Design against fluctuating load

Stress concentration factor, Notch sensitivity, Endurance limit, etc

1. DESIGN AGAINST FLUCTUATING LOAD

2. Stress Concentration: Causes and Remedies
•Stressconcentrationisdefinedasthelocalizationof
highstressesduetoirregularitiespresentinthe
componentandabruptchangesincross-sectionof
thecomponent.
•Irregularitiesmaybeoil-holes,grooves,keyways,
splines,screwthreadsandshoulders.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

3. •Toconsidertheeffectofstressconcentrationandfindout
localizedstress,afactorknownasstressconcentration
factorisused,denotedby??????
�
??????
�=
HighestValueofactualstressneardiscontinuity
Nominalstressobtainedbyelementary
equationsforminimumc/s
Or ??????
�=
????????????��
??????0
=
????????????��
??????0
;
where??????
���,??????
���arelocalizedstressesand??????
0,??????
0arestresses
determinedbyelementaryequationsatthediscontinuities.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

4. Causes of stress concentration
1.Variationinpropertiesofmaterials:itisduetointernal
cracks,flaws,cavitiesinwelds,airholesinsteelcomponents,
non-metallicorforeigninclusions.Thesevariationsactas
discontinuitiesandcausestressconcentration.
2.Loadapplication:inapplicationsasmeshingteethingears,
ballsandracesinballbearings,railandthewheel,crane
hookandthechain,aconcentratedloadisappliedovera
verysmallarearesultinginstressconcentration.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

5. 3.Abruptchangesinsection:Stepsareprovidedontheshaftin
ordertomountgears,sprockets,pulleysandballbearingsleadsto
changeinc/sandcausesstressconcentration.
4.Discontinuitiesinthecomponent:featuresofmachine
componentsasoilholes,oilgrooves,keyways,splines,screw
threads,resultindiscontinuitiesinc/s,andcausesstress
concentrationinvicinityofthem.
5.MachiningScratches:Machiningscratches,stampmark,
inspectionmarkaresurfaceirregularitieswhichcausestress
concentration.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

6. Charts for Stress concentration Factor
Fig.: Rectangular plate with
transverse hole
in tension or compression
Fig.: Flat plate with
shoulder fillet
in tension or compression
Fig.: Round shaft with
shoulder fillet in tension
S K Chand (Assistant Professor)
Government Engineering College, Raipur

7. Fig.: Round shaft with
shoulder fillet in torsion
Fig.: Round shaft with
shoulder fillet in bending
S K Chand (Assistant Professor)
Government Engineering College, Raipur

8. •Itisimpossibletocompletelyeliminatetheeffectofstress
concentration,itcanonlybereducedtosomeextent.
•Itisachievedbyprovidingspecificgeometryshapetothe
component.
•Inordertoknowwhathappensattheabruptchangesofc/s
oratdiscontinuity,understandingthe‘flowanalogy’isuseful.
Thereisasimilaritybetweenvelocitydistributioninfluidflow
inachannelandstressdistributioninanaxiallyloadedplate.
Remedies for reduction of stress concentration
S K Chand (Assistant Professor)
Government Engineering College, Raipur

9. S K Chand (Assistant Professor)
Government Engineering College, Raipur

10. 1)AdditionalNotchesandholesintensionmember:Severityofstress
concentrationisreducedbyuseofmultiplenotches,drillingadditional
holes,removalofundesiredmaterial.Methodofremovalofundesired
materialiscalledprincipleofminimizationofmaterial.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

11. 2)FilletRadius,UndercuttingandNotchforMemberinBending:A
transmissionshafthasshouldersforballbearings,gearsorpulleystobe
mounted.Theshouldersgenerateschangeinc/sresultinginstress
concentration.thestressconcentrationinsuchcasesisreducedbythree
waysas,
S K Chand (Assistant Professor)
Government Engineering College, Raipur

