespondingcalculatedvalues

ethod.Thisindicatesthat slightly


/ h t t p : / / w w w . h





S P R IN G

ringdeflectionsmaybe obtainedby

sfiguredbytheordinaryformulaby

ation52orF ig. 32. Itshouldbementioned,

Deviations

pringF ormula

TestCa lculated

eviation

usingfactor&

2 . 4

0— .7

ve,i.e.,deectionswereslightlylessthancalculatedfrom

ation7. E f fecto fpitchangleconsidered.

flectionformulaforhelicalsprings

mostpractical purposes.

I N MO D U U O F R I GI DI T

ofdeflectionsin helicalsprings,

ofmodulusofrigidity ortorsional

def lectionformulaEq uation7isneces-

s,theeffectivetorsionalmodulusofa

dbe usedinthespringformula)may

tobee pectedonthebasisof torsion

esamematerialwithgroundand

ngthereasonsforthisdifferenceare

hematerial,presenceofa decar-

e,and residualstressesresultingfrom

ng.

actors willbediscussedincon-

ablein theliteraturerelativetothe

lsprings andspringmaterials.Un-

wsthattheeffectivetorsionmodulus

varyfroman averagefigurebysev-

cases.

ng—Thevalueofmodulusof rigidity

ntbyoverstra iningthemateria l. How -

r apartofthis reductiontobelost

dfor sometime.A dams5foundthat

arsofhigh-carbonspringsteel in

pMemoirs, Iron& S tee lInstitute , 1937, Page1.


/ h t t p : / / w w w . h





L ICA L S PR IN G

ralper centinthemodulusof rigid-

thisdecreasecouldbe eliminatedby

heattreatmentaftertheoverstrain-

eobta inedbyP letta , Smith, andHarri-

ngsmade from%-inch diameterbar.

decreasesinthetorsionalmodulus

percentdependingonthe amountof

ncyofthemodulusto partiallyre-

thespringhasstoodfor aconsider-

carburization—A factorw hichiso f

nfi ingtheeffectivetorsionalmodulus

zationofthewiresurfaceinthe com-

A PP Y

N G .

lspringfordeflection

at,ifthereis adecarburizedlayer

whenthe springisstressedthis ma-

rbonsteelandwillyieldat arelatively

verstra inonC lose lyC oiledHelica lS pringsandtheV ariationof

eCo ilsw ithL oad byP letta , SmithandH arrison, Eng. E x p.

, V irginiaPo ly technicInst.


/ h t t p : / / w w w . h





S P R IN G

the springwill,therefore,acttosome

decarburizedmaterialwerenotpresent

flectionratewillbe roughlythatcor-

ehavingadiametere ualtotheactual

hic nessofthedecarburizedlayer.

ris usedinthespring formula,this

effectivemodulusofrigidity werede-

, aspringof% - inchhot-woundstoc may -

rburizedlayer e tending.01-inchinto

erloads,since thedecarburizedlayer

ngth,the load-deflectionratecorre-

of.5—2(.01)— .48-inchor4percent

d-deflectionratevariesasthe fourth

uation7, thismeansareductionof

entintheformer, oradecrease inef-

of16 percent.Inmanycases,spring

I N G, L B .

O N O N E S I D O F S PR I N G.

N D IA M TR I CA L L Y O PPO S I T S I D

deflectiondiagramforhelicalspring

)

modulusfigureofabout10per cent

pringsteel thanforhard-drawnma-

pleshowsthatadecarburizedlayer

e- inchdiameterstoc w ouldbesuf f icient


/ h t t p : / / w w w . h





L ICA L S PR IN G

modulusvalue.S uchalayermayeasily

materials.

Ingeneral themodulusofrigidity

ithincreasein temperature.This

ofa springunderagivenload willbe

I N G, L B .

N O N E S I D O F S PR I N G.

O N D IA M TR I CA L L Y O PPO S I T S I D .

deflectiondiagramforspringof

otethatmeandeflectionof tw o

theoreticalvalue

res.H owever,availabletestdata

urearelimited.O neofthefewin-

hislineis thatcarriedoutbyK eulegan

vestigatedthechangeinmodulusof

o f temperature(f rom—50 C . to

ingmaterials.Theseinvestigatorsfound

mostmaterialsthemodulusofrigidity

o imatelyby

gdityat0 C , m— temperaturecoeff i-

temperature,C.

es. . V o l. 10. 1933, Page305. S eea lsoB rombacherk

ch. R eportNo. 358, 1930.


/ h t t p : / / w w w . h





S P R IN G

ainedbytheseinvestigatorsfor

regiveninTable IV . F ore ample ,

intemperaturefrom0to50 degrees

thetorsionalmoduluse ualto

sofModulusofR igidityC

C.00025

98% Cr..24% V a.00026

8% Ni . 00040

0

c anandHauseman, B ureauofS tandardsJourna lo fR esearch.

.

ed,

0G- . 013Gor1. 3percentinthisrange.

cyisdesired(as ininstrumentsprings)

tant.

modulusofrigiditydropswith

—100 degreesF ahr.to600or 800

btainedfromthecurves ofF ig.54.These

smoothcurvesthroughtestdata

ndhiscollaborators8.S incetheactual

considerableirregularitiesandscatter,

sideredas givingonlyaroughindi-

uluschangewithtemperature.

N O F M O D U U O F R I GI DI T

methodswhichhavebeenused

ofrigidityfor springmaterials:

surementsofdeflectioninhelical

pression

mentsoftwistofa straightbarin

thod:Measurementsoftheperiod

mwhichby k nownformulasthe

determined.

ng thefirstmethod,deflections

. T . M. , 1930, PartII, Page356.


/ h t t p : / / w w w . h





L ICA L S PR IN G

icalsprings loadedintestingma-

eviouslyitisadvisableto measure

ncoilsinthebody ofthespringto

etotheeffectsof endcoils.A lsoto

avoidableeccentricitiesofloadingit

flectionsondiametricallyopposite

rageof theseva luesthenista en. If

ennturnsofthe springisfound,the

e foundfrom

dbysolvingE q uation7forG,the other

meaningsasbefore.S lightlyhigher

- F

effectonmodulusof

ls

cularlyforsmall inde es,bydividing

t a e n f ro m F i g . 3 2. H o w e ve r , fo r i nd e e s

aybeneglectedinmost cases.It


/ h t t p : / / w w w . h





S P R IN G

t,foraccurateresults,carefulmeasure-

sionsat manypointsarenecessary.

spring mustbecutup afterthetest

rediameter.

nsomereluctanceonthepartof

flectionmethodonthe groundthat

asmayintroduceun nownerrorsin

r sopinionthatthequestionsre-

ormulahavebeensettled,both e -

cally,andthattheresultsarereliable

edarecarriedout. Inaddition,the

advantagethatthemodulusofrigidity

tespring and,hence,maybemore

wounduptoformactualsprings.I t

besomedifferencebetweenthema-

gandheattreatedand straightbars

uiredbytheothermethodsdiscussed.

hedirectmethodfor findingthe

ht,roundbarof thespringmaterial

ngmachine.Toeliminatedisturb-

dsit isadvisabletomeasuretwist

hof thematerialundertorsionby

torsion measuringdevice.O nemethod

tachmirrorsonthe baracertain

deflectionsofthesemirrorsaremeas-

cale.If6is theangleoftwistin

nthe gagelengthIthemodulusof

ducingthetwist, Z = gagelength, d=bar

eoverallangularmovementof

achinearemeasuredaresub ectto

nindefiniteamountoftwistin the

endsof thespecimenandthismayintro-

e results.

lyderivedf romthek now nformulaforangulartw isto fa

eeS trengthofMateria ls—Timoshen o, PartI, Page261.


/ h t t p : / / w w w . h





L ICA L S PR IN G

od—Inthethird ortorsional

htissupportedbythe springwireand

uencyofoscillation/in cyclesper

omthisthetorsionalmodulusG maybe

ation

tofinertiaofpendulumbob,1= effective

rediameterasbefore.Thise uation

mthek now ne uation.

pingconstantof thew ire1 .

ylong toreduce,asmuchas

teeffectsduetoclampingatthe ends

est, o fcourse , themateria lissub ect

and tensionstress,thelatterbeing

endulumbob.

A summaryofava ilabletestdata

arbonspringsteelisgiven inTable

ereobtainedby variousinvestigators,

tmethodsdescribedpreviously,i.e.,

nalpendulummethod.Ineachcase

etherwiththemethodusedis indi-

hetable.

sentsanaveragefigurecalculated

rementsbetweencoilsofhelicalsprings

hemodulusfigurebeinganaveragefor

usingthetestdata,deflectionsbeyond

imitofthe springmaterialwerenot

sare seldomloadedtosuchhigh

orsiontestofa %-inch diameterbar

testedbyA dams,whileItem3refers

overstrainingat ator ueabout50

enHartog—Mechanica lV ibrations, McGraw -Hill , 1940, Page43.


/ h t t p : / / w w w . h





S P R IN G

uecorrespondingtotheproportional

at treatmentat228degreesCent.

frigidity from11.82to11.14X 10

by thistreatmentgivesanideaof there-

omstressinghelical springsfarbeyond

material.This investigatorshowed

igidityGforC arbonS pringStee ls

hearthQ & T

dandtempered.

a ined, mildheattreatment.


/ h t t p : / / w w w . h





L ICA L S PR IN G

ct thatnodecarburizedlayerwas

wasnotoverstrained.Thelower

thercasesareprobablydue largely

woeffects.Theselatterfiguresare,

ativeofthosetobee pected.

e landPhosphorB ronze summary

e modulusofrigidityofspring mate-

ls isgiveninTableV I.

roughaveragevaluesofthe

eta enas: 11.5X 10 poundsper

lloyS tee ls,

ndPhosphorBronze

igidityG

.in.y10 )Investigator

T .f % 1 1 .7 5 A d a m s

T. f . 14811. 2Z immerli, Wood

. 37511. 45B erry1

. . A . S rC . D. t. 14810. 5Z immerli, e ta l

D. t. 04- . 16210. 8Wahl

. D. t. 1259. 1Z immerli, eta l

rC . D. t. 096. 7Z immerli, e ta l

ayre

6.2BrombacherorMelton

dandtempered.

edandco lddraw n.

a , .2 1 % N i .

a .

.

pMemoirs,Iron& S teelInstitute,1937,Pages1-55.Directmethod

ained.

. M. , 1930, PartII, Page357. Directmethodused.

.E ngrs.,1938,Page460.Directmethodused.

Deflectionmethod.A verageof14springs.

M. . , 1934, Page556. Torsiona lpendulumtests.

ica lR eportN o. 358, 1930, Page568. Directmethodused.

vanadiumstee ls, 10. 6X 10 forsta inless

, 9X 10 forMonelmetal, 6. 4X 10 for

itshouldbe notedthatindividualtest

heseaveragesbyseveralpercent.

nyinvestigationshavebeen

durancepropertiesofhelicalsprings

oading.Theusualmethodof fatigue


/ h t t p : / / w w w . h





S P R IN G

testsofautomotivek nee-action

.Theresultsof suchtestsareof direct

ngineerswhoare responsibleforthe

ngunderfatigueloading.A nother

evalvespringused ininternalcombus-

eshowsthatthere areconsider-

urancelimitsor limitingstressranges

rtedbythevariousinvestigators.The

inthefactthat theendurancelimit

fspringis verymuchdependenton

spring wireorbar.S lightsurface

edecarburizationresultingfromthe

ayresultinaconsiderablereduction

nge.Thelow valuesreportedin

es maybeduetothis. A further

esults obtainedliesinthefact that

maybeused. Sincethesensitiv ityof

sconcentrationeffectsdueto bar

ferenceinresultswouldbee pected.

ChapterV I.

A mongthemore importantinvestiga-

calsprings,thetestsconductedby

mentioned.Thesetestsconsistedofen-

stressrangeson small-sizedhelical

tivevalvesprings.A typicalendur-

onchrome-vanadiumsteelspringsin

nF ig.56,thestressranges actually

theverticallines betweenthecircles

tress -.Thecircleswiththe arrows

imitsoftherange whichdidnot

ioncycles,while theplaincircles

d.O nthebasisof theseteststhees-

erangeisrepresentedbythe upper

diagramit maybeseenthatthe

areaboutasfollowsfor thismaterial:

00,40,000to 98,000,60,000to108,000

ssR angeforSmallH e licalS prings F . P . Z immerli, Engineering

niversityofMichigan,July1934.

agramhasbeenpreviouslydiscussedin ChapterI.


/ h t t p : / / w w w . h





L ICA L S PR IN G

. Thismeansthatthespringcouldbee -

elywithinanyofthese ranges.S imilar

notherspringmaterials.

sobtainedbyZ immerlitogether

erinvestigatorsisincluded inTable

ingk indofmaterial,heattreatment,

rs

tso fk neeactionhelica lsprings

sofrupture,yieldstrengthofthe ma-

wiresize,coil diameter,numberof

aregiventogetherwithvaluesofthe

hereseveralvaluesoflimitingen-

iagramssimilartothose ofF ig.56

fora givenmaterial.Inallcasesthe

q uation18w asusedtocalculatethe

atiguefailureofalarge helicalspringis

1.

ations,theresultsofwhichare


/ h t t p : / / w w w . h





S P R IN G


/ h t t p : / / w w w . h





/ h t t p : / / w w w . h





S P R IX G

arethoseby Johnson13atWrightF ie ld

directionof thespecialresearchcom-

ringsof theA . . M. . andreportedby

seinvestigationscoveredtestsonthe

around% -inchbardiameter)and

ange fromzerotoma imum.

ryof e pectedlimitingstress

edataof TableV IIandassuminga

elow10,000-20,000poundspers uare

0100000

, L B . / IN '

amofendurancetestsonhelical

umsteel.F romtestsbyZ immerli

smallerwiresizes.Thusthelimiting

old-woundcarbonsteel wiremeansthat

rangeof60,000poundspers uareinch

s,i.e..rangessuch as0to60,000,5,000

000pounds pers uareinch.Thefig-

matedfromtestsreportedbyvariousin-

rist icso fHelica lSprings , IronA ge, March15, 1934,

ressR eportN o. 3onHeavyH elicalS prings , Transactions

r1 9 37 , P ag e 6 09 .


/ h t t p : / / w w w . h





L ICA L S PR IN G

en thereareconsiderabledifferences

edbydifferentinvestigatorsforthe

Thereasonfor thisisthatthe q uality

gesSmall- izeHelica lS prings

mMinimumStressN earZ ero)


/ h t t p : / / w w w . h





S P R IN G

orthy.Bythisprocessit appearspos-

cerangetovalueswhichmaybee -

hedbars testedintorsion.Thus,

shot-blastedhelicalspringsof chrome-

nendurancerangein zerotoma i-

undspers uareinch.Thiscompares

ndby Johnsonongroundandpol-

diumsteelforarange fromzeroto

' andw ithava lueofonly70, 000pounds

gs withouttheshotblastingtreatment.

propersize ofshotandpeening

Manufacturersf re uentlychec the

ardA -typespecimen,3incheslongby

hthic treatedtoR oc w ellC 44-50.

aheavyplateandsub ectedtothe

tas thespring.A fterpeeningthe

easuredona1.25-inchchord.F rom

gesforLarger- izedHelica lC ompressionS prings

)

bleV III.

lmenthefollowingvaluesaresatis-

wrediametersprings, shotsize . 040and

25-inchchordwiththe standardA

sand torsionbarsof1.25-inchdiam-

e.060,deflection.012to.015-inchon

thic nessof.0938-inchbutis other-

ecimen. F orf la tsprings. 020- inchthic ,

003- inchonanA specimen1 .

hisisgiveninthearticles: Z immerli—MachineDesign, N ov .

Tra ilsinS urfaceF inishing , S tee l, July5, 1943, Page102.

nedS urfacesImproveE nduranceofMachineParts ,MetalProgress.

. " Improv ing F atigueS trengthofMachineParts , Mechanica l

43, Page553.

ivesmoredataonendurance limitso f springmateria ls(asdis-

springs).


/ h t t p : / / w w w . h





L ICA L S PR IN G 93

ehighendurancerangeswhichare

gcannotordinarilybeutilized inde-

000100000

. in.

mateendurancediagramsforgood

imitingstressrangeread

minimumstressandlines

tingma imumstress.Curva-

used

tressisused, e cessivecreeporload

ThisisdiscussedinC hapterV ). How -

blasttreatmentgreatlyreducesthe

that theproblembecomesmainlyone

etorloadloss.

E stimatedlimitingstressrangesfor

helarger-sizedspringsare summarized


/ h t t p : / / w w w . h





S P R IN G

agiveninTablesV II andIX ,

wingthevalueof enduranceranges

f romgood- ua lityhe licalcompression

Thesecurvesholdroughlyfor

between5and10. F orlargerinde es,

aybee pectedandviceversaforthe

ethesecurvesrepresentroughaverage

ationinindividualinstancesmaybe

ssconcentrationnearthehoo ends

hatlowervaluesofenduranceranges

57mayalsobe found.

O N S P R IN G W ITH F E W S T R E S S C C E S

obe availableintheliteraturefor

obut asmallnumberofstresscycles.

mentioned.Z immerli11foundon

springw ire( A E 6150, inde 7. 4)a lif e

esfor astressrangeof86,000,the

0poundspers uareinch(curvature

nd followingdata).E dgerton

0,000to 250,000cyclesforhot-

sofinde 5atastress rangeof100,000

(minimumstresszero). H. O . F uchs

fcenterlessgroundwire(446Brinell)

648-inchdiameterwire,inde 7.3.

m 170,000to409,000cyclesata stress

nimumto138,000poundspers uare

im ilarspringsof . 628- inchdiameterw ire,

to 178,000cyclesatastress range

00poundspers uareinch.

in thischapterapplyonlyfor

turewithnocorrosionpresent.. In

ginstolose itseffectivenessattem-

eesF ahr. 1\ Itshouldbeemphasized

ssrangesfoundinfatiguetests should

sdiscussedinChapterI, amarginof

ountunavo idableuncerta inties, isre uired.

.


/ h t t p : / / w w w . h





IN G UN D R S T A T IC L O A D I N G

cal springsbasedonelastic

tensivelyinChapterII.It shouldbe

calculationsbasedonproportionality

donot applyrigidlyaftertheelastic

materia lhasbeene ceeded. A lthough

terII areofbasicimportancein

rationofwhathappenswhenthe

lhasbeene ceedediso fva lue in

oadedhelica l

ablestressforhelicalsprings.In the

basisfor thechoiceofwor ingstress

dingbasedonsuchconsiderations

C hapterV Itheq uestionof fatigue

discussed.


/ h t t p : / / w w w . h





S P R IN G

alspringmaybedefinedas one

oadortoa loadrepeatedbutare lative ly

fthespring.A springloadedless

ngits lifewouldusuallybecon-

dincontrast tofatigueloadingin-

fcycles.

tantapplicationsofstatically-loaded

ybeenmentionedinChapterI.

esprings,springstoprovidegas et

ngs inmechanismswhichoperateonly

eotherapplicationsmightalsobe

sinlightningarrestersF ig. 58. Herethe

maintainadefinitespacerelation-

fthearrester,regardlessof temperature

ectto fatigueorrepeatedloading,

evelopmentofafatiguecrac which

of thespring( ig. 25). F ractureof

veroccurs,however,wheresprings

ads insuchcasesthedesignermustguard

porlossinload(whichta esplace if

sareused).Theseeffectsare particu-

edtemperatures.Ifasmallamount

eto lerated, ahigherw or ingstress

thecaseotherwise.

icularimportanceinthedesign

cttostaticloadingisthespringinde ,

diameterandwire diameter.Where

,thehigheststress isconcentratednear

enthe loadisca lculatedby ta ing

willbeseenlater,a highervalueis

springsthanwouldbethe case

es.

C U A TI O N S

stressisbe low theelasticlim it, in

sub ecttoastaticloadandnormaloper-

essdistributionalongatransverse

imatelyby the linebe inF ig. 59a. The


/ h t t p : / / w w w . h





O A D D H E L I CA L S P R IN G 97

binthiscase isca lculatedbyEq uation

ticconditions.

enthat, foraspringofsmallinde , the

hecoil ismuchlargerthanthe stress

eco il, i . e ., mosto f thehighstressiscon-

fstressalongtransversediameterofbarof helical

S mallinde ata , la rge inde atb

smeansthat aconditionofstress

sisshow ngraphicallybyF ig. 60aw here

ntratedinthe relativelysmallshaded

islarge,conditionsareconsiderably

g. 59b. Herethestressabonthe in-

A K

stributionof regionsofpea stressincross-sectionofbar

e ataandlarge inde atb(forthespringof

tressis concentratedneartheinsideofthe coilata)

ttle largerthanthestressa conthe

ea stressisgivenappro imate lyby

uation18. Thisfollowsf romEq uation19


/ h t t p : / / w w w . h





S P R IN G

forvery largeva lueof the inde c. F or

tstressesare locatedinthering-shaped

g.60binsteadofbeing concentratedin

arthe insideofthecoil asisthecase

ll( ig. 60b). Inotherw ords, inthecase

onlyarelativelysmallportionof

sub ecttostressesnearthepea , w hile

ndia -

m

- IN C H E S P R I N C H

f large inde , are lative ly largepart

ecttosuchstresses. Hence if the

ieldingoccursovertheentirecross-

tis clearthatthespringof small

rryamuchlargerloadthanw ouldbe

f thema imumstressca lculatedf rom

umespurelyelasticconditions.Thereason

asticlimitis passedandyieldingbe-


/ h t t p : / / w w w . h





O A D D H E L I CA L S P R IN G

tionwill beeffectiveincarryingload

sinceagoodshareof thecross-section

isa lreadysub ecttostressesnear

loadnecessarytoproducecomplete

tionwill notbesogreat asinthe

spring, w hereonlyasmallparto f the

cttostressesnearthepea .

lshaveconsiderableductility

hlessthanhavestructural materials).

nstress-straindiagramofatypical chrome-

orsmallhe lica lsprings isshow ninF ig.

shapeoftensilestress-straindiagram

aterialwith afairlysharplydefined

fthe usefulspringmaterialshave

percent in2inches,and stress-strain

ig.61,(althoughtheymayhavea

dpoint hasbeenpassed)itappears

sductile materials.Insuchcases,

ved,asbroughtout inChapterI,it

tstressconcentrationeffectsindesign .

augmenttobarcurvature(which

sconcentrationeffect)maybe

estressunder static-loadconditions.

Curvature—Tocalculatethe stress

gstressconcentrationduetobaror

ureisas follows:A ssumingahelical

ringundera loadPandneglecting

nsandpitchangle,the torsionmoment

w illbee ua ltoPrw hilethedirect

The distributionoftorsionstressalong

tothe momentPrwillbeasshown

a torsionstresst, duetothismoment

y theusua lformula( q uation4). Thus

uperimposedtheshearstressr.,due to

archBulletinN o. 26, UniversityofMichigan, R ivesother

ss e s — C . S . S o d er b er g , Tr a ns a ct i on s A . . M . . , 1 9 3 3, A P M 5 5 - 16 .


/ h t t p : / / w w w . h





S P R IN G

hichforourpurposesmaybe consid-

overthecross-section3.Thisstress

dasshownin F ig.62bandis

btainedby superpositionofthe

andbgiving aresultantdistribution

ma imumstress, T . , thusobtainedby

ationeffects,is

8

itten

illbecalled ashear-stressmultiplication

tionofspringinde cinF ig. 63.

pearslogicaltouseE q uation89

nthatstressconcentrationeffectsmay

formanideaofthe marginofsafety

ng,the stresscomputedbyE q uation

hthe yieldpointofthematerialin

ngmaterialsmaybe ta enasabout

yieldpoint.

M P E T Y I L D IN G

ent(andperhapsmorelogical)

fdesigninga springforstaticload

menon-uniformityinthe distributionoftheshear

veasimilareffect tothatofstressconcentrationdue

eglected.


/ h t t p : / / w w w . h





O A D D H E L I CA L S P R I N G 1 01

ionofstressesinhe lica lspring—stresscon-

tureneglected.A tais shownstress

stressduetodirect shear,andcsu-

wnata andb

basedonthe considerationofthe

completeyieldingofthe materialin

loadbeingthenta enasacertainper-

uiredtoproducecompleteyielding.If

tress-straincurvesimilartothatof

ectedthataf tere ceedingthey ieldpo int

ross atransversediameterwillbe

owninF ig. 64aforaspringofsmallinde

of large inde . A ctua lly itmaybee -

rialssomerise inthestress-strain

C - ^ f-

ngshearstressmultiplication

esintoaccounteffectsdueto

obarcurvature


/ h t t p : / / w w w . h





S P R IN G

ldpo intw illta eplaceduetothe

hatactua lly thecurvesofF ig. 64w illbe

dalinform.H owever,theassumption

n,whichlendsitself tosimplicityin

yaccurate.Becauseofthenecessity

direct-shearload,particularlywhere

stributionoftorsionstressunderplastic con-

rentinde es. A t a isshow nsmall

b. F orthesmallerinde estheareaA , is

ta ecareof thedirectshearload

ll, itmaybee pectedthatinsuchcases

terthanA , inF ig. 64a. O ntheother

e,thesetwoareaswill beaboutthe

tothiseffect, thepo intO' w herethestress

mounte f romthegeometrica lcenterO.

P atwhichcompleteyielding

n occursrepresentsaproblemin

remelycomplicatedforlowinde

adeterminationofthedirectionsof

t allpointsofthe crosssectionunder

aluatetheshear forceandmoment.

matesolutionbasedonreasonableas-

dasfollows:

ctionsoftheresultantshear stress

ofthecrosssection mayberepre-

sasshownin F itf.65a.Thecenters

displaced sothateachcircle intersects

ate ua linterva lsbetw eenPC andO' B

sthepo into f zerostress(the lineB B rep-

rsetothespringa is). Ifnostra inhard-


/ h t t p : / / w w w . h






heresultantshearstressista ene ua l

t atallpointsand thedirection

t ispossibletocalculatethe resultant

ra givendisplacementt ofthepoint

. F romthisthespringinde maybefound.

thecirclew ithcenteratA represents

65a. Onthebasiso f theassumption

oints ofintersectionofthecircles be-

B , t h e ra d iu s p o f t hi s c ir c le w i ll b e

po intDa longO ' B , aseriesofcircles

ig. 65a.

ntdA atradiuspand angle6,

lbe actedonbya shearforcerydA

ththeradiusOC . Thearea is

orce aboutthecenterO willbe

) pd A

his,

93)

diagramofresultantshearstressdirectionover cross

icalspringunderyielding.Pointof zerostressO '

icalcenterO aw ay fromspringa is


/ h t t p : / / w w w . h





S P R IN G

illbe theintegraloftheseelementary

rosssection.Thusthe momentMyfor

sectionbecomes

O (94)

aedfunctionofbothc , pand6and

ofEq uation94ingenera ltermsis

ssumingagivene anddandbydraw -

nF ig. 65a, theva lueofsinipcanbe

andradiusp.Drawinge ually

r ofthecrosssectionO asindicated

theva lueof t p2sin, palongeachradius

p, the integra lo fE q uation94maybe

hedeterminationoftheareaunderthe

plyingbya constantdependingon

radiiandaddingthetotal.In this

ora givene anddmaybeobtained.

overthecrosssectionmaybe

er.R eferringtoF ig.65cthevertical

rceactingon theelementdA is

9 ) dA

sincetheshearforceis considered

thee pressionfordA givenby

ppde(95)

ceP actingoverthesectionfor

eintegralof theseelementaryforces

ence

dO (96)

atedbydraw ingcirclesasinF ig.

pacedradia ll ines, measuringifz— Q , and

sin(i/ < — 0)a longeachradius, andf ind-

urve.Byaddingthesewithproper


/ h t t p : / / w w w . h






ingbya constant,theresultantshear

ndbardiameterdisfound.

er e r ec o il r a di u s,

M andP obtainedbygraphicalor

q uations94and96arek now n, the

foragivene andd. F romthisthe

a ined.

ringisessentiallya straightbar

mentM, = P ' r, directshearbeing

forconstantyield stresst is

onw ithEq uation3, it isseenthatthe

' )forcompletey ieldingofa large inde

thigherthanthat atwhichyielding

uation3by ta ingrm= ry ). Inactua l

ardeningandother effects,higher

e pected,however.

,/ ( q uation97)fora large inde

uation94)forasmallinde spring, anesti-

irectshearloadis possible.Theratio

comparabletothefactorK H ( q ua-

byneglectingstressc oncentration

pletesolutionofthisproblemobtained

4and97forvariousspringinde esisnot

eindicationsat presentarethatthe

co n si d er a bl y l es s t ha n K „ a s g iv e n by E q u a -

useofpossibleinaccuraciesintheas-

gtheshearstressdirections,F ig.65a,

eningeffectsnot considered,themore

givenbyE q uation90w illbereta ined

ntuis k nown,theloadatwhich

ringoccurs (wheretheloaddeflec-


/ h t t p : / / w w w . h





rTurnfor

e licalS prings0

ofSprings(inches)


/ h t t p : / / w w w . h





rings(inches)

enin. i

stressT,

ectionpe

1OO , OO 0


/ h t t p : / / w w w . h





S P R IN G

hatbelowtheactual loadobtained

enbyE q uation90isprobablysomew hat

eningeffectscomeintothepicture.

F F O R MU A S T O S P R I N GTA B E S

nofE q uations89and7 inthe

helicalspringsTable X ,hasbeen

sloadsanddeflectionsperturn ata

ers uareinchanda torsionmodulus

pers uare inchascomputedf romEq ua-

standardoutsidecoildiametersandwire

s usedforsizesup to.090andthe

rsizesf rom. 106to% - inch.

00, 000poundspers uare inchmayac-

cticalcases,itshouldbe notedthat

blefor convenienceonlyandisnot

ndedw or ingstress. Theactua lw or -

oadedspringshouldbee ualtothe

materialdividedbyafactorofsafety

IIIgivedataonthey ieldpo intsof spring

ssin tensionisk nown,theyield

a enasabout57percentof thatin

basedontheshear-energytheory

aluesoffactorofsafetyas usedin

to aslowas1.25in somecases.

alueofstressother than100,000

theloadsanddeflectionsgiven inthe

nthesameratio . A sane ample, as-

houtsidediameterand.135-inch

adanddeflectionperturn are103

pectively,at100,000poundsper

edf romEq uations89and7. If the

withanultimatestrengthof260,000

nthissizeanday ie ldpointintension

0poundspers uare inch, they ie ld

percent ofthisorabout 120,000.

etyof 1.5basedontheyieldpoint is

ingstressof120000/1.5= 80,000

Thismeansthatthe permissibleloads


/ h t t p : / / w w w . h






ndertheseconditionswould be80

hetable.In thecasecited,the

rnwouldbe.80(.141)= .113-inch

ouldbe.80(103)= 82.5pounds.

orintermediatecoilandwire

bleX , thechartso fF igs. 66and67

g.66theordinaterepresentsload al

R , IN C H E S

. 20

R , IN C H E S

000lb./s .in.Tofind loadatanyotherstress r,loadsgiven

0,000.N ottobeused forfatigueloading

ulatingloadsinstatically-loadedhelicalsprings


/ h t t p : / / w w w . h





S P R IN G

areinchtorsionstress(calculatedby neg-

due tocurvature)whiletheabscissa

achcurverepresentsagivenoutside

sfor awiresizeof .090-inchand

-inchtheloadat 100,000poundsper

bout61pounds.

per turnrepresentedbytheor-

twirediameterforvariousoutside

ire sizeof.106-inchanda coilout-

thedeflectionperturnis .035-inch.

mallerrorwillresult inreadingthe

igs.66and67 andforbestaccuracy

uld beused.Thesecharts,however,

rmostpracticalpurposes.

re uently , springtablesorchartsbased

chyie ldthepea stressincludingcurvature

pterV II).Thesetablesmayalsobe

sson whichthetableorchart isbased

w h er e K C = K / „ v al u es o f K a n d K „

uations19and90. Theva lueofK , .

entrationeffectdueprimarilyto wire

,representstheincreasedstressdue

a ia lload. Thisfo llow ssinceK — K C „ .

culation, va luesofK , -aregiveninF ig. 68

c. Thus, if a tableorchartisbased

undspers uare inchandif thespring

1. 15, thestressca lculatedbyneglecting

0/1.15or87000poundspers uare

be comparedwiththeyieldpoint.

e ampleof thisdesignprocedurefor

aspringmaybeconsideredofinde

lhavingatensionyieldstressofaround

areinchandayield stressintorsion

r110, 000poundspers uare inch.

f safetyof1.5basedonthe yieldstress

ewor ingstressforthe static-load

sngEq uation89w ouldthenbe

oundspers uare inch. F oraninde 3,

ig. 68) hencethea llowablestressas

tion18(whichincludescurvatureeffects)


/ h t t p : / / w w w . h






aluesshouldbomultipliedbyT/100,000.A lso,ifthemodulus

0 ,valuesshouldbemultipliedby 11,400,000/G

ulatingdeflectionsinstatically-loadedhelicalsprings

orsionmodulus, G= 11. 4x 10 Ib. /s . in.


/ h t t p : / / w w w . h





S P R IN G

98000poundspers uare inch. If

uationareused(suchasthosegivenin

ring proportionssuchasactiveturns,

eandsolid heights,aredetermined.

: A springhasaninde o f15w ith

asintheprev iouse ample . F oran

1. 06f romF ig. 68. A ga inassuming

byneglectingstressc oncentration,

ressintorsionor73000poundsper

blestressf iguredf romEq uation18w ould

oundsper s uareinch.Thisstress

valuein thepreviouse ample.This

pea calculatedstress(withcurva-

rywiththespring inde ,assuming

eingmaintainedbetweenthewor -

uiredtocausecompleteyielding.

L A X A T IO N U N D R E L E V A T D T M P R A T UR E S

he determinationofwor ing

helicalspringswasbasedon theyield

altemperaturesbeingassumed.A s

essisk eptw ellbe low thispo int, no

shouldbee perienced,providedthe

otmorethanabout200 degreesF ahr.

mperaturesitis notusuallysuffi-

ntheyieldpointor elasticlimitof

hodofdeterminingwor ingstresses

actua lcreeporre la ationtestsat

fortunately,thereisnot agreatdeal

amountoflossofloadwhich maybe

mprehensiveseriesoftestssofar carried

tionorlossofload inhelicalspringshave

immerli4.Thesetestsweremadeby

sbyagiven amountinaspecialtest

dspringwasthenput intoafurnace

evaryingfromthreedays forthe

tureonC oiledS tee lSpringsatV ariousL oadings —

clioniA . . M. . , May , 1941, Page363.


/ h t t p : / / w w w . h





O A D D H E L I CA L S P R I N G 1 1J

springstoten daysforthestainless

eating,thespringswereremovedfrom

1 1

13 1 4 1 5

C=

ngstress-concentrationfactorK c

ossinf reeheightdetermined. F romthis

permanentset)thepercentageloss

d.

eresultsobtainedbyZ immerlia re

orvariousspring materials.Thevalues

ntpercentagelossinloadin aperiod

eraturelisted,e ceptforthestainless

tswere runtendays.Thestresseswere

ection,E q uation18.Iffiguredwith-

esevalueswouldbeabout 10per

sweremadewith springswhichhad

usbluingor stressrelievingtempera-

minutes.Inthetablethe valuesof

ptimumbluingtemperaturesare

lowertemperatures,thevaluesof

iderablygreater.Itappearsthat at

uresnotallthe coilingstressesare

tterarecombinedwiththe loadstresses

ethanwouldbe thecaseotherwise.

rsthat,atstressesof100,000pounds

90,000figuredwithoutcurvaturecor-

loadlossmaybee pectedwithinthree

.6percent carbonsteelwire,when


/ h t t p : / / w w w . h





S P R IN G

sof350degreesF ahr. Somew hatlow er

dforthechrome-vanadiumsteel.S tain-

eshowedaloadlossof onlyabout

esF ahr.whichincreasedto11.5per

hr.atthesamestress(100,000poundsper

tertestsw ererunfortendays. F orvery

oad lossesmaybee pected.

Z immerliconcludedthattheusual

henstressedto notmorethan80,000

(ortoabout72, 000poundspers uare

turecorrection)attemperaturesupto

ovethistemperatureandupto 400degrees

aybee pected, w hileordinaryspring

temperaturesabove400degreesF ahr.

stainlesssteelsofthe18-8 typeresist

tterthanothers,e cepthigh-speed.

wnfromthisseriesof testswasthat,

t-treatedaftercoilingshowedno

undfrompretemperedwireproperly

g temperaturewasfoundtobeinthe

adforHelica lSpringsatE levatedTemperatures


/ h t t p : / / w w w . h





O A D D H E L I CA L S P R IN G

withoutob ectionableloweringof

rties.F orfurtherdetailsonthis the

riginalarticle1.

culations—S omeanalyticalmeth-

byN adai forca lculatingcreepand

resumeof thesemethodsw illbegiven:

ppro imationthatthespringis

etorsion.This willbeappro imately

ngs. L ettingP= loadonspring, r=

rediameter,and6theangleof twist

wire,thetwistingmomentwillbe

shearatadistancepf romthecentero f

n

problemstheunitshearstrain i

to fanelasticpartf andaplastic

q uation98,

(9 9)

followingrelationholds:

tradiusp( ig-24, C hapterII)and

ofthematerial.

on99withrespectto timet,

erentiationwithrespecttotime.

t= g(r)w hereg(r) isa functionof

q uation100andbysubstitutionin

UnderV ariousS tressC ondit ions * —-A . N adai, Th. ton

o lume, 1941. A lsobibliographygiveninthisre ference.


