(1.5.5)-2.5 Thermal Effects,Misfits,an材料力學材料力學

Chapter2AxiallyLoadedMembers

MechanicsofMaterialsThermalEffects,Misfits,andPrestrainsWelcometomechanicsofmaterials.Inthepreviousstudies,stressesandstrainsinastructurewerediscussedinducedbyonlyexternalloads.However,externalloadsarenottheonlysourcesofstressesandstrains.Othersourcesincludethermaleffectsarisingfromtemperaturechanges,misfitsresultingfromimperfectionsinconstruction,andprestrainsthatareproducedbyinitialdeformations.Thermaleffects,misfits,andprestrainsarecommonlyfoundinbothmechanicalandstructuralsystemsandtheyaremuchmoreimportantinthedesignofstaticallyindeterminatestructures.Wearegoingtodiscusstheminthissection.ThermalEffects-Changesintemperatureproduceexpansionorcontractionofstructuralmaterials,resultinginthermalstrainsandthermalstresses.α

-thecoefficientofthermalexpansionExpansionispositiveandcontractionisnegative.Thermalstrain:Unitsofα

:1/K,1/℃SignconventionforεT:Temperature-displacementrelationThermaleffects,refertothe*changesintemperatureproduceexpansionorcontractionofstructuralmaterials,whichwillinducethermalstrainsandthermalstressesinstructures.This*isasimpleillustrationofthermalexpansion.Whentheblockisheated,thedimensionsoftheblockincrease,becauseeveryelementofthematerialundergoesthermalstrainsinalldirections.Formoststructuralmaterials,thermalstrainεTisproportionaltothetemperaturechangeΔT;thatis*.Inwhich,αisapropertyofthematerialcalled*thecoefficientofthermalexpansion.thedimensionsofαcanbeexpressed*aseitherthereciprocalofkelvinsorthereciprocalofdegreesCelsius.And,thesignconvention*forthermalstrains,normallyassumethatexpansionispositiveandcontractionisnegative.AssumethatthematerialishomogeneousandisotropicandthatthetemperatureincreaseΔTisuniformthroughouttheblock,ifoneofthedimensionsisL,thedimensionwillincreasebytheamount*.Thisequation*isatemperature-displacementrelation,analogoustotheforce-displacementrelations.Itcanbeusedtocalculatechangesinlengthsofstructuralmembers*subjectedtouniformtemperaturechanges,Staticallydeterminatestructures:

uniformtemperaturechangesinthemembersproducethermalstrains,withnothermalstressesproduced.Staticallyindeterminatestructure:mayormaynotdeveloptemperaturestresses,dependinguponthecharacterofthestructureandthenatureofthetemperaturechanges.ThermalEffectsonstructuresNostressesTheprecedingdiscussionsofthermalstrainsassumedthatthestructurehadnorestraintsandwasabletoexpandorcontractfreely.However,manystrucutreshavesupportsthatpreventfreeexpansionandcontraction,thenthermalstressesmaydevelop.Toillustratesomeoftheseideasaboutthermaleffects,considerbothstaticallydeterminatestructuresandstaticallyindeterminatestructures.Forstaticallydeterminatestructures,takethisstructure*forexample.Undertheuniformtemperaturechange,allmembersinthisstructurearefreetolengthenorshorten,resultinginadisplacement*ofjointB.Therearenostresses*ineitherbar.Thisconclusion*appliesgenerallytostaticallydeterminatestructures.Thatisuniformtemperaturechangesinthemembersproducethermalstrainswithoutproducinganycorrespondingstresses.However,staticallyindeterminatestructures,*mayormaynotdeveloptemperaturestresses,dependinguponthecharacterofthestructureandthenatureofthetemperaturechanges.Forinstance*,nostressesaredevelopedwhentheentiretrussisheateduniformly.However,ifjustsomebarsareheated,thermalstresseswilldevelop,becausethestaticallyindeterminatearrangementofthebarspreventsfreeexpansion.Theanalysisofastaticallyindeterminatestructurewithtemperaturechangesisbasedupontheconceptsdiscussedintheprecedingsection,namelyequilibriumequations,compatibilityequations,force-displacementrelationsandtemperature-displacementrelations.Wewillseehowtosolvetheseproblemsthroughanexample.Example1:DeriveaformulaforthethermalstressσTdevelopedinthebar.AprismaticbarABoflengthLismadeoflinearlyelasticmaterial.ThebarhasamodulusofelasticityEandacoefficientofthermalexpansionα.

IfthetemperatureofthebarisraiseduniformlybyanamountΔT

.Solution:Degreeofstaticindeterminacy=1Thecompatibilityequation:RBRAEquationsofequilibrium:Force-dispalcementrelation:Temperature-dispalcementrelation:Example,toderiveaformulaforthethermalstressσTdevelopedinthebar.Byobservingthisbarwithbothendsfixed,thisstructureisone-degreestaticallyindeterminate*.ThecompatibilityequationfromthefactthatthechangeinlengthofthebarδABiszero*.δABconsistsoftwoparts,thechangeinlengthinducedbytemperaturechangeδTandinducedbythereactionforces,δRB*.TheonlynontrivialequationofstaticequilibriumisthatreactionsRAandRBmustsumtozero*.Introducetheforce-displacementrelation*andthetemperature-displacementrelation*forthisproblem.Then,combinealloftheequations,wecanobtainRA,RB*andthenormalstressσT*inducedinthebarbytemperaturechange.MisfitsandPrestrains②Staticallydeterminatestructures:Smallmisfitswillnotproducestrainsorstresses.①Staticallyindeterminatestructures:Misfitswillintroduceprestrainsandprestresses.

ABCD

lMisfits:Amemberdoesnotfitintothestructureinitsintendedmannerduetoitsmanufacture.Prestrains:Inducedbymisfits

inastructureandexistbeforeanyloadsappliedtothestructure.SupposethatamemberDAofthisstructure*,ismanufacturedwithitslengthshorterthanitsprescribedlength,withthedifferenceofδ*.Then,memberDAwillnotfitintothestructureinitsintendedmanner.Andthefinalgeometry*ofthestructurewillbedifferentfromwhatwasplanned.Situationsofthiskindarecalledmisfits*.Misfitswillinducestrainsinastructureatthetimeitisbuilt.Thosestrainsexistbeforeanyloadsappliedtothestructure,arecalledprestrains*.Accompanyingtheprestrainsareprestresses,andthestructureissaidtobeprestressed.Commonexamplesofprestressingarespokes*inbicyclewheels,whichwouldcollapseifnotprestressed,thepretensionedfacesoftennisracquets*,shrink-fittedmachineparts,andprestressedconcretebeams*.Misfitswillinduce*prestrainsandprestressesinstaticallyindeterminatesturctures,becausethesestructuresarenotfreetoadjusttomisfits,justastheyarenotfreetoadjusttocertainkindsoftemperaturechanges.However,ifastructureisstaticallydeterminate*,smallmisfitsinoneormorememberswillnotproducestrainsorstresses,althoughtherewillbedeparturesfromthetheoreticalconfigurationofthestructure.Thus,theeffectsofamisfitaresimilartothoseofatemperaturechange.Theanalysisofastaticallyindeterminatestructurewithmisfitsandprestrainsproceedsinthesamegeneralmannerasdescribedpreviouslyforloadsandtemperaturechanges.

aaQuestion1:Theupperandlowerendsofaladder-likesteelshaftarefixedattemperatureT1=5℃asshowninthefigure.Areasoftheupperandlowersegmentsare