12. 3)Drillingadditionalholesforshaft:Inthistwosymmetrical
holesaredrilledonthesidesofthekeyway,whichpressthe
flowlinesandminimizetheirbendinginthevicinityof
keyway.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

13. 4)ReductionofStressconcentrationinthreadedmembers:Fora
threadedcomponenttheforceflowlineisbentasitpassesfrom
shankportiontothreadedportion,whichresultsinstress
concentrationintransitionplane.
–Asmallundercutbetweentheshankandthethreadedportion
reducesthebendingofforceflowlineandstressconcentration.
–Ideallywhenshankdia.=coredia.Thereisnostress
concentration.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

14. S K Chand (Assistant Professor)
Government Engineering College, Raipur

15. Fluctuating Stresses
•Theexternalforcesactingonthem/ccomponentare
generallyassumedtobestatic.
•Inmanyengineeringapplications,thecomponentsare
subjectedtoforceswhichvarywithrespecttotime.The
stressesinducedduetosuchforcesareknownasfluctuating
stresses.
•80%failuresofmechanicalcomponentsaredueto‘fatigue
failure’whichresultsfromfluctuatingstresses.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

16. •Stresspatternisvariableandirregularinpracticalsituations
asincaseofvibrations.
•Simplemodelsofstress-timerelationshipareusedforthe
purposeofdesignanalysis.
•Popularlysinecurveisusedforstress-timerelationship.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

17. Mathematical Models for Cyclic Stresses
There are basically three types of mathematical
models to demonstrate cyclic stresses viz.
1. Fluctuating or alternating stresses
2. Repeated Stresses
3. Reversed Stresses
S K Chand (Assistant Professor)
Government Engineering College, Raipur

18. 1. Fluctuating or alternating stresses:
a)Thesestressesvariesinasinusoidalmannerwithrespecttotime.
b)Ithassomemeanvalueandamplitudevalue.
c)ItfluctuatesbetweentwolimitsMaximumandMinimum.
d)Thestresscanbetensileorcompressiveorpartlytensileorpartly
compressive.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

19. 2. Repeated Stresses: (??????
�??????�=0)
a)Variesinasinusoidalmannerwithrespecttotime,butthe
variationisfromzerotosomemaximumvalue.
b)Minimumstressiszerointhis,andhencetheamplitude
stressandmeanstressareequal.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

20. 3. Reversed Stress: (??????
�=0)
a)Itvariesinsinusoidalmannerwithrespecttotime,butithas
zeromeanstress.
b)Inthishalfportionofthecycleconsistsoftensileandthe
remaininghalfconsistsofcompressivestress.
c)Inthisthereiscompletereversalfromtensionto
compressionbetweenthesetwohalvesandsomeanstress
iszero.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

21. Referring to figure above,
??????
���=��????????????���������
??????
�??????�=�??????�??????���������
??????
�=����������
??????
�=����������??????����
We have
??????
�=
??????
���+??????
�??????�
2
??????
�=
??????
���−??????
�??????�
2
S K Chand (Assistant Professor)
Government Engineering College, Raipur

22. Fatigue Failures
“Fatigue failure is defined as the time delayed fracture under cyclic loading”.
1.Underfluctuatingstresses,thefailureofmaterialoccursatalower
magnitudeofultimateorsometimesyieldstrength.
2.Resistanceofmaterialtostressmagnitudesdecreaseswiththeincrease
innumberofcycles,andthisisthemaincharacteristicoffatiguefailure.
e.g.bendingandunbendingofwire,Componentssuchastransmissionshafts,
connectingrods,gears,vehiclesuspensionspringsandballbearingarecommonly
subjectedtofatiguefailures.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