/ h t t p : / / w w w . h





S P R IN G

by

wire.

latethe steadycreepofthespring

willbeassumedthattheshear stress

sisgovernedbya powerfunctionlaw.

reewithtestsoverlimitedrangesof

singEq uation101,

( i o4 )

isane ponentw hichdependsonthe

terminedbyactualcreep tests.

f t intoEq uation103, andinte-

ufficienttimehas elapsedsothat

becomeconstant,thee pression

on102. So lv ingfortheva lueof6byusing

bstitutinginEq uation104,

of stressoverthecross-section

oadPisindicatedby thecurvedline

calculatingt asafunctionofp from

ightlinewhichrepresentsthe initial

ing oflargeinde )isalso shown.

edistributionofstressgivenby

ursaftera considerabletimehaselapsed.

ibutionfor theintermediateperiod,

ow s: F romE q uation105sinceP is


/ h t t p : / / w w w . h






dr/ dtgivenbyEq uation102,

E q uation102andrearrangingterms

rentiale uationresults:

d f i- g (r ) ( 10 8 )

ationfort= 0istheusual formula:

isgivenbyE q uation106.

ulatethere la ationorlossinload

essed(ore tended)toagivenlength,

S

eadyF ig. 70—S tressdistribution

ring rela ationofspring

howingtime effect

s:S incethelengthofthe springdoes

ngleof twistperunitlength6

e6— 0. UsingE q uation102thefo llow -

isobtained:

unctionis againassumed:


/ h t t p : / / w w w . h





S P R IN G

nt.

uation109,

ctto timeanddeterminingthe

theconditionthatfort = 0,alinear

cross sectionoccurs,

he init ia lstressdistribution. So lv ingfort,

distributionof shearingstressesover

ustimest isillustratedbyF ig.70.

epea stressdropsconsiderablyas

essdistributiontendstoflattenout

rmdistribution.A fteraconsiderable

tressesapproachthevalue:

( 1 13

inga rectangulardistributionof

orloadPat largevaluesoftbecomes

obtainedfromrela ationtests.

nsforloadmaybeobta inedbyusing

uation112inE q uation103and

mericallyorgraphically.


/ h t t p : / / w w w . h





A R I A B E L O A D IN G O F H E L ICA L

ationalbasisfordetermining

ca lspringssub ecttostaticorinf re-

gwasdiscussed.Incaseswheresprings

orrepeatedloading, asfore ample inau-

omewhatdifferentapproachtothe

or ingstressisnecessary.In the

problemiscomplicatedbythefact

sub ecttoa load(orstress)w hich

auetoama imum. A sshow ninChap-

oaconstantorasteady loadonwhich

oralternatingload.Thus inthecase

ring,theconstantcomponentofthe

itial compressionofthespringwhile

determinedbythevalvelift.

lculatedbymeansofthe curva-

q uation18, conservativedesignw ill

pearsj ustifiedif thespringissub ect

tress infatigue.W herethereisa

mponentonwhichis superimposed

t,however,itappearslogicaltoneglect

tsduetocurvaturein figuringstresses

ofthe load.Inthisconnectionthere

amongspring engineersthatthe

esult intoo low valuesofw or ing

tthe stressescomputedthiswayaretoo

medtoa certaine tentbytheresults

defatiguetestsonsmallhelical

escarriedoutbyZ immerli1. These

ressrangein fatigue,whenfigured

ashigherforthesprings ofsmaller

ltsw erereportedbyE dgerton- inconnec-

M. . , January , 1938, Page43.

M. . , October, 1937, Page609.


/ h t t p : / / w w w . h





S P R IN G

r onheavyhelicalspringsbythe

esearchC ommitteeonMechanica lSprings.

dlater.

efullstress-concentrationeffectcor-

ecorrectionfactorK doesnotalways

ding)lies inthefactthat somema-

eto stressconcentration.Inother

lsaretestedbymeansof specimens

llets,thefatiguestrengthreduction

ofsuch" stressra isers isnotasgreat

basedontheoreticalstress-concentration

eoreticalstress-concentrationfactors

byanalyticalmeansusingthe theory

asurements,orbyphotoelastictests4.

stressconcentrationeffectsis in

inthesmallersized specimensandfor

eontheotherhand thefine-grained,

areverysensitivetosuch effects5.In

,thedecarburizationofthesurface

eattreatmentand theeffectsofshot-

epresentfurtherfactorswhichtend to

-concentration.

C A L C U A T IO N

viousdiscussionand thatof

luatingwor ingstressinhelical

dingbasedonthe followingassump-

effectsdueprincipallytobar orwire

maybe neglectedinfiguringthe

imitingvalueofthe staticandvariable

refollowsalinear law

ofE lasticity—Timoshen o, McGraw -Hill . A lsoNeuber

pringer,Berlin formethodsofdeterminingstressconcentration

ods.

toe lasticity, V o l. I, Wiley .

onon" CorrelatingDatafromF atigueTestsofS tress

ns , T imoshen oA nniversaryV o lume, Macmillan, 1938, and

ssC oncentrationF actorsinDesign , P roceedingsSociety forEx peri-

is, V o l. 1, No. 1, Page118, discussthis. A lsoarticlebyPeterson

ppliedMechanics,March,1938,PageA -15anddiscussionDe-

46.


/ h t t p : / / w w w . h





D IN G O F H E L I CA L S P R IN G 1 21

ertiesofthe materialarethesame

es,assumingthesamesizewire

ofloadingdue toendturnsare neg-

ducedbyheat-treatmentoroverstress-

glected.

sedmorefully later.F orthepresent,

alto stressconcentrationwillalso

ffectsofvariationsin thesensitivityin-

duetosurfacedecarburization, shotblast-

.

ressC oncentration—R eferringtoF ig.

typicale perimentalcurveoffailure

inationofstatic andvariablestress.

aluesofvariablestresswhichwill j ust

mposedinthestatic stressesshownby

gthatfatiguetestsaremadeon aspring

) s o t ha t K c = 1 , a nd l e tt i ng r ' d e no t e

erotoma imumstressrangeobtained

T ,

nofstra ightline law , tohe lical

mitt eforpulsatingloadappli-

m)andy ie ldstresstyarek now n

intPonthediagramis determined.

imumstress)boththestaticandvari-

ua ltore / 2. A s anappro imation, thee -

eplacedbythe straightlinePA drawn

bscissasatt , thetorsiona ly ie ldpo int


/ h t t p : / / w w w . h





S P R IN G

one sinceingeneralnostress should

Toapplythis diagraminactualdesign,

springisoperatingundera fatigue

totma w herethesestressesaref iguredby

orrectionfactorK = K , 3(seePage

mponentofstresst, is

esfullsensitivitytostress concentra-

tofstresst ,whenfiguredbyneg-

neffectsduetobar curvature,then

uationisdividedbyK c= K / „

adybeenusedinf iguringrma andt, , ;

ermining

for

1

ermining

rC trfor

1

arplydef ined, asanappro imationitmaybeta en

asticstrainis. 2percent. S ee" C oncerningtheY ie ldPoint

e lls, P roceedingsA . . T . M. , 1928, Page387.

rsioncould besubstitutedforj yifdesirable.In

results incloseragreementwithtests,butthe results

point will,ingeneral,beon thesafeside.


/ h t t p : / / w w w . h






sconcentrationeffectsduetoc urvature

ary. A nana lytica le pressionforthe line

f t a n d t i s

tionofastraightline passingthrough

titutingtheva luesof tvandt givenby

ermining

f o r t / t <

16inthise uation,

ma imumstress3tma intermsofK ,,

/ t c a n d ma y be w r it t en

f un c ti o n of K C , tu , t , i / t , u r a nd t u /t , ,

thevariablecomponentofstressisnotH rea Tthanthestatic

stressconditionscorrespondingtotheline PA inF ig.71an

alwaysthecase.


/ h t t p : / / w w w . h





S P R IN G

ofspringinde c, va luesofC w may

hartsforvariousspring inde es,and

' .

evalueof t, o givenbyE q uation119

fetyN sothatthewor ingstress

amountstoassumingaline CD,

nePA andintersectingthea iso fab-

f romO. A nycombinationofvariable

lls onthelineCDis thusassumed

.

lation,chartsshowingtherela-

n/ tma forvariousspringinde esat

aregiveninF igs. 72, 73and74. These

dusingthee pressionforCwgiven

assumingadef initeva lueof tu/ t, ' . Thus

o ista ene ua lto2. Thesechartsshow

mpermissiblestressincreaseswithin-

dthatthisincreaseis greaterforthe

.H owever,thisincreaseinallowable

eepandrela ationeffectsasdiscussed

ensitivetoS tressConcentration

maybeusedfor caseswherethespring

eto stressconcentration.Itwillbe

it ivity inde of thematerial(w hichisa

itivitytostress concentration)has

venmaterialandwire sizebyactual

it ivity inde q isdef inedas

ote5giveafurtherdiscussionof " sensit iv ity inde .


/ h t t p : / / w w w . h






trengthreductionfactor,i.e.,theratioof

ressconcentrationtoendurancelimit

present.H ereK cisagainthe theo-

nfactordueto barorwirecurvature.

yinsensitivetostressconcentration

ation122, q= 0. F ormateria lsfully sensi-

nceq = l. Thusthemoresensit ivemate-

nedalloysteelsinthelargersizes,

fq thanwouldmaterialsnotsensi-

n.A smentionedpreviously,thesur-

arwouldalsohaveaneffecton the

heinde .

effect oflac ofsensitivitytostress

rangetimu: — i- , i shouldbeca lculated

threductionfactor K /ratherthanthe

isk now n, byso lv ingforKt inE q ua-

att, u andtmi havea lreadybeenca l-

yusingthecurvaturecorrectionfactor

ore, if theva lueofmultipliedby

ueofvariablestresst w il lbeobta ined

countthesensit ivityef fect. Hence,

this,

e -l )

ons116and125inE q uation117,

l( fullsensit ivity )thise uation

18. F orq = 0(nostressconcentrationef -


/ h t t p : / / w w w . h





S P R IN G

hise uationisinef fectdiv idedbyK ,

ncentrationeffectsdueto curvature

eneglectedentirelybothfor thestatic

nents.Thisise uivalenttousingthe

ti o n 90 .

isobtained:

uation120whenq = l.

nctionofspringinde cfora

V i a n d fo r t / t , e u a l to 1 . 5, 2 . 0, a n d 2. 5 a re

77, respectively.Thesechartsare

estoshowtheeffectofa reductionin

ialas measuredbytheinde con

romthesef igures, itmaybeseenthat

tostress concentrationtheallowable

rablyhigherfor thespringsofsmaller

he largerinde springs.

To illustratetheapplicationofthe

inpractica lw or , thefo llow ingcondi-

springofV 4-inchwirediameterand


/ h t t p : / / w w w . h






er,i.e.,c= 3.F atiguetestsonsprings

hesamesizew iresub ectedtoapulsating

e ldava lueofendurance lim itt/ =

reinch,whiletorsiontestsshowa yield

0poundspers uare inch. F urtherthe

stressrangef romt, i to r w w here

hstressesbeingcomputedbyusingthe factor

sensitivityofthe materialisas-

t / T , / = 2, thecharto fF ig. 73applies.

n de c = 3 , C , = 1 . 53 w h en t m in / tm a — - ^ > -

ailure maybee pectedattma — Cw

0= 92, 000poundspers uare inch( q uation

rof safetyN — l. 5, thewor ingstress

tion121, t = C T/ / N= 92, 000/ 1.5-=

reinch(thestressbeing figuredbyusing

ginde w ere10insteadof3, thefac-

g. 73, andthew or ingstressrw —

6, 000poundspers uare inch, assuming

ductioninthesensitivityinde q ,

sofconsiderablysmallerinde inthis

eassumedtoshow avalueofq — ^k .


/ h t t p : / / w w w . h





S P R IN G

ig.76(t /2>' = 2, q = V z)C= 1.65

= % . Inthiscase, thea llow ablew or ing

afetyN= 1. 5, wouldbetw = C w tc / N=

6, 000poundspers uare inch.

fundamentallimitationinthe

discussedforspringsundervariable

ptionthatthestressconcentration

edin figuringthestaticcomponentof

under fatigueloading.A sbrought

theloadis purelystatic,thisappears

er, furthertestsw illbere uiredtoestablish

ionwhenappliedto combinations

s.The alternativemethoddiscussed

ge131) doesnot,however,involve

emethodis theassumptionofa

estaticandvariablestress components

failure,i.e.,astraightl inePA inF ig.

ctuallimitingcurve.Usuallythee -

omewhatabovethisline asindi-

ueof thezerotoma imumendurance

ctualtestsona springoflargeinde

nderconsideration,itappearsthat

basis,thecalculatedresultswillbe on

ectionitisnecessaryto determine

heactualwire sizeused,sincethe

angeconsiderablybetweensmalland

thatthelineoffailure PA tends

n torsion.Inmanycasesit will

proachestheultimatestrengthintor-

e lattercouldbeuser1insteadof t inap-

ulas.If thisisdone,however,a

uldbe usedthanotherwise.

Cv,it hasbeenassumedthat

ofthematerialdo notchangebetween

einde assumingagivenw iresize. A l-

areasonableassumption,thereare

ganE ngineeringR esearchBulletinN o. 26mentionedprev iously,

her.


/ h t t p : / / w w w . h





D IN G O F H E L I CA L S P R IN G

estrict ly true. F ore ample, inco ld-

re size,springscoiledtosmaller

ctedtothegreatestamountofco ldwor -

s maypossesssomewhatdifferent

nwouldotherwisebee pected.Ifa

tisgiven,thisdifferencemaybeslight.

ee pectedw herespringsareq uenched

ctof theheattreatmentmaybedif-

ntinde es.Theshot-blasttreatment

(ChapterIV )willprobablyalsointro-

tivityinde .

hemanufactureofcompression

pringsolid, thusgivingita permanent

ntroduceresidualstressesof op-

hespringisunderthew or ingload,

ducedby theamountofsuchresidual

thetendencywill beforthefatigue

causeof thepresenceofthesestresses.

vatureofthe springshavingsmaller


/ h t t p : / / w w w . h





S P R IN G

rtointroducesuchresidualstresses in

happenthat thefatiguestrength

de maybeincreasedbysuchtreat-

re tentthanisthe case,forsprings

effectsofeccentricityofloading

beenneglected.Theseeffectsmayin-

ressfrom4to 30percent,dependingon

endturns,andonthe totalnumberof

nof thisw illbegiveninC hapterV III.

F T H E O R E T ICA L A N D T S T R E S U T

ledgethemostcomprehensiveseries

theef fect. o f springinde onendur-

ngs werethosecarriedoutby

easeriesof testsonspringsof . 148-inch

wedishvalve-springwire,havingin-

. 5to12. Itw il lbeof interesttocompare

eobtainedby theapplicationofthe

d74 whichassumefullsensitivityto

l) . V a luesof theminimumandma i-

J-)of thelimitingstressrangesas found

ousinde esaregiveninthesecondand

I.

a imumendurancelimitj > ' ,

, thetestresultsforc= 11. 9areused

19, 000poundspers uare inch, tma —

are inch, andrmi / tm(U= . 21. A ssuming

1 . 5, f r om E q u a ti o n 11 9 , rm u — L l l t f o r

ing, r, = 91, 000/ 1. 1-= 82, 700poundsper

eyie ldpo intintorsionforthematerialw il l

spers uareinch(orsomewhatabove

, itmaybeassumedthatt / t ' =1. 5

owingtheuseof F ig.72.Usingthe

ndassumingthe minimumvaluesof

sgiven, the lim itingva luesofma imum

edusingthecharto fF ig. 72andE q ua-

luesoftma thusobtainedaregiven

M . . , J a n ua r y, 1 9 38 , V a s 4 3 .


/ h t t p : / / w w w . h






eX II. F orcomparison, va luesof the

—rmiH asfoundby testandasdeter-

venin thelasttwocolumns.

ntheselast twocolumnsindi-

culatedvaluesoflimitingstressrange

nt.Thisofferssomeindicationthat

uesofL imitingS tresses

im itingR angeinS tress

ests9C alculatedtmo —tmin

m l n t in n B y t es t C al c ul a te d

. )( lb. / s . in.)( lb. / s . in.)( lb. / s . in. )Ob. / s . in.)

095,50086.00081,500

096,00075,00077,000

93,50074,00074,500

92,00071,00073,000

091,00072,00072,000

usingthecurvaturecorrectionfactor K .

gw or ingstress, usingE q uation118

vitytostressconcentration(< 7= 1),

eementwithactualfatiguetests,at

dwiresizes.

ionedpreviouslyweremadeon

ledf rom% - inchdiameterbarstoc , one

eotherof inde 5. Theendurance lim its

econventionalformula,E q uation4,

orthe twogroupsofsprings,while

atedbyusingthe K factordiffered

ndicatethatforthesesprings neither

directshearstresshaveanyeffect on

ontrasttothepreviouslydiscussed

doshowthatthewire curvaturedoes

cerange eveninsmallsizesprings.

atawill bere uiredbeforedefinite

.

E M TH O D O F C A L C U A T IO N

methodofevaluatingwor ing

dervariableloadingisbasedonthe

ncentrationeffectsduetocurvature


/ h t t p : / / w w w . h





S P R IN G

atingthe staticcomponentofthe

hemethodproposedbyS oderberg

or ingstresses.A nalternative,how-

hatsimplermethodisthefollowing:

operatingbetw eenma imumand

ndP , , , thentherangeinloadwillbe

ntorsionstressrristhencomputed

ngthefullcurvaturecorrectionfactor

ataare availableregardingsensitivity

erials, thevalueofK maybereducedto

ndsonthesensitiv ity inde q andisgivenby

absenceofactua ltestdataore perience,

hatasensit ivity inde q e ua ltounity

henca lculatedf romtheloadPma using

actorK . If thispea stressisabovethe

atterva lue ista enasthema imum

nnearlyallcaseslocalized yielding

stothisvalue.Then theactuallimiting

sthetestva luew iththepea stresse ua l

n. H ow ever, if thepea stresstma is

oint,thenthe actualendurancerange

sta enasabasis. Thismaybefound

ofthetypeshownin F ig.56orfrom

nTableV II,Page88.

at, if thepea stressdoesnot

torsion,therewillnotbe muchvaria-

urancerangeforvariouspea stresses.

ses,anappro imatefigureoflimiting

ew iththepea stresse ua ltothey ie ld

c onservativefigure.Theallowablestress

factor K ,wouldthenbethis limiting

bythefactorof safety12.

essivepermanentsetthestressat

W, ca lculatedbyneglectingcurvaturecor-

hapterV , shouldnote ceedtheallow -

g.This alternativemethodofde-

pearstobe promisingandissome-

of thismethodw asgiveninapaperon" H elicalS pring

tandardC ode TransactionsA . . M. . , Ju ly1942, Page476.


/ h t t p : / / w w w . h





D IN G O F H E L I CA L S P R IN G

ussedpreviously.F urthertestdata,

bletodifferentiatebetweenthetwo

either wouldbesufficientlygoodfor

ampleof theuseof thismethod: A car-

soutsidecoil diameter,%-inchbar

hree, sub ecttocontinuousa lternating

mof1700poundsanda minimumof

uation18thestressatthepea load

correctionwillbe82,000pounds

somewhatbelowthetorsionalyield

dof1700—1200= 500poundsthe

00poundspers uare inch. F rom

heat-treated,carbon-steelspringthe

oma imumloadapplicationwith

poundspers uare inch, maybees-

oundspers uareinch.Thisgives

0/24,100= 2.9onthestress range

tressfiguredwithoutcurvaturecorrection,

0poundspers uare inch. S incethee -

tofthis materialshouldbeabout

are inch, TableV IIPage88, thefactor

lding wouldbe110,000/61,000=1.8.


/ h t t p : / / w w w . h





L E CT IO N A N D D S I GN O F H E L I CA L

S P R I N G

fthischapterto presentdataon

as chartsandtables,whichmaybe

cilitatethepracticalselectionofhelical

ns.A lthoughthemethodsofevaluat-

cribedinthe precedingtwochapters

htothe designproblem,inmany

involvedbytheuseof thesemore

rranted.Thisis particularlytrueif

incharacteristicsarere uired,for

venmechanism,plentyofspacebeing

use ofthespringtablesgivenhere

ry.

maybecaseswhentheproper

ng isvitaltothe successfulopera-

hileatthesametime,theavailable

se,aconsiderableamountoftime

g re uirementsonthebasisof the

andV I w ouldprobablybe j ustified. Even

used,however,thechoiceofthe proper

useofthechartsand tablesgivenin

E S S E S U E D I N PR A C TI C

eswhereaq uic selectionof

ewof suggestedwor ingstressvalues,

rces,isdesirable1.Inutilizing these

hatfor bestresults,aconsiderable

re uiredandforthisreasoninimportant

withaspringmanufacturerusually

arcdiscussedfurtherinauthor sarticle , TransactionsA . . M. . ,


/ h t t p : / / w w w . h





F H E L I CA L CO M PR E S S I O N S P R I N G 1 35

nofw or ingstressesusedasa

esignbyWestinghouseE lec. & Mfg.

ingstressvalues,whichshouldbe used

ngselection,applytospringsmade

chasmusic oroil-temperedwire,hot-

edafterforming.Inmostcases the

h e a r -H e l i c al

tee l

eServ iceA verageServ iceL ightServ ice

. in.)( lb. / s . in.)

3,000

085,000

074,000

065,000

056,000

50,000

uality springstee l. A llstressesbasedontheuseofacurva-

tabledoesnot holdwherecorrosioneffectsorhightem-

rphosphorbronzesprings50per centandforrust-resisting

valuesareused.

TableX IIIwillbe foundtobecon-

eincreasedaftera carefulstudyof

nbasedon thestresseslisted

Pages138to149are given.

beseenthatlow erstressesareused

ndforsevereserviceinaccordance

ce.Theclassificationofparticularap-

ge,orlightservicedependsto acon-

udgmentof thedesigner. Ingenera l,

cttocontinuousfatiguestressingin pul-

eretheratioof minimumtoma i-

ss, asinvalvesprings,for e ample,

reservice.O ntheotherhand,a spring

plicationsof loadduringitslif eortoprac-

ormaltemperaturewouldbelight

nswherespaceisat apremium,

ressesaresuggestedby theS. A . . War

ingC ommittee(Manualon" Designand

andS pira lSpringsforOrdnance ). S ug-

gstressincompressionspringsof music


/ h t t p : / / w w w . h





S P R IN G

or.015-diameterto140,000for.15-

erablelowervaluesfortensionsprings.

essionsprings,hotwoundandheat

estedvaluesofstressvaryfrom 116,-

to80,000for1-inchdiameterbar. These

rHelica lS prings

Ma i m um S o l i d

ss,

0

00100,000

50,00080,000

00080.000

0

0,000

houtcurvaturecorrectionandapplytc

ofthesestressesassumesthat coldsetting

nsareusedto obtainma imumstrength.

houldnot beusedwherelonglifeor

uired.

fw or ingstressesusedinpractice ,

IV aresuggestedinapamphlet

on-RaymondDivisionofA ssociated

uesrefertoma imumw or ingstress

tress. Wherepossible it isfurthersug-

thespringbelimitedto % to2/3the

ress.

Wire CompanyintheirManualof

ggestforspringsofplain carbonsteels

mumTorsiona lDesignS tressesforHelica l

derA verageS erviceConditions

3inches

3inches


/ h t t p : / / w w w . h






ditionsvaluesofma imumdesign

.

herspringmaterials, sa fewor ing

estedby thiscompanyasfollows:

uare Inch)

0

esesuggestedstressesandthose

erageserviceconditionsdefinedasnon-

rmaltemperatures,andwithslowly

ndividualcases,wherefatigueor

nt,lowervaluesofstress willbere-

caseshighervaluesmaybepossible.

S

fsprings foragivenpurpose,

mputed,basedonthe stressesof

ce.TableX V Iappliestocarbon-

alitysuchasmusicor oil-temperedwire,

gs.TheallowableloadsPare based

hilethe deflectionsperturnywere

dsusingamodulusof rigidityof

rs uare inch. Thislatterva lueappliesto

fficientaccuracy.Thetotal deflection

ebee ualtothedef lectionyper

berof activeturns.F oradifferent

sG thedeflectionsgiveninthetable

1. 4X 108/ G. F orw or ingstressesother

loadsand deflectionsmaybeta en

IIIsim ilartabulationsaregivenfor

orbronzehelicalcompressionsprings.

steel springsisbasedon stressese ual

fTableX V Iw hileTableX V IIIf o r

basedon stressese ualto50per

eel.Theshearmodulususedinc om-


/ h t t p : / / w w w . h





erIurny forC arbon

are inch


/ h t t p : / / w w w . h





elica lS prings


/ h t t p : / / w w w . h





S P R IN G

nperturny forC arbon

are ineh


/ h t t p : / / w w w . h






elica lS pringsf


/ h t t p : / / w w w . h





S P R IN G

urn yforS tainless

are inch


/ h t t p : / / w w w . h





F H E L I CA L CO M PR E S S I O N S P R I N G

elica lS pringsf


/ h t t p : / / w w w . h





S P R IN G

erturn_v forS ta inless

are ineh


/ h t t p : / / w w w . h






elica lS pringsf


/ h t t p : / / w w w . h





S P R IN G

rturn yforPhosphor

are inch


/ h t t p : / / w w w . h






elicalS pringsf


/ h t t p : / / w w w . h





S P R IN G

nperturny forPhospho i

u a re i n ch


/ h t t p : / / w w w . h





F H E L I CA L CO M PR E S S I O N S P R I N G

elicalS pringsf


/ h t t p : / / w w w . h





/ h t t p : / / w w w . h





S P R IN G

sta inlessstee lw as10. 5X 10 w hilethat

V IIIforphosphorbronzesprings

ers uare inch. F orotherva luesof

sperturn giveninTableX V IIshould

lO' / G, those inTableX V IIIby

erageorlightserv icethese loadsanddeflec-

proportiontowor ingstress(see

useof thespringtables: A stee lcom-

edforamechanismto give160poundsat

spaceavailableis suchthatanout-

maybeused. If thespringissub ectto

X V Ifor.263-inchwirediameterand

allowableloadis161poundsandthe

rn.124-inch.Toobtain.8-inchdeflec-

24= 6. 45, say&k , activeturnsorabout

apterV IIIdiscusseseva luationofend

aspringofabout1. 1(8. 5)(.263)4-

ngth, a llowing10percente tra lengthfor

henthespring iscompressedataload

at aloadof160 poundswouldbe2.46

about 10percentless orabout2.2

olidlengthwouldbeabout10 percent

he tableisbasedor52,800pounds

tve ly low stress. A furtherdiscussionof

imumstresswhenthespringis com-

inChapterV III.

T

preparedwiththecurvature

dareshownonF igs. 78and79. F or

sarebasedona valueof100,000

w or ingstressandatorsiona lmodulus

ndspers uare inch. Itshouldbeempha-

essisusedmainlyfor convenience

recommendedwor ingstress.The

edbyW allaceBarnesCo.inTheMainspringforJune

rereproducedthroughthecourtesyof thiscompany.


/ h t t p : / / w w w . h






erange inloadbetweenoneand

79,between100and10,000pounds.

tesrepresentloadat100,000

orsionstress,theabscissas,inches de-

er activecoil.Thustheabscissa,

rofactiveturns willyieldtherecip-

ntinpoundsperinch.The setoflines

sto thehorizontalinthesecharts

s,whilethesetinclinedat about30

ecoildiameters.Theintersectionof

ato f theothersetf i esthe loadat

areinchstressandthe deflectionper

n. Thus,fore ample,ifthewire

tsidecoildiameter' /i-inch,theload

uareinchstresswill be9.3poundsand

load perturnwillbe .0025-inch.If

hespringconstantwillbe 1/.025= 40

t anyotherstresstdifferentfrom

are inchw illbe indirectratio tothe

alow ablestressof60, 000poundspers uare

e amplebecomes9.3(60,000/100,000)

stressarek now nthere uiredspring

m thechartsofF igs.78and79.Thus,

essof60, 000poundspers uare inchis

ngloadof30pounds, thew or ingload

uare inchw illbedirectratio tothe

00)= 50pounds.F romthechartit is

sizeswillyieldthis valueofload.

izeof .100- inchandanoutsidecoildiam-

loseto it.Inthis sizethespringwill

002-inchper poundofloadperactive

t30poundload,assuminga steel

10 poundspers uare inch. If , say,

uiredat30poundsloadthenumberof

uldbe.25/.06orslightlymore thanfour.

dif ferentf rom11. 5X 10 poundsper

nindeflectionmaybemadetota ethis

gthedeflectionby 11.5X10V G.


/ h t t p : / / w w w . h





/ h t t p : / / w w w . h





oil— Courtesy,WallaceBarnesCo.

bpers in. )

ge100to10, 000pounds)


/ h t t p : / / w w w . h





S P R IN G

78and79aseriesofdashedlinesat

orizontalareshown.Theserepresent

dastheratioofthe deflectionat

areinchstresstothe netsolidheightof

g.Itis clearthatspringswitha large

ealargedeflectioncomparedto the

Thus aspringof.100-inchwireand

f% - inch(asusedintheprev iouse -

ationratioofalmost 100percentat

are inch, F ig. 79. Thismeansthatthe

ndspers uareinchstresswill beabout

t. A t 60, 000poundspers uare inchthe

bout60per cent.

erewillalwaysbeasmall in-

esigncharts.This errorshouldnot,

ent andwillusuallybewithin2 percent.

dbenotedthatbecauseofmanufac-

nsinwiresize,and incoildiameter,

entestandcalculatedresultswill

rcent,unlessspecialprecautionsin

ta en.F orafurtherdiscussionof this,

esevariables.Thismeansthat thecharts

dbesufficientlyaccurateformostprac-

r,incaseswherema imumaccuracyis

emadeusingE q uations7and18,or

rX V IIImaybeused.


/ h t t p : / / w w w . h





G CO N S I D R A T IO N S — H E L I CA L

S P R I N G

siderations, otherthanwor ing

tindesigninghelical compression

ssedin thischapter.Theseinclude

cesforend coils,effectsofeccentric-

iationin springdimensions,variation

sat solidcompression.Theeffect

atera lloadingtogetherw ithbuc ling

in thefollowingchapter.

T O E N D T UR N S

mployedinhelicalcompression

.80.Themostcommontype— endsset

dicated inF ig.80a—has theadvan-

entricityofloading(andhencea lower

anwouldbethecase wheretheends

ig. 80fo , cord. InF ig. 806theends

closed, w hile inF ig. 80c, theendsare

ng.Thistype ofspringwouldgive

ntricityofloading.The springof

thatatce ceptthattheendshavebeen

2turnateachendisf la t. InF ig. 80e, a

ssetupisshow n.

onofdeflectioninhelical compres-

hattheef fecto f theendturnsbeesti-

curacy.S omee perimentalandan-

1indicatesthatfortheusua ldesignofend

andground, F ig. 80a, thenumberofactive

mberofcomplete ly f reeco ilsplus% .

thiscaseis determinedbythe

pcontactpoints.)Thus ifacom-

iv e C oi l s in H e l i c al S p r in g s , T r an s ac t io n s A . . M . . , J u n e, 1 9 34 ,


/ h t t p : / / w w w . h





S P R IN G

coilsand12total coils(tiptotip of

numberofactivecoils wouldbe10J/2,

einactiveateachend.H owever,when

is someprogressiveseatingofthe

berofcompletelyfreecoilsdecreases

reasesthenumberofinactiveturns.

rison2madeaseriesof carefultests

ngaspecial setuptodeterminethe

softhesetests indicatethatatzero

eturnsw ase ua lton - f -% w heren

yfreec oilsatzeroload.A stheload

umberofinactivecoilswasfoundto

the endcoils,theamountofincrease

rnatusua lw or ingloads, w ithan

inceforcalculationpurposesthe

turnsinthe rangefromnoloadto

ary interest(thisw ouldbeusedinthe

earsreasonabletosubtractabouthalf

umber ofactiveturns.Thisgivesa

urnsvary ingf romn ton + V t. B e-

f turnsisn + 2fortheusua ltypeof

eansthatthe inactiveturnsfoundin

out1% to2withanaverageof1.85.

. C . K eysor3indicatesthatthetotal

thespringisappro imatelye ua lto

so lidturns basedonthecommonly

thenumberof " so lidturns e ua ltothe

orwirediameter.S incefortheusual

0a , the" so lidturns aree ua ltothe

edf romtiptotipofbar, m inusV - iturn,

e uivalenttoadeductionof1.7 turn

inactiveturnswasgiven by

eresearchof theS pecialR esearchC om-

pringsofA . . M. . Theaveragevalue

as1.15asadeductionfrom " solidturns

rstra inonC lose lyCo iledHelica lSpringsandtheV ariation

eC oilsw ithL oad , V irginiaPo ly technicInstitute , Engineering

onB ulletinNo. 24.

lasticC urveofaHelica lC ompressionSpring —H . C . K eysor,

.,May,1940,Page319.

Desig , December, 1939, Page53.


/ h t t p : / / w w w . h





I D R A T IO N S — H E L I CA L S P R IN G 1 59

mtotalturns.

eseinvestigationsasabasis,it ap-

signofendcoil thenumberofin-

about1.65 to2consideredasa deduc-

blya meanvalueofP/iinactivecoils

asany touse inpractice . F orthe

resomewhathighermaybe j ustified,

usedforlowerloads.Theseating

easesalsotendsto produceaslight

ctiondiagram.F orfurtherdetails,

einvestigationbyPlettaandhis

asbeenconcernedonlywith the

stresultsconcerningtheothertypes

N D G R O UN D O R F O R C D (b )S Q U A R E D O R C O S E D E N D

T P ) N O TGR O U N D

( d) P A I N E N D GR O U N D

E T U P

urnsasused inhelicalcompressionsprings

. 80are lac ing, butappro imateva lues

low s: F orpla inends, F ig. 80c, active

totalturns forpla inendsground,

ren— 1. / / 2% turnsateachendareset

80etheactiveturnsmaybeta en

F L O A D IN G

usualdesigniscompressedbe-

sin atestingmachine( ig.23),it

raltheresultantloadisdisplaced from

allamounteasindicatedinthisf igure .


/ h t t p : / / w w w . h





S P R IN G

loadingistoincreasethe stresson

meteranddecreaseitonthe other

ple, by the load-stressdiagramofF ig. 49

son onesideofthespring thanon

of thiseccentricityofloadingbased

salsobeencarried outbyK eysor

tyoftheanalysis,it willnotbegiven

resultsofthe analysisaregiveninthe

natesrepresentingratioe/r between

usrandthe abscissasbeingthenum-

tipcontactpoints.The totalnumberof

gnw illbee ua lton + 2. It isseen

uctuatesbetweenzeroand ma i-

esoccuringappro imatelyatn =

Theoretically itshouldbepossibletoget

eccentricity)bychoosingn toconform

ver,becauseofvariationsinactual

ecauseof errorsintheassumptions

nnotingenera lberea lizedinpractice .

refore,theenvelopeofthec urveasin-

beemployed.

oe/rthefo llow inge pressionsgiven

d

- ( 1 29 )

so lidco ils. Thisw illbeappro imately

nthenumberofco ilsn betw eentipcontact

1. 5. B yusingthesee uationstheratio

sanappro imationitmaybeassumed

e isfa irly largethestressw illbe in-

e/rascomparedwiththestress for

adebythe writerwhichgivea

ula.Thesewerecarriedout incon-

perimentsmadebyPlettaandMaher— " H eli Warping

prings , TransactionsA . . M. . , May1940, Page327.


/ h t t p : / / w w w . h






whereit wasdesiredtoobtainas

loadonahelicalcompressionspring.

all helicalspringsusingaspecial

resoarrangedthatthe eccentricityof

ed.E ssentiallythisconsistedofaflat

achingdeadweights120 degrees

hene ualloadsw ereappliedate ua l

T UR N S B T W E E N T I P CO N T A CT PO I N T

tw eeneccentricityandco ilradiusasafuntionofn

ysor

ndthattheloadingplanesat each

tparallel.Theloadswerethenad-

smof these loadingplanes f romthe

edloadstheeccentricityofloadingc ould

aresummarizedinTableX IX ,

r,wirediameter,numberofturnsn ,

pringstestedhadgroundendcoils of

columnthevaluesofthe ratioe/r

coilradiusas calculatedfromE q ua-

en.F orcomparisonthetestvaluesof

arioussprings arealsogiveninthe

.

cases theagreementbetween

s sufficientlygoodforpracticaluse,

edthatthetestspringswerehand made

sta eninformingtheendturns.


