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NormalStressDistanceforaBeaminPureBending

Subject:

S&SA

StudentNo.:

SH-08-0652G

StudentChineseName:

张晗

StudentName（Pinyin）:

ZhangHan

StudentEnglishName:

Vincent

TeacherName:

MarkCaulfield-browne

Date:

4/24/2010

Abstract

Theaimofthisexperimentistomeasurethenormalstressdistributionacrossthecrosssectionofarectangularbeamindifferentpurebendingwhichareproducedbyabendingbeamexperimentaldevice.YJ-4501Astaticdigitalstraingaugesareusedtomeasurethestrainandtheresultscanbereadfromastraingaugeoutputdevice.Themeasuredexperimentalstressvaluesaretocomparedtotheoreticalvaluesandtheerrorcalculated.

Introduction

Whenexternalforceisappliedonobjectmadeofelasticmaterial,therewillbeachangeofsizeandshapeoftheobject.（R1）Strain,representedbytheGreekletterε,isatermusedtomeasurethedeformationorextensionofabodyduetoexternally-appliedforces.（R2）Itcanbeclassifiedasnormalstrainandshearstrain.Normalstrainisgenerallydefinedasthechangeinlengthdividedbytheinitiallength.

ε=ΔL/L

Inthisexperiment,onlynormalstrainwillbemeasured.Stressistheinternalforceassociatedwithstrain.AccordingtotheFigure1.below,itcanfoundthatthereisalinearrelationshipbetweenstressandstrainwhenelasticdeformationhappens.Andtheequationσ=Eεcanbegot,whereEisthematerialpropertythatrepresentsthestiffnessofthematerialcalledYoung'sModulus.

Figure1.

Inengineering,it’simportanttomeasurestrainasitcanrepresentthehardnessofmaterial.Ithelpsengineerstochooseappropriatematerialwhiledesigning.Inthisexperiment,normalstresswillbemeasured.Andthemeasuredstressvaluewillbecomparedtothetheoreticalvaluesandtheerrorcalculated.

Background

Straingaugeisametalstripthatcanbeusedasameasuringelementforphysicalforceifthestressiskeptwithinitselasticlimit.Ifthestripisstretched,itwillbecomeskinnerandlongerleadingtoanincreasingofitsresistance.Onthecontrary,ifitisplacedunderacompressiveforce,itwillbroadenandshortenanditsresistancewilldecrease.（R6）Thusappliedforcecanbeinferredfromitsresistance.

In1856,therelationshipbetweenstrainandresistanceofwireconductorwasfirstreportedbyLordKelvin.Andinearly1930s,bondedresistancestraingaugewasfirstusedtomeasurevibratorystrain.Thenadiscoverythatsmalldiameterwiresmadeofelectricalresistancealloyscouldbebondedtoastructuretomeasuresurfacestrainwasmadein1937.It’sabreakthoughthatfromthenonstraingaugemeasurementswereadoptedforuseinaircraftdevelopmentprogrammes.Lateron,theideaofmakingastraingaugebyetchingthepatternforthegaugefromathinfoilwasdevelopedbySaunders-Roein1952.（R7）

Thetypesofstraingaugematerialsarevariousincludingpiezoresistiveorsemiconductorgauge,carbon-resistivegauge,bonded-metallic-wire,andfoil-resistancegauges.Andthemostcommontypeofstraingaugeisthebonded-wirestraingaugewhichisoftenusedinpressuresensor.（R8）Itlookslikethis:

Figure2

Itsusuallyresistancesrangefrom30Ωto3kΩ.Thisresistancemaychangeonlyafractionofapercentforthefullforcerangeofthegauge.Soaquarter-bridgestraingaugecircuitisdesignedconsideringprecisiondemanding.ThecircuitisshowninFigure3.

Figure3.

Normally,therheostatarmofthebridgeR2issetatavalueequaltothestraingaugeresistancewithnoforceapplied.Toensurethebridgewillbesymmetricallybalancesandthevoltmeterwillindicatezerovoltswhichrepresentingnoforceonthestraingauge,R1andR3aresetequaltoeachother.Whenthereisaforceappliedonthestraingaugemakingiteithercompressedortended,itsresistancewilldecreaseorincrease.Asaresult,itwillproduceachangeinthevoltmeterwhichhastherelationwithappliedforce.

Theory

AstraightbarofhomogeneousmaterialissymmetricalaboutY-Yasthefollowingfigureshows.

Figure4.

Assumethatthetraverseplanesectionsremainplaneandnormaltothelongitudinalfibresafterbending.Ifthebarissubjectedtoamomentatoneendandanequalbutoppositemomentattheotherend,theveryclosesectionABandCDwillnolongerbeparallel.ACwillhaveextendedtoA'C'andBDwillhavecompressedtoB'D'astheFigure5.shows.

Figure5.