23. 3.Phenomenonoffatiguefailurebeginswithacrackatsomepointinthe
material.Thiscrackgenerallyoccursatdiscontinuityregionsaskeyways,
oil-holes,screwthreads,Scratchesonsurface,stampmark,inspection
markwhichrepresentirregularityregions.Defectsinmaterialsasblow
holesetc.
4.Suchregionsaresubjectedtocrack,whichspreadsduetofluctuating
stressesuntilthecross-sectionofthecomponentissoreducedthatthe
remainingportionissubjectedtosuddenfracture.
5.Therearetwodistinctareasoffatiguefailureasregionindicatingslow
growthofcrackwithfinefibrousappearanceandregionofsudden
fracturewithacoarsegranularappearance.
6.Fatiguefailureissuddenandtotal.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

24. Designprocessiscomplexasitinvolvesfactors
suchasnumberofcycles,meanstress,stress
amplitude,stressconcentration,residual
stressescorrosionandcreep.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

25. Endurance Limit
•Fatigueorendurancelimitofamaterialisdefinedas
maximumamplitudeofcompletelyreversedstressthatthe
standardspecimencansustainforanunlimitednumberof
cycleswithoutfatiguefailure.
•10
6
cyclesisconsideredassufficientnumberofcyclesto
definetheendurancelimitasthetestcannotbeconducted
forinfinitenumberofcycles
S K Chand (Assistant Professor)
Government Engineering College, Raipur

26. •Thetermfatiguelifeiscommonlyusedwithendurance
limit.
•Fatiguelifeisdefinedasthenumberofstresscycles
thatastandardspecimencancompleteduringthetest
beforetheappearanceoffirstfatiguecrack.
•Inlaboratory,theendurancelimitisdeterminedby
meansofarotatingbeammachinedevelopedbyR.R.
Moore.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

27. S-N Curves
Arotatingbeamm/cdiagramisshowninfigurebelow.Specimen
actsasrotatingbeamsubjectedtobendingmoment.Henceitis
subjectedtocompletelyreversedstresscycle.
1.Thestressamplitudescanbevariedbychangingthebending
momentbyadditionorremovalofweights.
2.Withtheaidofelectricmotorthespecimenisrotated.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

28. Fig.: Rotating beam fatigue testing machine
S K Chand (Assistant Professor)
Government Engineering College, Raipur

29. 3.Usingarevolutioncounter,thenumberofrevolutionsbeforethe
appearanceoffirstfatiguecrackarerecorded.
4.Ineachtest,tworeadingsarenotedviz.Stressamplitude(�
�)and
numberofcycles(�).
5.Thesereadingsareusedtoastwoco-ordinatesforplottinga
pointonS-Ndiagram.
6.Thispointiscalledfailurepoint.
7.TheresultsofsuchtestsareplottedusingS-Ncurves.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

30. Fig.: S-N curve for steel
S K Chand (Assistant Professor)
Government Engineering College, Raipur

31. 8.S-Ncurveisthegraphicalrepresentationofstressamplitude(�
�)
versusthenumberofstresscycles(�)beforethefatiguefailure
onalog-loggraphpaper.
9.Eachtestonfatiguetestingm/cgivesonefailurepointontheS-N
diagram.
10.Thesepointsarescatteredinfigureandaveragecurveisdrawn
throughthem.
11.ItisalsocalledWohlerdiagramafterGermanEngineer,August
Wohlerwhopresentedthismethodin1870.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

32. NOTCH SENSITIVITY (�)
“Theextenttowhichthesensitivityofamaterialtofractureisincreasedby
thepresenceofasurfaceinhomogeneitysuchasanotch,asuddenchangein
section,acrack,orascratch”.
•Notchsensitivityfactor(�)isdefinedastheratioofincreaseof
actualstressovernominalstresstoincreaseoftheoreticalstress
overnominalstress.
•Mathematically,??????
0=Nominalstressobtainedbyelementary
equations.
??????
�×??????
0=??????�����������&??????
�×??????
0=�ℎ�����??????���������
S K Chand (Assistant Professor)
Government Engineering College, Raipur