/ h t t p : / / w w w . h





S P R IN G

entricityof

prings


/ h t t p : / / w w w . h






ctioncharacteristicmaybebrought

.Ifsprings aretobeheld torelatively

oallowsomeleewayonthecoil diam-

rofturns,sinceotherwisethe cost

s—Inwindingspringscold,there

bac " . Inotherw ords, the insidediam-

inC ommercia lSpringWireS izes

-227-39T\ i

wire. .A -229-39T> <

31-39T I(

gwire....A -230-39T^ i

pringwireA -232-39T/I

ndingwillbeslightlygreaterthan the

sa conse uenceoftheelasticand

terial.A lthoughthiseffectmaybe

aslightlysmaller mandrel,fordiffer-

pectedthatsomevariationsincoil

ualvaria tionsinco ildiameterstobe

tolerancesgivenby onespringmanu-

ableX X I. Itmaybeseenthatthese

thespringinde D/dandonthe

mall variationsincoildiameter

estimatedq uantitativelyasfollows:

mulaforhelicalsprings( q uation7)

he nominalmeancoilradius

tively.S upposingthatthetruemean

ngineeringpublishedA mericanS tee landWireC o., Page97.


/ h t t p : / / w w w . h





S P R IN G

t er a r e r = r ( l + « ) a n d d = d ( l- f A )

uantities,relativetounity,thetruede-

medthateandA aresmallre la tiveto

igherpow ersmaybeneglected. Hence

ttenwithsufficientaccuracy(since

/( l + A ) , sr l — 4 A ) :

0)

ectionS 1ismerelythe nominal

term l+ 3e—4awhich dependsone

w thattheactua lmeanco ildiameteror

terthanthenominal,i.e.,e= .01,

truewirediameterisoneper cent

= —. 01. Puttingtheseva luesofc

13 0 ,

c onditionswithaonepercent

andwirediameterfrom thenominal

nwillbe 1.07timesthenominalde-

ter.

acticale ample , anactua lcasee amined

ssed.Thisspringwasmadeof

= .5625inch.A ftercuttingupthe

imensions,itwasfound thatthe

s.551-inchwhichwouldcorrespond

of(.5625— .551)/.5625= 2.04per

ssumingthetruemeanco ildiameterof this

enominal, i. e ., thatc= 0, thenf rom

edef lection8tw ouldbe1—4A =1. 08times

ctualcoil diameter,however,had

t lessthanthenominal,whichmeant

hisvalue inE q uation130thetruede-

esthenominaldeflection.Thus,the


/ h t t p : / / w w w . h






enmadeslightly smallerthanthe

gma ertocompensateforthede-

reused.

beshownthatoneper centvari-

rmeansappro imatelya3percent

aonepercent changeinthecoildiameter,

tress.Usually,thestressdoes nothave

ilDiameters*

icalS prings)

ationsinDiameter

1 2

5 ± 00 5

0105

0130

18

8

S tee l& WireC o.

itsas thedeflection however,acon-

ommercialvariationsin wiresize

for certainhighlystressedsprings.

O DU U O F R I GI DI T

ofthe deflectionofactualsprings

eef fectiveturnsbek now n, buta lso

of thespringmaterialbek nown

ndicatedinthe discussionofChapter

reportedintheliteraturevaryconsider-

ctof adecarburizedlayeronlya

emodulusbyseveralper cent.

ultsgiveninTablesV andV I agood

sofrigidityfor carbonandalloysteel

ers uare inch. How ever, forhot-wound,


/ h t t p : / / w w w . h





S P R IN G

-rolledmaterialinthelarger sizes,

mmendamodulusfigureof10.5X 10

. On thebasiso f thetestdatagivenin

odulusofR igidity

1.5X 10

0. 5X 10a

averagef iguresgiveninTableX X II

ring materials.Itshouldbenoted,

ofseveralpercentmayoccur.

O L I DCO M PR E S S I O N

ionspringsitisdesirableto choose

enthespringis compressedsolid,no

twilloccur.Thereasonfor thisis

espring mayattimesbecompressed

condit ionsitta esaset, the loadat

bechanged.Thusthe springwillno

cteristics.

mpressionspringisinitiallywound

ygreatsothattheelastic limitofthe

henthespringiscompressedsolid,the

a diameterofthecrosssectionis

pringof large inde 7. A tlow loadsbe-

chedthedistributionisappro imate-

. 82a. A f terthe loadisre leased, the

w illbe li e thatinF ig. 82c.

de theseresidua lstressesmaybe

lyfromtheconditionthatthemoment

ythe triangleobcaboutpointo must

ofthe stressesrepresentedbythearea

sagainappliedthe resultantstress

hodsof calculationofloadsforcompleteyielding,see


/ h t t p : / / w w w . h






g. 82d. It isclearthatthema imum

nreducedbytheoverstressing,since

tesignare inducedandthesesub-

uetothew or ingload. H ow ever, in

or overstressing,thefreelengthhas

initial freelengthofthespring ismade

reelengthby theproperamount,

heldtothespecifiedvalue.A tthe

s overstressingprocess,ahighercal-

pressionmaybe permissible.

dacertainlimit,there willbe

this process.Inotherwords,be-

ength,thefinallengthafterthe set-

same.Thereasonforthisis thatthe

oflattenout ( ig.61)sothat ahigher

fstressesovercross

f large inde (a),

alloadbeforecold-

aboveelasticlim it (c)

etlingwithloadre-

mal loadaftersetting


/ h t t p : / / w w w . h





S P R IN G

essesatSo lidC ompressionfor

cluded.

preciablygreatertorsionmoment.If

co ldw or andlossofductil itymayoccur.

f fectwhichoccursw henthistypeof

w hatisk now nas" recovery . Thusim-

eoperationonacompressionspring,

ngwillbea certainvalue onstanding

thesettagestress istoohigh,thefree

.Thisagainwill changetheload-

ofthespringand isob ectionablein

ample,instrumentsprings).


/ h t t p : / / w w w . h





R A L A N D A X I A L L O A D IN G B UC L I N G

CO M PR E S S I O N S P R I N G

madetoolongrelativetoits

at atacertainload,a suddenside-

This phenomenonisessentiallysimi-

ongslenderco lumnwhenthe loade -

hedesignofhelicalcompression

guardagainstthislateralbuc lingby

tionsinsuch awaythatthecritical

ysbe greaterthananyloadencoun-

otdone,somesort oflateralsup-

eforaguide)must beprovided.

ngloadforhelicalsprings maybe

e samemannerasthatusedin

er,theanalysisinthecaseof thespring

rdinarycolumntheoryinthatit is

ecreasein lengthunderload.In

column,ontheother hand,thisde-

eneglected.Thereasonforthislies in

tyof moststructuralmaterials,which

engthfromnoload tofullloadis

nt.Thisisnot true,however,fora

a verylowmodulusofelasticity.Be-

underload,itis alsonecessaryto

softhespringdue tolateralshearing

eassumedthatthespringis close

glemaybeconsideredas small.

sionofco lumntheorysecTheoryofE lasticS tabil ityby

ill, 1936.A discussionofbuc lingofhelicalspringsis

it- ona lre ferencesonthebuc lingofspringsseearticles

i ( . V r r . d . In . . V . 5 4 , P ae e 1 38 , 1 91 0 b y R . G r a mm e l. Z e i t A n e ru .

e384, 1924 andbyB iezenoandK och, Z ettA ngew Math. Mcch.


/ h t t p : / / w w w . h





S P R IN G

ngthofspring / —lengthofspringaf ter

berofactiveco ils r= meanco ilradius x ,

ompressive, f le ura l, andshearingrigidit iesof

dcondition-anda,fi," arethesame

ssionofthespring.

eshownfromcolumntheorythat

nsina columnwithhingedendsare

dis

earingrigidityof theco lumnandPristhe

I/ t2, EIbe ingthef le ura lrigidityof the

lloadis thatfiguredbyconsidering

andneglectingshearingdeformations.

stoaspring withshearrigidity7

e criticalloadbecomes

gthecompressive, f le ura land

inverselyproportionaltothenum-

h( q uations135, 141, and144). Hence

7 . .- , - ' 1 3 3 )

on133inE q uation132,

pterII,compressiverigiditybecomes

yisme(inttheratio ofloadtodeflectionper unitoflength

direct compression.F orabarof cross-sectionalareaA

thecompressiverigidity ise ua ltoA E . L i ew isethe

tioofbendingmomenttocurvaturefor abeaminpure

modulusofelasticitytimes momentofinertiaofthecross-

tyise ualtotheratio ofshearingforcetoshearing

hand forabeamise ualtomodulusofrigiditytimes cross-

pliedbyaconstantdependingon theshapeofthesection.

age140.


/ h t t p : / / w w w . h





D I N G— C O MP R E S S I O N S P R I N G 1 71

w henthespringiscompressedtothecrit ical

uationholds:

E q uation135,

rgivenbyE q uations134and137, thefo l-

nthise uationmaybereducedto

0 ( 1 3 8)

C a lculationof thef le ura lrigidity / o f

ishedbydeterminingthe angular

spring undertheactionofa moment

he coilasindicatedin F ig.83.This

ingaq uarterco ilsub ecttoamoment

ig.84.Themomentishere repre-

crosssectionatanangle< f> thebend-

os< f> whilethetwistingmomentM,

sideringa lengthds= rd< f> , thetota lcom-

outthea isofthemomentwillbe


/ h t t p : / / w w w . h





S P R IN G

rethef le ura landtorsionalrigiditiesof

ectively.Thetwistdue tothebending

pliedbycos < f> toobtainthecomponent

hemoment thatduetoM< mustbemulti-

ectedtotransversemoment

momentof inertia inbendingof thesection

sionasIp. S ubstitutingMb= Mcos< £ ,

d, t= rd< f> inE q uation139gives

< t > + s m < p d < t >

oracompleteturnwill befour

betw een< f> = 0and4> =ir/ 2. Thus:

henumberof turnsperincha ia l

meansthatthe angulardeflectionin

llben6/ l . Thisw illbea lsoe ua ltothe

ng1 = 2Iforacircularcrosssectionof

lrigidity / „ , w hichistheratioofbend-

sseento be


/ h t t p : / / w w w . h





D I N G— C O MP R E S S I O N S P R I N G

calculate theshearingrigidity,the

g(or coil)underashearforce Q is

nsideringthedeformationoftheq uar-

b,thebendingmomentatan angle$

lundermomenttrans-

thisdividedbyEIandmultipliedbydsw ill

d6in alengthds.H ence

isy— ywillbethis anglemulti-

ngturnundershearforce


/ h t t p : / / w w w . h





S P R IN G

gives, usingEq uation142,

, integratingbetween0andr/2,and

shearingdeflectionyfora complete

w O r

turnsperincha ia llength, thetota l

cha ia llengthw illbe

gidity,or ratioofshearingforceto de-

comes

onsinEq uations135,141, and144fora„

ti o n 13 8 a nd t a i n g G= E / 2 (1 + " ) w h er e » = P oi s -

0(145)

uationhasonerealposit iveroo l

calvalueofz atwhichbuc lingoccurs.

n,thecorrespondingcriticalloadis,using

( 14 7 )

uation145,maybee pressed:


/ h t t p : / / w w w . h






stantofspringor loadperinchdeflection,

ndingontheratio l / rbetweenf ree

r= crit ica lorbuc lingload.

dendsaswasassumedinthe deriva-

nby the low ercurvea inF ig. 86. A

opivotsasindicatedin F ig.87bmight

— I 1— I 1 — I — I — I— I — I I

L E N G TH

A D IU

dngbuc lingloadfactorC i> . C urvea

ds curvebforhingedends

atelyasaspringwithhinged endspro-

ssmallcomparedtothe freelength.

s—W heretheendsof thespringmaybe

milaranalysismaybe carriedout.The

thesameformasE q uation148e cept

actorC a isnow tobeta enf romtheupper

seof built-inendsissimulatedwhena

dbetweenparallelplatesas indicated

eof incompletef i ityo f theendsandof

ngloadmaybelowerthancalculated .

cularw irethespringconstantC is

noandKoch, loc. cit . fo ra furtherdiscussionof thisproblem.

investigatorsshowgoodagreementwiththeanalysis

ofcoils isnottoosmall andthatthecoilsdo nottouch

lsocommentsinMachineDesign,July1943,Page 144.


/ h t t p : / / w w w . h





S P R IN G

uation7, E q uation148maybow rittenas

consideredastheratioof thecritical

lingoccurs)tothe freelength.Thus,

aybee pectedatadef lectione ua lto . 4I.

testsshow agreementw ithEq uation

meinaccuracymaybe e pecteddue

nsionsandthe effectofendturns.

e a m pl e o f th e u se o f t he b u c l i ng l o ad

buc lingload,asteelhelical

efollowingdimensions:F reelength

diusr= .75-inch,outsidediameter==

hf i edandhingedends

rd= .25-inch,activeturnsn— 12.

79(C hapterV II)fo rthesedimensions

142poundsperinch. F romF ig. 86the

unde ualto . 64forZ „ / V = 6/ . 75=8. It

ring iscompressedbetweenparallel

arecompletelyrestrainedfromrota-

ig. 86forf i edendsmaybeused. Then


/ h t t p : / / w w w . h






eca lculatedbuc lingloadPcrisC B Z cC =

nds. A ssumingama imumw or ing

ers uare inchtheactua lloadonthe

echarto fF ig. 79, C hapterV II)P=

ercent ofthevaluefor100,000pounds

heseconditionsthereis aconsider-

w or ingloadandthebuc lingload.

spring werehingedasindicatedin

raintduetorotationoccurs,thenusing

onstantC M= . 2. InthiscasePcr~

170pounds. Hence, w iththistypeof

bedangerof suchaspringbuc ling

of190 poundswasreached.

L A N D L A T R A L L O A D IN G

ngsaresometimescalledon towith-

s,butalsotransverseloadsas indicated

e Q representsthetransverseload.

icationsarecertaintypes ofrailway

ngunder

ia lload


/ h t t p : / / w w w . h





S P R IN G

calspringsmusttransmitlateralloads

ds.Incertainrefrigeratormechanisms

pportedonhelical springs,these

bsorblateralforcesduetothe un-

echanismaswellasa ialloadsdueto

ral deflectionsofaspringunder

nedeffectofthe a ialloadPand the

nsidered.Ingeneralthe largerthe

ebuc lingloadthe largertheef fecto f

n.

dbotha iallyandtransverselyas

beconsideredas acolumnundercom-

seloads5.Itis alsoessentiallythesame

r combineda ialandtransverseload-

I. Theprocedure inca lculatingsucha

tthelateraldeflectionof thespringis

ew erenoa ia lload. A ssumingthe

swayis 8 ,thecriticalload Prrisfound

rthecaseofbuilt- inendsusingcurveb

ia lloadactingonthespring, theratio

il lbeshow nlaterinC hapterX V Ithe

tionduetothe a ialloadwillbe given

nofP / P, , aregiveninF ig. 156of

actuallateraldeflectionwill be

8 whichwouldoccurifno a ial

sof beamtheorymaybeused.The

oadedbya latera lloadQ ( ig. 147

onsideredastwo cantileversoflength/,2.

to bendingof

lasticS tability,Page4.

methodofdeterminingthisfactorisgiveninthereferenceof

thattheappro imatee pressionissuf ficientlyaccuratefor


/ h t t p : / / w w w . h





L A N D L A T R A L L O A D IN G

ura lrigidityof thecantilever.

lyloadedhelicalspringofF ig.

/ 3givenbyEq uation141isused, ta ing

ssedlengthIunderthe loadP.Thus

deflectionduetodirect shear

videdbytheshearingrigidity7 and

. Tof ind7E q uation144isusedta ing

shearingdeflectionbecomes

sthesumof 8,and8 .Thus,usingE q ua-

15 4)

ngs, E = 30X 10 poundspers uare

poundspers uare inch. Usingthese

4andsimplif ying, thee pressionforthe

ia lloadbecomes:

1 . 0 6r = ) ( 1 5 5)

sedin E q uation151tocalculatethe

auseofthis lateraldeflectionthere

stress.A naccuratecalculationofthis

sandwould beverycomplicted.A s

smay becomputedasfollows:The

a ialloadPwill bePr.Theeffective

byanamount8/2dueto theeccentric

sionmomentdueto Pbecomes

dit ionthe latera lforceQ producesa


/ h t t p : / / w w w . h





S P R IN G

sresultsina totaltorsionmomentof

themomentMtwillbeobtained

tion(4c—l)/ (4c—4)fortheef fecto f

pringinde , asindicatedinEq uation

6

l \

eto thedirectshearloade ualto

mthe secondterminthebrac etsof

ectshearstressatthe insideof theco ildue

lbezero . Hencethema imumshear

onsideredas veryrough.S ince

uchgreaterthant , itmaybee pected

adingistoincreasethe stressinthe

9

ceP ismuchlargerthanQ, andfor

esareofprimaryimportance.Ina

ressesmaybecalculated.

ample: A stee lspringhasthefo llow -

diameter— 5inches,meancoilradius

meterd= ? 4- inch, f ree lengthl0=9 A

ringof% - inchw ireand5inches


/ h t t p : / / w w w . h





L A N D L A T R A L L O A D IN G 181

of 1100poundsthedeflectionper

chor1.5inches for8activecoils.

= 1100/1.5=732poundsper inch.

3= -4.48andfromF ig.86thebuc ling

gtheseva luesinE q uation148the

(9 . 5) 7 32 = 4 8 60 l b

a la ia lloadPonthisspringis2400

00/4860= . 494. F romEq uation150, the

actorC,is2.Tocalculate thedeflec-

5isused. Theva lueof Iusedinthise ua-

minusthedef lectionduetoa loadof

i l lbe2400/ C = 2400/ 732= 3. 28

3. 28=6. 22inches. A ssuminga latera l

bysubstitutioninEq uation155,

04 ( 6- 2 2) 2 + 1 -0 6 (2 ) , -r " 2 7 4

thedeflectionata latera lloadof200pounds

. 5 48 i n ch e s

casetheactualdeflectionwitha ial

calculatedbyneglecting-theeffectof

ssatana ia lloadof3050poundsis

areinchwithcurvaturecorrectioncon-

sa ia lloadthestressw ouldbe

8, 500poundspers uare inch. F rom

orC fordeterminingthe increase instress

2 5 f or « = . 5 4 8 ,Q = 2 0 0 lb

ein stress.to1.25(78,500)=98,000

maybee pectedduetothe latera lload.


/ h t t p : / / w w w . h





S P R IN G

eatmanydifferentspringswas

C haplin, andSheppard7. Thetests

asteelplate ortableonfour springs.

a ialloadwhilethetransverseload

le.Inthismanneressentiallythe

.88wereobtained.R esultsofthese

lescatterbetweentestvaluesof lateral

atedbyusingE q uations151and

nin thetestpointswasfrom about

tof thetheoreticalvalues,mostofthe

gwithin20percent ofthecalculated

tbyLehrandGrossinGermany

d20percentor morebetweencalcu-

Byta ingspecialprecautionsin

springandbyaccuratelydetermining

mberofturns,these investigators

goodagreementbetweentheoryand

loadedcompressionsprings.F orthis

chiefcausesofthediscrepancybe-

n (1)imperfectclampingoftheend

ghtrotationunderloading whileno

theory,and(2)inaccuracyinde-

mberofturnsandlength ofthespring.

ompressionspringswiththe

dturnsthedesignermay,therefore,

nsasmuchas20percentandevenmore

oflateraldeflectionas obtainedfrom

calSpringsunderTransverseLoading byB urdic , C haplin

cionsA . . M. . , October, 1939, Page623.

edbyV . D. I. , Berlin, 1938, Page100.


/ h t t p : / / w w w . h





IN G F O R

E F F I CI N C

uentlyarisesinhe licalspringdesign

gwithgivenloadand deflectionchar-

ablebeinglimited.S inceloadtimes

oenergy,thismeansthat acertain

storedwithinthe givenspace.A con-

bleenergystoragevarieswith spring

erest.

G

sproblemis tocalculatetheenergy

ringwhen thecoilsj usttouchand

ssatsolid compressionbeingassumed

wablevalue.Theamountofenergy

calculatedforvariousspring inde es.

ace occupiedbythespringthe amount

oredw illbeama imumatadef inite

ow ever, thisoptimumvaluewillde-

ghas variableorstaticloading1.

htV o lume—F orstatica lly loadedcom-

springsare compressednearlysolid

e occupiedbythespringwill ap-

htvolume,i.e.,thevolumeoccupiedby

ualtotheoutsidespring diameterand

e heightofthespring(this neglects

endturns).H ence,forapplications

springs,theuseofsolid-heightvol-

s acriterionofefficiencyof space

and,wherethe springisundervari-

ma imumstressrange,thespace

hissimilar lothatusedby J.Jennings( ngineering,

34).H owever,themethodusedbytheauthordiffersfrom

atadistinctionis madebetweenstaticandvariable

eflectionformulais usedinsteadoftheW oodformulaused

eris q uiteaccurate(seeChapterIV ).


/ h t t p : / / w w w . h





S P R IN G

nunloadedmaybe muchlarger

me.Inthiscase thefree-heightvol-

edbythe activepartofthespring

samorerepresentativecriterionthan

owever,itshouldbementionedthat

on offree-heightvolumeiscompli-

lueofallowablestressat solidcom-

.Ifa lowstressisused,the difference

edbyusingthe free-heightandthose

eightvolumewouldbemuchless than

igh stress.

nswherespringsare sub ect

finitial compression,acriterionof

atebetweenthefreeandsolid-height

mostrepresentative.Thisvolume

mountofinitialcompressionandon

dheight.I naddition,forbestac-

dbe considered.Toavoidallthese

criterionofsolid-heightvolumewill

marilyas aconvenientguidefor

fspaceutilization.

Toapply thiscriterion, thepotentia l

duetoaload Pandadeflection8

storedenergyis

ingisgivenbyE q uation7.S ub-

q uation160thestoredenergybe-

eusualmeanings.

the loadactingonthespringisvari-

anautomotivevalvespring,the

hespring asfiguredusingthecurvature

uation19, may, asaf irstappro imation,

sub ectbyE. L atshaw , MachineDesign, March, 1942.

nanumberofcurvesbasedon free-heightvolumeanda

ers uareincharegiven.


/ h t t p : / / w w w . h





- F F I CI N C

heloadcarryingability .Thisstress

venbyE q uation19.

PandsubstitutinginE q uation161

denergybecomes

rion ofsolid-heightvolume

energymustbestoredin avolume

derw ithadiametere ua ltotheoutside

nda lengthe ua ltotheactive length

atterisfully compressed.This

angle,whichissmallforpractical

olumeisthus

- = — — ( c+ 1 )

.

on164in163ane pressionfortota l

pendingonthespringinde candis

t,foragivenvolumeof spaceoc-

stress,theenergystoreddependsonly

twhichin turndependsonlyonthe

morecompletediscussionof this.


/ h t t p : / / w w w . h





S P R IN G

fC , areplottedagainstspringinde c

.89.Thiscurveshowsthat forvariable

ngthema imumenergy isstoredin

tagivenpea stressif thespringinde is

C ~ J

of f icients forvariable loads

er,itshouldbenotedthatif thefree-

enasabasis thisoptimumvalueof

eensomew hatless . Thus, if astressof

areinchisassumed,theoptimuminde

hefree-heightvolumeisusedas the

aceutilization.

orage—Wherethe loadisstaticorre-

ringthe servicelifeofthespring as

. L atshaw , loc. cit .


/ h t t p : / / w w w . h





- F F I CI N C

indicationsarethatcurvatureeffects(but

ear)maybeneglectedincalculating

stresst0 shouldbecalculatedby

\ . ,

q uation90andta esintoaccountthe

ct shearload.

eadofE q uation162andproceeding

,theenergystoredbecomes

C = * r

f f icientsforstaticloads


/ h t t p : / / w w w . h





S P R IN G

thespringinde c,thelower

ined. Thiscurve indicatesama imum

alestpracticalspringinde . Thismeans

nvolved,ma imumenergystorage

beobtainedinagiven spacebyusing

eofthe inde (whichwillusuallybe

creasingtheamountofenergy

ivenspaceisto useaspringnest,

rmore springstelescopedonewithin

nestof

ig. 91. A practica le ampleof the

wninF ig. 92w hichrepresentsanend

for alocomotivetendertruc .


/ h t t p : / / w w w . h





- F F I CI N C

ystoragetheso lidlengthsofa llthe

tshouldbe thesame.A ssuminga

ngs,thismeansthat

sefollowingthesubscripts1 and2refer

ngsofthenest,respectively.In addi-

atthefreelengthsof thespringscom-

ocomotiveWor s)

nestforlocomotivetender

esame.Thismeansthatthe total

thesameatanygivenload,i.e.,that

orvariable loadingthedef lectionis

ions7and18.


/ h t t p : / / w w w . h





L S P R I N G

r t ni ( 1 71 )

e esclandc, thesee uationsmay

-- * * * * . ( 17 2)

stressineachspringisassumed,

d1= n2dif romE q uation169thismeans

c,andc-. (andhencealsothecurvature

K . , )shouldbethesameifS l= S „ . If

adethesame,theenergy coefficients

heinde es)willbethe samefor

uation165thismeansthatthe total

by

thevo lumesenclosedby theouterand

whencompressedsolid.S ncc

ationmaybew ritten

)

uch,theouterdiameterof the

a ltothe innerdiameterof theouter

2+ d2. UsingEq uation164thisgives

= — n , 2 22 , J (c , - 1) 2

d n1 < 2 1 = n d 2 , fr o m th e se e u a ti o ns i s o b-

uation173, foratwo-springnest,


/ h t t p : / / w w w . h





- F F I CI N C

osedbyouterspring.

e f ficientC , areplottedagainst

9. F romthisit isseenthatforatw o-spring

g,thema imumenergystorageisob-

saround5to 7.Thesevaluesaresome-

ainedfora singlespring however,

wouldbethecaseif thefree-height

asabasis. Thusfore ampletheana lysis

previously2indicatesanoptimumvalue

nneste ua ltoabout4, basedonanal-

poundspers uare inch. F ora low er

ofspring inde wouldbehigher.

donsolid-heightvolumemaybe

st.Thisgives,for energystored,

nbyEq uation166.

aga instcinF ig. 89show thatfora

eassumptionsofe ualdeflectionand

ma imumenergy isstoredinagiven

shav inginde esaround6to8are

swillbe lowerifthefree-heightvolume

oraspringnestsub ecttostaticloading

n e actlythesamewayasbefore,

isf iguredf romEq uation89w hich

ntdueto curvature,andinsteadofC,

on168)isused. R esultso f thisana ly sisare:


/ h t t p : / / w w w . h





S P R IN G

aticallyloaded,thestored energy

1 (1 80 )

taticallyloaded,storedenergyis

( : ) i ( >

" areplottedaga instspringinde cin

tappearsthatforstaticloadsthema imum

pate willbehadbyusing springswith

ratw o-springnestandw ithinde esof4to

Theseoptimumvaluesaresomewhat

forspringsunder variableloading,

olid-heightvolume.However,if the

umeisusedfor springsundervariable

weentheoptimumvaluesofinde for

willbe small.Inanycase,theresults

imumenergystoragewithina

onsideration,aratherlowvalueof

yaround3 forasinglespring andabout

ree-springnests.

A ssumingthedesignerre uiresa

anddeflectionforagivenapplication,

edis fi ed.Byusingtheformulasof

amountofspacere uiredforthespring

ea stress. A ctua lly , o therpractica l

elargerspacere uirementsthanthose

sshouldgivea roughindicationof


/ h t t p : / / w w w . h





IN G

dcombinationtension-com-

mthatof compressionspringsinthat

inreducingallowablestressshould be

S I O N S P R IN G

eninmanufactureafairlysharp

aratthepo intw herethehoo j o ins

, F ig. 93b, mayoccur. Thiscurvature

ss concentrationwhichisnotcon-

dof stresscalculationforhelical

ahalfendturnis bentupsharplyso

elysmall,whichtendstoresult ina

s.F orthisreason,mostfailuresof

ch pointsandthisis onereasonwhy

ngstressis usuallyrecommendedfor

edwithcompressionsprings.This

ctmaybereduced,andthestrength

endturn sothattheminimumradius

possible.

stbeconsideredin tensionsprings

on.Bycertainmethodsofcoiling

bringthecoils togetherinsucha way

eappliedbeforethecoils willbeginto

hisinitialload islimitedtoa value

round6000 to25,000poundsper

neglectingcurvatureef fects), thee act

pringinde . A f terseparationof the

heload-deflectiondiagramisthesame

tainedforaspringwithno initial

smstheinitialtensionis important.

s A lthoughane actca lculationof

elicalspringsis complicated,arough


/ h t t p : / / w w w . h





S P R IN G

nin F ig.93(whichhasa sharpbend

maybemadeasfo llow s: Thebendingmoment

harpbendbegins)duetothe loadP isP rap-

albendingstressatthispoint willbe

hesectionmodulusof acircularsec-

gwithhalf-loopcoilend

s greatasthenominaltorsionstress

momentPr).Thema imumbend-

emultipliedbyafactorK , w hereK,

actordependingontheratio2r../d,the

atthestartof thebendinthe plane

dinF ig. 93. A nestimateofK , maybe

fF ig. 180, ChapterX V II, f o rtorsion

undwire inbending.Tothisbend-

dedthedirecttensionstress4P /V d- .

bendingstressatpointA ' e ualto

93bnearthepo intw herethebendj o ins

pringthestressconditionis prin-

maybemadeasfo llow s: F romE q ua-

caseofpuretorsion actingona

matee pressionforstressconcentration


/ h t t p : / / w w w . h





IN G

)w here inthiscasec, istobeta en

gtheradiusofcurvatureof thebend,

umstressduetothetorsionmomentPr

rstresspresentat pointA dueto

tshear,however,doesnot actatthe

thetorsionstressgivenbyEq uation

uently , E q uation184maybeta enas

essionforthema imumstressatpo intA .

small,andtheq uantityintheparen-

gwithfullloopturnedup.

eappro imatelye ua ltothe

ng, E = B / 3appro imately ,

n = numberof turnsindimension/

psbegin. Wor ingturns

maybelarge.Thismeansa highstress

showstheadvisabilityofk eepingr,

ndA therewillbeacombinationof

i ma te ly


/ h t t p : / / w w w . h





S P R IN G

eswhichdependonthe shapeofthe

diir, andr2. Sincethepea sof these

ssesdonotoccur atthesamepoint,

presentsconsiderablecomplication

here . F orpractica lpurposesE q ua-

bablysufficient.

ypeoftension springwithafull

inF ig.94,theminimumradiusof

blylarger,andthe stressconcentra-

he casewiththespringshownin

pisturnedup.Dimensionsas commonly

alsoindicatedinF ig.94.

Def lectionofTensionSprings—To

ctiveturnsina tensionspring,the

pointswheretheloop beginsisdeter-

edthedeflectionduetothe endcoils.

dicatethatahalf coilturnedupto form

g. 93ise uivalentto . 1full- coilasfar

Thus,if aspringhadn turnsbe-

psstart,the totalactiveturnswould

a. 2turnbeingthee uiva lento f two loops.

wnanalyticallyasfollows.Thehalf

oopise uiva lenttothequarter- turn

loadedasshow n. UsingEq uation143

arter-turnbecomes

emeanradiusof the loop(ta ene ua l

inceformostspringmaterials

2. 6G, appro imately , w hereG—

uationmaybewritten

ately , w here$ isthedeflectionperturn

up, F ig. 94, thee perimenta lwor

f lectione ua lto , 5-turn. Inthiscasethe

M. . , 1934, Pngc558.


/ h t t p : / / w w w . h





IN G

sw ouldben + lw heren isaga in

tswheretheloopsbegin.

ntof initialtensionwhichcan

dsprimarilyonthespringinde

thelowertheinitialtensionvalues.

pondingtopractical valuesofinitial

IV w erepublishedbyWallaceB arnes

ca lculatedfromEq uation4, curvature

encetheinitialtensionloadmay be

ofstressusing theformula

stress.

lationforf indinginit ia ltensionload:

nch outsidediameterandy4-inchwire

2r/ d= 7. Whenw oundwithma i-

pondingtoInitialTension

T e « ° " S t re ss

u a re I n ch

tess, fromTableX X IV , dueto init ia l

00poundspers uare inch. F romEq ua-

loadbecomes

of winding,valuesofinitialtension

, 1941.


/ h t t p : / / w w w . h





S P R IN G

obtained.

Usua lly theendturnsof tension

mpleformsindicatedinF igs.93and

fullturn isbentupto formaloop.

ons,however,awidevarietyofend

hareshownin F ig.95,maybeused.

particularlywheretheloopor hoo

considerablygreaterstressinthe

donthebasiso fana ia lload, E q ua-

E

D N D

E

H O L D

L E Y E

I TH

T

pesofendloopsfortensionsprings

atthe side,themomentarmofthe

mumtorsionstressinthe springdepends

ichw oulde istif apure lya ia lload

pring.This meansadoublingofthe

oadingoftension springs,evenif

isused,theline ofactionoftheload

considerableamountfromthea is

considerableincreasein stressmay

onsideredbythedesigner.

madewithplain ends,F ig.95d.

d" springends areattachedtotheseasin-

enusingthese,thespring isclosewound

arespreadapartbyscrewingthe

iscaseaninitial stresscorrespondingto


/ h t t p : / / w w w . h





IN G

urns neartheendsis setup.This

ndtoa loade ua ltothe init ia lten-

oadcorrespondingtothe distance

secondtypeofspringendis shown

wedintotheendsofthe springcoil.

educestressintheendco ilsare indi-

8. InF ig. 97thediameterofendco ils

etheend loopisformed.Then,when

emomentarmofthe loadatthepoint

wire isthesharpestwillbe small.

hespring isreducedaccordingly.S uch

pensivethantheusualformofend loop,

tresses areunavoidable.

cingthestressin thespringisto

nghoo sateachendtofitoverthe

is arrangementasharpcurvature

pringendsfor tensionsprings

rreducedstressintensionspring


/ h t t p : / / w w w . h





S P R IN G

pointsof highstressisavoidedand

hespring reduced.Inadditionthis

uentlyofadvantageinmechanism

ecttoaw hippingactionas, fore ample ,

gisattachedtoa bell-cran which

andstopssud-

ofthespringis

selfhas avelocity

orsuchapplications

bythedesign

e inreducingthe

d,a springfast-

aplug wouldbe

stressatthesepo ints

.

F ortensionsprings

ndloop,F igs.93

remaybe rather

tedthatthestrength

anfor compres-

materialandheat

rlytruewhere

fatigueorrepeated

hestressconcen-

harpcurvature

lsoitis difficulttopresettension

ertheinitial tensionwillbelostor

sultbetw eenturns. B ecauseof lac o f

smorecreeporloadloss may,there-

orproperlypresetcompressionsprings.

ductionofw or ingstressto75or80per

onspringsisfre uentlymade.

E S S IO N S PR IN G

aspringtoe ertbothtensionand

ase inpointisacran - typefatiguetest-

gfull-sized impulseturbineblades .

nof thisseeTheMainspring, A ugust, 1939.

roon—" TurbineB ladeF atigueTesting , Mechanica lEngi-

31.


/ h t t p : / / w w w . h





IN G

pressionspring.Thespringis clampedatboth

andcompressionloadsmaybee erted

avingtheshapeshowninF ig.99are

heryoftwocircular platesasshownin

platesismovedbac andforthbyacran

oacrosshead.Bythismeansan al-

weentensionandcompressionisap-

specimenwhichisheatedatthe same

emperatureandvibrationconditions

ingsuchsprings,careshould be-

ilshavea gradualtransitionbetweenthe

straightportionofthe end.Inthis

eend turnsmaybereducedandthe

tionminimizedasmuchas possible.

ngementofF ig.100isthat adefinite

st specimenevenifdeflectionsof

riouscauses.

pressionsprings,itshouldbe

ressiscompletelyreversed,thestress

faspringsub ecttopulsating(zero

thesamepea va lue. Thusif theendur-

o f


/ h t t p : / / w w w . h





S P R IN G

dspers uare inchforspringstestedina

(thisisan averagevalueforspringsof

ee pectedendurance limitw ouldbe

ers uare inchinthecaseofatension-

samematerial.A dditionalstresscon-

endturns wouldtendtoreducethe

svalue.O ntheotherhand,the en-

stressmaybe somwhatgreaterthan

zerotoma imumstressrange. In

arsthatusuallyawor ingstressabout

ra zerotoma imumrangemaybe

.

g.100,whichhaveawire diameter

of3. 5, wor ingstressesof±25, 000

figuredwithcurvaturecorrection,have

nservice.


/ h t t p : / / w w w . h





D R E CT A N GU A R - W IR E CO M PR E S S I O N

-wirecompressionortensionsprings

applications.F ore ample,inthede-

es, arectangularcross-sectionen-

na morenearlylinearload-deflection

cationofthis typeutilizingrectangular

utysca lesofF ig. 1C hapterI. A nother

chspringsisthe setofinterchange-

ig.101madefora testingmachine.

eof thes uareorrectangular-barsec-

maybepac edintoagivenspacefor

epossible forroundwire.H owever,

nullifiedbythe factthattheefficiency

lfor therectangularsectionisnot as

areinvolvedandsprings arecold-set,

butionoccurs andthisdifference

dandrectangularwiremaybe small.