ThelineEFwillbelocatedsuchthatitwillnotchangeitslength.  ThissurfaceiscalledneutralsurfaceanditsintersectionwithZ-Ziscalledtheneutralaxis.ThedevelopmentlinesofA'B'andC'D'intersectatapointowithanangleofθradiansandtheradiusofthecurveE'F'=R.Ify（E'G'）isthedistanceofanylayer（H'G'）originallyparalleltoEF.Thentwoequationscanbegot

H'G'/E'F'=（R+y）θ/Rθ=（R+y）/REq.1

AndthestrainεatlayerH'G'is

ε=（H'G'-HG）/HG=（H'G'-HG）/EF=[（R+y）θ-Rθ]/Rθ=y/REq.2

Theacceptedrelationshipbetweenstressandstrainisσ=Eε.（Eiselasticmodulus）

Therefore

σ=E.ε=E.y/REq.3

σ/E=y/REq.4

Asthebeamisinstaticequilibriumandisonlysubjecttomoments,theforcesacrossthesection（AB）areentirelylongitudinalandthetotalcompressiveforcesmustbalancethetotaltensileforces.  Sotheinternalcoupleresultingfromthesumof（σ.dA.y）overthewholesectionmustequaltheexternallyappliedmoment

Theforceoneachareaelement=σ.&A=σ.z&y.Eq.5

Theresultingmoment=y.σ.&A=y.σ.z&y.Eq.6

ThetotalmomentM=∑（y.σ.&A）=∑（y.σ.z&y.）Eq.7

Usingσ=E.y/R

M=E/R.∑（y2.&A）=E/R∑（y2.z&y.）Eq.8

∑（y2.&A）istheMomentofInertiaofthecrosssection.

Fromtheabovethefollowingimportantsimplebeambendingrelationshipresults

Eq.9（R9）

Equipment

1.Abendingbeamexperimentaldevice

Itisusedtoproduceaforceappliedonthebeam.

2.YJ-4501Astaticdigitalstraingauge

Itisusedtomeasurethestrain.

3.Astraingaugeoutputdevice

Itisusedtodisplaytheresultsofstrain.

ExperimentalProcedure

1.Setaloadof0.5KNonthebeam.

2.Resetthestraingaugeoutputdevicetomakestraingaugereadingstozero

3.Increasetheloadby1KNandrecordthereading.

4.Repeatthisforatotal4setsofstraingauges.（1KN,2KN,3KN,4KN

Results

Table1.Themeasureddata

StrainGauge

DistanceFromNeutralAxis（mm）

StrainGaugeReadingsForAppliedLoad（uε）

1KN

2KN

3KN

4KN

1

y1=0

0

0

0

0

2

y2=10

-37

-73

-110

-146

3

y3=-10

36

71

107

141

4

y4=15

-55

-109

-163

-216

5

y5=-15

56

109

162

215

6

y6=20

-72

-144

-215

-286

7

y7=-20

74

146

218

288

Thetableshowsthereadingdatarecordedfromtheexperiment.Asσ=Eε,experimentalstressesfordifferentloadscanbegot.

Table2.Experimentaldata（106Pa）

1KN

2KN

3KN

4KN

Stress

0

0

0

0

-7.77

-15.33

-23.1

-30.66

7.56

14.91

22.47

29.61

-11.55

-22.89

-34.23

-45.36

11.76

22.89

34.02

45.15

-15.12

-30.24

-45.15

-60.06

15.54

30.66

45.78

60.48

I=bh3/12=0.02x（0.04）3/12=32/3x10-8m4

M=（F/2）.C,whereC=150mm.SoM1=75N.mM2=150N.mM3=225N.mM4=300N.m

σ=（M/I）y

Table3.Theoreticalstress（106Pa）

1KN

2KN

3KN

4KN

Stress

0

0

0

0

-7.03

-14.06

-21.09

-28.13

7.03

14.06

21.09

28.13

-10.55

-21.09

-31.64

-42.19

10.55

21.09

31.64

42.19

-14.06

-28.13

-42.19

-56.25

14.06

28.13

42.19

56.25

Accordingtotheequationsabove,theoreticalstressescanbecalculated.

Figure6.

Figure6.showsthediagramofstressdistributionagainstdistancefromtheneutralaxisforexperimentalandtheoreticalvalueswithaload1KN.Theequationforthetheoreticalvalueis

y=-1.42x,andtheequationfortheexperimentalvalueisy=-1.3x

Figure7.

Figure7.showsthediagramofstressdistributionagainstdistancefromtheneutralaxisforexperimentalandtheoreticalvalueswithaload2KN.Theequationforthetheoreticalvalueis

y=-0.71x,andtheequationfortheexperimentalvalueisy=-0.66x

Figure8.

Figure8.showsthediagramofstressdistributionagainstdistancefromtheneutralaxisforexperimentalandtheoreticalvalueswithaload3KN.Theequationforthetheoreticalvalueis

y=-0.47x,andtheequationfortheexperimentalvalueisy=-0.44x

Figure9.

Figure9.showsthediagramofstressdistributionagainstdistancefromtheneutralaxisforexperimentalandtheoreticalvalueswithaload4KN.Theequationforthetheoreticalvalueis

y=-0.36x,andtheequationfortheexperimentalvalueisy=-0.33x

Discussion

Byusingthebendingbeamtestingdevicetoproducealoadonthebeam,strainsfordifferentbendingmomentscanbegot.Accordingtotheequationσ=E.ε,theexperimentalnormalstressdistributionacrossthecrosssectionofarectangularbeamcanbecalculated.Also,accordingtotheequationσ=（M/I）y,thetheoreticalvaluescanbeobtained.

Comparethetheoreticalvalueswithexperimentalvalues,the%errorscanbecalculatedout.Heretheslopeofthelinerepresentingthedeformabilitywillbemadeasthecomparisonobject.

The%errorforload1KN

%error=

=8.45%

The%errorforload2KN

%error=

=7.04%