33. • Increase of actual stress over nominal stress = (??????
�×??????
0−??????
0)
• Increase of theoretical stress over nominal stress = ??????
�×??????
0−??????
0
�=
(??????
�×??????0−??????0)
(??????�×??????0−??????0)
�=
(??????
�−1)
(??????�−1)
The above equation is rearranged as ;
??????
�=1+�(??????
�−1)----(A)
S K Chand (Assistant Professor)
Government Engineering College, Raipur

34. Following are the observations are from above equation
•Whenthematerialhasnosensitivitytonotches
�=0& ??????
�=1
•Whenthematerialisfullysensitivetonotches
�=1& ??????
�=??????
�
•Themagnitudeof�variesfrom0to1.
•Incaseofdoubtdesignershoulduse�=1&??????
�=??????
�andthe
designwillbeonsafeside.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

35. Fig.: Notch sensitivity charts (for reversed
bending and reversed axial stresses)
Fig.: Notch sensitivity charts (for
reversed torsional shear stresses)
S K Chand (Assistant Professor)
Government Engineering College, Raipur

36. •Fatiguestressconcentrationfactor(??????
�);itisapplicableto
actualmaterialsanddependsuponthegrainsizeofthe
material.
•Theoreticalstressconcentrationfactor(??????
�);itisapplicableto
idealmaterialswhicharehomogenous,isotropicandelastic.
??????
�=
Endurancelimitofthenotchfreespecimen
Endurancelimitofthenotchedspecimen
S K Chand (Assistant Professor)
Government Engineering College, Raipur

37. Endurance Limit-Approximate Estimation
Notationsusedregardingendurancelimitare(�
�
′
)and(�
�);
Where
�
�
′
=Endurancelimitstressofarotatingbeamspecimen
subjectedtoreversedbendingstress(MPa),
�
�=Endurancelimitstressofaparticularmechanical
componentsubjectedtoreversedbendingstress(MPa)
S K Chand (Assistant Professor)
Government Engineering College, Raipur

38. Relationsbetweenendurancelimitandtheultimate
tensilestrengthofthematerial
�
�
′
=0.5�
��;ForSteels
�
�
′
=0.4�
��;ForCastIronandCastSteels
�
�
′
=0.4�
��;ForWroughtAluminiumalloys
�
�
′
=0.3�
��;ForCastAluminium
Alltheaboverelationshipsarebasedon50%reliability.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

39. •Thereisdifferencebetween(�
�
′
)and(�
�)duetothefactthat
therearestandardspecificationsandworkingconditionsfor
rotatingbeamspecimen,whiletheactualcomponentshave
differentspecificationsandworkunderdifferentconditions.
•Differentmodifyingfactorsareusedtoaccountforthis
differenceandthesefactorsarecalledderatingfactors.
•Thederatingfactorsreducetheendurancelimitofthe
rotatingbeamspecimentosuittheactualcomponent.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

40. Therelationshipbetween(�
�
′
)and(�
�)isgivenas
�
�=??????
�??????
�??????
�??????
��
�
′
Where,
??????
�= Surface finish factor
??????
�= Size factor
??????
�= Reliability factor
??????
�= Modifying factor to account for the stress
concentration.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

41. •SurfaceFinishFactor:Surfacefinishfactortakesintoaccount
thepoorsurfacefinishandgeometricirregularitiesonthe
surfacewhichserveasstressraisersandcausestress
concentration.
•Italsoconsidersthereductioninendurancelimitduetothe
variationinsurfacefinishbetweenthespecimenandthe
actualcomponent.
•Followingfigureshowsthesurfacefinishfactorforsteel
components.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

42. Fig.: Surface finish factor for steelsS K Chand (Assistant Professor)
Government Engineering College, Raipur

43. •ForsteelcomponentsNollandLipsondevisedanequationas
??????
�=�(�
��)
�
; [??????�??????
�>1,���??????
�=1]
•Values of �and �are given in table below
•For grey cast iron components ??????
�=1.
Surface Finish � �
Ground Machined 1.58 -0.085
Cold Drawn 4.51 -0.265
Hot Rolled 57.7 -0.718
Forged 272 -0.995
S K Chand (Assistant Professor)
Government Engineering College, Raipur