A R G IN D X , S MA L L PITCH A N G E

arge,a helicaltensionorcom-

reorrectangularw ireandof large inde ' -

allyasabar ofs uareorrectangular

oatorsionmomentPr, F ig. 102w hereP is

meancoilradius. Toca lculatethetor-

arectangularbarundertorsion,

e a na l og y m a y be u s ed ' .

hisanalogymaybebriefly de-

etchedmembranehavingarectangular

formtension atitsedges,combinedwith

onof thispointtogetherwiththeoreticalresultssee

orest—" N ew SpringF ormulasandNew MaterialsinPrecision

presentedattheA nnualA . . M. . Meeting, December, 1934.

maybeconsideredastheratio 2r/abetweenmeandi-

wirecross-sectionbtii.102.

ofE lasticity,McGraw-Hill, 1934,Page239,givesamore

sanalogy.


/ h t t p : / / w w w . h





S P R IN G

causingit tobulgeout.ThePrandtl

imumslopeofthemembraneat

earingstressat thecorresponding

andthatthevolumeenclosed

ami S ons

changeablerectangularbarspringsmadefor atest-

y f rom1/ 10to1 poundsperinchdef lection

neofitsedges representstor ue.

hatthedeflectionsofaloaded

epartialdifferentiale uation:

nofthemembraneatanypointhaving

, q isthepressureperunito fareaof the

nsionforceinthe membraneitselfper

.In additionthedeflectionzmust

hichrepresentsarectangularcross

w here(b> « ), thedeflectionzof the

essedinseriesformas follows:

, Page240.


/ h t t p : / / w w w . h





- W IR E CO M PR E S S I O N S P R IN G 20 5

nofyonly. B ypropercho iceofY„ , the

186)maybesatisfied.

ation187inE q uation186ande -

q uation186intheformofaF ourier s

orY„ maybedetermined. Thef ina lre-

alogy,theshearingstressatany

slopeofthemembraneatthe cor-

mq/ be ingreplacedby2Gi9w here

pringof

y loaded

nd6 theangulartwistperunit length.

ation188andta ingy= 0, x =a/ 2, the

he mid-pointsofthelongsides)becomes


/ h t t p : / / w w w . h





S P R IN G

, f romTableX X V .

terminedin termsoftheangular

hevolumeunderthe deflectedmem-

ectangularBarsinTorsion

e ingaga inreplacedby2GH . S incetw ice

8 f or z a n d ta i n g q / = 2 G h th e e p r es -

s

t an h \

itten


/ h t t p : / / w w w . h






atio b/aandmaybeobtainedfrom

of6givenbyE q uation191in

imumstressrmmaybedeterminedin

enfromTableX X V .

helica lspringof large inde and

ngmomentMt = Prwherer= mean

ma imumshearingstressthenbecomes

se uationassumesa large inde ,

eto curvatureanddirectshearis neg-

arbarsection

swillbe consideredlater.

gsw herea=b, thise uationre-


/ h t t p : / / w w w . h





S P R IN G

theangulartwist6perunit length

t= Pr. Thetota langulartw istw illbe

,thisvaluemultipliedbythe coil

nr-9, orusingEq uation191

T ab l e X X \ r .

gsw herea= bthise uationre-

sdef lectione uationa large inde

e actcalculationbasedonelastic

nde greaterthanfourthe errorwill

sin contrasttothestressformula

yinerror evenforinde esgreater

largedeflectionsatheorymaybe

irespringssimilartothatdescribedin

springs.

E S PR I N G O F S MA L L IN D X

simplyasas uarebarundera

of themembraneanalogyyieldsan

umshearstressw hichisgivenbyEq uation

de esthisvalueofstressw illbeappro i-

llor moderateinde estheerrorwill

ses,as forroundwire,amoree act

inthe ma imumstress,theordinary

ult ipliedbyafactorK dependingon

ountforcurvatureanddirectshear.

ThefactorK' maybecomputedf rom

erechnungZ y lindrischerS chraubenfedernV . D. I. V o l. 76,


/ h t t p : / / w w w . h






wayasforthecase ofroundwire,

analysis showsthatforsmallpitch

aterthanthreeane pressionforthefactor

percent, is

ginde .

onstressfors uarewirebecomes

ottedasfunctionsofspringinde 2r/ a

ofthisfigurewith thecorresponding

g.30ChapterII) showsthatthevalues

C= Z %

utiplicationfactorK' f o rs uare-w ire

r/ o> 3)

undertheK va luesforroundw ire , thedif fer-

ntforaninde o f3, and4percentfor

re-w irespringisw ound, asacon-

272.


/ h t t p : / / w w w . h





S P R IN G

rmationduringcoilingthe sectionbe-

atedinF ig.105.Insuch casesan

hadby ta inganaveragevalueofa

nde e ua lto2r/ a, forf indingK ' . The

ectangular-wirespringstobediscussed

hodis sufficientlyaccurate.

s uare-wirespringsofsmall

6. derivedonthebasiso fastra ightbarin

ect towithin4per centforspring

applicationof themoree actca lcula -

oy y ie ldsthefo llow inge pression

ndsmallpitchangles:

thesesrepresentsdeflection8 based

tbar, E q uation196, w hilethef rac-

stheeffecto f thespringinde . F oran

s.963whichmeansa deflectionabout3.7

uredf romtheusua lformula inEq ua-

of4thedeflectionw illbeabout2per

d79 whichapplytoroundwire

usedforanappro imatecalculation

uarewirespringsatgivenstresses.

alculatetheloadanddeflectionat

sinthe correspondinground-wirespring,

utsidecoil diameter,numberofturns

a ltotheaveragesideof thes uare

usfound aremultipliedbythefactor

.738to findthoseforthes uarewire

s orbestaccuracy, how ever, Eq ua-

beusedinsteadofthe chartsmentioned.

E x actTheory—A pplyingthemore

rmannerasoutlinedinC hapterIIto


/ h t t p : / / w w w . h





- W IR E CO M PR E S S I O N S P R IN G

thefo llow ingmoree actformulashave

a eintoaccountthe effectofpitch

s

2 p/

rethe inde c=2r/ a> 3anda< 10

maybee pressedas

Eq uation197orF ig. 104.

egreesandc< 3,thefollowing

umtorsionstressho lds:

fhe lica l

rew ire

alin

ingcoiled

wire.

m-

on

derivedbyGoehner,loc.cit..Page271,using methods

n ChapterII.


/ h t t p : / / w w w . h





S P R IN G

choccurs asaresultof the

tedusing curved-bartheory,ta ing

ua ltoPrsinaandusingasim ilarpro-

apterII.Thebendingandtorsion

en becombinedinasimilar way

uiva lentshearstress. Ingenera l, f or

uivalentstresswillnotbe greatlydif-

umtorsionstresst, .

pressionforcalculatingtheef fecto f

aybe derivedinasimilarwayas

rings,E q uation51.Theanalysis

is givenby

4)

onofs uare-w irespringf iguredf rom

uation196, and

tana(205)

uationissim ilartoE q uation51

re ample, if c= 3 anda=10de-

eoffromE q uation205is.97,which

willbe about3percentl owerthanthat

epitchangle andcurvatureeffect.

- W IR E S P R IN G

f rectangular-w irespringsiscompli-

urvatureinincreasingthestress is

arlytrueif thespringiscoiled flat-

herethe longsideof thecrosssec-

nga isandw heretheratiob/ a is

ig. 102, anappro imatee pression

llpitch angles)is:


/ h t t p : / / w w w . h






06 )

f romE q uation197orthecurveofF ig.

reb/ a isbetween2. 5and3this

saccurateto withinafewpercent forin-

cisbetw een3and4theerrormaybeas

herectangleisparallelto the

andZ > >3a, anappro imationforstress8is

ictheoryhavealsobeen developed

angular-barsprings .Itshouldbe

mtorsionstressina rectangularbarunder

midpointofthelongsidesof the

rue wherethebariscoiled inthe

nde . F orsmallerinde es, however,

ndstooccuratthe insideof theco ilbe-

ctshear effects.Thustwoopposing

nto play.W herethespringisc oiled

ea stressmayoccure itherontheshort

ndingonthespringinde andthe

ress—F orpracticalcalculationsof

rings thecurvesofF ig.107,based

iesec e f romGoehner se uationsfor

rings maybeused.

sonsshow nonthes etchesinF ig.

ringstresst, inthespringisgivenby

erpendicularandparalleltospring

meancoilradius, / J= a factorto

7dependingontheratioa / borb/ aand

. D. I., 1933, V o l. 77, Page892.

Page425.


/ h t t p : / / w w w . h





L S P R I N G


/ h t t p : / / w w w . h






2r/ a . E achcurverepresentsagiven

onisusedfor intermediatevaluesofc.

hiscaseb alwaysrepresents

ralle lto thea iso f thespring hence,

e.

e a mp l e of t h e us e o f th e c ha r t of F i g .

imumstressina rectangular-wire

atwiseasindicatedonthe figure.

ctorC,forrectangularbarsprings

% - inch, a / b= 2, r= 1. 5inch, c— 2r/ a

nds, f romF ig. 107fora /b= 2andc= 6,

ema imumstresst, becomes

Tocalculatedeflectionsin rec-

glargeinde es(saygreaterthan8),

edontorsionofa straightbarof

eldresultsaccurateto withinafew

ngleisnotlarge:


/ h t t p : / / w w w . h





S P R IN G

sthe longsideandathe short

laforrectangularwirespringsof

ing1

2 0 9a ( ta n h + - 00 4 ) ] ( 21 1 )

ducesto

herectangleisparallelto the

102, Eq uation210w illy ie ldresultsac-

centevenforinde esaslow as3,

ththespring inde .Ifhigherac-

ofF ig.109shouldbe used.

edf la tw iseasinF ig. 106, Eq uation

ccuratetowithinafew percent,for

hanfive.In thecaseofsuchsprings

andforlargerinde esw herehigher

olow inge uationmaybeused1 :

ndingonb/ aandisgivenby the

trmC isgivenbyE q uation211or

enthattherightsideofEq uation213

ndingterminEq uation210div idedby

c= 3andb/ a= 4, C 2= 1. 18, i. e .,

Page271. It isassumedthatthepitchangle isunder12


/ h t t p : / / w w w . h






uation210forsuchcasesmaybearound

argeva luesofcthefactorC be-

dE q uations210and213become

ninrectangular-wiresprings

be simplycarriedoutbythe useof

Inthiscasethema imumdef lection

.1933, Page892.

alculatingdeflectionsinrectangular-wirehelical

byLiesee e, V DI, 1933P. 892)


/ h t t p : / / w w w . h





S P R IN G

ndson theratioa/borb/a,F ig.109.

sta enasthesidepara lle lto thespring

pendicularthereto . If thespringis

f sc uare-w irehelica lspringinposit ion

bis ta en,whileifitis coilededge-

.

sco iledf la tw isew itha= % - inch,

h, inde c= 2r/ a=6, P=300pounds,

5, G= 11. 5X 10 poundspers uare

. 109theconstant7= 6. 7fora / b= 2,

ti o n 21 4 ,

3 . 3 7X 5 . ,

i n ch .

1 0

beenthepracticeinspring

odulusofrigidityG forrectangular-

hatused incircular-barspringsof

ethereisno goodreasonwhythe

bedifferentfor springsofthesame


/ h t t p : / / w w w . h






sbeendoneto compensateforinac-

yusedempiricaldeflectionand stress

arsprings.Comparisons12ofsomeof

cticewith themoree acttheoryfor

showconsiderableerrorsupto100 per

ob/a.It istheauthor sopinion

rewill yieldmoresatisfactoryresults

springsthanwilltheempirical form-

beenused inthepast.

ne actca lculationofstressinrec-

largepitchanglesis complicatedand

owever,anappro imationsuffi-

20000

T R E S S , L B / Q U A R E IN C H

scurvesforsemico il, c= 3. 07

racticalpurposesmaybe obtained

.107andneglectingthepitch angle.

nsforrectangular-wirespringsthe

pressionta espitchangle intoac-

e inMachineDesign, July , 1930forfurtherdeta ilso f thiscom-

e271.


/ h t t p : / / w w w . h





S P R IN G

by

T R E S S , L B / Q U A R E IN C H

sscurvesforsemicoil, c— 4. 14

tion211or212andEI= f le ura lrigidityof

outana ispara lle lto thespringa is.

UA R E -WIR E S PR IN G

ula,E q uation198,fors uare-wire

rementsweremadeonsemicoilscut


/ h t t p : / / w w w . h






springs.Thestrainmeasurementswere

as thosecarriedoutonround-wire

semico ilw ascutf romanactua ls uare-

armswereweldedonasindicated

entsthesemicoilin positioninatesting

ownhavesphericalpointssothatthe

a lloadasisthecase inacomplete

heHuggenbergere tensometerusedto

wnin positionontheinsideof the

mstressoccurs.S hearingstressesplotted

edbythefull linesinF igs.Illand

of twotestsontwocoils,one ofin-

f inde 4. 14. F orcomparison, dashed

resscalculatedbythe moree act

8,arealsoshown.It maybeseenthatthe

ismoree acte pressionagreew ell

rcomparisonacurverepresentingthe

inarytorsionformulafor s uare

hichneglectscurvatureeffects)isalso

ttercurveshowsconsiderabledevia-

F F O R MU A S T O S T A TI C

O A D IN G

eformulasforstressgivenin

ndrectangularbarspringsare based

erefatigueloadingisinvolvedthe

a imumstressrangewhichis of

er,forstaticloadingtheseformulas

cflow withresultantincreased

ryload.Insuchcases ananalysis

V forroundw irew ouldbere uired

escopeofthis boo .Intheabsence

on,useofrectangular-barformulas

nddirectshear( q uation206ta ing

ablybe j ustifiedw herestaticloadingonly


/ h t t p : / / w w w . h





D S U R G IN G O F H E L I CA L S P R IN G

stressanddeflectioninhelical

edthattheloadis applied(orthe

wrate1 sothatadditionaldynamic

vibrationdonotoccur.In mostprac-

sassumptionis probablyrealized

herearealarge numberofapplica-

namiceffectsduetosurgeorvibration

p.

iveva lve

headditionalvibratorystressesthus

toaccountby thedesignerif fa tigue

hemostimportante ampleofsuch

neinwhichthe timeofapplicationoftheload is

ralperiodof vibrationofthespring.


/ h t t p : / / w w w . h





D S U R G I N G

r automotiveenginevalvespring.

picalautomotivevalvespringand

3, whileas etchofatypica lva lve-gear

mentisshowninF ig.114a.

alve-liftcurveshowingvalveliftplotted

ig.114b.Thislattercurvealso rep-

ftheendof thespring(beyonda

cva lvespringandgearfor

e

againsttime.It isclearthat,if

asuddencompressionoftheendof

mpressionwavetotravelalongthe

edfrom theend,thetimefor the

dof thespringtotheother being

re uencyofthespring.Thisphe-

ongaspringmayeasily bedemon-


/ h t t p : / / w w w . h





S P R IN G

gf le iblespring, suchasacurta inrod

hedbetweenthe twohands.Ifone

movingonehand,a compressionor

seentotravelbac andfortha longthe

esame assurgingofvalvesprings.

icationofdynamicallyloaded

pefatiguetestingmachineshowninF ig.

emachinesofthistypeare tooperateat

onshouldbeconsidered.

I D R A T IO N S

signofspringssub ecttoarapid

hasvalvesprings),it isimportant

ble,resonancebetweenthefre uency

fthe endofthespringand oneof

ofvibrationofthespring. Usuallythe

cy isof themostimportance. F oraspring

allelplatesthefirstmodeofvibration

stnaturalfre uency)willconsistof

iddleportion ofthespringwiththe

.Thesecondmodeofvibration(cor-

uency)willhaveanode(or point

)inthemiddleof thespring,while

co ilsoccursatpo intsV \ and% of the

end ofthespring.Thenatural

ngtothesemodesofvibrationmaybe

cussedlater.

F ore ample, if aspringissub ect

ymeansof asimplecran arrange-

.115,providedtheratior/l between

ectingrodlengthisnottoo large, thee -

acementfromits positionattop

sufficientaccuracybythee uation2:

c o s 2 wt J ( 2 17 )

hecran inradianspersecond

ica lV ibrations, SecondEdit ion, McGraw -Hill , 1940, Page

on.


/ h t t p : / / w w w . h





D S U R G I N G

h er e N = s p ee d i n re v ol u ti o ns

tforspringssub-

eansofasimple

cipalf re uencies

:

encyof rotationasrep-

ermofE q uation217.

hisva luerepresentedby

q uation217.

resonance,

noughsothat

ncy,calculated

sconsiderablyhigher

cyof rotationof the

tedbyacamas

ft curvey—f(t)

mplesinefunction

ortwo termsas

insteadacompli-

beassumedto

ofsinusoidal

ries). Thusthee -

aybewrittenas

n (U - - < t > i ) + C i s i n ( 2w t + < t > : ) +

) + * ( 2 18 )

dofthespringmaybe considered

v ingacircularf re uencyw e ual

speedi nrevolutionspersecond,on

monicsof2,3, 4....timesthis fre-

desof theseharmonicsarec,, c3, c> . . . .

eachof theseharmonicsmaybeas-

y.Inpracticeharmonicsas highas

beconsidered.


/ h t t p : / / w w w . h





S P R IN G

enera l, itshouldbenotedthatthe

esehigherharmonics,representedby

creaseastheorder oftheharmonic

efounddifficult toavoidresonance

drangesbetweenoneofthe higher

e uency(usuallythelowest)ofthe

splace,severevibrationorsurgingoccurs

ndthis mayincreasethestressrange

tor more.Thisistrueeventhough

ularharmonicin resonancemaybe

eisa largemagnificationofthe

ons.

suchresonantvibrationsinvalve

reopen.Inthe firstplacethenatural

maybeashighaspossiblesothat

orthe higherorderharmonics(which

ude).H enceanimprovementisob-

t upbyresonancewiththesehigher

as thosesetupby resonancewith

ramplitude.

cingsurgestressesin valvesprings

ursoasto reducetheamplitudesof

importancein thespeedrangewith-

eused. F ore ample , itm ightbe

th anoperatingrangefrom2000to

tethetenth,eleventh,andtwelfth

ewiththel owestnaturalfre uency

speedrange.H enceacamcontour

forthese harmonicswouldbe

n manycasesitispossible bya

toreducethemagnitudesofthe

thincertainspeedranges4.

pitch ofthecoilsnear theends

entoftenmaybeobtained.The

onanceoccurswithone harmonic,

pthuschangingthe naturalfre uency

of Iheamp ; tudesof thevariousharmonicsmaybede-

lysisforanygivenvalvelift curve.

" S chwinffungenS chraubenfoermigenVentilfedem

ertation,T.H .Berlin,publishedbvDeutschenV ersuchsanstalt

dlershof.1938.


/ h t t p : / / w w w . h





D S U R G I N G

othrow itouto f resonance . F rict ion

ree-prongeddevicewiththeprongs

rcoils ofthespringhavealsobeen

oscillations .

R V I BR A T IN G S P R I N G

uencyandvibratorycharacteristics

arytoderive thedifferentiale uation

lementA (showncross-hatched)

ingcom-

plates

g.116compressedbetweenthetwo

dered.Itis assumedthatwhenthe

elementA oflengthds,isat adis-

of thespring.Theactivelengthof

f thespringista enas/ , w hile the

eneglected.Thedeflectionofthe

meanposition(or positionwhenthe

et willbedesignatedy.Ifwd:/4

ofthe wire,7theweightofthe spring

and gtheaccelerationofgravity,

is- rrdrtds/ A gandtheforcere uired

willbe

ndL ehr,Page115.publishedbyV .D.I.,Berlin,1938.

ineV a lveSprings, bySw anandS avage, S p. R ep. N o. 10,

esearch, London.


/ h t t p : / / w w w . h





S P R IN G

uationforcee ualsmasstimesaccelera-

of thiselementisd2ydt2.The par-

yisa functionofbothsand t.

cedswill be(dy/ds)ds.F or

e willbeA (/=27i-r dy/dswherer—

otalforcePactingat adistancex

pring deflectionformula,E q uation7,

gidity.

na lengthdswillbe(dP/ds)ds,

ceF (,actingtoacceleratethe ele-

ntia tionofEq uation220, thisforcebe-

* y .d ( 22 1)

dditiontherearedampingforces

sesincluding:

springmaterial

intheendturns

energyinthesupports.

a inga llthesesourcesofdamping

elesslycomplicated.F ormathemati-

ngforceisassumedproportionalto

Thismeansthat,ifc isthedamping

ewireperunitof velocity,thedamp-

ticforce F i H ence,frome uilibrium,


/ h t t p : / / w w w . h





D S U R G I N G

ions219, 221and222inthisanddiv id-

- y dy

ere I= active lengthofspring, and

ns

r2 n2 d -

sevaluesin E q uation223andre-

wingdifferentiale uationisobtained:

2 2 4)

w eightofactivepartcf spring(225)

t(226)

asureof thee uiva lentdamping

willvarywith suchfactorsask ind

motion,designofendturns, rigidity

determinedbyactualtestsonvi-

mpingiszero , b= 0andE q uation

ewell- nowne uationforlongi-

rt iclebyC . H . K ent, MachineDesign, October, 1935, fora

oreferencesoffootnotes4and6.


/ h t t p : / / w w w . h





S P R IN G

inprismaticalbars,abeing theveloc-

a longthebar .

E Q U N C

uencyofaspring,it isper-

gsincethe smallamountofdamping

es notaffectthenaturalfre uency

ispurposethe simplerdifferential

used. Toso lvethise uation, the in-

tanypointof thespringisassumed

nctions,oneafunctionof x only,

ftonly.Thus,

(2 30

/ < (f )arefunctionsofx andtrespective ly . Then

• d > ( ) — • l ( t)

uation229,

) d -

esatisfiedif bothmembersofthis

aconstant, say— w 2. Then

ationsare

t c os a t ( 23 3 ) -

- B 2 c os ( 2 34 )

ionProblemsinE ngineering,S econdE dition,Page309,V an


/ h t t p : / / w w w . h





D S U R G I N G

arearbitraryconstantsdependingonthe

venproblem.

tions233and234inE q uation230a

satisfiesthedifferentialE q uation

d—Ifbothendsof thespringareas-

ed,thismeansthatregardlessof the

0 f or x = l .

sre uiresthatB.,= 0,whilethe

wl/a= 0.Thismeansthatthefollowing

• e tc .

/ isanaturalf re uencyof thespring,

hatthenatura lf re uenciesare inthe

nbyEq uation227, thee pression

yofthespring(in cyclespersecond)

orderofthevibration,i.e.,m= l

esecondmode,etc.

orthespringconstantk andspring

uations226and225, the lowestnatura l

omes


/ h t t p : / / w w w . h





S P R IN G

usuallyofmostimportancein practice.

tforagiven materialthenatural

ng isproportionaltothewire diameter

totheproductof thecoildiameter

oils. F ortheusualsteelsprings where

spers uare inchandt= . 285poundsper

lowestnaturalfre uencyreducesto

ree—F oraspringwithoneendf reeand

westnatura lf re uencywouldbee ua l

wiceaslong butwithbothends

ingEq uation237maybeusedif the

nastw icetheactua lnumberinthe

ampleof theuseof thesee uationsin

ency,assumingasteelspringclamped

nch, r— 1- inch, n= 6andusingEq ua-

lfre uencybecomes

rationthespringfre uencywillbe

persecond.

e ighted—F oraspringw ithaw eight

wninF ig.117,thelowestnaturalfre-

aybecalculatedasfo llow s: It isk now n

ctedbyacertainamount8 underits

e springbeingsmallcomparedto

ural fre uencyincyclespersecond

.

anica lV ibrations, Page45.


/ h t t p : / / w w w . h





D S U R G I N G

ssissupportedbyahelical spring

dicatedinF ig.117ithas beenfound

hirdofthespring isaddedtothatof

heca lculateddef lectionmaybeusedforf iguring

W isthespringw eight, E q uation225,

tin poundsperinchdeflection,thefre-

nd becomes

N G IN E V A L V E S P R IN G

utomobileenginesrunat variable

viously,itispracticallyimpossibleto

oneofthehigher

urve anda

espringatsome

nthisoccursthe

dtheresulting

dprimarilyon

onicwhichis in

untofdamping

bythe damping

4).

essin the

itisassumed

inF ig. 116, say

an amplitude

n

eamplitudeoftheparticularharmonic

chisinresonancewith anaturalfre-

g, usua lly the low est. Thecircularf re -

aharmonicista enasw = 2w f . The

ressdueto thisharmonicmaybe

edif ferentia lE q uation224incon unc-

aryconditions.Thisstress isthen

cstressdue tothevalveliftas indicated


/ h t t p : / / w w w . h





S P R IN G

dot-dashlinerepresentsthestressdueto

imumvaluebeingt . O nthisis

uencyvibrationrepresentedbythe

sonancewitha givenharmonicofthe

4forthesteadystate condition,the

fromtheendofthe springisas-

4 1)

ef u nc t io n F ( ) i s a f un c ti o n of x o n ly . Th i s

entoscillationsdueto suddenspeed

oncernhere.Theassumptionrepre-

41isj ustifiedsince, fora forcedv ibration

partso f thespringmustv ibrateatthis

re:F orx = 0F ig.116,y= 0re-

neendof thespringisassumedf i ed

c sinw tsincetheotherendof thespring

monicmotionofamplitudec produced

uation218w hichisinresonancew iththe

ferentia lEq uation224satisfy ingthese

eenobtainedbyH ussmann10.

pingsuch asoccurinpractical

sto therelativelysimpleform:

U + 4 > ) ( 24 2)

mumamplitudeofmotioninthespringand

angle . F romthisso lution, forsmalldamp-

blestressobtainedis

staticstressinducedbycompress-

ntc . T in se uationindicatesthat

otnote4.


/ h t t p : / / w w w . h





D S U R G I N G

g.118isinverselyproportionaltothe

tlyproportionalto thefre uencyf

atter, inturn, isproportiona ltothe

ticularharmonicinresonance.

dampingfactorh inactualsprings

magnificationof100to300times occurs,

o 300timest8cIt hasalsobeen

ngfactorbvaries withtheamountof

TR E S S D U

I T.

tionofvibrationstressesonstresses

orvalvespring

pringand thatitincreaseswiththe

monicin resonance.Thisisreasonable

internal dampingofthespringma-

lbe lower. A lso fore tremelyhigh

ervaluesofbare found,resultingfrom

tbetweenturns.A tmediuminitial

edampingfactorarelowerwhile,at

esevaluesagainrisebecauseofdamp-

ingoftheend turnsfromthesup-

about 1sec-1atloweramplitudesof

ehigheramplitudeshavebeenob-

stva luesbe ingbetween2and4.

useofEq uation243assumingthat

ency / o f thespringise ua lto200

thecamshaftspeedis1200revolu-

nsthat resonancebetweenthisnatural


/ h t t p : / / w w w . h





S P R IN G

nd thetenthharmonicofthevalvelift

mingalsothattestson springsunder

ownadampingfactorb = 5sec-1,

mpressionofthespringbyan amount

ndif thea lternatingstressduetothe

curve is, fore ample, . 002t (asfound

011001200

P E D , R .P.M .

peolresonancecurveforvalvespring.

ateach resonancepea

healternatingstressdueto resonance

e251X . 002t, < orabout. 5t . This

e stressrangewillbeincreasedto 2

ration.

a lstressrangeinthespringis

tialcompressionofthespringwith

sition,therangeinstresswill befrom

t 1 — t t o a m a i m u m v al u e tm * = t 1 +

nwithendurancediagramssuchasthose

relativemarginofsafetyofthe spring

ybeestimated.

esimilarto thoseobtainedbyactual


/ h t t p : / / w w w . h





D S U R G I N G

howninF ig.119.Inthis curvethe

hemiddlecoil ofavalvespringis

peed.Itisseen thatthiscurvecon-

pacedatinterva ls, eachpea be ingdue

hevalveliftcurve(indicatedby the

pea mar ed10isduetothetenth

rve,i.e.,to avibrationfre uency

spersecondforacamshaf tspeedof1200

eamplitudesofthesepea svarysince

thec sofE q uation218)of thevarious

DI N T

b ecttorapidreciprocatingmo-

the followinge pedientsareoften

hnatura lf re uency

snearend ofspring

ssibleofresonancebetweennaturalfre-

dane cit ingf re uency

soastoreduce themagnitudeofthe

urve withincertainspeedranges

n,incon unctionwithtestre-

ngesin actualspringsundervibra-

eand inthiswaythe marginof

redetermined.


/ h t t p : / / w w w . h





E D DI K (B L L E V I L E ) S P R IN G

thedirectionofloadapplication,

dis springsisf re uentlyofadvantage.

lso k nownasBellevillesprings,consist

so fconstantthic nessandhavean

suitable variationoftheratioh/t

htanddis thic ness, it ispossibleto

veshavingawidevarietyofshapesas

fF ig. 121. F ore ample, re ferringto

ncharacteristicfora ratioh/t— 2.75

dbyC urveA . S uchashapemaybe

ngdeviceisbeing designed.Byre-

aload-deflectioncurvesimilarto

ypeofspring, k now nasthe" con-

aconsiderablerangeofdeflectionwithin

lyconstant.S uchacharacteristicis

pplications,suchas fore ample,

eddis (B e llev il le )spring

iedto agas etisnecessary.Byvary-

ediateshapes,F ig.122,ispossible.

f theapplicationofsuchspringsisthe

ero iderectifierstoprovidea con-

astac o f rectif ie rdis stogether.

espringsforproducinggas etpres-

pacitorsandincondenserbushings

t.Insuchcases,springswiththe char-

BF ig.121havebeenfoundadvantage-


/ h t t p : / / w w w . h





ndingdeflectionfactorC,forinitially-coneddis

ncharacteristiccurveswillbesimilar


/ h t t p : / / w w w . h





S P R IN G

llsupplyappro imate lyconstantgas et

evariationofdeflectiondue totem-

rcauses.S uchdeflectionchanges

amplebytemperaturechangesasacon-

R A L L E L

R I S

stac inginit ia lly -coneddis springs

nce ine pansioncoeff icientsbetweenthe

ecopperpartsof electricale uipment

ringsmaybestac edinseriesorin

. 123. Bystac ingthespringsinpara l-

edforagivendeflection stac ingthe

argerdeflectionatthegivenload.

stac edinseries, ratiosofh/ tbetween

ssgreaterthanabout1.3shouldnot

tabilityor snapactionisaptto occur,

ctioncharacteristicwillthenresult.

hematicaltheoryofelasticityto

coneddis springsise tremelycom-

practicalsolutionoftheproblemmay

heassumptionthatduringdeflection,

eoryofP latesandS hells—S . T imoshen o, McGraw Hill,


/ h t t p : / / w w w . h





P R IN G

dis srotatewithoutdistortionas

ine inF ig. 124c. If theratior / r, be-

erradius isnottoolarge,tests show

llyield sufficientlyaccurateresults

atleastas farasdeflectionsarecon-

tionsonflatcircular plateswith

nmadeon thebasisofthis assump-

sectionsrotatewithoutdistortion,

odresultswhencomparedwiththose


/ h t t p : / / w w w . h





S P R IN G

ory2.F ore ample,comparisonbe-

ppro imateso lutions1show sthatwhere

outerandinnerradii isnotover3(w hich

es)theerror indeflectionmadeby

verabout5percent,while theerrorin

coneddis springwhichfol-

mptionofrotationofradialcrosssec-

ndisduetoA lmenandL aszlo . These

medeflectiontestswhichindicate

sfactoryforpractical use.

initially coneddis springcut

btendinga smallangled6,F ig.124a,

ternalload,aradialcross sectionro-

sindicatedby thedashedoutlineofF ig.

entarystripoflengthd atadistance

tionundertheseassumptionsconsists

acementdrand arotation< f> ,thelat-

nofthecrosssectionabout pointO

lacementdrcausesauniformtan-

entd (thesmalldifferencesipradial

fthespring betweentheupperand

entd arehereneglected).Thero-

tangentialbendingstrainwhichis zeroat

ma imumattheupperand lower

stresses duetothemovementsdr

ducemomentsaboutpointO whichresist

aldisplacementdrmaybecalcu-

circumferentiallengthofthe ele-

onis

ngle.A fterdeflectionthislength

sdescribedinC hapterX V .

obo, Jr.—" S tressesandDeflectionsinF latC ircularP lates

ransactionsA . . M. . , 1930. A . P . M. 52-3. A lsoS . T iinoshen o—

anN oslrand, Part2, 1941, Page179.

t io n D is S p r in g , T r an s ac t io n s A . . M. . . 1 93 6 , Pa g e 30 5 . A

taperedspringsis givenbyW .A .Brechtandthewriter

dDiscS pring , TransactionsA . . M. . , 1930, A . P . M. 52-4.


/ h t t p : / / w w w . h





P R IN G

) d 9

) - co s 0

s i n < t > — x c o s 0 ( 1 — c os 4 > ) ( 2 45 )

thattheanglesfiand < f> aresmall

springs)sothat

s in 4 > = < t > ; l — c os 4 > — ~ ^

cos< f> is obtainedbyusingthecosine

sabovetheseconddegree.

on245,

(2 46 )

tangentialdirectionwillbe,

heradial stresseswhichareas-

ss isobtainedbymultiplyingthisby

modulusofe lasticityand/ =Po isson s

onlytotheradialmotiondr becomes

gnifiescompression.Thefactor1— > r

tractionor e pansionofelementsof

thise pressionmayalsobeobtained

aforcalculatingstressesfromstrainsin

stress5.

T imoshen o—S trengtho Materials, 1941, Part1, Page52.


/ h t t p : / / w w w . h





S P R IN G

duetothetangentialstresses

w illbe

s i n( 0- < t > )

t > ) ~ / 3 — < t > f o r sm a ll a n gl e s an d u si n g E q u a -

' d

c — r t o x — c — r , ,

. ,

0 -r . )+ r f o g, — | ( 2 49 )

stressdue tothebendingstrain

o f thee lementd , F ig. 12A b, it isneces-

in curvatureinthetangentialdirec-

w hichis

uralrigidityinthe caseofbeams.L et-

curvatureoftheelementduringdeflec-

mentactingon elementd duetothis

— - c x ( 25 1)

tureoftheelementin theun-

mate ly sinf i/(c—x )w hilethecurvature

ssin(/ J—< £ )/ c—x . Thechangeincur-

t >

251,

,PartII, Page120,givesafurtherdiscussionof platerigidity.


/ h t t p : / / w w w . h





P R IN G 2 45

atthesurfacewillbe thisvalue

cionmodulusof thee lementd w hichis

q uation252,

eneutra la is(ta enposit ive inan

sswillbe

used tosignifycompression.

mentdM2whichacts inaradial

uation252,

c—r tox = c—n, themoment

ar motionoftheelementsofthesec-

= E t W e l og i n. / r. )

2 (1 - ,. )

ointO thusbecomes,usingE q ua-

25 6)

pointO tothea isofthedis is


/ h t t p : / / w w w . h





S P R IN G

atthe sumofallforcesover the

becausethereisnonet e ternal

o f thedis . S incethebendingstresses

nt,onlythestress a/duetothe radial

nsidered.Thus

8

gforc

on257inE q uation256,

r ,) 2

2 < : s ( 25 8)

ualtothee ternalmomentonthe

hichis


/ h t t p : / / w w w . h





P R IN G

nof thespringandusingE q uation

2 -> + < H . • ( 26 )

(- ) < 2 62 >

tC , = C . fo rpractica lpurposes.

+ ' ]

asfunctionsof theratioa= r / ri

uation263itmaybeseenthatthe loadP

deflection8.Byusingthise uation,

sticsmaybedeterminedforvarious

22. TheapplicationofE q uation263

4. 550

terminingfactorC2


/ h t t p : / / w w w . h





S P R IN G

facilitatedbythemethoddescribed

ressata pointatadistancex

of thestressesa, anda . H ence, using

54

6 4)

tthe uppersurfaceofthespring

r, andt/ = t/ 2. Ta ingtheseva luesto-

r , ) , < f > = S / ( r — r < ) a n d s u bs t it u ti n g th e

uation257inE q uation264thisma imum

inneredgeofthe springisob-

1/ 2inE q uation264. Thisy ie lds

( 26 8)

ativevaluesignifiescompression,a

I G N

ofE q uations263and264in

lly coneddis springsthefollowing

ordeterminingtheload-deflection

.E q uation263maybewritten:


/ h t t p : / / w w w . h





P II IN G

1

on theratiosh/tand S /twhile

— r / T ionlyandmaybeta enf rom

rf romE q uation261.

putations,valuesofC,havebeen

tforvariousva luesofh/ t inF igs. 122

S S

rdeflectionfactorC , forB e llev il lesprings

e ight: thic ness

g.126maybeused toobtaingreater

luesof h/t.

ecurvesofF igs.122and126also

characteristicsforspringshavingvari-

alcone heightandthic ness.Thisis

tlyproportionaltotheconstant Cr

ndependentof theratior r, (andhence


/ h t t p : / / w w w . h





S P R IN G

apeoftheload-deflectioncharac-

teriallyonlyby alteringtheratioh/i

htanddis thic ness. A th/ t— 1. 414,

6,thecurvehasa horizontaltangent

determiningstressfactorK , for

/ r(= 1. 5

ethe springrateisvery low.F or

vengreaterrangeof low springratebut

slightlyafterreachinga ma imum.

of about2.8theloaddropsbelowzero

that permanentbuc lingofthespring

orintermediatevaluesofh/tmaybe

cyformostpractical purposes.

ofstress a,E q uations265and

ows


/ h t t p : / / w w w . h





P R IN G

gvalue:

beforetheconstantC.. thestressin

espringisobtained,whileusing the

essat thelowerinneredge.Itis

safunctionof r / r( , h/tandh/ t.

omputations, va luesofK thavebeen

atioh/tforvariousvaluesof h/tin

ratiosa= r / r(e ua lto1. 5, thecurves

vevalue ofK 1representingtension

determiningstressfactorK , for

/ r, = 2to2. 5

presentingcompression.Itwasfound

nby thecurves,forvaluesofh/t

imumstresswillbecompressionat the


/ h t t p : / / w w w . h





S P R IN G

8< 2/ . F ordef lection8e ua lto2h, the

edgee ualsthecompressioninthe

tensioninthe loweredgebecomesthe

sshownby theuppercurveforh/ t= l,

2tthenthecompressionandtension

sbecomee ual, andfor8/ <> 2theupper

F ormostpracticalcaseswhereh/t

a imumstressesw illbeobta inedby

urves. Interpolationmaybeused

h/t.Indoubtfulcases where8> 2fi,

edbyusingE q uation271. A further

nofstressin thesespringsisgivenon

sdistributionofstress inatypicalcase.