44. •SizeFactor:Thesizefactortakesintoaccountthe
reductioninendurancelimitduetoincreaseinsize
ofthecomponent.
•Values of size factor are given in table below;
Diameter (d) mm ??????
�
d ⩽7.5 1.00
7.5 < d ⩽50 0.85
d > 50 0.75
S K Chand (Assistant Professor)
Government Engineering College, Raipur

45. •ReliabilityFactor:dependsuponthereliability
thatisusedinthedesignofcomponent.For
50%reliabilitythereliabilityfactorisone.To
ensurethat50%ofthepartswillsurvive,the
stressamplitudesonthecomponentsshould
belowerthanthetabulatedvalueof
endurancelimit.
•Reliabilityfactorisusedforachievingthis
reduction.Reliabilityfactorsbasedona
standarddeviationof8%aregivenintable.
Reliability
R (%)
??????
&#3627408516;
50 1.000
90 0.897
95 0.868
99 0.814
99.9 0.753
99.99 0.702
99.999 0.659
S K Chand (Assistant Professor)
Government Engineering College, Raipur

46. •ModifyingfactortoaccountforstressConcentration:Itisdefined
as
??????
&#3627408465;=
1
??????
&#3627408467;
&#3627408454;
&#3627408480;&#3627408466;= Endurance limit of a component subjected to fluctuating
torsional shear stresses
According to maximum shear stress theory;
&#3627408454;
&#3627408480;&#3627408466;= 0.5&#3627408454;
&#3627408466;
According to distortion energy theory;
&#3627408454;
&#3627408480;&#3627408466;= 0.577&#3627408454;
&#3627408466;
For axial loading; &#3627408454;
&#3627408466;&#3627408462;= 0.8&#3627408454;
&#3627408466;
S K Chand (Assistant Professor)
Government Engineering College, Raipur

47. Reversed Stresses -Design for Finite and Infinite life
•Designproblemsforcompletelyreversedstressesaredividedinto
twogroupsasdesignforfiniteanddesignforinfinitelife.
Design for Infinite life
•Whenthecomponentistobedesignedforinfinitelife,the
endurancelimitbecomesthecriterionoffailure.
{Theamplitudestressinducedinsuchcomponents}<{endurancelimit
in-ordertowithstandinfinitenumberofcycles}.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

48. •Suchcomponentsaredesignedwiththehelpof
followingequations;
??????
&#3627408462;=
??????&#3627408466;
(&#3627408467;&#3627408480;)
& ??????
&#3627408462;=
??????&#3627408480;&#3627408466;
(&#3627408467;&#3627408480;)
•Where(??????
&#3627408462;)&(??????
&#3627408462;)arestressamplitudesinthe
componentand&#3627408454;
&#3627408466;and&#3627408454;
&#3627408480;&#3627408466;arecorrectedendurance
limitsinreversedbendingandtorsionrespectively.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

49. Design for finite life
•ForsuchcaseS-Ncurveisused.Arepresentativecurveisshown
infigbelow,whichisvalidforsteels.
•ItconsistsofastraightlineABdrawnfrom(0.9&#3627408454;
&#3627408482;&#3627408481;)at10
3
cycles
to(&#3627408454;
&#3627408482;&#3627408481;)at10
6
cyclesonalog-logpaper.
•Designprocessforsuchproblemsisasunder
–LocatepointAwithco-ordinates[3,&#3627408473;&#3627408476;&#3627408468;
100.9&#3627408454;
&#3627408482;&#3627408481;]since&#3627408473;&#3627408476;&#3627408468;
10(10
3
)=3
S K Chand (Assistant Professor)
Government Engineering College, Raipur

50. ̶LocatepointBwithco-ordinates[6,&#3627408473;&#3627408476;&#3627408468;
10&#3627408454;
&#3627408466;]since
&#3627408473;&#3627408476;&#3627408468;
1010
6
=6
̶LinejoiningA&Bisusedasacriterionoffailureforfinite-life
problems.
̶DependinguponthelifeNofthecomponent,drawaverticalline
passingthrough&#3627408473;&#3627408476;&#3627408468;
10&#3627408449;ontheabscissa.ThislineintersectsABat
pointF.
̶DrawalineFEparalleltotheabscissa.TheordinateatpointEi.e.
&#3627408473;&#3627408476;&#3627408468;
10&#3627408454;
&#3627408467;,givesthefatiguestrengthcorrespondingtoNcycles.
❖Thevalueof(&#3627408454;
&#3627408467;)obtainedbyaboveprocessisusedfordesign
calculations.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

51. Fig.: S-N curve
S K Chand (Assistant Professor)
Government Engineering College, Raipur

52. Cumulative Damage in Fatigue
(Miner’s Equation)
•Incertainapplications,themechanicalcomponentis
subjectedtodifferentstresslevelsfordifferentpartsofthe
workcycle.
•ThelifeofsuchcomponentisdeterminedbyMiner’s
equation.
supposeacomponentissubjectedtocompletelyreversed
stresses(??????
1)for(&#3627408475;
1)cycles,(??????
2)for(&#3627408475;
2)cyclesandsoon.Let&#3627408449;
1
bethenumberofstresscyclesbeforefatiguefailure,onlyifthe
alternatingstress(??????
1)isacting.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

53. Onestresscyclewillconsume(
1
&#3627408449;1
)offatiguelifeandsincethere
are&#3627408475;
1suchcyclesatthisstresslevel,theproportionatedamage
offatiguelifewillbe(
&#3627408475;
1
&#3627408449;
1
).Similarly,theproportionatedamageat
stresslevel(??????
2)willbe(
&#3627408475;2
&#3627408449;
2
)andsoon.Addingthese,weget
(
&#3627408475;1
&#3627408449;
1
)+ (
&#3627408475;2
&#3627408449;
2
) + ……. + (
&#3627408475;&#3627408485;
&#3627408449;
&#3627408485;
) = 1
Above equation is known as Miner’s equation.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

54. •Attimesthenumberofcycles&#3627408475;
1,&#3627408475;
2,……..Atstresslevels??????
1,
??????
2,……..areunknown.Supposethat??????
1,??????
2……areproportionsof
thetotallifethatwillbeconsumedbythestresslevels??????
1,??????
2……..
etc.
•LetNbethetotallifeofthecomponent,then
&#3627408475;
1= ??????
1&#3627408449;&#3627408475;
2= ??????
2&#3627408449;
•UsingtheseinMiner’sequation
??????1
&#3627408449;1
+
??????2
&#3627408449;2
+……
??????&#3627408485;
&#3627408449;&#3627408485;
=
1
&#3627408449;
Also ??????
1+??????
2+??????
3…..+??????
&#3627408485;=1
S K Chand (Assistant Professor)
Government Engineering College, Raipur

55. Soderberg, Gerber and Goodman Lines
•Whenacomponentissubjectedtofluctuatingstress
thereismeanstress(??????
&#3627408474;)aswellasstress
amplitude(??????
&#3627408462;).
•Meanstresscomponenthaseffectonfatiguefailure
whenitoccursincombinationwithalternating
component.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