E x ample1: To il lustratetheuse

6to128 inpracticaldesigna spring

llowingconditions:Thespringis to

plicationw herethe loadistobeheld

at6000poundssothat thetypeofload-

thatfor h/t— 1.5,F ig.121.The

usingan 8%-inchoutside diameter

espring mayvarybetweenand

adandthema imumcompressionstres

on271istobe lim itedto200, 000pounds

auehasbeenfoundbye periencetobe

taticorrepeatedbuta fewtimes7.

ches,ri= 2V & inches,a= r /ri—2h from

K 1 ^ l / r , w he r e K , = — 6 . 7 f r om F i g . 12 8 f or

ma imumvalue. So lv ingthisfortand

undspers uare inchcompression,

1. 5, C , = 1. 68onthef la tparto f the

fora= 2, C 2= 1. 45. F romE q uation269

e

wor ingstressseePage25U.


/ h t t p : / / w w w . h





P R IN G

ired,it willbenecessarytouse 5

l giveappro imatelytherightload.

enthatforh/ t= l. o , C , isappro imately

to5/ f=2. 1. S incef= . 134thismeans

mate lyconstantf romS ~ . 75(. 134)= . l-

ichisaboutw hatisre uired. If thema i-

i -

A C

ch, thema imumvalueofS/ fw il lbe

mF ig. 128for8/ f= 2thefactorKtw ill

6.7whichmeansthatthecalculated

lightlyless than200,000poundsper

e calculatedloadfrom6500to6000

f thedis maybereducedabout2percent.

esuchasthatshow ninF ig. 122for

snapactiondevicetooperatein such

reachesacertainpointrepresented

,thesystembecomesunstableand a

hresultingsnap-action.A lso,a

520poundsis desired,spaceisavail-

rdis , andastopisprov idedsothat

-inchbeforecomingagainstthestop.

eflectionwhenthespringis againstthe

dbythepointonthe curveofF ig.126

ingto8/f= 3. S inceatma imumde-

smeansthatt= . 25/3= . 0833- inchand

8- inch. A ssuminga= 2, f romF ig. 128,

3, h/ t= 2. 5. F romE q uation271onPage250,

efollowingrelation:


/ h t t p : / / w w w . h





S P R IN G

0poundsper s uareinch(compression),

g, r = 3. 8- inchsay3% inches. F rom

1. 45andf romF ig. 122thema imum

gtoma imumload)forh/t=2. 5is

269thepea loadis

5 l b .

520poundsweredesired.To geta

helatterf romE q uation269increases

ua l)thethic nessmaybereduced

• = . 935. Thusf= . 0833X . 935= . 078-

eofcurveh/ tmustbek eptthesame

078=. 195- inch. A tY 4- inchdef lection

2whichwillbesomewhatbeyondthe

3, whichinthiscaseis permissible.

tressw illnotbechangedappreciably since

2. 5and8/ f=3. 2, thefactorK, is practic-

. F romF ig. 122for8/ f=3. 2, h/t=

4. 6atthepea load, the loadwhenthe

willbereducedto 1.3/4.6X 520=147 _

I GN F O R C O N S TA N T - O A D

5,aload-deflectioncharacteristic

type isobta inedasindicatedbyC urveB

vefor7i/ f= 1. 5ofF ig. 126suchthatthe

5percentfrom adeflection8= .8fto

whichareof particularvalueinmany

ybedesignedina simplemanner

mallow ablestressisgiven. L ettingDbe

spring,then theconstantloadPand

ssf toobta inthisloadaregivenby

stedbyR .C.BergvalloftheW estinghouseCompany.


/ h t t p : / / w w w . h





P R IN G

dK dependonthema imumallow able

or(r /r()betweenouterandinner

eseconstantsmaybeta enf romthe

000180000200000

L By Q I N A T D F L E C TI O N < T = 2 .2 5t -

131. Itshouldbenotedthatthethic -

dtotheva luegivenbyEq uation274

dcharacteristic.Inall casesthema i-

medas2.25timesthethic ness.


/ h t t p : / / w w w . h





S P R IN G

esofconstantloadP , thic nesst, and

25faretabulatedforma imumstresses

100,000poundspers uareinch,and

gf r m1. 25to2.5. It isassumedthat

oadB ellev il leS prings

a e n a s 30 x 1 0 l b ./ s . i n .

hichthemodulusofelasticityE may

p o un d s pe r s u a re i n ch .

fE q uations273and274as wellas

V Imaybeca lculatedasfo llow s:

it ispossibletoso lveforthethic nesst


/ h t t p : / / w w w . h





P R IN G

q uation274inthis,

uation273.

dagiven stress,thefactorC.may

25w hileK w illbefoundf romE q uation

T IO N £ = 2 .2 5t -

= S P R IN G TH I C N E S S .

M T R

80000200000

, a / O J N . A T D F L E CT IO N < 5 = 2 .2 5t

determiningthic nessforconstant-

s

ppro imatelythatgivenbytheflat

1.5,F ig.126.Usingthesevaluesthe

igs. 130and131andtheconstantsin

mputed.


/ h t t p : / / w w w . h





S P R IN G

E DW ITH TH E O R Y

estshavebeencarriedoutbyA lmen

yconeddis springs. Thesetests, made

rtions,showcurvessimilar tothose

hat themethodofcalculationdevel-

eformost practicalpurposes.H ow-

sarenote actand, inpractice , devia -

uchas10to15percentmaybee -

alculatedvalues.F orhighestac-

testsshould thereforebecarriedout.

,aload-deflectioncharacteristic

ourdis sinpara lle lisshown. These

about1.45andshowclearlythe constant-

tiallargedeflectionwasdue to

I N IN C H E S

stof stac o f fourstee lB e llev il lesprings, r = 4V 4.

48

esbetweenthedis s.Inspiteof the

eindividualspringswereslightly

esisloopbetweentheloadingand

ned,indicatingconsiderablefriction

singagroup ofspringsinparallel the

imatelyinproportiontothenumber

eappro imatemethodofca lculation


/ h t t p : / / w w w . h





P R IN G

results asfarasdeflectionsare con-

ctedthatdeviationsbetweentestand

stressthanfordeflection.Compari-

atedbythetheoryweremadeby

ltso f sometestscarriedoutbyL ehrand

lculatedS tresses

thic Bellevillesprings.Theresults

ents,obtainedwithaspeciale tensom-

elength,arecomparedwiththere-

eX X V II.

stedbythese writersthedeflec-

hgoodagreementasTableX X V IIindi-

ectedforlargedeflectionsrelativetothe

pected, how ever, thateveninsuch

ultsmaybeobtained.

E S S E S

ere init ially coneddis springsare

or toaloadrepeateda relativelyfew

e perienceshowsthatstressesof

spers uareinchascalculatedby

eusedeventhoughthey ie ldpointo f the

is madeisonly120,000poundsper

. A lthoughthisstressseemse tremely

esign,May,1939,Page47.

, 1936, Page66.


/ h t t p : / / w w w . h





S P R IN G

eredthatit islocalizedneartheupper

ig.129.Conse uently,anyyielding

ributethe stressandallowthere-

ngtota eagreatershareof the load.

ssiscompressioninthe usualapplica-

oramorefavorablecondit ion. A lsodue

themanufactureofthe spring,

tesignare inducedandthesereduce

owthatcalculated.

t ingw or ingloadsforstaticloading

t-loadtypeofspringis tofigurethe

A ssumingthatthespringis flattened

ameterofthe verticalreactionacting

enceofthespring willbe(P/2)2r /ir

yo fasemicircleof radiusr isata

hediameter).Themomentofthevertical

lf-circumferenceaboutadiameter

. ThenetmomentMactingonadi-

gwill bethedifferencebetweenthese

78 )

endingstressesacrossthedi-

efiguredfrom theordinarybeam

hestress-concentrationeffectsofthe

pressionforma imumstress>rbecomes

ivenbyE q uation278)

onmodulusof adiametralsectionis

a m pl e o f th e u se o f E q u a ti o n 27 9 : A

calculatedstressof 200,000pounds

f romEq uation271andhasanoutside

romTableX X V Itheconstantload

ndsforD/ d—1. 5. Thethic nessw ill

. 0297- inches. UsingP= 74poundsand


/ h t t p : / / w w w . h





P R IN G

ation279thecalculatedstress(neglecting

mes

0 0 lb . /s . i n .

guredinthis way(i.e.,stress

cted)thestressisonly about80,000

ndspers uareinchfigureobtained

oy . Thise pla insw hyspringsdesigned

sfactorilyunderstaticloadssincethe

reinchfigureis wellbelowthetension

.In thecaseofthe constant-load

ded,it maywellbethatthe stress

tion279w illy ieldabetterpictureof the

rry load. How ever, themoree act

sconcentrationintoaccountshould

peatedloadingis involved.

herefatigue loadingof init ially -

olved,considerablylowerstressesthan

cloadingmayhaveto beused.S o

e fatiguetestdataavailableby

strengthof suchspringsmaybe

mateofthefatiguestrengthofthis type

wever,asfollows:

goperateswithina givenrangeof

essin theupperandlowersurfacesof

byusingE q uation271. A ssumingno

ntheusual casetherangein theupper

intermediatevaluetoama imumin

thelowerinner edgewillbefrom

sion,orfrom anintermediatevalueto

esidualstressespresentinactualsprings

s.

gtensionstress willbemore

ingcompressionstressonly.H owever,

latedcompressionstresswillbebe-

dpointofthe material.H ence,yield-

uallyoccureitheronfirst loadingorin

withtheresultthat tensionstresseswill


/ h t t p : / / w w w . h





S P R IN G

range.In practice,therefore,itappears

sisfordesignthema imumrangeof

willusuallybea rangeincompres-

uation271.Theactualstressrangein

comparedwiththeendurancerange

eddis springsareheattreatedaf ter

ationofthesurfacematerialwillno

etheendurancerangetobe usedfor

obtainedonspecimenswithunma-

icatedinC hapterX X IIIthisla tterva lue

s ofthevalueobtainedonmachined

thefatiguestrengthof Belleville

edasvery roughandthefinalanswer

ual fatiguetests.


/ h t t p : / / w w w . h





A T DI K S PR I N G

conedor Bellevillesprings,initial-

ntagewherespaceislimitedin thedi-

whileat thesametimehighloadsare

e Bellevillespring(whichmaybede-

of load-deflectioncharacteristics)the

aload-deflectiondiagramwhichis

andconcaveupwardfor largeones.

gbecomesstifferasthe loadincreases.

dewith acrosssectionof constant

ig. 133orw itharadia lly -taperedcross-

1341. A s-w illbeshownlatertheuseof

ethic nessisproportionaltothe

niformstressdistributionandhence

ofthe springmaterial.S uchsprings,

nesstypeorofthe radially-tapered

sshowninF ig. 135. B y thismeansthe

ncreasedwithoutreducingthe load-

metime,the springassemblyisen-

w ellasvertica lloads.

A P R E D S P R IN G

noftheradially-tapereddis spring

chrepresentsasectionalview ofacom-

ailwaymotors.Thefunctionofthe

se istosupplyaconstantpressureforho ld-

gether,whileatthe sametimeallow-

othate pansionsduetotemperature

ew ithoutproducinge cessivestressin

rbars. F orthispurpose, thedis spring

F oranappro imateca lculationof the

B rechtandtheauthor, " TheR adia llyTaperedDis S pring ,

.,1930A .P.M.52-4.


/ h t t p : / / w w w . h





S P R IN G

dis springs,asinthecase oftheap-

viouslydiscussed,itwillbeassumed

thespringrotateduringdeflection

c ness, init ia lly - f latdis spring

risonwithamoree actsolutionas

this appro imationissatisfactoryfor

videdtheratior /r(betweenouter

3.F orstress,theagreementbetween

apereddis spring

tetheory isnotsogoodwheretheratio

usingtheappro imatemethod, how ever, it

nonlinearload-deflectioncharacteristic

thisisfarmorecomplicatedif thee act

temethod,anelementofaradially-

ially-taperedspringismeantaspring havingathic ness

t anypoint.


/ h t t p : / / w w w . h





R I N G 2 65

cutout bytwoneighboringradial

angled6 asshowninF ig.137,the

wninF ig.134.UndertheloadP the

anangle < f> asshownbythedashed

tac ingradia lly -tapereddis springs

erotationisassumedtota eplaceabout

cecf romthespringa is.

nitiallyata distancex fromO

heneutralsurface(ormiddle plane)

ngthofthiselementis

anangle< f> thefinallengthbecomes

y s i n < t > ) d 9

iselementduetodeflectionof the

weenthesee pressions.Thus

— x ( l — c o s$ )

ssmall, sinand1—cos< f> = < f> - / 2,

bethisdifferencedividedbythe


/ h t t p : / / w w w . h





L S P R I N G

altotheunit elongationmultiplied

yE (effectsduetolateralcontraction

ngthise uation,thestressbecomes

ngusedinra ilw aymotorcommutator

ifthe springdeflectionissmall,theterm

glectedandthestresswillbegivenbyE <f> y / r.

mwillthenoccur wheny= t/2=k r.Thus

ade,itisseen thatforsmalldeflections

onstantalonga radiusforthistypeof


/ h t t p : / / w w w . h





R IN G

ctingon theelementG,F ig.137,

perpendiculartothepaperw illbe

edbyta ingtheradialcomponentof

multiplyingbythe leverarmy.

nthe slicecutoutof thedis will

ementaryforcesoverthecross-section,

nthe limitsc—r andc—uandybeing

t s — k ( c — x ) a n d - - ( c — x ) .

essionforagivenbyEq uation280in

lmomentbecomes

ee ualtothemomentdueto the

testhatthemomentactingon the

oplanesatanangle d6willbed6/2ir

section,radially-taperecldis spring

uetothe loadPw hichisP (r —r().

fM givenbyE q uation283in


/ h t t p : / / w w w . h





L S P R I N G

tingandso lvingforP ,

+ ^ - W + r ^ + r . ) ( 28 5)

n8isgivenby

2 86 )

w oe uationsdeterminePasa

q uation286forandsubstitutingin

+ r < 1 ) ( 28 7)

ns, E q uations281and287maybereduced

ms

s, Eq uation287maybewritten

„ .

l ) J


/ h t t p : / / w w w . h





R IN G

rac etsisusuallynotgreatlydifferent

imationfor8willbe obtainedbyusing

hisva lue inE q uation293, acorrectedva lue

ained.Thiscorrectedvaluemayagain

93forasecondappro imation, ifdesired.

toconvergerapidly.A nothermethod

avalueof8and calculatethecorre-

q uation292.Byassumingseveral

tioncurvemaybe plottedreadily.

hec theaccuracyof theresultsob-

atemethoddevelopedpreviously,itis

ee actflat-platetheorytothisprob-

ations3whichmustbesatisfiedat any

atewhichis loadedbyaloadP uni-

sedges,are

+ - -f = 0 (2 94 )

5)

96)

naradialdirectionat radiusr,inch-pounds

tangentialdirectionat radiusr,inch-pounds

teatradiusr

lope inradia ldirectionatradiusr

usr

2 k 7 3 ( 1 - A » ' ) f o r a ra d ia l ly - ta p er e d di s w h er e

on295withrespectto rgives

u b \ / d d > d > \

differentia le uationsisgivenbyT imoshen o—S trengthof

o l. 2, Page135, V anN ostrand a lsobyA . N adai—Ktastische

ger, Page52.


/ h t t p : / / w w w . h





L S P R I N G

onsform2anddm drgivenby

97inEq uation294y ie ldsthefo llow ing

t > P

ise uation4is

- i " « > - — — — ( 29 8)

re:F orr= r ,m^ O sinceno

theedge hencefromEq uation295,

a n d

ablethedeterminationofthetwo

uation298. Theseva luesare

l, ,r - l ( 29 9)

~ * - r rw r. -

s + M )

)

-

oDr. A . N ada io f theWestinghouseR esearchL abora-

rdingthemethodofintegrationof thise uation.


/ h t t p : / / w w w . h





R IN G

heplate willoccuratthe inneredge

tinneredge(tb= 2 ri) .

298,299,and300,andthe derivative

u a ti o n 30 1 , an d t a i n g r= r f t h e m a i m u m

) r( 1- ) 1

(M-8-_)

torK dependsprimarilyonthe

outerandinnerradiusandonPo isson s

sofK areplottedasfunctionsof the

ioe ua lto . 3, whichisappro imately true

nsiderablechangeinPoisson sratio

butslightly.

mthee uation

givenbyE q uation298andintegrating


/ h t t p : / / w w w . h





S P R IN G

3 - ) P

sisdeterminedfromthe condition

roattheouter edgeoftheplatewhere

tion305thiscondit iongives

Cr „ (- " - > P

306)

q uation305andta ingr— r the

ecomes

deflectionconstantChavebeen

r / rif o rPoisson sratio^= . 3.

sC , C, K , K ' f iguredby thee actand

q uations308, 291, 303, and290, are

II.

istableshow sthatforva luesof the

uter diameterslessthan3thereis

dCwithinabout10per cent.This

ctions(say,lessthan abouthalfthe

greementwithinthis percentagebe-

redby thee actandby theappro imate

nt accuracyisusuallysufficientfor

appro imatemethodmaybeusedin


/ h t t p : / / w w w . h





R IN G

tiosa lessthan3.H owever,theagree-

od,sincethe differencebetweenK

0percentforva luesofagreaterthanabout

e e acttheorybeingsomewhathigher

pro imatetheory.

q uations302and307w erederived

deflections.H owever,whenthede-

ndK ' fo rV ariousV a luesofa

ee actf la t-platetheorybecomese -

nimprovementinaccuracyforcalcu-

btainedbymultiplyingE q uation307

mthemoree actplatetheorybythe

uation293. Thisgives

1 )-

eativetothic nessEq uation309re-

7sincetheterminthebrac etsbecomes

ns,thee uationcorrectsfortheeffect

rt ionasisdoneintheappro imate

sedontheappro imatetheoryshow s

dediameterandstressthedeflection

byA . N adai, Pade284presentsafurtherdiscussionof this.

ote1givesadditionaldetails.


/ h t t p : / / w w w . h





S P R IN G

mesama imumforva luesofdiameter

proportionsalsoresultin bettercondi-

forgingthandis soflargerratios.

etterinpracticeto usevaluesofaaround

dictateotherwise.

as—Inthepracticaluseof radially -

appliedasmalldistanceinsidethe

.140.Insuch casesitisadvisableto

ostressconstantK forra -

gs

theloadwereappliede actlyatthe

sthen multipliedbytheratio

dial distancebetweenpointsofcontact,

eductionofstresscausedbythis effect.

ationsincethemomentperinch ofcir-

hisproportionaltostress)has been

hedeflectionofthe inneredgeofthe

outerwillalsobe reducedintheratio

flectionof thepo intsof loadapplicationw ill

rethanthisor appro imatelyinthe

seofthesecorrections willimprovethe

.

applicationinpractica lca lculationof


/ h t t p : / / w w w . h





R IN G

ischapter:It isdesiredtodetermine

f lectionforaradia llytapereddis spring

mutatorapplicationinF ig.136.The

: r = 9inches, r(= 6inches, a= 1. 5

A D IU

E R R A D IU

Thema imumloadP is90, 000pounds.

39 o r T ab l e X X V I I I C = . 2 71 a n d K -

omE q uations302and307,

s . i n .

rectedbecausein theactualde-

cationisV fe-inchinsidetheedgeasin-

splacedin-

spring

sthedistanced=23A inchesandr —r4=

ueof stresswillbe110,000X 2.75/3=

areinchandthedeflectionat thepoints

ll be.157(2.75/3)2= .132-inch.


/ h t t p : / / w w w . h





S P R IN G

aremuchsmallerthanha lf thethic -

dthattheload-deflectioncharacteristicof

imatelylinear,beingmodifiedonlyby

ysupportingthe springattheneutral

ppededge( ig. 136)thisf rictionmaybe

W I TH T H E O R Y

eorygiveninthischapter, sometests

rrangementshownschematicallyin

ment,theloadwasappliedinatest-

cylindersto aheavyringwhichap-

oundtheouter circumferenceofthe

springw assupportedbyacylinderrest-

H E A D O F T S T IN G MA C H IN E

A D O F T S T IN G MA C H IN E

ent

rn g,

t e n-

etestingmachine.S ufficientclear-

ntheedgesofthe dis spring,thecylin-

ely,to ma ecertaintherewouldbe

onofthe load.Inmostofthe tests

ring wasbeveledasshown,greatly

epointofload applicationwasdefinite.

ometers,placedalongtheinneredgeofthe

measurementofma imumstress.A t

ibletoreadthee tensometerswhilethe


/ h t t p : / / w w w . h





R IN G

asesstrainmeasurementsweremade

sidesofthespring todeterminewhether

l.F ormeasuringdeflectionsadialgage

atotalofsevenradiallytapered

erdiametersvary ingf rom3.8to4 inches,

om1% to2% inches,andminimum

% inches.

dload-stressdiagramsasobtained

by thefulllinesof thecurvesof

thetheoreticallydeterminedcurves

and309(correctionbeingmadeforthe

epointofload application)areshown

tresseswere determinedfromthe

ofthespringwherethe ma imum

thepointwherefailurestarts as

sts).A modulusofelasticityE —

uare inchw asusedinconvertingthe

ntwasfoundbetweentest andcal-

tthelowerloadsfor deflectioninall

oundtobe q uitegood,butatthe

therewasasmalldeviationbetween

blytothe factthatthepointof appli-


/ h t t p : / / w w w . h





S P R IN G

t on d i s s p ri n g A '

moveinwardsothatthedistanced

thusma ingthespringslightly stif f er

r,theagreementinallcasesbetween

entlygoodfor mostpracticalpurposes.

CO N S T A N T TH I C N E S S

ingofconstantthic ness,F ig.133a,

r manneraswasdonein thecase

spring.

F oranappro imatetheory7theas-

atradial cross-sectionsrotatewithout

e< f> asindicatedbythedottedoutline

ecaseof theradia lly - taperedspringfor

sin anelementGata radiusrfromthe

f romthemiddlesurfaceof thedis w il l

uation280). Themomentof theforces

boutana isthroughO w illbeasbefore

attheelementGiscutoutby tw oslicesat

ThetotalmomentM w illbethe integra l

ngthofMaterials,PartII,S econdE dition,Pago179,V an

00100000 120000

0 IN A T IN N E R E D G


/ h t t p : / / w w w . h





R I N G 2 79

entsta enovertheareaof thesection.

glimits

ngonthespringise ua ltoP , the

ontheelementwillbe thesameasthat

4fortheradia lly - taperedspring. Eq uating

yEq uation312tothatgivenbyEq ua-

f> ,

a)

nisthen

it isclearthatthestressw illbeama i-

dr= r, . Usingthesevaluesandtheva lueof

ation312ainEq uation311andsimplif ying, the

mes


/ h t t p : / / w w w . h





S P R IN G

e acttheory forca lculatinginit ia lly -

stantthic nessisbasedonthek now n

f> betheslopeatany radiusro facir-

oaded,thenfromplatetheorythefol-

tionmustbesatisfied:

Q

t712(l-M:)

ceperunitcircumferentia llengthatany radiusr.

ig. 133w herethe loadP isdistrib-

dges

uation318andintegrating

ationconstantsto bedeterminedlater.

y radiusr, theslope< / > = — duy f / r. Inte-

w ithrespecttor,

~ C J og . r+ C , ( 3 20 )

Cl,C2,C3arefound fromthe

heouterand inneredgeswherer= r

ndingmomentsm, mustbezero. F rom

eansthat


/ h t t p : / / w w w . h





R IN G

0

nablethedeterminationofthethree

q uation320.

thetangentia lbendingmomentm2per

momentisama imumw hen

mstressisthen

on319andta ingr= r(intheresult-

andd< t> / dr, substitutinginEq uation321the

mes(for^ = . 3):

23)

n8isobtainedfromE q uation320

henbecomes(for/ i= . 3)

e pressionsforstressanddeflec-

masthoseobtainedbythe appro imate

4and316).Comparisonsofthenumer-


/ h t t p : / / w w w . h





S P R IN G

onstantsforDis S pringsV a luesofa

' obtainedby thee actandappro i-

TableX X IX .

esshowsthatupto aratiofora of

IU

D I U

terminingconstantsCand

ofconstantthic ness

ntbetw eenthee actandappro imate

lation,valuesofC andK areplotted

ain F ig.144.

ssesnndDeflectionsinF latC ircularP latesw ithCentralH o les

w riter, TransactionsA S M , 1930, A . P . M. 52-3givesafur-

ar plateswithvariousloadingandedgeconditions.


/ h t t p : / / w w w . h





R IN G

E C TIO N S

uslydiscussedforcalculatingini-

basedon theassumptionthatdeflec-

er halfthespringthic ness,forrea-

heredeflectionsarelarge,thee act

orpractical use however,theap-

nit ially -coneddis springsC hapterX IV

taccuracyfor mostpurposes.Itisonly

foraninit ia lly - f la tspring)inE q uations

efollowinge pressionfortheloadP:

ation261orF ig. 125.

safunctionof theratioS / tbetween


/ h t t p : / / w w w . h





S P R IN G

sinthecurveofF ig. 145. Thiscurve

urvedeviatesfromastraight lineafter

eaterthanabouthalfthe thic ness.

aybeobtainedfromE q uations271

Thisgives:

0 (3 29 )

venasfunctionsof< x —ro/ rt inE q ua-

esofK, area lsogivenby thecurvesfor

128fordif ferentratiosof r / r< .

C U A T IO N

d-deflectioncharacteristicofthe

isdesired, theca lculationof re uired

ametersbecomessimple . B yproceeding

neforthecase ofinitially-coneddis -

ectiondia -

/ t= 1. 75

N

N E S S

maybecalculatedfor steelsprings

dbyK .C.BergvalloftheW estinghouseCompany.


/ h t t p : / / w w w . h





R IN G

30X 10 lb. / s . in. )asinTableX X X .

load lessthanthema imumload

foundbyusingthecurveofF ig. 146.

are lative ly simplemethodofdesignfor

swithagivenload-deflectioncharac-

pro imatemethodinvolvestheas-

atS tee lDis S prings9

urveofF ig. 146w henma imumdef lectionJ= 1. 75t.

sectionsrotatewithoutdistortion,but

ttheresultsare sufficientlyaccurate


/ h t t p : / / w w w . h





A F S P R IN G

trm" f la tsprings isagenericterm

striporbarstoc madeinaw ideva-

heshapeswhich arepossibleforthis

discussionis beyondthescopeof

ronlythefundamentalprinciplesunder-

simplerformsofflatspringssuch as

igs.147and148)and theirapplica-

lbeconsidered.Inadditionthe ef-

tressconcentration,andcombined

will alsobetreated.Morecomplicated

analyzedbysimilarmethods.

cantileverspring overthehelical

espring maybeguidedalongadefi-

usthespringmayfunctionas a

asan energyabsorbingdevice.By

particularshapeitmaybepossibleto

s,thussimplifyingdesign.F ore ample,

maybedesignednot onlytoabsorb

carry latera lloadsand, insomecasesto

u e a s we l l.

S P R IN G

ngisa flatstriporplate ofrectangular

ectionas showninF ig.147.A ssuming

ndandloadedattheother, thema i-

the well- nowncantileverformula:

g,E = modulusofelasticityofthema-

nertiaofspringcross-section.(J--

w idthofspring, h= thic ness.)


/ h t t p : / / w w w . h





A F S P R IN G

antilever

e

nsofthedeflectionshowthati fb

sforspringsofcloc -springsteel)the

ratio. S inceformostmateria lsnisaround

his e uationisabout10per centless

uation350.Thereasonforthis difference

ring ofrelativelygreatwidthcompared

pansionorcontractionofe lementsnearthe

evented,whichresultsinaslightlystiffer

amtheory.Thisstiffeningis ta en

1—n2)inEq uation331. Inmany

tionsthedeflectionwill probablybe

alculatingdeflections

pezoidalprofile


/ h t t p : / / w w w . h





S P R IN G

tedbyEq uation331thantothatca l-

301.

built-inedgeO ,F ig.147,isgiven

eseformulasarebasedonusual

essmalldeflections.Thecasewhere

consideredlater. Where variable

areinvolved,thestressesmustbe mul-

tionfactorswhichdependon thecon-

e.A discussionofthiswillalso be

gs—In manycases,leafsprings

wninF ig.149may,forpracticalpur-

deredascantileversprings oftrapez-

ig. 148. S uchaprof ilema esamore

lthan doestherectangularprofileof

loadPthema imumstressisagaingiven

re inthiscaseb isthew idthatthebuilt-

nordinarybeamtheory,however,shows

reasedoverthose obtainedinthe

rectangularprofileby anamount

betweenwidthatthefree andbuilt-

analysisshowsthatin thiscasethe

givenby

M

-, , :)

ert iaatbuilt- inends. ThefactorKl de-

eta enf romthecurveofF ig, 148. It

IV givesafurtherdiscussionof thiscorrection.

ainedbydividingthebendingmomentbythe section

rnet section.


/ h t t p : / / w w w . h





A F S P R IN G

ctionofatrapezoidalprofilespring is

ngularpro f i lespringP l:< / E I mult iplied

om1forfo / fo = l (rectangularpro f i le )to

rprofile).Theoreticallythemosteffi-

whereb/b 0sinceotherthingsbe ing

imumdef lectionforagivenva lueof load.

owever,usuallydictateavalueofb/bn

sesw hereb islargecomparedtothe

arytomultiplythedeflectiongivenby

ctor(1—/ 2)ase pla inedpreviously.

smentionedpreviously,thebeam

tions331to333are basedassumessmall

gise uivalenttoacantilever

e

pringlength.In somepracticalcases,

ctionscannotbeconsideredsmall.This

0. Whenthespringisdef lectedbyan

heorywillhold. H owever,whenthede-

,S 2itmaybe seenthatthemoment

isconsiderably lessthanthe lengthIo f the

creaseinbothstressand deflection

fromE q uations332and333.

the deflectionislargecompared

oreaccuratemathematica le pressionfor

ofthebeamisused.I fx isthe

endO of thebeam( ig. 150)and8

ce,then bye uatingthecurvature

rnalbendingmomentdiv idedbyE IX ,


/ h t t p : / / w w w . h





S P R IN G

esults:

inertiaatdistancex .

widthisalinear functionofthe

nessthemomentofinertiaat adistance

appro imate lyby

essioninE q uation334, thefollow ing

tion1utilizingtheboundarycondi-

atx —0, y= 0, anddy /d = 0, thereduc-

E CTI O N

ring withdeflections

onbelowthosecalculatedfromE q ua-

e pressedasfunctionsofthedimen-

/ I andtheratiob/ b betweenw idth

atbuilt-inedge.In F ig.151aregiven

uation335forestimatingthepercentage

rossandL ehr,publishedbyV .D.I.,Berlin,1938,Page133,

gration.


/ h t t p : / / w w w . h





A F S P R IN G

erspringsof trapezoidalprofilefor

va luesofc=P l2/ Iaascomparedw ith

eusingE q uation332. Whereb/ b ~0

dand,in thiscase,thestressreduc-

stimatingstressreduction

antileversprings

12percentfor valuesofcbetween0

ore ample , if c= landthestressis

t ion332, theactua lstressw illbe12per

pringsofrectangularprofilethe varia-

tw ithinarangec—0toc= l. InF ig.

o estimatethepercentagereductionin

ca lculatedbyusingEq uation333. F or

hiscorrectionvariesf rom0to8percent

(b/b = l)andf rom0to18percent

b/ b 0). It isclearthatthecorrec-

becomessmaller.A lthoughinmost

secorrectionsmaybeneglected,for

rlywhereb/b issmall,theyshouldbe

ample, to il lustratethepractica lutil iza -

,a cantileverspringoftrapezoidalpro-

efollowingdimensions( ig.148):1=

b = . 2, b = 6in. , P= 200lb. Themateria l

0 poundspers uare inch. F orthis

cbecomes


/ h t t p : / / w w w . h





S P R IN G

M ) 3

thenomina lstressis

0 l b. / s . i n .

1, forc= . 77, b/ b = . 2, there isa6V 2

ssasaconse uenceofthelargedeflec-

R

iecalculationoflarge

prings

is96000(1-.065)= 89,700pounds

m F i g . 1 4 8 fo r b /b . 2 , K , = 1 . 3 1 an d t he

q uation333,

orrectedby thepercentagegiven

2forc= . 77andfo / fo = . 2w hichindicates

estimatedlO MtpercentifEq uation

eflectioniscorrectedby afactor(1—


/ h t t p : / / w w w . h





A F S P R IN G

.105)= 9.05inches.Inaddition

hb islargecomparedw iththethic ness

eductionbymultiplyingby (1—/ i2)= . 91

cussedpreviously).Thisgives afinal

1)= 8.25inchesisconsiderablyless

uredby thesimpleformulaofEq ua-

S P R IN G

etchofasimple leafspringloaded

dsupportedbya force2Patthe bot-

ffrictionasafirst appro imation,this

sa simpletrapezoidalspring.Ifthere

andwidthb1andif thereareatotal

ocomotiveWor s

aspringforlocomotive

g. 149, b= nib1andb =nbi-Theratio

hecurvesofF igs. 148, 151, and152may

falargeleaf springasusedin locomo-

ig. 153.

S P R IN G UN D R C O MH I N E D L O A DI N G

ngw hichf re uentlyoccursinprac-

gwithoneendrigidlybuilt inandthe

rallybut restrainedfromrotation

ype shownschematicallyinF ig.154.

eightin theverticaldirection,this

by thea ia lforceP . S uchcasesoccur

edcasesofe ll ipticleaf springsandthosesupportedby lin s

risreferredtotheboo byGrossandL ehr, loc. cit .


/ h t t p : / / w w w . h





S P R IN G

supportedbyspringsofthistype,the

byacran arrangement. A n e ample

wninF ig.155whichshowsanappli-

apermachine.Theverticalstripsvisible

sofMicarta( now nassha esprings)

fthe table.

cttoessentially

wnschematic-

ia lloadF is

hebuc ling

essmayeasily

eamtheory.

are

m( ig. 154),

moment

w idth, h— thic -

ton, a= nomina l

ss concentrationneglected).When

tigueloadingas mentionedpreviously

be multipliedbyafatiguestrength

ntoaccountthe stressconcentrationat

altestdatarelativeto thevaluesofsuch

P, F ig. 154isnotsmallcomparedtothe

tions336and337no longerapplyaccurately .

rateanalysisshowsthatthestressand

ymultiplyingtheresults calculated

y factorsC , andK , w hichdependonthe

nthisPcr= E Iw 2/l2istheEulercrit i-

ngedends.These factorsare:


/ h t t p : / / w w w . h





A F S PR I N G 2 95

husbecome:

luesofC 1andK 2areplottedaga inst

ppro imatemethodofca lculating5thefac-

Y -

espringsofcantilevertypeonF ourdrinierpaper

combinedlatera landa ia lloading

heactionofthe loadsPandQ the

e curveoftheform

dinthe beamis(usingthise ua-

ethodisappliedtothecaseofasimplysupportedbeaminTheory

T imoshen o, McGraw -Hill , 1936, Page28.


/ h t t p : / / w w w . h





S P R IN G

eflectionS increasesbyasmall

uation342thepotentialenergyV changes

)A 8/ 16/ 3, neglectingsmallquantit iesof

orceQ doesthew or @A 8anditmay

lmovementof thea ia lforceP is

8/ 8Z w hichmeansthatthew or doneis

culatingthedeflectionofacantileverspring under

ia ltypeof load

atingthechangeinpotentia lenergy tothew or

, thefo llow inge uationresults

.2 I

ameasE q uation341. Ma imumstressis

modulus(Z = foh2/ 6).

essionforQ obta inedfromEq uation


/ h t t p : / / w w w . h





A F S P R IN G

ectionfactorforcantileverspringsunder combined

ottedagainstloadratios

q uation340isobtained.

owsthatwherethe a ialloadis half

ad, thefactorC , = 2, i. e ., tw iceasmuchde-

edthanifEq uation336basedonordinary

ntheotherhandforratiosP / P r, e ua l

ig.157indicatesthatthe stressformula

ssthan10percentinerror. Inaddit ion,

casewherea cran arrangementis

thestressesfiguredfromE q uation

higherthantheactualstresses.

twhenthespringwidthis large

ss(asitusuallyis) thecalculateddeflec-

t ion341shouldbemultipliedby1—n2as

stressasgivenbyEq uation340di-

Po isson sratio . F ormostmateria ls

percentincreasein stress.

G

noftheuse oflargeflatspringsin

chematicallyinF ig.158whichrepre-

ringssupportingtheframeof a60,000


/ h t t p : / / w w w . h





S P R IN G

phasealternating-currentturbo-generator.

ementisto absorbtheperiodictor ue

gle-phasegeneratorofthis typewith-

ectionablevibrationtothefoundation.

T R A M :

carrangementofplatespringassemblyfor

ting-currentgenerator

his typeofspringis shownbythe

romthebeamtheory,thedeflection

edas

ectionofaplatespring

imensionsshow nonF ig. 159andP is

rverywidespringsin relationtothe

nshouldbemultipliedby1—/ <» ase -

a imumstressisgivenby


/ h t t p : / / w w w . h





A F S P R IN G

normally sub ecttore lative ly low

evereshortcircuitconditions(which

thedesignstressesare reached.

methodsforcalculating nominal

variousflatand leafspringapplications

ses,however,theeffectofthese nom-

bystress concentration,whichmaybe

mpededges,sharpbends,sudden

thespringisunder apurelystatic

epeatedarelativelyfewtimes,such

tsmayusuallybeneglected,inprac-

erialsof goodq uality.H owever,

ho leunderbending, ho le

omparedto strip

euppercurveofF ig.

d tofindtheoretical

r

oadingoccurs,carefulconsideration

stressra isers by thedesigner, since

artat localizedpointsofstressconcentra-

etefailureofthe spring.

tionintoaccount,it isnecessaryto

alstressto whichthespringis sub-

ththediscussionof ChapterIthe

ameanstressa andavariablestress

ethema imumandminimumnomina l

(f fm< w + amm)andac=^ (o-ma —amin)-

C N TR A T IO N E F F E CT


/ h t t p : / / w w w . h





S P R IN G

ss a,afatiguestrengthreduction

To beonthesafeside, intheabsence

ueofK fmaybeta ene ualtothetheo-

nfactorK t(seePage125).In some

ctorK fmaybeappreciablybelowK t,

factorsmaybenearlye ua l. F atigue

articularlytrueofthefine-grained,

sedforsprings.F orthisreasonthe

rationfactorsgiveninthe following

tisfactoryforuse untilmoretestdata

it isdesirableornecessary toprov idea

dingthespring inplaceorfor manu-

S S

R

R

ress concentrationfactorsfor

g.N otethatthelowercurve

circularnotches

mmone ampleisthesemiellipticauto-

uallyis providedwithaholein the

isholeholdsthe leavestogetherand


/ h t t p : / / w w w . h





A F S P R IN G

etweentheleaves.

lessmall indiameterrelativeto

sshow ninF ig. 160(thismayoccurin

gs),it appearsreasonabletoapply

estson tensionbarswithholes 5.The

gw ithhole large indiametercompared

whered/hislargeconsiderably

factorsmaybee pected.F orsmall

3comparedto1.85forsmall d/w

howsvaluesoftheoreticalstress con-

rminedphotoelasticallyasafunction

iameterandplate width.Thuswhere

orK tapproaches3, awellk now nresult

wever,forsmallhofesitis possible

ngthreductionfactor K Iwillbecon-

the" sizeeffect maybepronounced.

hinsprings(fore ample, thosemade

wheretheholediameterd islargecom-

esshasshow ninF ig. 162, bothtests

derably low erva luesofKt w ille istthan

all.A mathematicalanalysisby

mall holeinawidestrip underpure

hered/h islargethisfactor K (— 1.85.

mallerthanthe valueof3foundfor

/ hissmall. Sincew hentheholediam-

idtha factorof1maybe e pected,

, A ug. , 1934, Page617, andMechanica lEngineering, A ug. ,

scriptionsofsuch tests,togetherwiththeoreticalstresscon-

, V . 22, 1936, Page69. Thisw or hasa lsobeenchec ed

montusingstrain-measurementsonalargealuminumplate.

hnica lN oteNo. 740. V a luesofK(= 1. 59w erefoundforrf / t = . 145


/ h t t p : / / w w w . h





S P R IN G

urve isthatshowndashedin F ig.161

stressconcentrationfactorsisil-

ig.163whichshowsanestimateden-

high q ualityspring-steelstripina

tion(thic ness.006-inch).Theen-

alwas foundbytests(carriedout

of theWestinghouseR esearchL abora-

poundspers uare inchinreversedbend-

gthwas275,000.It maybee pected

ub ecttoastressrangebetw eena , i

acharacterist icappro imatelyasshownby

Thusthestress rangeforcompletere-

andBorbetween-(-130,000and

uare inch. F orarangef rom0toma i-

aeGandH f rom0to180, 000. N ow , ifa

trip, assumingthatd/ w issmalland

actorK, — 1. 85. Sincethismaterial

tiveto stressconcentration,itmay

thatthe fatiguestrengthreduction

atelye ua ltothisvalue. Onthisbasisa

owadiagramasindicatedby the

thepointC andDrepresentingnominal

±70,000.Theordinatesbetweenthe

andeithertheupperorlow erfull l inearea lso

o toma imumstressrangeforthe

esentedby the lineEF or0to112, 000

Inthis case,therefore,thestrength

cationhasbeenreducedbythe pres-

,000to112,000poundspers uareinch

112,000= 1.6.Thisisconsiderablyless

eductionfactorassumedfor completely

1.85.Thisdifferenceis aconse uence

ssconcentrationeffectsmaybe

ticcomponentoftheappliedstress

tualtestswouldprobablyshowa

ncelimitforthestripwith aholethan

thisw ay , dueto" sizeef fect forsuch

se,however,themethodofcalculation


/ h t t p : / / w w w . h





A F S P R IN G

fordesign.A lsotheassumptionthat

tsmaybeneglectedincalculatingthe

maynotbe entirelycorrect( ee

roducea furtherdeviationbetween

ults.

presentsthevaluesto bee pected

ecurvesforahigh-gradespring-steelstrip,

facegroundandpolished

h-gradestripmaterialinavery thinsize

acein goodcondition,i.e.,ground

hic ersections, suchasareusedinleaf

surface intheconditionleftby rolling


/ h t t p : / / w w w . h





L S P R I N G

0000OO C OO

S

urancecurvesonleafspringmaterial

reatmenton thebasisofavailabletest

esofendurancelimit maybee -

nF ig.163.Theresultsofendurance

sedinleafspringsare indicatedin

low ercurvesA andA ' representthe

steelplatesasusedin leafspringswith

ethedecarburizedlayerleft byheat

eshadedarea representtestresults


/ h t t p : / / w w w . h





A F S P R IN G

B atsonandB radley , andHoudremont

springsw iththesurfacesuntouchedafterheat-

pectedthat,forleafspringsthe results

somewherewithintheshadedarea

weringoftheendurancerangetofrom

rmachinedorgroundspecimensistobe

uenceof thedecarburizedsurface layer

hnotchesinbending. Where

urveofF ig. 166. Where

wercurveofF ig. 161

owever,forhighq ualitymaterialsand

eatmentsitispossible thatimproved

erthevalues indicatedonF ig.164.

tressconcentrationeffects(holes,

thevaluesoflimiting stressrangeas

ucedstill lower.

ificandIndustrialR esearch, (B rit ish)Spec. R eportN o. 5,

onofNfechanica lEngineers, 1931, Page301.

, publishedinSta ll \ indEiscn, July7, 1932, Page660.


/ h t t p : / / w w w . h





S P R IN G

sitisnecessaryforvariouspractical

si nthesidesof flatsprings.W hen

rcularform,the stressconcentration

sfollows:Ifthe stripisrelatively

/hbetweennotchdiameterandspring

dicatedinF ig. 165a, itappearsreasonable

toelastictestsonnotchedbarsin ten-

esofthetheoreticalstress concentration

R

IP

ress concentrationfactors

thic strips,smalld/h

swayareplottedas afunctionofthe

ameterandplatewidth.

sinthin stripmaterialsasshownin

ratiod/hislarge, it isreasonabletoe pect

bepracticallythe sameasthatfora strip

ameterdandthesamewidth w.Itis,

thelowercurveofF ig.161mayalso

ationforthiscase.Thedashedcurve

note6discussresultsof suchphotoelastictests.


/ h t t p : / / w w w . h





A F S P R IN G

ore,representtheendurancediagramfor

hsemicircularnotchesunderbending

all).

ngflatsprings,sharpbendsarefre-

ampleof thisisthebendatA inthespring

pshow ing

tdue

at A

ecauseoftheirsharp curvature,these

tressconcentrationeffectwhich may

rrepeatedloadingby usingastresscon-

vedfromcurvedbartheory1-.V alues

ousvaluesoftheratior/h betweenmean

nessofmaterialaregiveninF ig. 168.

ctorK tincreasesrapidlyasr/h ap-

tanceofavoidingverysharpbendsin

whensub ecttorepeatedloadingis

nwasmadepreviouslyof theeffect

atespringsincertainapplications.

uentlynecessarybutbecauseof theclamp-

ountof stressconcentrationissetup

erfatigue loadingthisw illresult inare-

belowthatobtainedwhenthe spring

nbythedashedlineb inF ig.169which

ssconcentration.Inaddition,under

soa certainamountofrubbingat

,whichresultsin thisso-calledrub-

t.This latterresultsina further

ngthwhichmaybe particularlygreat

erials.

tobelittle dataavailableonthe

ppliedtospring steels,someresultsof

edbyH an ins . Thesetestsw eremade

edmorefully inChapterX V II.


/ h t t p : / / w w w . h





S P R IN G

springsteel(% - inchthic )w iththe

erheattreatment,theloadingcon-

oseofF ig.169.W henthe,specimens

rmwidththeendurancelimit incom-

tressfora.48 percentcarbonsteel

O F B N D

N E S S

centrationfactorsK tfor

the specimenswithoutstressconcen-

evalueof± 47,000.Thefatiguestrength

fore,was47,000/33,600=1.4.S imilar

onspringsteel showedareductionin

00to45,000anda fatiguefactor

eregivenastotheactualclampingpres-

uldbenotedthatfor higherstrength

cehasbeengroundafter heattreat-


/ h t t p : / / w w w . h





A F S P R IN G

D TH

gw ithclampedend. Duetoclamp-

ntrationoccursatA

ofK f thanthesearepossible . F or

usedinrollerand ballbearingswhere

ts, va luesofKfashighas3to4havebeen

aftsareused.F oramedium-carbon-

-inchdiametercarbon-steelshaft

m1.4to 2,dependingonfitpressure,

O F F L A T S PR I N G

aratherunusua lapplicationw herethe

edoutadvantageously isthe latera le -

ig.170.Thepurposeofthis instrument

)isto measureminutelateralcon-

modelofBa elitewhenstressed.

001-inchtotal toobtainsayoneper

arytomeasurethesemovementsto

ownthattheselateralcontractionsare

sumof theprincipalstresses15atany

nderload.Bythusdeterminingthe

nypoint,anddeterminingtheirdif-

photoelasticmethodsthecompletestress

sonandWahl, " F atigueofS haf tsatF ittedMembers , Journal

5,PageA -lgivesfurtherdetails.

,1939,presentsamorecompletedescriptionofthesee -

simaginedascutoutofa f la tspecimenunderloadand

dtoberotateduntilno shearingstressesactonits sides,the

pendicularplanesare calledprincipalstresses.


/ h t t p : / / w w w . h





L S P R I N G

tensometerutil izesf la tsprings

menmay befound.

ale tensometerisshow ninF ig. 171.

nd P arepressedlightlyagainstthe

thepressuree ertedby thef la tspring.

holdthepointsat adefinitelocation

wholeassemblyis supportedbythe

ig.170,astatic balanceoftheinstru-

ansofthe weightsshown.Becauseof

tyofthesupportinghelicalsprings, slight

ofthetestspecimenmayoccur without

of latera le tensometer


/ h t t p : / / w w w . h





A F S P R IN G

f thepointsPandP ( ig. 171)resting

beseenthatany latera lcontractionof

ea relativemotionofpointsPand P

otionofthe barBwithrespectto the

cordedontheH uggenbergere ten-

areheldagainstthe instrumentbya

ereflatspringshavebeenusedto form

theshortgage-lengthe tensometershow nsche-

cs etchforarrangementof

someter

sinstrumentisused inthedetermina-

nwherea shortgage-lengthisneces-

rumentconsistsoftwolight, hollow

feedgesB andB . Thesek nifeedgesarc

ngstee lstripsS ( ig. 172a)w hichare

ung, " A nInvestigationof theC ross- pringPivot , presented

. M. . meetinggivesdesignformulasfore lastichinges. Thesemay

esas highas45degrees.


/ h t t p : / / w w w . h





S P R IN G

ee tensometerpointsarepressed

eansofa specialclampingarrange-

R eferringtoF ig. 172aw hendeforma-

urspo intB movestosayB . Thiscauses

erarmT asindicatedbythedashed

mationofthespringsteel strips.The

semblypivotsaboutthepointO ,and

e leverarmsaconsiderablemagnifica-

5in thiscase).Thusanyrelativemo-

sthuscommunicatedtothetargetsA a f ter

erratio.The relativemotionofthe

bymeansofa microscopewithameasur-

odahigh magnificationisobtained.

( ig. 172a)itw asnecessary to

sothatbuc lingduetotheclampingload

sametimesufficientfle ibilitymust

reciablerestrainttothedeformationof

Byusingflat springsinthisway,the

heirdisadvantageswasavoided.

tspringsis showninthespecial

orholdingthee tensometerpo ints

Theclampingpressureis suppliedby

wn.TheflatspringsDand D allow

amp whenthescrewE isturned.

maybelocatedaccurately.A definite

tensometerbycompressingthehelical

t.Itis importanttomaintainadefinite

hightopreventslippageof thepoints

causebuc lingof thef la tsprings. A

o f


/ h t t p : / / w w w . h





A F S PR I N G 3 13

eclampingpointP (notshown)isalso

isessentia llya f la tspringw hileA isa

eof thisarrangementistoallow

tensometerrelativetotheclamp,

tensometerandmagnetclamping

sitiononshaftfillet

test specimen,bothlaterallyand

ultinginlateralforceswhichmay

umentor slippageofthegagepoints.

ctswould produceerrorsintheresults.

espossibleathree-pointsupportforthe

e sametimeslightdistortionsofthe

BandCmaybe ta enupbydeflection

gappreciablelateralload onthe

atsprings itwasagainnecessaryto

ndertheactionof theclampingforce.

someterclampedontoa largeshaf t

4. ThemicroscopeM, thetargetA , the

ntsB , andthemagnetN are indicated.


/ h t t p : / / w w w . h





I

S I O N S P R IN G

veessentiallythesameshape as

sionspringse ceptthattheends are

thespringmaybesub ecttotor ue

useofthemodeof stressingsuchsprings

ural,incontrastto thehelicalcompression

primarystress istorsional.S ome

orsion springsareshowninF igs.175

ng endismadeprimarilyfromthe

nge ternaltor uetothespring. Tor-

arnesCo.

poftorsionsprings

widevarietyofapplicationsamong

oorhingesprings,springsfor starters

sforbrushholdersin electricmotors.

methodof loadingatorsionspringisin-

erethespringissupposedtobew ound

beingfastenedtothe rodwhilethe

pro ectingoutward.Ifthisarmis


/ h t t p : / / w w w . h





S I O N S P R IN G

adiusR f romthea isinsuchamanner

nthemomenttendingtotwistthe spring

the figure.Becauseoffrictionbetween

actualmomentmaydecreasealong

actcalculationbecomesinvolved.

gsare woundcold,itisadvisableto

hatthespringtendsto windupasthe

y lesofendsusedintorsionsprings

for thisisthatthe residualstresses

ceof theco ldw indingare insuchadirec-

epea stressduetotheloading,pro-

esamedirectionas thatinwhichthe

ngsub ectto forcePatradiusR

ection ofloadingissuchas toun-

abletoheattreatbymeansof ablu-

uchresidualstresses.

loadingthe springinthisman-


/ h t t p : / / w w w . h





S P R IN G

177fora loadtendingtow indupthe

againstthearborandthepea bend-

w illbePR . H ow ever, if the loadisin

atshown,the pea momentwillbe

meancoilradius. Thismeansaconsider-

ticularlyif risaboutas largeasR .

manycasesw heretheendsof the

pecialendsareused,somestressc on-

ctedneartheends.Thesestressconcen-

yconsideredparticularlyifthe spring

ading, orif it istobesub ecttoa large

duringits life.O ntheotherhand,

cationsissmallduringthe springlife,

smayprobablybeneglected.

nspring (forusualapplications)

,itsdiameterdecreases.Indesignit

clearancebeallowedbetweenthe

e springiswound,andtheinner

s isnotdone,thespring maybind

andhighstressesmaybeset up.The

beestimatedfromthecalculatedde-

springasgivenbyE q uation367. Thus,

90degreesorV 4-turnandthespringhas

changeintheratioofV * to8orabout

wedforin design.

beandis loadedsoastounwind,

beallowedbetweentheoutsidediam-

nsidediameterofthe tube.This

edinthesame mannerasbefore.

esquite longtorsionspringsareused.

s alwaysthepossibilityoftorsional

oidedby providingsuchspringswith

es.Byproperlyclampingtheendsor

nload,it ispossibletoavoidbuc ling

se ofguides.

formanufacturingreasons,torsion

wire. H owever,wherema imum

redinagivenspace, theuseofs uare

advisable.Thereasonfor thisisthat


/ h t t p : / / w w w . h





S I O N S P R IN G

rrectangularsectionhasa largerpropor-

tedtostressesnearthepea va luethanis

. Conse uently , forthesamepea stress,

aybeobtainedfora givenvolumeof

ctangularwirethanfor circularwire.

nde , morematerialmaybecompressed

meterin thecaseofrectangularwire

mountof energywhichmaybestored

X IIdiscussesthisfurther.

signed,mosttorsionspringsmay

cttoa purebendingmomentaboutthe

eusua lsmallpitchangles, thespringmay

ngclementactedonbybendingmomentM

vedbarsub ecttoabendingmomentM,

bartheorymaybeapplied1.

bed,F ig.178,cutfromthe spring

passingthroughthecenter ofcurva-

andincludingasmallangled > , the

f thebar, V > , themeanco ildiameter, is

ngthofMateria ls. Part11. SecondE dit ion, page65givesa

eory.


/ h t t p : / / w w w . h





S P R IN G

rta inradiusr therewillbenostressin

cecorrespondingtothisradiusisk now nas

tMactson thiselement,ifitis as-

tionsnormalto thecenterlineremain

nbc deflectsthroughasmallangle

e s u p th e p os i ti o n ef .

alelementshownshadedata dis-

surface is(r —y)d< f> beforebending

helengthincreasesbyanamounty (A d< / >).

sbe

willbethis elongationtimesthemodulus

material.Thus,

tdiestressdistributionacross thesec-

sindicatedinthe diagramofF ig.179.

wnradiusr oftheneutralsurface,

ardef lectionA d< f> , tw oe uationsart-

edfromtwoconditions:

orcesactingoveraradial crosssection

e ternalforce(butonlyane ternalmo-

tsoftheelementaryforcesabouttheneutral

thee ternalmomentM.

areaoverw hichthestresso> acts

ularcrosssectionw herehisthewidthof

mthefirst conditionmentioned,

on,


/ h t t p : / / w w w . h





S I O N S P R IN G

ybewrittenas

- ( A d < t> )

_ M

-yI

hisgives

eofthecentroidofthe sectionfrom

uation347,

345, thee pressionforstressaa

ndrectangularcrosssectionsthe

urat theinsidesurfaceofthe spring

y= ruthe insideradiusof thebar, F ig. 178.

sevaluesinEq uation348, e pressionfor

. ratthe insideof thecoil, F ig. 179, becomes

utsideofthe coilisobtainedbyta ing

r 2 i n E q u a ti o n 34 8 . Th u s,


/ h t t p : / / w w w . h





S P R IN G

gnifiescompressionforthedirec-

ated inF ig.178.

B A R T O R S I O N S P R IN G

sectionofw idthb, F ig. 178, dA = bdy

uation346,

tion, thevalueof r maybedetermined.

nisfacilitatedbythefollowingmethod

o2. L etting / , = / - (-<? w here(/, isthe

crosssectionfromthe centroid,then,

a n d e~ t — r , f r o m E q u a ti o n 34 9

_ f M_ r -d A - = Q ( 3 50 )

,

on351,

- dA = A a + » > ( 3 52 )

tions351and352inE q uation350,

forarectangularcrosssection,the


/ h t t p : / / w w w . h





S I O N S P R IN G 3 21

ation351maybee pressedinseriesform

. . + ( ^ y , + . . )

/

uation351, ta ingdA = bdy l, andso lving

y , + _ ,

1 r 3 /

2r/h,thevalueofmbecomes

h ( 35 4

2)—eforrectangularsection, bysubstitution

q uation353,

ubstitutingE q uations353and355in

i n g r , = r — h 2 a n d A = " b h , th e ma i m um

tion becomes

thespringinde c= 2r/ 7iforarectangu-

rec isoverthree,theseriesof

vergesveryrapidly andsufficientaccuracy

beobtainedby ta ingtwoterms.


/ h t t p : / / w w w . h





S P R IN G

thise uationisverysmallcompared

springs,thismaybewritten:

orminE q uation356, thema imumstress

ssimplythe ordinaryformulafor

sub ecttoabendingmomentM (i.e.,

f actorK, greaterthanunityw hichde-

c

candw hichmaybeconsideredasa

r.V aluesofK .,areplottedasfunctions

nthat, asw ouldbee pected, thevalue

ninde , sincethespringbarthenap-

straight barunderabendingmoment.

de o f3, thecorrectionfactorK. , ~ 1. 30.

stressisabout30per centgreaterin

dbytheusualformulain whichthe

cted.

he deflectionofatorsionspring

e-sectionw ire , theusua lbeame uation


/ h t t p : / / w w w . h





S I O N S P R IN G

ctingeffectsoffrictionandassumingthat

constant momentMasbefore,theangu-

tof thespringoflengthds willbe

nertiaofthewire section.Thetotal

radians)willbetheintegral ofthis,or

elength ofthespringwireand nthe

R -

c - T- O R ^ -

ssconcentrationfactors

andrectangularw ire

ngthis valueof/,theangulardeflec-

ualbeame uationw ithmoree actresultsca lculatedfrom

hatthedifferenceisnegligiblefor practicalpurposes.Thisis

ressionsprings,wheretheusuale uationfordeflectionis

practicalpurposes,althoughderivedin anelementaryway.


/ h t t p : / / w w w . h





S P R IN G

pringindegreeswillbe 57.3timesthis

ngissub ecttoaforcePattheend

sindicatedinF ig. 177, themomentM—

ialdeflection8atthe endofthisarmwill

uation358thedeflectionatradiusR is

T O R S I O N S P R IN G

lthoughtherectangular-wiretorsion

cientuse ofmaterial,springsofciicular

yusedforreasonsofeconomy.Thestress

uredinasimilarwayas before.Ifd

singE q uation348thema imumstress

ssedasfo llow s:

oceduremaybeusedasin thecase

rng. F romE q uation360, ta ing

c= = 2r,dandA -* • (/-4forcircular wire,

ritten

uation352isused.Ta ing


/ h t t p : / / w w w . h





S I O N S P R IN G

s

/

pressionundertheradicalmaybe written

uation362, ta ingA = (ir/ 4)d-andsolv -

rarectangularcross-section,theinde

sectionc= 2r/ d)

herethe inde c> 3, thisseriescon-

msaresufficient forpracticaluse.Thus

hebrac etsof thise pressionissmall

ttenwithsufficientaccuracyas

q uation361,andsimplifying,the

umstressbecomes

o llow inge pression


/ h t t p : / / w w w . h





S P R IN G

uation364representsthestressfig-

afor astraightcircularbar,i.e.,stress

noversectionmodulus. ThefactorK,

asedueto thehyperbolicstressdistri-

esofK, areplottedasfunctionsof thespring

tisseenthatforaninde o f3thestress

about1.33.It mayalsobenotedfrom

sofK, donotdiffermuchf romthoseofK . ,

hesameinde .

he deflectionofatorsionspringof

pressionmaybeusedasthat usedfor

tthatthemomentof inertia / ista enas

vesfortheangulardeflection

ghl= 2irrn, thise pressionmaybewritten

berof turns, sinceoneturn= 27rradians,

(3 6 7)

toamomentsetupbyaforcePat

radiusR , F ig. 177, thenM= PR andthe

atthisradiusw illbee ua ltoR < f> ,

radians. H ence, usingEq uation366, the

mes

A sane ample , thedesignofabrush-

catedin F ig.177willbe considered.

dtoapplypressureonthe carbon


/ h t t p : / / w w w . h





S I O N S P R IN G

ssumingthefollowingdimensions:

meancoilradiusr=3/ 16- inch w irediam-

ginde c= 2r/ d—9. 4, the load' ? atthe

* pounds. F romF ig. 180thefactorK ,

q uation364thema imumstressis,

25X . 875, BB nMt. . .

lb . /s . i n .

s,fromE q uation367thedeflection

10 poundspers uare inch,

X . 187X 10

5 4 tu r ns

)

ctionof slightlymorethan180degrees.

tressesforTorsionSprings

n H i - in c h W i r e S i z es % t o V i - i n ch

M a i m um M a i m um

al W o r i n g To t al

sStressS tress

00180,000215,000

00215,000140,000180,000

00180,000120,000155,000

,000180,000115,000145,000

5,00048,00070,000

0060,000

or ingstressesfortorsionspringsof

diameteraresuggestedbyW allace

1.


/ h t t p : / / w w w . h





S P R IN G

oodguidein designingspringsofrea-

neraluse.Theseareshownin Table

Wire Company(ManualofS pring

vesvaluesofma imumdesignstress

asindicatedinTableX X X II. These

foraverageserviceconditions,defined

ere,temperaturesnote ceeding150

velyslowlyvaryingorstaticloads. The

mumDesignStresses

tions)

ucedin generalforthelargersizes of

esarealsogivenforstainless steel,monel

ybesatisfactorywhencomputed

or364whichta e intoaccountthestress

andprovidedthespring issub ectto

ssduringits life.

ecttorepeatedorfatiguestresses

ge,ingeneralit willbenecessaryto

sesthanthosesuggestedinTablesX X X I

lsotrue if thespringissub ecttoe levated


/ h t t p : / / w w w . h





II

G

stingessentiallyofflat stripwound

nyadvantagesfromthestandpointof

itedspace,particularlyif thespring

r ue. Inaddit ion, suchspringsarere la -

re.Becauseoftheseadvantages,

edin cloc s,watches,electricalin-

ces.O therapplicationsinclude

g.182,phonographmotors,etc.A nun-

sho lderspringtormotor

ringasan energystoringdeviceis

ta lcircuit-brea ermechanismofF ig. 183.

undthatindividualturns donot

sisfor thespringmaybecarried out

cy. S uchane ample isprov idedby

ntheotherhand,if theturnsofthe

gether,asis trueofaphonograph

rtof analysismustbemadebecairse


/ h t t p : / / w w w . h





S P R IN G

menta lmeehanismutilizesspira lsprings

hesecaseswill thereforebetreated

rposeofthischapterbeinga discussion

alspringcalculations.

M A N Y T UR N S W I TH O U T CO N T A C T

hefirstanalysisit willbeassumed

pringisclampedby amomentM,as

so willbeassumedthatthespring has

hichare,however,separatedsuffi-

tturnsdonotcome incontactduringde-

he springisfastenedtoan arbor

andisactedonbyator ueM .

, attheouterendA o f thespring

alforceR (passingthroughO )and

ee ternaltor ueM is

ntatanypointof thespiralhaving

y , thenf romthestaticalcondit ionsofe ui-

( 3 70 )

TheoryofE lasticity , Ox ford, ClarendonPress, 1936, Page66.


/ h t t p : / / w w w . h





G

ation369andsubstitutinginEq ua-

ortlengthdsofthe springacted

mordinarybeamtheory2

felasticity andIthemomentof inertia

erestripmaterialisused asindicatedin

urateresultswillbe obtainedbyreplacing

/ = P o i ss o n s r a ti o .

hespringis

enoverthetota llengthof thespira l.

statesthatthepartialderivative

hrespecttoastatically indeterminate

trengthofMaterials,PartI,S econdE dition,V anN ostrand

ng

e


/ h t t p : / / w w w . h





S P R IN G

oesnow or , mustbezero . Sinceneither

entM, dowor asthespringdef lects,

ddifferentiatingundertheintegral

hofthe spiral.

dconstant, f romthesee uationsthefo l-

,

ogetherw ithE q uation371inE q ua-

M, - R x ] ^ - = 0 ( 37 7)

ations371and375,

M, - R x ] x d s = 0 (3 78 )

alsostatesthatthepartial derivative

hrespecttoane ternalmomentgives


/ h t t p : / / w w w . h





G

tothis moment.Thustheangularde-

rnalmomentM becomes(usingE q ua-

on371withrespectto M,

and380in379angulardef lection< f> becomes

T) ~ M r ~ H ( + 7 ) ( 38 1)

ew ritten

—-ds- fA f,—ds-C R x 2ds=0 (382)

largenumberof turns,thefol-

o ldw ithsuff iciente actitudeforprac-

0 Cx yds=0

hreeintegralsofE q uation382are

tbezero itfo llowsf romthise uationthat

attheouterendA o f suchaspring, F ig.

i l lbezero . F ora smallnumberof turns,

ver.

,

s - f M ^ d s - C R - - ds = Q


/ h t t p : / / w w w . h





S P R IN G

O a n d / ( y / r ) ch = 0 f o r a la r ge

uationreducesto

annotbezero , Eq uation383showsthat

= M X . U s in g t hi s c on d it i on i n E q u a ti o n 36 9 ,

hetangentia lforceattheendA , F ig.

onditionassumed.S inceR isalsozero,

ation371showsthatM= M w hichmeansthat

ngthe lengthofthespring.

= 0, E q uation381reducesto

erof turns/ (y / r)ds= -0and/ ds~ l

ection< f> becomes

dians(or degreesdividedby57.3).

sthattheangular deflectionofaspiral

of turnsandalengthI withbuilt-in

hatofa straightbeamoflengthIbuilt

yamomentat theother.

stantalong thelengthofthespiral

ctingcurvatureeffects)for thecase

nby

crosssectionandli= thic nessofstrip.

sconcentrationat theclamped

ueorrepeatedloadingispresent(as

h),inaccordancewiththediscussion

ouldbeta enintoaccountbymultiply ing

q uation386bya stressconcentration

ionswherethenumberof repetitions


/ h t t p : / / w w w . h





G

espringis small,stressconcentration

ever.

f turnsisinvo lved, Eq uations385

das discussedlater.

re uently inpractice , formanufac-

dof aspiralspringmaybe heldwith

mped.N eglectingfrictionnomoment

andtheloadingconditionswillbe

. Inthiscasethee ternalmomentMn

o nottoucheachother,the mo-

piralhavingthecoordinatesx andy

)

hise pressionmaybewritten

orem,asbefore,

uenceofthefactthat theradialforceR

def lectionof thespring. ThereforeEq ua-

on388w ithrespecttoR andsubsti-

3, thefo llowinge pressionisobtained:

mberofturns,thefirst twointegrals


/ h t t p : / / w w w . h





S P R IN G

encethise uationgives

ntf romzerothismeansthattheradia l

ozeroforthepin-endedcase. F ig. 185.

otation< f> isgivenbyEq uation379,

byE q uation389.Differentiatingthe

ttoM andsubstitutingtheresultto-

389inE q uation379,

( 1+ 7 )

bezerothissimplif iesto

t/ yds=0 fora largenumberof turns,

o f turns

te.UsingitinE q uation391,thee -

sto

uation385it isseenthat, forthe

M,,aspiral springwithahingedouter end

tmoreangulardeflectionthanthecor-

pedouterend,providedad acentturns

inthespringwill occurwheny= r

ngy= rinE q uation389, sinceR —0, this


/ h t t p : / / w w w . h





G

eM= 2M . Thema imumstressisthen

momentM thisstressistw icethatfor

uteredge( q uation386). How ever,

ccursata pointoppositetothepinned

ssconcentration.Ifthearbordiam-

r,themomentattheinner clampedend

ameasthatforaspringw ithclamped

the stressatthisend willalsobethe

lwayssomestressconcentrationatthis

hatinsomecasesthis isthelimiting

hecoils, asmayeasilyoccurin practice,

essgivenbyEq uation393. F oramore

spiralspringswithlarge numbersofturns

hearticlebyV andenB roe 1.

rsionspringhav ingapinnedendA F ig.

t e rn a l to r u e M e u a l to 2 5 i nc h -p o un d s.

ches,thebarsectionis " 2by.06-inch,

ches. R e uiredthestressandthede-

odulusE = 30X 10 poundspers uare

forapinnedend

, J .

1. 73radians

0 -

ularrotationof 1.73(57.3)=99 de-

thema imumstressis

endO wherestressconcentration

r oc — " S p i ra l S p r i n gs T r an s ac t io n s, A . . M . . , 1 9 3 1, 5 3 -1 8 .

yTheory, Wiley , 1942.


/ h t t p : / / w w w . h





S P R IN G

hisor84, 000poundspers uare inch

ersmallcomparedtor. H owever,as

tterstresswill beaugmentedbystress

toclampingofthe end.

erelarge tor uesareinvolvedit

ivelyheavycrosssectionforthespiral

oflargerdiameter.Thismeansthat

springmayberelativelysmallso

edtheory(basedona largenumber

owever,ananalysisofthiscase

lar methodstothosedescribedpre-

bebrieflyoutlined.

small numberofturnsasshown

edthatthespringisclampedorbuilt in

hile theouterendA maymoveinan

ngulardef lectionof theendA ( in

movementofA a longthearcdiv ided

t onthespiralhavingthe co-

egivenbyEq uation370, usingr, forr,

x ( 3 94 )

uantitiesM PandR inthise uation

hreee uationsobtainedbyusingthe

cethepointA isassumedconstrained

rcaboutO, thew or doneby theforce

ansdU/ dR — 0andE q uation375ho lds.

ation394partia llyw ithrespecttoR ,

on394inE q uation375, thefollow inge -

" S pira lSpringsw ithSmallN umberofTurns , Journal,

o l. 225, 1938, Page171.

E W T UR N S


/ h t t p : / / w w w . h





G

dof thespringthroughanangle

w illa lsomovethroughthesameangle . F rom

isconditiongives

, dM/dM^ l. UsingthisinE q uation396,

on394,

M ds = ^ j J ^ P (r t+ y ) + M 1- R x ] d s (3 97 )

soobta inedfromtheC astigliano

thetotaldeflectionin thedirectionof

a ltodU/dP . SinceP isa lw aysassumed

rcofmotionof theendA , F ig. 186, this

H ence

4, dM/dP~ r2- -y . Usingthistogether

q uation398,

) + M, -R x ] ( r + 3 ) rf s( 3 )

ringista enintheformofaspira l,

nd399may

llength

eous

f rom

titiesmay

e,the

point

uation394.

tionof

themoment

btained

a lueofF igise—S pira lspringw ith

M, . IfM smallnumberof rums


/ h t t p : / / w w w . h





S P R IN G

thisisa lsothema imummomentfora

nda largenumberofturns)the ratio

mummomentande ternalmomentmaybe

entrationfactor.V aluesofthisratio

8409201000 1080

G E O F CO L ( IN D GR E E S )

centrationfactoraforspringwithfew turns

87asfunctionsof thetotalspiralangletI

functionA =(r2— r1)/ rihavebeenob-

venport1. Theva lueof6ista enasthe

us vectorintravelingfromoneend

enthatthestressconcentrationef fect

urnsissomewhatgreaterforthe larger

hestress concentrationvaluesare

angle 6isnear 360,720or1080de-

turns.This isshownbythedips inthe

isofadvantagewhendesigningspiral

awholeratherthana fractional,num-

omF ig. 187it isa lsoseenthatasthe

s,thema imumvaluesoftheratio

.e.,thestressconcentrationeffectde-

for0= 1080degrees(3turns)andA —

stil le ua ltoa lmost1. 2whichmeansthat

be almost20percenthigher thanthat

6, derivedontheassumptionofa large


/ h t t p : / / w w w . h





G

ress concentrationeffectmaybe

ectsofimperfectclampingat the

thatbecauseofthe smallnum-

omewhatstiff erthanwouldbee -

simpleformula( q uation385)de-

fturns.Themoreaccurateanalysis

ectionata momentM isgivenby

sgreater thanunity)depends

spira landontheratioA . V a luesof this

188. H ow ever, forratiosA between. 4

0000O f lC

E O F CO L C I N D G R E E S )

actorpforspira lspringw ithsmallnumberof turns

woturnsofthe spiralthisfactordiffers

ercentand mayusuallybeneg-

es,i.e.,theusual formula( q uation

w ever, fromF ig. 18 itmaybeseen

nvolvedin theusualformulawhere

honlyoneturn.

ctionat variouspointsalongthe

ingeneralthedesignershouldtry to


/ h t t p : / / w w w . h





S P R IN G

560 640720800880960 1040

O N G CO I C IN D GR E E S )

ndingtheradialdeflectionofcoils ofspiral

suredf romouterend

hduring deflection.Theseq uantities

asfunctionsof theanglealongthespiral

sand forvariousvaluesoftheratio A ;

ig.189.Theordinatesrepresentradial

f> r. , . Inthis< f> istheangulardeflectionof

ternalmomentw hile \ p istheanglemeas-


/ h t t p : / / w w w . h





G

theouterend.Byusingthesecurves

weencoilsmaybewor edoutfor

ersofturns.F orfurtherdetailsthe

aperbyK roonandDavenport4.

ngsw herethethic nessisfa irly

entrationentersdue tothefactthat

rvedbar.Thisstress concentrationis

eterminedfor agiventhic nessand

gcurves givenfortorsionspringsin

II.

nd188apply onlytospiralsprings

heretheouterendis pinconnected

,ananalysismay becarriedoutusing

ee acte pressionsfordef lectionand

pressionsarerathercumbersomeand

eris referredtothepublicationby

E S S E S

ssesinspiral springsmayrunas

0poundsper s uareinchormorewhere

afactor. F ore ample, anordinary

femaybesub ectto lessthan5000cycles

dmuchhigher thanwouldbethecase

einvolved.W herefatigueconditions

mpleinthespiral springforthebalance

s rangeshouldbek eptwellbelow

material,stress concentrationcondi-

being considered.S omedataonen-

forspring materialsaregivenin

E CTIO N — CO I S I N CO N T A CT

asbeenbasedonthe assumption

toucheachother.Thiscondition,

manycases,as fore ampleinthe

wer springofaphonographwherethe

eF edern, Page73, 1938, V . D. I. , Berlin.