56. S K Chand (Assistant Professor)
Government Engineering College, Raipur

57. Fatiguediagramforgeneralcaseisshowninfigureabove;Inthis
diagram
1.MeanStress(??????
&#3627408474;)isplottedonabscissa
2.Stressamplitude(??????
&#3627408462;)isplottedonordinate
3.Magnitudeofforceactingonthecomponentdeterminesthe
valuesofmeanstressandstressamplitudes.
4.Whenthestressamplitude(??????
&#3627408462;)iszero,theloadispurelystatic
andthecriterionoffailurebecomes&#3627408454;
&#3627408486;&#3627408481;or&#3627408454;
&#3627408482;&#3627408481;andtheselimits
areplottedonabscissa.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

58. 5.Whenthemeanstress(??????
&#3627408474;)iscompletelyzero,thestressis
completelyreversingandthecriterionoffailureis
endurancelimit(&#3627408454;
&#3627408466;)plottedonordinate.
6.Whenthecomponentissubjectedtobothmeanstress(??????
&#3627408474;)
andstressamplitude(??????
&#3627408462;),theactualfailureoccursat
differentfailurepointsasshowninfigure.
7.Thereexistsaborder,whichdividessaferegionfromunsafe
regionforvariouscombinationsof(??????
&#3627408462;)&(??????
&#3627408474;).
S K Chand (Assistant Professor)
Government Engineering College, Raipur

59. 8.Variouscriterionsareproposedforconstructionof
borderlinedividingsafezoneandfailurezone.They
include
a)GerberLine:Itisaparaboliccurvejoining&#3627408454;
&#3627408466;ontheordinate
to&#3627408454;
&#3627408482;&#3627408481;onabscissa.
b)SoderbergLine:Itisastraightlinejoining&#3627408454;
&#3627408466;onordinateto
&#3627408454;
&#3627408486;&#3627408481;onabscissa.
c)GoodmanLine:Itisastraightlinejoining&#3627408454;
&#3627408466;onordinateto&#3627408454;
&#3627408482;&#3627408481;
onabscissa.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

60. Discussions……
•Gerberparabolafitsthefailurepointsoftestdatainthebest
possibleway.
•Goodmanlinefitsbeneaththescatterofthisdata.
•Gerberparabola&Goodmanlineintersectat(&#3627408454;
&#3627408466;)onordinate
to(&#3627408454;
&#3627408482;&#3627408481;)onabscissa.
•FromdesignconsiderationsGoodmanlineismoresafeasit
liescompletelyinsidetheGerberparabolaandinsidethe
failurepoints.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

61. •Soderberglineismoreconservativefailurecriterionandthereisnoneedto
considerevenyieldinginthiscase.
•EquationofSoderberglineisgivenby,
??????
&#3627408474;
&#3627408454;
&#3627408486;&#3627408481;
+
??????
&#3627408462;
&#3627408454;
&#3627408466;
=1
•EquationofGoodmanlineisgivenby,
??????
&#3627408474;
&#3627408454;
&#3627408482;&#3627408481;
+
??????
&#3627408462;
&#3627408454;
&#3627408466;
=1
•Goodmanlineiswidelyusedasthecriterionoffatiguefailurewhenthe
componentissubjectedtoboth??????
&#3627408474;&??????
&#3627408462;.
•GoodmanLineissafeasitiscompletelyinsidethefailurepointsoftestdata.
•Equationofstraightlineissimpleascomparedwithequationofparabolic
curve.
•Ascalediagramisnotrequired;aroughsketchisenoughforconstructionof
fatiguediagram.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

62. Modified Goodman Diagrams
Componentssubjectedtofluctuatingstressesaredesignedby
constructingmodifiedGoodmanDiagram.Fromdesign
considerationsproblemsareclassifiedintotwocategories:
1.Componentssubjectedtofluctuatingaxialorbending
stresses;
2.Componentssubjectedtofluctuationtorsionalshear
stresses.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

63. 1. Components Subjected to Fluctuating Axial or
Bending Stresses Figure
Fig.: Modified Goodman diagram for axial and bending stressesS K Chand (Assistant Professor)
Government Engineering College, Raipur