/ h t t p : / / w w w . h





S P R IN C

de ahollowcaseasindicatedin F ig.

how nwounduponthearbor. Whenthe

againsttheinsideofthecase asindi-

redbysuch aspringmaybe

yasfo llow s : If Iisto ta llengthofspring

ess,thetotalsectionalareaof thewound

F ig. 190thisisa lsoe ua lto

ngthatthe coilsarewoundtightlysothat

dneglectingthe turnsconnectingthe

Thus,

acentco ilstouch, thenumberof

fd2givenbyE q uation401, thee -

hespringiso iled, ad acentturns

hic nessof theo ilf i lmandthisw ill in-

e uation.

whenthespringis unwoundas

n bethenumberof turnsof theunwound

ngtheturns connectingtheinsideof

arbor,

sDeliveredbyF latC o iledSprings , TheMainspring, S pring7,

publishedbyWallaceBarnesC o.


/ h t t p : / / w w w . h





G

area ofthewoundportionmustbe

on406in404,

deliveredbythespringin un-

ositionofF ig.190tothe unwound

ng

bethedifferencebetweennandn .

nsdeliveredbecomes:


/ h t t p : / / w w w . h





C A L S P R I N G

ngthe woundpartofthespring

neglectedinthis derivationtheresults

8aresomewhathigh.Toobtainmoreac-

shouldbemultipliedwithacorrec-

ity.V aluesofthiscorrectionfactor are

f drumareaminusarborareadivided

usva luesofmassuggestedbyWallace

TableX X X III.

m between5and1.5areduction

thatca lculatedf romEq uation408of

aybee pected.

ssconcentrationduetocurvature

risusuallymadearound15 to25times

pringisw oundf romastrip% - inch

nd100incheslong.Thearbordiameter

ousV a luesofm

85

iameterof thecase2V 4inches. The

mberofturnsdeliveredfromthesolid

sh= M5, Z = 100, D,= 2. 25, ^ = . 375.

,

I, by interpo lationform= 2. 58, f c= . 76.

h, I, D., anddlinE q uation408

becomes19.6turns.Thismustbe mul-

tor fc— .76whichyieldsavalue


/ h t t p : / / w w w . h





G

forthenumberofturnsdelivered.

sa largenumberof turnsthee ua-

and deflectionbecomesimplepro-

eincontactduring deflection.Where

allandtheouterendis clampedthe

ctionmaybecalculatedbymeansof the

dtightly,as inthecaseof the

relativelysimplee pressionsgiven

turnsdelivered.Becauseoffriction

thiscase adeterminationoftor ue

e angleornumberofturnsis rather

iscussedhere.


/ h t t p : / / w w w . h





da relativelylargeamountof

, a typeofspringk now nasthe" ring

onsiderationbythe designer1.Thisis

cationis onewherealargeamount

e,such asfore ample,draftgear

ringspringconsistsessentiallyofa

al surfacesandassembledasindicated

. Whenana ia lloadisapplied, slidingoc-

c-

ces withtheresultthatthe inner

heouter ringse tended.Inthisman-

niformdistributionofcircumferential

nnerandouterrings.Becauseof this

ofstressdistribution,thering springis

" T il R ingS pring , Mechanica lEngineering, F eb. , 1926, Page

csof theR ingS pring , A mericanMachinist. F eb. 14, 1924.

ra ftGearS prings—PastandPresent , R a ilua9Mechanica l


/ h t t p : / / w w w . h





essentiallyasabar insimpletension

glyhighefficiency(consideredon the

torageperpoundof metal).A ctual-

nstressesatthe conicalsurfacesof

be aslightnonuniformityine uiva-

hereatensionandcompressionstress

iscase, thee uivalentstress—onthe

sheartheoryofstrength—w illbethesum

esofthesetensionand compression

a lthic nessof theringsisap-

nuniformityincircumferentialstress

eathic cy linderunderinterna lore -

stsprings,however,thisnonuniformity

ributionwillnotbe largeandhencethis

elativelyhighefficiency.O ntheother

at thedampinginthisspring isob-

ofacerta inamountofwearonthesliding

accordingto usualpractice.

gineeringandF oundryC o.

gandrotatingbloc for

springisutilized


/ h t t p : / / w w w . h





S P R IN G

C U A TI O N S

hysteresisloopforthering spring

romthisitmaybeseenthatonthecom-

igherspring constant(intermsofpounds

nedthan forthereturnstro e.This

sontheconical facesoftheringswhich,

dded totheelasticforcescaused by

foradecreasingloadaresubtracted

IN C H E S

-deflectiondiagramofringspring

sloopduringloadingandunloading

sa largehysteresisloopisobtained

energyabsorptionpercycle.

orpracticalpurposesof analysiseach

springmaybeconsideredassub ect

distributeduniformlyaroundthecir-


/ h t t p : / / w w w . h





f o rceF = / JV (w hennisthecoef ficient

actsinthedirectionshownwhen the

d,andin anoppositedirectionwhen

nded. TheseforcesN andF produce

cingonelementof ringspring

thering,althoughthere isatthe same

gtobendli eabaronelasticfounda-

owever,maybeneglectedfor practical

be assumedthattheringthic ness

meandiametersothatthenonuniform

nduetothethic cylindereffectmay

ngthe innerringofF ig. 195andas-

compressedsothatthefrictionforce

n,thetotalradialforceactingwill be

sina)w herea istheangleof taperof the

lloadp perinchofthe circumferen-

willbe thetotalradialforcedivided

eanradiusofthe ring.H encethisload

se uationbecomes

trengthofMateria ls, V anNostrand, SecondE dit ion, Part2,

cit . , Pnge236.


/ h t t p : / / w w w . h





S P R IN G

pressivestresswillbe

reaof the innerring. S ubstitutingE q ua-

thespringduringthecompression

ngthecomponentsofN andF a longthe

c e

a = N ( s i n a + n co s a )

substitutinginE q uation412, thecir-

estressa,-intheinnerrings becomes

a

ducedtothe simplerform:

1 4)

f E q uation414,valuesofK are

for variousfrictioncoefficientsnin

sofF ig. 196. Wherema imumac-

tionshouldbemadeby usingE q ua-

calculatingthecircumferentialten-

ingsis used.Thisgives


/ h t t p : / / w w w . h





sectionalareaofouterringandK isgiven

enotedthatEq uations414and416give

asafunctionof loadforincreasing

dicatedpreviously , toobtainthee uiva-

gs,thecompressivestressesdueto the

95shouldbeaddedto thecircumferential

dfromE q uation416.Thesecompres-

utedasfollows:F ortheouterring

fmeancircumferenceis obtainedfrom

insteadof r, . Thus

vs

tensionstress,fromthering

ation417.

ro ecteda ia llengthofcontactareaatthe

.192(whichlengthmaybe obtainedfrom

fthespring anditsgeometricalpropor-

ompressionstress< » inthecontactre-

)/2= meanradiusof innerandouterrings.

areaoverwhichtheforceN actsis

utinginthisthevalueofN givenbyEq ua-

becomes


/ h t t p : / / w w w . h





S P R IN G

ctionof thespring,theradialde-

firstbefound.F ortheinnerring the

ppro imatelye ua ltoacrm/ w hereE

The a ialdeflectionofthespring

betwotimes theradialvalueacrm/

r twoisusedsince therearetwo

enceifn,is thetotalnumberofin-

ngof halfthefullsectionbeing con-

deflection8< duetotheseringsis

a ldef lection8 duetoouterringsis

ro fouterrings. Thisw illalsoe ua l

to ta lnumberof " e lements inthe

stingofahalf innerandahalfouter

ations421and422,

° < + » c ) (4 23 )

givenbyE q uations414and416

ngthecompressionstro emaybee -

adP.

uation415orF ig. 196.

ccuringduringthe unloading

eanalyzedinasim ilarmannerbycon-

thedirectionof thef rict ionforcesF , F ig.


/ h t t p : / / w w w . h





3 55


/ h t t p : / / w w w . h





L S P R IN G

stheload and3,thecorresponding

rnstro e,

edforconvenience inthe low ergroup

P,(return stro e)andtheload

tanygivendeflectionis obtainedby

uations424and425. Thisgives

fthespringconstantsfor thereturnand

spectively it isonlynecessary tota ethe

ivenva luesof / ianda. Thisistruesincethe

rtionaltothe respectiveloadsatany

A T IO N

applicationof theseformulasinprac-

ofthefollowingdimensionsastested

considered: A , = A , = . 584in.2, r(=4. 42

= 4 . 58 i n . ta n a = . 2 5 , a — - 14 ( a pp r o . ) ,

n , = 9 , n = 1 8 . T e st s o n th i s ri n g sp r in g i nd i -

on^ = . 12. F romF ig. 196, forx = 14

onK = . 095andK , = . 031.

dP , = 100, 000pounds, f romEq uation

thedeflection8is

. 0 95

tro eforthissamedeflection

pliedby theratioK , K . Thisgives


/ h t t p : / / w w w . h





32, 700lb

compressionstro eis

eis:

00X -^ -= 10, 000lb/ in.

eouterringat P,= 100,000pounds

tion416,

iscase, f romEq uation414, theforego ing

ecompressionstressinthe innerring.

er,theinnerring areaA ,ismade

hasbeenfoundf rome periencethat

maybeused incompressionthaninten-

aftgearspringshavebeendesignedfor

tressof125,000poundsper s uareinch

aredwithacompressionstressof 210,-

inchininnerringsw henspringisso lid .

nproportionsof(he springareso

tedcontactlengthb, F ig. 192, is. 79- inch

ds, thenf romEq uation420thecom-

thecontactareais

onstressat= 143,000poundsper

valentstressintheouterringbecomes

900poundspers uare inch. Thisisa

wouldbeobtainedif thecontactcom-

glected.

dicationof loadsanddeflectionspos-

g. TableX X X IV , isuseful5.

w aterS tee lC o.


/ h t t p : / / w w w . h





S P R IN G

tings


/ h t t p : / / w w w . h





I G

essentiallyofa relativelywide

blade,whichhasbeenwoundtoform

sectioninF ig.197.Beforewinding,

howninF ig. 199. A f terw indingbut

springwill haveeitheraconstant

eandavariableco ilradiusasshownin

ngoperationusuallyemployedwill,

ngeintheheli angledistribution

scondition.W henloadeda ially

behavesessentiallyas acurved

ipalstresses beingtorsional.

t

ring

cation

yin

gthese

ng:

nufac-

yfric-

pring

h de-

otect

d.

tially

ble

e

rthe

gth.

n

the

o ilsabovea


/ h t t p : / / w w w . h





S P R IN G

atbeyonda certainloadsomeofthe

portingplate, increasingthestiffness.

stressdistributionthe thic ness

aperednearthe innerendof theco il .

L I X A N G E

calculatestresses anddeflectionsin

entofthecoil may,forpracticalpur-

ntiallyasaportionof ana iallyloaded

coilradiusandhavinga rectangular

dthusneglectsfrictionbetweenad-

pingsuspensionforM-5tan

certainsecondarystresseswhicharediffi-

fthesestressesarisefromthe factthat

197 ingenera lw il lnotbea ia lasas-

ut willbedisplacedfromthea isof

toadditionalstressescausedbythis

certainstressesk nownasconeandarch

maymodifytheresults1.To simplify

iscussionofvolutespringcalculationsincluding theeffect

thic nesssecarticlebyB. S terne, " C haracterist icso f the

rna lS . A . . , June1942, Page221. Seea lsopaperbyH. O . F uchs,

ryS tressesinV o luteSprings, T ransactionsA S M , July1943, Page

gnMethodforV o luteSprings , JournalS . A . . , S ept. 1943, Page317.

arcgiveninarticlebyB. S terne, TransactionsA . . M. . , Ju ly

P roceedingsS ociety forE x perimentalS tressA na lysis.

ults ofstrainmeasurementsandeccentricitydeterminations


/ h t t p : / / w w w . h





I G

lutespring

eeheli anglewillfirstbeassumed.

ehe li anglewillbetreated.

eferringtoF ig. 200w hichrepresents

fthe bladeofavolutespring,the

atedbytheline A B,abeingthefree

onstant).Inthistheordinaterepresents

nterline,andtheabscissathe distance

moderateloadsbeforetheouterend starts

engthwillberepresentedby the

eatheavy loadsw henaportionA C of the

developedlengthisrepresentedby

chwillbe calledtheinitialbot-

uter coilj uststartstobottom,the

sticwillbeastraightline asindicated

sload, astheco ilsbottom, thespringbe-

ndthe load-deflectioncharacteristic

ttomingloaditwillbe assumed

O N G CO I —

tofcenter lineofvolutespringfor constant

theavy loadsspringbottomsbetweentoandr


/ h t t p : / / w w w . h





S P R IN G

angle 6fromthebuilt-inouter endA

pesentedappro imate lybyaspira l. This

eformost practicalpurposes.Thus

)

beginningandend,respectively,ofthe

g( ig. 197)andnisthenumberof

ahelicalspringofnarrowr ec-

erethelongside ofthesectionis paral-

dw herethew idthb( ig. 197a)isgreater

ecaseinvolutesprings,is givenwith

fo llow inge uation :

g/ i= thic nessofblade, b= bladewidth,

coilradius.

rementofthedeflectiond will

pliedbydd/2w . Hence, usingE q uation430

usatangle6( ig. 197/7).

m ingof theouterco ilmaybee pected

heli anglewhenthesloped / dsatthe

ig. 197)ise ua ltothetangentof the

npracticalvolutesprings thetangentof

8.


/ h t t p : / / w w w . h





I G

entaccuracybeta ene ualtothe

gds—rd6andtanoc= a , thiscondit ion

uation431, andsubstitutinginEq uation

oadP,for constantfreeheli angle

eacise pressedinradians(degreesdivided

fdeflectionwillbe discussedfor

heretheloadsarelessthan initialbot-

f lection

ng

eyaregreater.

ethedeflection8for loadsPthat

ottomingloadP E q uation428and

itutingthevalueof rgivenbyE q uation

tofdeflectioninsmallangled6 becomes

) ^ ' -


/ h t t p : / / w w w . h





S P R IN G

helimits0= 0and6 — 2irnwhere

oils,thetotal deflection(forloadsun-

ad)maybe e pressedas

, ) ,w h en P < P , ( 43 5

uation433, and

tedasfunctionsof fi- (r — ri) / r in

andr, inthise pressiondependonthe

ig. 197). Wherethese latteraretapered

three-fourthsturnateachend isfre-

ndingfactorK, asfunctionof$

tive,butthisfiguremaybe changedas

vailable.

natany loadPlessthanP, the

ation435maybeused, thevalueK 1being

ig. 202. ItshouldbenotedthatE q uation


/ h t t p : / / w w w . h





C

caseofa variablefreeheli anglepro-

oils occurs.

he loadP isabovethebottomingload

constantf reeheli anglemaybecon-

woparts, e. g. , apart8 ( ig. 200)dueto

tomedportionA Cofthespring and

lectionof thef reeportionC D. A ssum-

ttomedtoaradiusr andangle6 asin-

nf romthecondit iond / ds— aatthe

oceedingasbeforethefo llow ingisob-

s p ri ng i nd e a t r~ r .

a n d 6 = 6 i n E q u a ti o n 42 8 , th e a ng l e

asfo llow s:

s p r i ng i n de a t r = r , ,.

ttana= a , thedeflection8 isgiven

thisandintegrating,

40)

33, 437and438thise uationmaybe

eratioP/ P , asfo llow s:

obtainedbysummingupthe ele-

tweenthe limits6= 6 and6=2w n. Thus,


/ h t t p : / / w w w . h





S P R 1N G

/b)

gandaddingtothe valueof8

1, thetota ldeflection8becomesforP>P1:

2 n r a ( - l - — )( 44 3)

f theratioP / P ,.

>

enasfunctionsofP /P inF ig. 203. B y

ofF ig.202,thedeflectionatany load

In thismannerthecompleteload-

fthespringfor aconstantinitialheli

hallcoils bottomtheprocedureis

uation437, byusingthee pressionforP ,

3,

swill occurwhenr,= r(and

o n 44 5 a nd t a i n g / = ( r — r , ) / > „ t h e fi n al

es

oad P,isobtainedbyta ing6 —2n-n

ntegrating, usingtheva lueof rgivenby

ogivesthe differencebetweenfreeand

,m U ( - -r ) (4 47 )

ei angle intermsof is


/ h t t p : / / w w w . h





I G

he ightsof thespringarek now n,

ians)maybeca lculatedf romthise ua-

hich initialbottomingoccursisob-

435ta ingP= P1. Thisgives

mateload-deflectioncurveforany

/ t>

indingfactorK: f romloadratioP/ P ,


/ h t t p : / / w w w . h





S P R IN G

tialheli angleitisonlynecessary

nd8, f romEq uations433, 446, 447and449.

awnbetweentheoriginandpointA

ig.201).PointB(representingP,and

yasmoothcurveconcaveupward.F or

d,additionalpointsonthis curvemay-

uation443.Thustheload-deflectiondia-

simply.

calculatethe stress,theformulasfor

ngswillbe used,modifiedtoapply

mentionedpreviously,thesestresses

asfirst appro imationsbecauseaddi-

uallywillbepresent.

he loadP islessthanthe initia lbot-

stressw illoccuratthema imumradius

imatee uationforarectangularbar

hthelong sideoftherectangleparal-

discussedprev iously4, thema imumshear

ecomes

s p r in g i nd e a t r = r ( l. W h e r e h/ b i s sm al l ,

ecomparedto thethic ness,theterm

nasunit) . Thisgives, appro imately ,

he loadisgreaterthanthe init ialbot-

umshearstresstw illoccuratr= r

tw hichbottomingoccurs. ThusEq ua-

tt ingcv= cfw herec ~ 2r / h=spring

g i ve s

P i

.


/ h t t p : / / w w w . h





I G

n 44 5 , c = c \ / P JP , b y su b st i tu t io n o f

thestressatany loadP(forP>P, )becomes

curs,theloadP = P,.Usingthevalue

tion446inE q uation453thepea stress

oadP , maybee pressedby

ginde atr= r, .

valueofP . , givenbyEq uation446

IN C H E S

000

ectionandload-stresscurves,

tomingloads

byE q uation433, thestressr2forf ina l

mplee pression


/ h t t p : / / w w w . h





S P R IN G

eeffectofbar curvature.Ifitis

ct wherestaticloadingispresent,the

singthesame e uations(450to455),

sionintheparenthesisofthe numerator

ation450, todothisc ista eninsteadof

or). F ormostvolutespringsthisw illnot

ference,however.

Design—A sane ampleof theuse

cticaldesigna volutespringwitha

gleandwiththefollowingdimensions

2% - inch, r, = lV 4- inch, h~V 4- inch, b=5-

e ight—7Mi- inch, c — 2r / 7i= 20,

numberofactiveco ils, / 3= (r —r()/ r =

bethedifferencebetweenthe

hus8. , = 2% - inch. F romE q uation448

forsteel,thepea shearstresswith

f romEq uation455,

5 3 1X 1 1 1 , .

P,is, fromE q uation433,

3X . 0531X . 969

a lbottomingloadP , fromEq uation450is

s i n .

thef ina lloadP , isfound:

nitial bottomingloadP1isgivenby


/ h t t p : / / w w w . h





distributionofhe li angleas

usz values

heva lueofK , = . 47givenbyF ig. 202for

2 Jr X 4 X 2 . 5X . 0 5 3 1X . 4 7 = 1 . 57 i n .

bottomingloadP.,willbethe difference

height,i.e.,8.,= 7.5— 5= 2.5-inch.

dP , a loaddef lectioncurvesim ilartoF ig.

sparticularspring.A similarload-

d,since thestresswillvarylinearlywith

gloadP,.Thestressatany loadbe-

lculatedf romE q uation453. Inthis

-stressandload-deflectiondiagrams

eobta inedforthiscase. F romthesedia-

deflectioncurvemayalsobe plotted.

E L I X A N C E

y,theprocessofmanufacturein

anglewhichincreasesfrominsideto

mountof thisvariationinheli


/ h t t p : / / w w w . h





itionsobtainingduringthe cold-setting

of winding.A nanalysisbasedon

riationoffree heli angleislinear

rradiushasbeencarriedout by

nmaybee pressedby

theheli anglesatradiir, r, andrrespec-

lativevaria tionof theheli anglemay

berzw here

ept. 1943, Puge317. Thisa lsodiscussesdesignofpresetting


/ h t t p : / / w w w . h





I G

constantf reeheli angle isobta ined

matelyconstantbottomingstresswille ist

ocurvature). R atiosof ( / a areplotted

. 205, forthevariousvaluesofz. Thisgivesan

nin freeheli anglewithradius,for

sumingelasticconditions,similarload-

obtainedforall springswithgivenval-

actualload-deflectiondiagrammaybe

ven" typecurve bycertainsca leratios.

rvesforva luesofz= 0, V * , % and1have

hs andaregiveninF igs. 206, 207, 208, 209,

curvesaredraw nforrj r e ua lto . 3, . 4,

gtosprings withsmall,mediumorlarge

diameters.Initialandfinalbottoming

circlesoneachcurve.The abscissas

R A T IO f

forz—% ,

R A t IO j

forz=


/ h t t p : / / w w w . h





S P R IN G

in termsofthema imumpossible

atesintermsofa loadP . , w here

d z is g i ve n b y E q u a ti o n 45 7 . V a l u es o f K 3

curveofF ig. 211forvariousvaluesofz.

o tt e d in F i g . 21 2 a ga i ns t < 7 = r , /r . F o r

referredtothearticleby F uchs5.


/ h t t p : / / w w w . h





I G

d-deflectioncurveforanyspring

dq theordinatesofthecorresponding

pliedbyP , andtheabscissasbyS b. In-

necessary.

einitialbottomingloadP,maybe

,

s

on433usinga= a . F ortheusualspring

arstresst atwhichbottomingstarts

ation450ta ingP= P1. Thisgives

/ h a t r= r . I f t he v al u e of P , g iv e n by

utedin thistheinitialbottomingstress


/ h t t p : / / w w w . h





S P R IN G

lectedtheterm unityinthe

ressionisdropped,obtainingthesimple

albottomingloadisgivenby

ga= a(andusingr, insteadof r,. . Thisgives

t )

rend(r= r, )a tf inalbottomingload

55usinga, inthiscase insteadofa :

e atr= r(,

lectedthis reducestothesimple

sane ampleof theuseof these

ngr = 3. 75, r, = 1. 93, fo= 7. 50, h—

10 lb, s . in. , a - - . 076, a , -= . 060,

mEq uation457. F romF ig. 211, forq — . 515,

, andf romF ig. 212, ^ = . 053. UsingE q uations

)3 X . 0 6 0X . 0 5 3 X 5 . 5

0 X 5 . 5 = 4 . 95

nobtainedbyusingthe curvesfor

i g s. 2 0 7 an d 2 08 ) f or q = . 5 d oe s n ot a m ou n t

yload,whichisverysmall compared

enceeithercurvemaybeusedforcon-

teload-deflectiondiagram.


/ h t t p : / / w w w . h





I G

ngstresst ,neglectingcurvature,

. Thisgives

rvaturethisstressismultiplied

4asafunctionof rl/ r

w he r e c = 2 r / h = l 8 . 7. T h is g i ve s

= 94, 000poundspers uare inch.

mingstresst,,neglectingcurvature,

. Thisgives

urethe stressthusfoundismul-

reC i= inde 2rf /7i=9. 7. Thestress

137,000(10.7/9.7)=151,000poundsper s uare


/ h t t p : / / w w w . h





G A N D M O UN T I N G

bberspringsincludehighenergy

dthe possibilityofformingincom-

sereasonssuchsprings havefound

brationisolatorsformachinery,

tomobileandaircraftengines,mount-

drichC o.

ringsappliedtocompressormounting

ibleelementsincouplings,andmany

of theapplicationof rubberspringstoa

ountingisshowninF ig.213.

t isso largethatane tensivetreatment

ntingsis beyondthescopeofthis boo ,

hatatleast thefundamentalprinciplesof

should beemphasizedthatthee -


/ h t t p : / / w w w . h





G

fthebehaviorofrubberunderstress is

he moreusualspringmaterialssuchas

horbronzeandthe li e . C onse uently ,

ringsby availablemethodsisatbest

mongthereasonsforvariationsbetween

haviorofsuch springsarethefol-

shearmodulimayoccuramongdifferent

houghofthesamehardnessreading.

ionspringsofrubber,frictionbetween

varythroughwidelimitsthusaffecting

hererubberpadsarcbonded to

swill notoccur,however.

micmoduliof elasticitywilldiffer.

sare deflectedbyrelativelylargeamounts,

oredifficult tocalculateaccurately

ssanddeflectionof varioustypes

stbe treatedafterwhichthefunda-

inthechoiceof fle ibilitiesforsuch

d.

lyusedtypes ofrubberspringsis

oc showninF ig.214whichrepresents

r compressedbetweentwosteelplates.

esdevelopedatthesurfacesof thebloc

iffnessofsucha springisfarlarger

dtotheothertwodimensions)than

S P R I N G

n


/ h t t p : / / w w w . h





S P R IN G

berisnot bondedon,thestiffness

differentamountsoffrictionbetween

usif theplatesarelubricatedwith

muchlargerdeflectionsmaybee -

wouldbethecaseif drysurfaces

L A S TI CI T I N C O MPR E S S I O N ( L B / O , IN .

ourna lo fA ppliedMechanics. 193 .

lasticityofrubberincompressionas

dnessnumber

uncertainamountoffriction present,

lculations ofdeflectioninsuchcases

yappro imateonly.

ethodwasdevelopedbyS mith1,

rageofaconsiderablenumberof tests.

percentagedeflectionofslabofrubber

sectiona lareaofslab, / ff—ratioof

thic ness, £ —modulusofelasticity

veragecurveofthevariationof modulus

e" R ubberMountings —J. F . Dow nieS mith, Journalo f

ch, 1938, PageA 13 and" R ubberS prings—S hearLoading

urnalo fA ppliedMcch., Dec. , 1939, PageA 159. Otherarticles

" R ubberSprings —W . O . K eys, Mechanica lE ngineering. May ,

lasticB ehavioro fV ulcanizedR ubber —H . H enc y , Transactions

45: " R ubberCushioningDevices —H irschf ie ldandP iron, Trans-

g. 1937, Page471 " TheMechanica lC haracteristicso fR ubber —

ransactionsA S M . F eb. , 1939, Page149. " UseofR ubberinV i-

. H . H ull, Journa lo fA ppliedMechanics, S ept. 1937, Page109.


/ h t t p : / / w w w . h





G

nwithdurometerhardnessis givenin

ulusofelasticityo f rubberhaving55durometer

ectionofa1-inchcubeof 55durom-

maybeta enf romtheaveragecurveof

k now n. Thenanempiricale pressionfor

ofrubberslabsis

ampleof theuseof thise uation, assum-

ometerhardness,asectionalarea

thic ness= linch, andload=10, 000pounds,

40 3 0 6 0 60 1 0 0

F H N CH C UB P R C N T

eflectioncurvefor55durorubber

2. F romF ig. 215themodulusofe las-

bberisE= 310poundspers uare inch.

erE= 430poundspers uare inch. The

00/ 32=312poundspers uare inch.


/ h t t p : / / w w w . h





S P R IN G

f lectionn o fa1- inchcubeof55durometer

be57percent.Using thesevaluesin

centagecompressionat10,000poundsload

therloads,aload-deflectiondiagram

R S P R IN G

earinvolvesnovolumechange,fric-

contact suchasmayoccurin com-

entand betteraccuracyincalculation

bbershearspringswhichhaveessentially

17consistingoftwo rubberpadsbonded

sed forvibrationisolationandmachine

icationisshow ninF ig. 213.

calculatedasfollows:A ssuming

diansisproportionalto shearstress

ubbershearspring

tothemodulusofrigidity G(accord-

ptiongivesthebetteragreementbetween

theshearstress t=P/2A and

areaof eachpad.Thefactor2is used


/ h t t p : / / w w w . h





G

etw opadssi/ b ectedtothe loadP .

greestheva luegivenbyE q uation469

).Themodulusof rigidityGdepends

0

R I GI DI T , L F A / Q . IN .

gidityasfunctionofhardness

softherubber andmaybeestimated

18.

8of thespringofF ig.217,ifthe

f thepadand7=theanglefiguredf rom

e (saybelow20degrees),for

ngentmaybeta ene ualtotheangle .

nd470the deflection8thenbecomes

prings aremadeinitiallyobli ue


/ h t t p : / / w w w . h





S P R IN G

esofF ig.219,thepositionof therub-

ingindicatedby thedashedlines.By

anner,itispossibleto introducead-

sesduringdeflection.Thesestresses

ubberspring

ds.

om-

rload

ndpointofthe adhesionofthebond

esteel.If thefinaldeflectionleaves

lyflatsothatc/his notlarge,asisusually

ttoogreat, itmaybeshownthatEq ua-

enoughaccuracyforpracticalpurposes2.

S H E A R S P R IN G

ype ofshearspringconsistsessen-

dedtoasteel ringontheoutside and

sideasindicatedinF ig. 220a. A load

isasshown. Theshearstresstatany

sameradius( ig. 2206)theslope

negativevalueof theshearangle.The

inceydecreasesw ithincrease inr. Since

a ltor/ G(appro imate ly )andthederiva-

entof theshearangle , usingEq uation472,

nalo fA ppliedMechanics, December, 1939, PageA -159.


/ h t t p : / / w w w . h





G 3 85

4 7 3)

henusingthek now nseriese pression

e,

= r(andr=r , thetotaldef lection

s(w hereb/ r < . 4)thetermsof this

be neglectedwithoutseriouserror.

imationthefollowingformula:

bber springofconstantheight

inga longa is


/ h t t p : / / w w w . h





S P R IN G

thic nesshofacy lindricalrubber

onaltothe radiusr,asindicatedin

twillbeconstant andbetterutilization

ubber

stress,

verse ly

ained.Thisfollowsfromthee uation

tutioninthisformula , PPP

ssatouterradiusr , F ig. 221. S incethestress

heshearanglev = t/Gwillalsobe con-

e

fort

essthan20degrees)wherethe


/ h t t p : / / w w w . h





G

ppro imatelye ua ltotheangle , thede-

tlyaccurateformostpractical uses

T O R S I O N S P R IN G

nthiscasethethic nesshof thespring

222w hileamomentMisassumedtoact

sindicated. Theshearstresstatradiusr

umshearstressw illoccurw henr=r(and

gulardef lectionaboutthespringa is

sion

essh

ressacting ontheelementalring

2b,then

entialdeformationoftheouter cir-

twithrespecttothe innerisdrtan 7


/ h t t p : / / w w w . h





L S P R I N G

t/G.Dividingthisby ryieldstheele-

nd(). S ince i= t/ G, byusingE q ua-

hG, thise pressionbecomes

( 48 3 )

ntseriesasbefore,theangularde-

utinglimitsthise uationreducesto

-77)l^+ - - - J- (484)

hatthis seriesconvergesrapidly

lywillbesufficient.This gives

cylindricalrubberspringis madeso

g. 223variesinverse lyasthes uareof the

uation480theshearstressw illbecon-

thedepthista ene ua lto

f rubberatthe insideradiusr=r(, F ig.

uation480thestressbecomes

e,thentheangular deflectiondueto


/ h t t p : / / w w w . h





G 3 89

hadedinF ig.223bwillbe,as before,

andr= r , thetotalangulardef lection

4 8 8)

sion springwithconstantstress

of theanglee ua ltotheangle(w hichis

ectionis nottoolarge),

ingacy lindrica lrubberspring(60durom-

oadedasinF ig. 222underator ueof

onsasfo llows: r = 3 inches, r(=2

mE q uation481thema imumstressat

dspers uare inch, is


/ h t t p : / / w w w . h





S P R IN G

i n.

^

er, fromF ig. 218themodulusof rigid-

uare inch. Usingthef irsttermof the

4theangulardeflectionbecomes:

10.2degrees

uation484isused, thisresultw il l

nt.

onstantstresstype( ig. 223)with

2inchesf romE q uation487thestressisthe

ously . F romE q uation489assuming

ulardeflectionat10,000inch-pounds

ees

og, ,. 1. 5—. 405.

88wouldgivearesult e ualto17.2

entgreaterthanthatfoundfrom

t asmentionedpreviouslythe

houldbeconsideredappro imateonly

ndstressesarelarge.

S TR E S S E S

fdatain theliteratureonallow-

rubbersprings. Hirschf ie ldandP ironJ

gstressesinrubbershear springs(and

nds)belimitedto25 to30pounds

re ceptincasesw hichhavebeen

sostatethatunderfavorablecondi-

hstandconsiderablyhigherunitloads,

ndspers uare inchhav ingbeenused.

, A ug. , 1937, Page489.


/ h t t p : / / w w w . h





G

ywiththoseofK eys4whostatesthat

dspers uare inchareusedonmeta lto

suggestedthatthethic nessofashear

anone-fourthofthe smallerofthe

esultsoflong-timecreeptests on

terhardness.W hentestedinthe form

g. 217, at50poundspers uare inchshear

reep ofabout.017-inchfor5/16-inch

gtoashearangleof .055-radian)after

rature. A t140degreesF ahr. , a tthe

epwas.15-inchi n5/16-inchor.48-radian

thertestson1- inchthic rubbersand-

ound(38durometer)at 40pounds

essand atnormaltemperatureshowed

hearangleof .055-radian)after500

p-timerelationwasappro imately

showedthatthecreepmaybe greatly

peratureandthatwidevariationsmay

dsareused.The necessityfork eep-

sesife cessivecreepistobeavo ided

these tests.

ainsonrubbercompressionsprings,

andP iron1suggestama imumcompression

ntof thef reethic ness, andanupper

se ua lto700poundspers uare inch

aushalter alsosuggeststhatthede-

ringsbelimitedto 15to20per cent

p.Where fatigueloadingisinvolved(as

gsforcouplingsin electricmotorap-

usstartingandstoppingisinvolved)

ntagecompressionsshouldbeallowed^

chesforusein shearorcompres-

esirabletok eepthema imumthic -

possiblebelow1inch.This isdonein

curingduringvulcanization.Ifmore

stac o f rubbersandw ichesinseries

ring,May,1937,Page347.

, F eb. , 1939, Page157.


/ h t t p : / / w w w . h





S P R IN G

s,caremustbeta entoavoidinstabil-

hiscanhappennot onlyincompression

prings.

D S H O C I O L A TI O N

sbeenconcernedprimarilywith

stressesanddeflectionsof rubber

s.H owever,thedesignerisalsofaced

ingwhatfle ibility,ordeflection,is

venmounting.In doingthishemustbe

plesforvibrationandshoc isolation.

tantofthese principlesparticularlyas

ntingsformachinery,andmilitary

brie fly considered6. F orpurposesofdis-

dedinto twoparts:

n.

shoc .

redas avibrationwhichlastscontinu-

onsideredasamotionwhichdies

me.

ion—Thisk indofv ibrationisencoun-

atingelectricalmachineryasa conse-

herotatingparts. Itmayalsobeset up

lternatingcurrents.A furthere ample

aircraftstructuresduetoengineun-

plosionforces7.Inmachinerymount-

nbalancedreciprocatingmassesis

e.

mountingforsteady-statevibra-

siderthevariouspossiblemodesof

anycasesthesystemcanbe simplified

sasingledegree-of-freedomsystem,con-

e spring-mountedmassonavibrat-

nF ig. 117ofC hapterX III) . Thisisthe

scussionofthegeneralvibrationproblemis giveninMechani-

enHartog, McGraw-Hill , S econdEdit ion1940, andV ibration

—S . T imoshen o, V anNostrand. SecondE dit ion1937.

oche, Mechanica lEngineering, A ugust1943, Page581presents

cussionofrubbermountingsforaircraftand militarye uip-


/ h t t p : / / w w w . h





G

taininstrumentmountingsinaircraft.

anatura lf re uencygivenby

ofthe massundergravity,inches,

cyofsystemincyclespersecond. Inprac-

stantoftherubbermountingshould

t / willbeconsiderablylowerthanthe

ationofthesupport.F ormachinery

e ua lto1/ 3to1/10thenormaloperating

condare usedinpractice .

ninvibrationrealizedby arubber

atthesupport( ig. 117, ChapterX III)

namplitudegivenbya sinw tw heret=

f re uencyof thee ternalv ibrationin

ectingdampingthedifferentiale ua-

y betweenthemassandthesupport

..

= m a t i na t( 49 1)

springconstantof spring. F orasteady -

ntothise uationis

fthespringsupportedmasswill be

q u a ti o n 49 2 f or y a n d ta i n g < , > / < » „ = / / / n

dR ubberC ompoundsinIsolatingMachineryV ibration -—B . C .

M , A ugust1943, Page619.

Page41.


/ h t t p : / / w w w . h





S P R IN G

f/ f approachesunity , thise uation

chcasestheeffectof damping(which

ation)is veryimportant.H owever,

danappro imationforf re uenciescon-

sonance,andinsuchcases,it maybe

motionof themasshasbeenreduced

scomparedwiththeamplitudee peri-

mounted.Theaccelerationstowhich

willa lsobereducedinthesameratio . Thus

ngineering, A ug. 1943.

ficiency intcimsof f re uencyandstaticdef lection

tionf re uency, f—30cyclespersecond,

so chosenthatthenaturalfre uency

d, f / /n= 3andf romE q uation493, the

emassw illbea / (32—1)— aa/ 8. This

vibrationamplitudeandacceleration

obtainedwitha fle iblemounting.

sedonE q uations490and493,and

beusedtoestimatethe percentagere-

edbythe useofrubbermountings

ction8 .Theordinatesofthis diagram

e uencyincyclesper minute,while

estaticdeflectionof themounting.


/ h t t p : / / w w w . h





G

toa definiteamountofreductionin

theshadedarearepresentsaregion

plitude.Thus,fore ample,ifthe

say,800cyclesper minuteandthestatic

eweightofthe mountedapparatus,

f80percentnormallyw ouldbee -

mphasizedthatEq uation493and

basedon theassumptionthatdamping

mostpracticalpurposesthis willprob-

lycloseto actualconditions.H ow-

mpingmustbeta enintoaccount.The

isproblemis toassumethatthedamp-

onstantctimestheve locity . Thisis

ermcdy / dttothe le f tsideofEq uation

enbesolvedfor steady-statecondi-

ctualtests,however,showthatthein-

s afunctionoffre uencyandinmany

pro imatelyasinverse lyproportionalto

ercompounds1 . Inaddition, ithasalso

usofelasticity ofthematerialand

asusedinEq uation491may insome

.Thisresults,forsomerubber

bledeviationbetweencalculatedand

vesbasedonconstantvaluesofc andk .

oseagreementbetweentheoretical

cand k values,andtestcurveshave

range offre uencies8.

onsanalternatingload P=Pn sin

tedmass,thesupportorfoundationbe-

previouscaseconsideredwherethe

vbrate . A ne ampleofsuchanapplica-

ountedonrubberbushings,theforce

alternatingload inq uestion.Inthis

wsthat ifdampingisneglectedtheload

ionisgivenby

F uchs, TransactionsA S M , A ugust1943, Page623 paper

tion byS . D. Gehman, Journa lA ppliedPhysics, June1942, Page

ynamicalP ropertiesofR ubber byC . O . H arris, JournalA pplied

-132giveadditionaldata.


/ h t t p : / / w w w . h





S P R IN G

tthereductionin vibrationfortheno-

thesamee pression(/ V / nl)—1asthat

v ibratingsupportinE q uation493. If

naltovelocityis assumed,ananalysis

sthattheratioof theforceP (transmitted

ssedforceP isgivenby

tngfactor, cc= 2V n fcdef inedas

. Insomecasesva luesofc/ ccaround. 02

butthis mayvaryconsiderablyfordif-

8.A curveshowingtheeffectofdamp-

(i. e. , ra tioP , / P ) isgiveninF ig. 2257.

ffectwithno dampingpresent,

calculatedontheassumptionofacon-

hiscurve alsoindicatesthat,forcon-

sesthetransmissibilityabovef/fn=

wthis value.Becauseofvariationinc

dtrue inallcasesparticularlywhere

cyisinvolved12.Thusforaneffective

ibilityshouldbechosenso astoobtain

above1.4.Ifthemountingis madetoo

ructuremaynotfunctionsatisfactorily.

ustusuallybereached.V aluesofratios

igherhavebeenusedintan anda ircra f t

notedthatastructuresuch asanair-

fferentfre uenciesofvibration(dueto

or ue, fore ample)w hilethesemay

ngeatvarious speeds.Consideration

,loc.cit..Page87.

: u c hs , F o o t no t e 10 .


/ h t t p : / / w w w . h





G

nciesisnecessary if resonanceande -

avoided.

hisproblemisparticularly important

mountingsforprotectinge uipmentin

omthesuddenmotionsresultingfrom

epth charges,orenemyaction.

trumentina shipmaybefle ibly

n spiteofthetransientmotionscaused

T UR I I N G F H O U i N C TT O N A I U A l F M O U N C T ( : h > )

ngineering, A ug. 1943.

ampingontransmissibilityatvariousratios f/fn

uencyandnatura lf re uency

Ingeneraltheimpacts ore plosions

scillationscombinedwithhighfre-

uchloweramplitudes,thefre uencies

dbythestructuralcharacteristicsof

ghf re uencymotionsmayresult in

uipmentrigidlyattachedtothe

iblemountingsthedamagingeffect

escanbeentirelyeliminated.H owever,

alsothelowfre uencylargeampli-

ountingssothatthemotionacrossthe

duringtherimeofthe transientto


/ h t t p : / / w w w . h





S P R IN G

emotionacrossthe mountingise -

achcasemust beconsideredwith

tionthatise pectedandthedesign

ble,forinstanceatlow speeds,two

beemployed.Thefirstis tointroduce

rmeans,whilethe secondistouse a

d-deflectioncharacteristicsothatthe

withincreaseddeflection.S uchmount-

anbe obtainedbyusingstopsorother

cillationsoccuratagiven fre-

nancew iththenaturalf re uency for

tudewilltendtobuild up.A sitbuilds

hecurvedload-deflectioncharacteristic,

ncreasesandwithit thenaturalfre-

throwthesystemoutofresonanceand

hemountingmust,of course,bede-

re uiredcharacteristicsatthegiven

untingsforagivenmachineit isde-

ethemin oneplanecoincidingwith

ismannerthetendencyofdifferent

mecoupledis reducedwhileatthe

yisobtained.

ibil ityhasbeendeterminedforagiven

ngsmaybedesignedbyusing the

eofthemountingmust besochosen

ntherubberare avoidedundertheanti-

tudeofmotion,whileatthesametime,

tismainta ined.


/ h t t p : / / w w w . h





I

R A G CA P A CIT O F V A R IO U S P R IN G

factorsmustbeta enintoaccount

efora givenapplication,totheprac-

mountof energywhichcanbestored

ofprimaryimportance.Thisis true

anddeflectionaregiven,whichmeans

agivenamountofenergy.Thisis the

edesignoflandinggear springsforair-

hespringsmustbeable toabsorbthe

ssof theplanefallingthroughacertain

presentchapter istocomparevari-

ashelical,leaf,cantilever,etc.,fromthe

agecapacityperunitvolumeofmate-

imumstress.Thiswillgivethe de-

mumvolumeofspaceneededforagiven

lumemaybemuchgreaterdepend-

tness).Itmaybenotedthatthis ap-

omewhatdifferentfromthatofChapter

fspring(i.e.,thehelical)wasdiscussed

lspaceoccupied.

O N - BA R S PR I N G

edas anidealspring,consisting

funiformsectionsub ecttoana ia l

thebarisloadeda ia lly, thestressdis-

is uniformandforthisreasonthis

umconditionfromtheviewpointof

geperunitvolumeofmaterial.If Iis

ss-sectionarea,thestressa willbeP/A .

a/ ,whereE isthemodulusofelas-

e totalelongation8willbeal/ .

ualtothearea undertheload-deflec-

ension-bar springtodistinguishitf romthehelica lten-

nownasatensionspring.


/ h t t p : / / w w w . h





S P R IN G

nergy17= %P8.This conditiongives:

vo lumeofmateria l.

staticloads, thetensiony ie ldpo int<rv

ngstress. Thecriterionofenergy

thereforebethevalueofU when

theotherhand, if thespringissub ect

ng,thestress, attheendurancelimit

ningenergy-storagecapacity.Because

chisusually presentneartheendsof

ped,itisals necessarytointroduce

nfactor2.Ifthis fatiguefactorisK t,

atfor variableloadstheenergystored

be

tbecauseofthepresenceof stresscon-

ragecapacityforrepeatedloadingwill

dealvalue(whichwoulde istifno

present).

S P R IN G

Thisspringconsistsessential-

frectangularprofileandconstantthic -

V I). F orsmalldef lections, thedeflec-

sing thenotationofChapterX V I,

by thefollowinge uation:

factorshavebeendiscussedinChapterV I.


/ h t t p : / / w w w . h





R A G CA P A CIT

heenergystoredmaybee pressedas

this, andta ingthevolumeV of thema-

hl, theenergyUbecomes

staticloads, thema imumenergystored

e ua ltothey ie ldstresso- w il lbe , from

uation496, it isseenthatthisva lue

ergywhichmaybestoredin theideal

samevolumeatthe yieldpoint.H ow-

t, whenthee tremefiberisstressed

tileverspringwillstillhavea consider-

lyieldingoverthecross sectionoccurs.

d energybyafactorofabout twoto

ethetension-barspringdoesnot have

9mentionedpreviouslyispessimistic

gecapacityofthesimplecantilever

thisreason,probablyabetter basisof

sis theenergystoredata loadproduc-

thesectionat thebuilt-inendofthe

aybemadeasfollows:

gulardistributionofstresse ists

completeyielding.Thismeansthat,

tressisaconstanttensione ua ltothe

nthecompressionside, thestress3ise ua l

angularsection, thee ternalbendingmoment

ngsteelsthe stresswilltendtor iseaftertheyieldpoint

agramofF ig.61).H owever,theassumptionofaconstant

presentpurpose.


/ h t t p : / / w w w . h





S P R IN G

followingrelation:

of load50percentabovethe value

chisbasedona linearstressdistribution

oad-deflectioncurveup tothisload

rbecauseofyieldingeffects.H owever,

lbe appro imatelylinearandfor

onedirection,the energystoredwill

fromE q uation500,usingthehigher

msreasonabletousethe lattere uation,

condition,asabasis forcomparison.

ulatedfromE q uation503andsubsti-

0thee pressionforenergystoredbecomes

for theidealcase.( q uation496).

erethe loadisvariable , thestressgiven

bemultipliedby afatiguestrengthreduc-

e intoaccountthestressconcentrationatthe

r.The storedenergyattheendurance

threductionisassumedfor the

t inasimpletension-barspring,the

ntheformerforfatigueconditionswill

er1.Thereasonfor" thismaybefound

verspringof rectangularprofileonly

totalvolumeofmaterial(i.e.,that

ub ecttoanythingapproachingthema i-

ss,inspiteofitsrelative inefficiencyin

springstill findsafieldof useparticu-

pringmustfunctionasa guideinaddi-

rengthreductionfactormaybeconsiderablylessin the

alvaluedependingonthe design.


/ h t t p : / / w w w . h





R A G CA P A CIT

onfunction.

eafS pring—Theefficiencyof utiliz-

antileverspringmaybeincreasedby

ngularor trapezoidalformsothatthe

ngthof thespringwillbeappro imate-

nditionintheusualtypeof leafspring

ngularprofile,thedeflectionattheend

henotationofC hapterX V I,

hesameas thatgivenbyE q uation499.

hestoredenergymaybee pressedas

E q uation507andta ingthevolumeV = =

mes

staticloads, theenergystoredw henthe

y ie ldpo inta is

is aboutone-thirdthatforthe

o int, Eq uation496. F orcompletey ie ld-

spring,theloadwill beabout50per

enbyE q uation509andforthiscondi-

springofrectangularprofile)the

maybeassumedtoincreaseroughly

Thisgives

gecapacityisabout25 percentbelow

lcase.

fatigue loadingtheenergystoredinthe


/ h t t p : / / w w w . h





L S P R I N G

larprofileis

strengthreductionfactorattheclamped

va luesofK f , thisvalue isappro imately1/ 3

case.

S I O N S P R IN G

ssumingahelica ltorsionspringof rec-

oaconstantmomentM( ig. 178), the

uation357,

ependsonthespringinde tota e into

edueto curvatureofthebar.F rom

gulardeflectionof theendof thespringis

scasewillbe halftheproductof

.Usingthis e uationtheenergybe-

nbyEq uation512inthisandta -

ae ua lto2w rnbh,

staticloads, thecurvaturecorrection

eredasa stress-concentrationfactorand


/ h t t p : / / w w w . h





R A G CA P A CIT

Theenergystoredat theyieldpointay

uation509foracantileverspring of

mpleteyielding,conditionswillbethe

antileverspring henceEq uation510

ase.

variable loadingtheenergystoredw ill

ngattheends inatorsionspring,

entrationeffectwhichmayberepre-

hreductionfactor K f.Ifthisfactor

refactorK 2,F ig.180,theformershould

15insteadof the latter.

orsionspringofcircularw irethe

tion364)is

correction factor.

diansis, fromE q uation366,

efore)

ir-rnd2/2 andusingE q uation516

edenergybecomes


/ h t t p : / / w w w . h





S P R IN G

astaticload, theenergystoredw henthe

nt< tvis,neglectingthecurvaturefactor

theenergystoredinthe ideal

yieldpoint.H owever,anincrease

elyintheratioof 1:1.7isnecessaryto

verthecrosssection foraround-wire

shownas beforebyassumingyield-

integratingoverthecircularcross

pondtoanincrease inenergytoabout

q uation519.Thus,forcomplete

centbelowthatcorrespondingto the

orvariable loads, theenergystoredw ill

ion518,

ctionfactor K 1attheclampedend is

mershouldbeusedinthise pression.

ueofKf forbothcases. E q uation521thus

oadingtheround-wiretorsionspring

sthe ideal(tension-bar)spring.

G

iporrectangularbarmaterialwith

llhavethesameenergy-storageca-

aterialas thetorsionspringofrec-

attheendsare clampedandthatthere


/ h t t p : / / w w w . h





H A G CA P A CIT 407

ntheturns sothattheydonot come

V III) . Insuchacaseasshow npre-

ro imatelyconstantalongthelengthof

aseofahelical torsionspringunder

fatigueloading,however,thestresscon-

different,dependingonthemethod

eends.

D R T O R S I O N

sub ecttoatorsionmomentMt

onstresstisgivenbyE q uation4, ta ing

f> ,inradians,ofa roundbaroflength

odulusofelasticity.

ne-halfthe productofthemoment

ence

valueofMtgivenbyE q uation522

ofmateria lV =w d- l/ 4,

staticloads, whentheshearstresstj ust

torsiont ,theenergystoredbecomes


/ h t t p : / / w w w . h





S P R IN G

atio . F ormostmaterialsmaybeta enas

/ 2. 6. A lso f romtheshear-energy

earingy ie ldpo intt maybeta enas

y ie ldpo int»> . Thus

uation524,forstaticloadingthe energy

ustreachestheyieldpointis

hanthat storedineithertheround

icaltorsionsprings( q uations514and

benotedthattheselatter springshave

weentheloadatwhich yieldingstarts

ingoverthe section.Thismargin

f thetorsionallyloadedround-bar

ofthesectionisstressed tovalues

henceforstaticloading comparisonon

s519and526isprobably toofavorable for

fairercomparisonmaybemadeby

ngoverthesectionasfollows:

utionofstress overthecrosssec-

ouldoccuraftercompleteyielding

sta enplace), themomentMtisgivenby

uation522thismeansthat,for com-

tise ualto4/3timesthe valuewhen

gthofMateria ls, PartI, Page57, S econdEdit ion.


/ h t t p : / / w w w . h





R A G CA P A CIT 409

rj ustreachestheyieldpoint intorsion.

ent-anglecurve(whichwillberealized

firstloadapplication)the energystored

na ltothes uareof themoment. Thus,

uation526w illbe increasedintheratio

3percentlowerthan thestored

ndshowsthatthehelicalc ompression

inde (w hichappro imatesacondi-

tivelyefficientasfarasenergystorage

is concerned.

orastra ightbarintorsionunc erfatigue

uation522maybeused. How ever, in

e somestressconcentration(at

e),andhencethevalue ofMfgivenby

videdbya fatiguestrengthreduction

isintoaccount. Ta ingtcastheendurance

uation523theenergystoredthus be-

droughlythat theendurancelimit

tothatintensionaediv idedbyV 3. Using

ingG= E / 2. 6asbefore, thefo llow inge -

andassuminge ualfatigue-strength

gy-storagecapacityofastraightbar

hatlessthan halfthaiofa simpleten-

er,aspreviouslyindicatedthiscompari-

differencesin theactualfatigue

ondesignand material.

ear-energytheory.


/ h t t p : / / w w w . h





S P R IN G

PR E S S I O N O R T N S I O N S P R I N G

spring,a iallyloaded(from

edbecomes

staticloads, thestressisca lculatedf rom

earstressmultiplicationfactor, F ig. 63.

uation531, ta ingthevolumeofma-

= ir2d2rn/ 2, andthestresse ua ltothe

hisgives

torfora straightbarintorsion,

e factorK0. A ga inta ingt, e ua ltothe

dby \ / 3andta ingE = G/ 2. 6this

33 )

or completeyieldingoverthesec-

multipliedby (4/ 3) asinthecaseofa

. Thisgives

I giveamorecompletediscussionofhe licalspringsunderstatic


/ h t t p : / / w w w . h





R A G CA P A CIT

fatigueorrepeatedloading, forarough

hodmaybeappliede ceptthatinstead

ctorK ( ig. 30, C hapterII)which

ffects duetocurvatureandtodirect

hisgives(f romEq uation533), ta ing

eadof ay

3and535it isseenthatthe largerthe

he energystorageperunitvolumeof

decreasew ithincrease inthe inde .

sonismadeonthe basisoftotalvolume

efoundthattheuse ofmoderateinde

imumeff iciency (C hapterX ).

uationsforspringcapacityasprev ious-

edinTableX X X havebeencom-

ergy-storagecapacityasfractionsof

nsion-bar)spring.Thusforstatic

ergystoragewhenfirstyieldingoccurs

gwillabsorb anamountofenergy

hatof the idea lspring. If comparisonismade

rageatfirstyielding,it appearsthat

ngisfar moreefficientthanother

,if comparisonismadeonthemore

agefor completeyieldingoverthe

deal springisnota greatdealmore

Thus,fore ampleonthisbasis(from

e),thetriangularcantileverspring,

gularsection,and theroundbarin

percentofthec apacityoftheideal

ding. F ora he lica lspringof inde 5,

astco lumnofTableX X X V , theenergy -

S T O R A C C A PA C I TI S


/ h t t p : / / w w w . h





S P R IN G

fvolumeon thisbasiswillbe

atoftheideal spring.H owever,when

basis ofenergy-storagecapacityfor

efatiguestrengthreductionfactors

ngis farmoreefficientthantheother

icethesefatiguefactorsmayvarycon-

torageCapacityforDifferentTypesofS prings

nsof thatinthe idea lcaseofasimpletension-barspring )

angu-TorsionStra ightC ompres-

pringR oundsionor

al(R oundBarinTension

gSpringB ar)TorsionSpring

11. 33. 33. 25. 43. 43/ , 2

. 77/ ,

-

. 33 / , 1 3 3/ , . 25 / , . 43 / , . 43 /1

torcompletey ie lding(staticloads)foridea lcasea2V / 2 -

ance lim it(variable loads)foridealcasece V / 2J .

gthreductionfactorsK finthis rowwillvaryamongthe

. Endurance lim itintorsionta enasf fe / y3.

nthe figuresgivenshouldbeconsid-

ationofthecapacitiesof thevarious

atin thechoiceofspringtype by

factors areinvolvedbesidesenergy

husabilityto fitintoamachineor

mountimportance.H owever,the

sasgiveninTableX X X maybehelp-

toformsomej udgementastothe

undergivenconditions.


/ h t t p : / / w w w . h





II

I A L S

concernedprimarily withadis-

antspring materials,theirproperties,

ticularreferencewillbemadeto

chmaybere uiredasaconse uence

eemphasiswill,however,beplaced

aterialitselfratherthanonthose of

terials,itshouldbeborne inmind

rictions,whereverpossibleplain

c wire,oil-temperedwire,hot-rolled

beusedinsteadof alloysteels,

rousmaterials,allofwhichutilize

als.Inmanypracticalapplications

sedwithoutlossof essentialproperties.

P R T I S O F M A T R I A L S

importantpropertiesofthedifferent

n T ab l es X X X V I , X X X V I I a nd

X X V Igivesatabulationof thecomposit ion

ordingto specificationsoftheA meri-

Materia ls. TablesX X X V IIandX X X V III

calproperties,includingultimate

sion,modulusof rigidity,modulusof

orferrousandnonferrousspringma-

asizedthatin individualcases,spring

mewhatfromthevaluesshown.

summaryofavailabledataonendur-

alsas foundintheliteratureis given

ndX L , theformerapplyingtotorsionand

propertiesofhelicalsprings werediscussedinChapterIV .

in, IronA ge, A ugust16, 1934, Page18, " Mechanica l

byWallaceBarnesC o. , 1944, and" ManualonDesignandA p-

pira lS pringsforO rdnance , publishedbySA . . WarE n-

eadditionaldataonspringmaterials.


/ h t t p : / / w w w . h





S P R IN G

nentinformation,includingk indof

surfacecondition(i.e.,whetherground

d),ultimateandyieldstrengthsin ten-

ndyieldstrength intorsion,andlitera-

ogetherwiththelimiting endurance

urancerangefrom0-110,000pounds

pringMaterials

pecifications)


/ h t t p : / / w w w . h





I A L S

endurancediagramofF ig.226has

sentswhatmaybe e pectedforgood

gmateria linthic nessesaroundV * toV i-

en thatforgroundandpolishedspeci-

endurancelimitsmaybee pected

erva luesmayalsobee pectedforhigh-

als.

causeofstressconcentrationeffects

mpededges,etc.,theactual endurance

atspringsingeneralare considerably

wninF ig. 226. Thisisshow nby thetests

s reportedbyBatsonandBradley

L II. F orthesetests, asshow nonthis

stress inthemasterleafof thespring

40, 000poundspers uare inch. F or

alS pringS teels


/ h t t p : / / w w w . h





S P R IN G

ncentrationeffectspresentintheactual

nC hapterX V Iasmallho le inaf la t

ancerangeabout 50percent.

F S P R IN G W IR E S A N D M A T R I A L S

tiesanduses ofthemoreim-

re andspringmaterialswillbebriefly

eferencetothedata giveninTables

I I , an d X X X V I I I. P e rt i ne n t da t a of i m po r ta n ce i n

mateendurancediagramsforgood

ngmaterials

ationanduseof thevariousmaterials

.

ua lity carbonstee l, thisw ire isw ide ly

springs,particularlythosesub ectto

hehighstrengthofthe materialis

ofabout.70to1.00per centcarbon,

gtosize.The compositionasspecified


/ h t t p : / / w w w . h





I A L S

sgiveninTableX X X V Iandtypica lphysica l

V II. Thisspecif icationa lsoca llsfor

mtensilestrengthvaluesasshownby

esofF ig. 227. A sw illbeseenf romthese

sofS tainlessS teelandN on- errousMetals

stionanceModulus

of

nchesinhendingrigidity

< ■ * < * > l T

0. 000toTO . 000. 026^ 10.

mPer 160. 000to110. 000to26 10 3-850.0009. 5 10

40,000

" V " ' 1 80 ,0 00 t o 13 0, 00 0 to 3 0x 1 0 5 -1 0 11 ' » > '

0170,0006to

O t o 6 t o

8 0 .0 0 0 to I , 1 0 5 - O ^ , .

ependingonheattreatment.

ofmusic wiremayvaryfrom255,000

orthelarger wiresizesto440,000pounds

smaller.Thecarboncontentofthisma-

the wiresize,thesmallersizes hav-

omespecificationscallfora limited

carboncontentfora givenwiresize.

sedforspringslargerthan about%-

canbesuppliedinlarger sizeson

f musicwire,thewindingisdone

erwinding,itis advisabletogivethe

eheattreatmenttorelievecoiling

mentmaycallfor heatingthesprings

500 degreesF ahr.foronehourfor

to30minutesfor thesmallersizes.

smadelargelyfromS wedishsteels

tyandhighqua lity . A t present, A meri-


/ h t t p : / / w w w . h





S P R IN G


/ h t t p : / / w w w . h





I A L S

toma ethismaterialwithsatisfactory

btainedwithcadmium-platedsur-

recorrosionisa factor(intheapplica-

ustbeta entoguardagainsthydrogen

happlications,cadmium-platedsprings

bstitutefor18-8stainless steelwire.

e—Thisisagood- ua lity , high-

theopen hearthorelectricfurnace

080910IIJ2. 13J41516

R , IN O C

mandminimumtensilestrengthcharacter-

rioussizes, f romA S TMA 228-41

co ld-woundsprings. A S TMA 229-41

istedinTableX X X V I, C omposit ionA

er7/32-inch.

iscold drawntosizeandthen

werlimitsfor tensileultimate

wireasgivenbythe foregoingspeci-

stwiresize inF ig.228.F urtherlimita-

h callforavariationin tensilestrength

poundspers uare inchinasingle lo tin

notmorethan25,000pounds per

ve. 120- inch. Itw il lbenotedthatthe

e aresomewhatbelowthevaluesfor


/ h t t p : / / w w w . h





PhysicalPropertiesofS pringMaterialsinBending


/ h t t p : / / w w w . h





I A L S

sicwire,F ig.227.O therphysical

ableX X X V II.

wire,springsmadefrom oil-tempered

dand thengivenathermaltreatment

Thismaybe donebyheatingat500-

-hour.

O f low erqua lity thanmusicor

terialisutilizedin caseswherethe

highdegreeofuniformityisnot es-

posit ionasspecif iedbyA S TMA 227-

T R , [ N O E S

mandminimumvaluesofult imatetensilestrength

mperedwire,A S TMA 221-41and229-41

X V I. Thisspecif icationfurtherre uires

lotofmaterialshall notvarybymore

manganesebynotmorethan.30per

rboncontentsare usedforthelarger

valuesoftensilestrengthinthese

mvaluesofultimatetensilestrengths

eninA S TMA 227-41areshownby

228asa functionofwirediameter.A

hisspecificationisthat theultimate

llnotvarymorethan 40,000pounds

below.072-inch,norbymorethan30,-

inchforsizesabove. 072- inch. Windingis

rmaltreatment.


/ h t t p : / / w w w . h





S P R IN G

wire—W ithhighductilityin thean-

sutilized incaseswheresevereform-

ryinthe manufactureofthespring

torsionspringswithcertainshapesof end

dPhysica lP ropertiesofL eafand

nB ending


/ h t t p : / / w w w . h





I A L S

gs,heattreatedafterforming—F or

prings(overabout% to% -inchwire

altowindthesprings cold.Insuch

woundhot fromeithercarbonoralloy-

eated.F orcarbonsteelbars,the com-

T M A 6 8 - 39 i s g iv e n in T a bl e X X X V I .

rings, A S TMA 125-39ca llsforheat-

00degreesF ahr.andcoilingona pre-

gsarethenallowedto cooluniformly

ll ipticLeafS prings

e llStressR ange

Ib/ s in. )

4453, 000to32, 000t

3492.500to46,000

3424,100to43,000

dB radlev , Dept. o fSci. & Ind. R esearch(Brit ish)S pecial

neffectsacttoreduce strength.Theseareduetoclampsused

.

hichtheyareheat-treatedtoatempera-

greesF ahr. andq uenchedino il. A f ter

retemperedbyheatingto 800degrees

willgivea hardnessaround375to425

opertiesofthismaterialare shown

re— Inthepastthis alloy-steel

ly specifiedw hereahigh- ua litymateria l

eraturesaresomewhathigherthan

eforautomotivevalvesprings.Because

loysteels,however,its useshouldbe

thisconnectionit shouldbenoted

Z immerli4didnotshow amar edsuperior-

eelcomparedtocarbonsteelsas faras

a ationatelevatedtemperaturewas

e-vanadiumvalvespringq uality

tureonC oiledS tee lSpringsUnderV ariousL oadings , Trans-

1941, Page363.


/ h t t p : / / w w w . h





S P R IN G

M A 2 3 2 -4 1 i s li s te d i n Ta b le X X X V I . F o r

msteelspringwire(asdistinguished

w ire), somewhathigheramountsofphos-

ur(.05% )areallowed.

btainedeitherintheannealedor

on.W henwoundfromannealedwire,

eatedaftercoiling.A fterwinding

e-vanadiumwire,alowtemperature

00-700degreesF ahr.shouldbegiven,

turesbeingpreferredforapplications

ratures.Othertensile propertiesofthis

eX X X V II.

ingwire— S tainlesssteelshavinga

rcent chromiumand8percent nic el

sub ecttocorrosionconditions.They

ted-temperatureconditions.A typical

ialcallsfor thefollowingcomposition:

wireisdevelopedbycolddraw-

,000to 320,000poundspers uare

zeasshow ninTableX L III.

lpropertiesofthistype ofsteelare

I I I .

steelwire arewoundcoldandmay

heattreatmentata temperatureof

minutestoan hour,theshortertimebe-

resizes.

dingitsgreatestusein caseswherea

conductivityis desired,phosphor

plicationswherecorrosionresistanceis

presentbecauseofhightin content(5to

elyrestricted.A possiblesubstitute

desiredis berylliumcopper.Typical

veninTableX X X V III.

, m in.


/ h t t p : / / w w w . h





I A L S

analloyconsistingessentiallyof

mandtherestcoppertogetherwith

ys5.Ithas theadvantageofhavinga

ywhilenotre uiringanytinin its

pringWire

wiremadefromthismaterialis

egreesF ahr. andthencolddraw nto in-

tercoiling,itis heattreatedtoincrease

isheattreatmentmayalsobe varied

asticity ortheamountofdrift orcreep.

ertiesof thismaterialaregivenin Table

nalloycomposedof about70percent

ncwhichiscold rolledtogiveit high

esare listedinTableX X X V III. B e-

stressesmustbek eptmoderateifthis

ver, ithastheadvantageofnotre uiring

,whileatthesametimepossessing

yandcorrosionresistance.

pper-nic e la lloy tow hich2to4per

added.A typicalcompositionis:

c e l66percent a luminum2.75percent.

en asolutionheattreatmentandthen

g,springsaregivena finalheattreat-

essand strength.Bythismeans,ulti-

C arson, " S pringsofB ery ll iumC opper, A eroDigest, July

oysforS prings, P roductE ngineering, June1938, giveaddit iona l


/ h t t p : / / w w w . h





S P R IN G

0,000to200,000poundspers uareinch

rdataonphysicalpropertiesare given

pringsof thisa lloyareusedforcorrosion

cetoelevatedtemperatures0 .

ion-resistanta lloycontainingabout98

erialalsohasgood mechanicalproper-

ub ecttoelevatedtemperatures.Be-

fterheattreatment,springsofthis

ndfromheattreatedwire.F urther

X V III.

onsforcriticalspring materials—

eredfor useasasubstitutematerial

itionsandstaticloadingare involved

springuse,this materialconsistses-

steelhavinga thincoatingofcopper

osion.Tensilepropertiesinthe heat

btainedwhichapproachthoseof hot-

ted.W herefatigueloadingisin-

otbeadvisabletouse thismaterial

uefailureof therelativelywea surface

tpresentfor staticloading.

mglass rodsandtemperedhave

undercorrosionconditions7.The

cingsurfacecompressionstressesby

herebygreatlyincreasingthetensile

wtensilestrengthof glassrelative

naftertempering),muchlowerwor -

acticaldesign.S incetheenergy-storage

esas thes uareofthestress,other

meansthattheglass springwillusually

anacorrespondingonemadeofspring

ughthemodulusofelasticityofglass

owever,wherespaceisavailablefor

smaybek epttolowvalues,this ma-

foruseparticularlywhere corrosion

. . T ransactionsA S M , July1942, Page465givesdata

ofthisandother nic elalloysatelevatedtemperatures.

hael,MachineDesign,A ugust1942,Page85,givesfur-

ula s, asw ellasarticlebvT. J. Thompson, P roductEngineer-


/ h t t p : / / w w w . h





or ingstress)

rome-vana-

el,etc.)

85, 165

18,420

, 341

wire

he licalsprings. . 30

gs(seeV alvesprings)

ation.260

5b

ulations.261

6

239,258

0

59

60

glected260

262

6

rings361,375

s

s316

soF latS prings)

93

(seeE nergy-

modulus80

8,420,422

18,420,422

ar-barsprings214,217

ded..151,154

111

4

4

re


/ h t t p : / / w w w . h





S P R IN G

ory,helical

0

berspring. . . 379

cal(seeH eli-

lutespring.... 360

springs(see

thic ness)

4

materials....22

ncipa lf re uen-

5

culation..115

calsprings

callyloaded

mplitude....12

bber384

ubber.... 387

5

8

ntings...395

y397

t20,304,420, 422

idity77

springs...151,154

,254,256

1

ring.. .386

prings388,390

thic ness. 279, 285

gs196

94

ndlatera l

gs

ralloading178

6

changle , e -

changle . . . 47

49

springs


/ h t t p : / / w w w . h





3

ation,modu-

ned(seeBelle-

t

27b

m284

given

84

ement.274

73

theory276

elicalsprings

60

tings,inreduc-

ation,helical

calsprings...26

diagram....16

gs193

springs157

5

seea lsoEndur-

terials..304,416

s13,14

91

ued)

2

2

7,420,423

alsprings..88,89,91

ts)

sspringfunction2

es399,412

410 P u b l i c D o m a i n , G o o g l e - d i g i t i z e d

/ h t t p : / / w w w . h





S P R IN G

tra lloading. 293

3

ects299

calsprings171

terionforenergy

3

pe

gs

68

ttheory43

5

111

vaturein-

ralloading.. 177

y44

4

59

ation183

o157, 196

y183,410

gs(continued)

alculation131

28

22

n414

ature113

163,165

65, 166 P u b l i c D o m a i n , G o o g l e - d i g i t i z e d

/ h t t p : / / w w w . h





springs(continued)

prings

210

0

10

rgystorage189,191

eeTorsionsprings)

sconcentration

gs

steel

ngs350

frontwheels4

210

42

s...213,219

208

elicalsprings 96

rings18,-94

ngs193,197

gs(seeBelle-

s(seeDis

83

lsprings


/ h t t p : / / w w w . h





S P R IN G

5

415

ation....82

n77

165

6

,417

f80

odofde-

6,415,417

bbersprings)

19

7,420

rings335

ction

s219

48,51,56

212

s

s211

48,54

211

2

ran arrange-

licalsprings

rgystorage

ting,.7% car-

sts

blestress17

essconcentra-

21

gs50

tation62

rings166

modulusof


/ h t t p : / / w w w . h





84 386

7

andcalcu-

0

ocentero f

24

entration,fa-

ngs120

urface22

9, 91, 420

ntensity...92

ngpeening

ot-blasting)

82

onfactor100

springs173

erionforen-

mounting.... 398

rings199

41

ncontact. .343

7,418,420,423

417,418,420

de , spring)

ngs P u b l i c D o m a i n , G o o g l e - d i g i t i z e d

/ h t t p : / / w w w . h





S P R IN G

ia landlat-

berofturns

193

ging234

ncerange, En-

oading)

8

375

or o ingstress)

8,121,124

tors,flatsprings

rvature

24

25

eeStra inmeas-

1

e(seeE ndurance

)

gs

gs,methodsol

nt,frontwheels

ancediagram

t20,304,420

idity77

ertiesanden-


/ h t t p : / / w w w . h





tionstress

vibration226,234

ress

5

o lutespring. .. 371

lsoF atigue load-

ndur-

sF atiguestress,

urance

s

rigidity76

ofreducing

antages..359

0

nued)

0

371

onfortan . . 360

lf re uency. . . 232

a inlessstee l

35

ngs202

helica lsprings,

129

100

eModulusofelas-


/ h t t p : / / w w w . h





/ h t t p : / / w w w . h





L I BR A R I S

mechanical springs wahl

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