64. •Goodmanlineismodifiedbycombiningfatiguefailurewith
yieldingfailure.
•Inthis,theyieldstrength(&#3627408454;
&#3627408486;&#3627408481;)isplottedonbothaxes,anda
yieldlineCDat45
0
toabscissaisconstructedtojointhese
pointstodefinefailurebyyielding.
•LineAFisconstructedtoJoin(&#3627408454;
&#3627408466;)onordinatewith(&#3627408454;
&#3627408482;&#3627408481;)on
abscissai.e.GoodmanLine.
•PointofintersectionoflinesAF&CDis‘B’.
•AreaOABCrepresentstheregionofsafetyforcomponents
subjectedtofluctuatingstresses.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

65. •RegionOABCiscalledModifiedGoodmanDiagram.
•InthisABistheportionofGoodmanlineandBCistheportion
ofyieldline.
•LineOEwithslopetan??????isconstructedforproblemsolving
processas
tan??????=
??????&#3627408462;
??????
??????
;
We know that
??????&#3627408462;
????????????
=
??????&#3627408462;/??????
????????????/??????
So tan??????=
??????
&#3627408462;
????????????
S K Chand (Assistant Professor)
Government Engineering College, Raipur

66. •Magnitudesof??????
&#3627408462;&??????
&#3627408474;aredeterminedbymaximumand
minimumforceactingonthecomponent.Similarlywehave
tan??????=
&#3627408448;
&#3627408463;&#3627408462;
&#3627408448;
&#3627408463;??????
•LinesABandOEintersectatpoint‘X’whichindicatesthe
dividinglinebetweensaferegionandtheregionoffailure.
•Co-ordinatesofpointX(&#3627408454;
&#3627408474;,&#3627408454;
&#3627408462;)representlimitingvaluesof
stressesusedtocalculatethedimensionsofthecomponent.
•Permissiblestressesarecalculatedas
??????
&#3627408462;=
??????
&#3627408462;
&#3627408467;&#3627408480;
??????
&#3627408474;=
??????
??????
&#3627408467;&#3627408480;
S K Chand (Assistant Professor)
Government Engineering College, Raipur

67. 2.Components Subjected to fluctuating torsional shear
stresses
Fig.: Modified Goodman Diagram for Torsional Shear Stresses
S K Chand (Assistant Professor)
Government Engineering College, Raipur

68. •Inthistorsionalmeanstressisplottedonabscissawhilethe
torsionalstressamplitudeonordinate.
•Thetorsionalyieldstrength(&#3627408454;
&#3627408480;&#3627408486;)isplottedonabscissaand
theyieldlineisconstructedwhichisinclinedat45
0
to
abscissa.
•Uptoacertainpoint,torsionalmeanstresshasnoeffecton
torsionalendurancelimit,soalineisdrawnthrough&#3627408454;
&#3627408480;&#3627408466;on
ordinateandisparalleltotheabscissa.
S K Chand (Assistant Professor)
Government Engineering College, Raipur

69. •PointofintersectionofthislinewiththeyieldlineisB.Area
OABCrepresentsregionofsafety.
•Itisnotnecessarytoconstructfatiguediagramforfluctuating
torsionalshearstressesaslineABisparalleltoX-axis.
•Afatiguefailureisindicatedif,??????
&#3627408462;=&#3627408454;
&#3627408480;&#3627408466;andastaticfailureis
indicatedif??????
&#3627408474;&#3627408462;&#3627408485;=??????
&#3627408462;+??????
&#3627408474;=&#3627408454;
&#3627408480;&#3627408486;
•Permissiblestressesare??????
&#3627408462;=
??????&#3627408480;&#3627408466;
&#3627408467;&#3627408480;
and??????
&#3627408474;&#3627408462;&#3627408485;=
??????&#3627408480;&#3627408486;
&#3627408467;&#3627408480;
S K Chand (Assistant Professor)
Government Engineering College, Raipur

70. S K Chand (Assistant Professor)
Government Engineering College, Raipur