chapter 1 torsion of circular shafts - aceec.ac.in · 1 t 2 1.1 assumptions in theory of pure...

AACE Engineering College : Ankushapur, Ghatkesar, Telangana 501301 (EAMCET Code: ACEG)

1.0 Introduction

T

Figure (a)

Figure (b)

Figure(a)showsaprismaticshaftwithlongitudinalfibres.WhenatorqueTis

applied in a plane perpendicular to the longitudinal axis of the shaft, longitudinal fibresundergoangulardisplacementsasshowninfigure(b).

InsuchacaseshaftissaidtobesubjectedtoTorsion.

SomepracticalexamplesofmemberunderTorsion.

• Shaft transmitting power• Lbeams• Curvedbeams• Porticobeams• Closed coiled springs (small helix angles)

DifferentWaysofTorqueDepictions:

P1

P1

P2

P2 d2

d1

Axisof bar

T1 = P1d1 T2 = P2d2

T1T2

Chapter Torsion of Circular Shafts1


2 Strength of Materials

For Micro Notes by the StudentT1 T2

1.1 AssumptionsintheoryofPureTorsion:• Materialofshafttakenishomogeneousandisotropicandelastic.• Theshaftshouldbeperfectlystraightanduniformincrosssection.• Sectioniscircularwhichmaybesolidorhollow.• Planesectionbeforetwistingremainsplaneaftertwisting.• Allradiiwhicharestraightbeforetwistremainstraightaftertwist.

Torsionalformulae

γmax

θ

r

c′

b′

b

c

T T

a

b

l

where T-Torqueapplied θ-Angleoftwistofcross-section t-Shearstressduetotorsionataradialdistanceof‘r’fromaxis. L-Lengthofshaft J-Polarmomentofinertia


3 Torsion of Circular Shaft

For Micro Notes by the Student

1.2 DerivationofTorsionEquation:




1.3 PolarMomentofInertia(J): Itisdefinedasthemomentofinertiaoftheareaaboutanaxiswhichisperpendicular

totheplaneofthefigureandpassingthroughthecentreofgravityofthearea.

• For solid circular shaft

τmax

O

T

J =




• For hollow circular shaft

τmaxτmin

O

T

J =

1.4 PolarsectionModulus(Zp):

Itisdefinedastheratioofthepolarmomentofinertiatotheradiusoftheshaft.

Zp = where, J-Polarmomentofinertiaofcrosssectionofshaftaboutlongitudinalaxis. rmax-radiusofshaftsection

• For a solid circular shaft

• For a hollow circular shaft




1.4.1StrengthoftheShaft: Itisdefinedasthemaximumpowerortorquethatcanbetransmittedthroughthe

shaft.

1.4.2TorsionalRigidity Torquerequiredtoproduceunitangulartwist. GJ-Torsionalrigidity FromTorsionformula

GJT,��

Units: Nm2/radian

1.4.3Torsionalstiffness Itisdefinedasthestiffnessrequiredtoproduceunitangleoftwistinthedirection

oftwist.

KGJ

,=

Unit:N-m/radian

1.4.5AngleofTwist Angulardeflectionoflongitudinalfibresgivenby‘θ‘ Unit: radians

1.5 PowerTransmittedbyShafts: InS.I.System: Power (P) is measured in watts (w)

where,T=averagetorqueinN-m N = shaft speed rpm

• Metricsystem: PowermeasuredinH.P(Horsepower) 1H.P=746watts≈0.75kW




1.6 HollowCircularSectionShaft




1.7 MembersubjectedtoCombinedBendingandTorsion

L

a

bA

P1

Letashaftbesubjectedtoabendingmomentof‘M’andtwistingmoment‘T’.Atanypoint‘A’onthesurfaceofshaftwhichisatadistanceof‘b’fromfreeend.

Bending:

Torsion:




1.8 CombinedTorsionandEndThrust:

Torsion:

Endthrust:

CombinedEffect:




1.9.1Shaftsconnectedinseries

l1l2

G1J1 G2J2

T

• Totalangleoftwistatthefreeendofcompositeshaftissumofangleoftwistsofindividualshafts.

1.9.2 ShaftsconnectedinParallel

l

G1J1

G2J2

T

Forsuchacaseangleoftwistforboththeshaftsisequalatanysection

Totalappliedtorqueisdistributedamongtheshafts




Thusangulartwist:

G J G J

T

1 1 2 2

,��

StrainEnergyinaMembersubjectedtoTorsion Strainenergystoredintheshaftonwhichtorque‘T’isapplied

U T2

1 ��

UGJ

T L

2

12

=

or

UGV

4

2��

where,τ-torsionalshearstress

G-Modulusofrigidity

V-Volumeofshaft= d L4

2 #��b l



For Micro Notes by the StudentJNTU Objective Questions

01. Whenashaftissubjectedtopuretwistingthetypeofstressdevelopedintheshaft is

(A) Bending stress (B) axial stress (C) shear stress (D) normal stress

02. Theratioofthemax.shearstressinanopencoiledtoclosecoiledhelicalspringof same mean coil diameter and wire diameter under same axial load is (A) 2 (B) 3

(C)1 (D)4

03. ThenatureofStressesdevelopedinashaftsubjectedtotorsionis (A) Shear Stress (B) Compressive stress

(C)TensileStress (D)Noneoftheabove

04.Ashaftturnsat150rmp,underatorqueof1500N-m.Thepowertransmittedis(A)15pKw (B)10pKw

(C)7.5pKw (D)5pKw

05.Themaximumtwistingmomentofashaftcanresist,istheproductofpermissibleshearstressand

(A) Moment of Inertia (B) Polar moment of Inertia (C)ModulusofRigidity (D)PolarModulus

06. AlongshaftofdiameterdissubjectedtotwistingmomentTatitsends.Themaximumshearstressactingatitscrosssectionisequalto (A)Zero (B)16T/πd3

(C)32T/πd3 (D)64T/πd3

07. Ashaft10cmdiaand2mlongissubjectedtotorqueof800Kg.m.ThemaximumshearstressinKg/sq.cmdevelopedis (A)308.6 (B)407.6

(C)128.6 (D)216.2

08. Theratioofmaximumshearstressdevelopedinsolidshaftofdia‘D’andahollowshaftofouterdia‘D’andinternaldia’d’forthesametorquegiven

(A) (D2+d2)/D2 (B) (D2-d2)/D2 (C) (D4+d4)/D4 (D) (D4-d4)/D4




09. Whencloselycoiledhelicalspringissubjectedtoanaxialload,thedeflectionofthespringisdirectlyproportionalto (A)Modulusofrigidityofspringmaterial(B)Diameterofspringwire

(C)Meandiaofspring (D)Numberofturnsofthespring

10. InCarriageofspringsinadditiontochangingthenumberofplates,tohaveuniformstrength..

(A)Thethicknessofeachplatewillbechanged (B)Thewidthoftheplatewillbegraduallydecreased (C) Additional rivets are provided (D)Weldingisdone

FILLINTHEBLANKS1. Alongshaftofdiameter‘d’issubjectedtotwistingmomentTatitsends.The

maximumnormalstressactinginthec/sisequalto…………………2. Acarriagespringsubjectedtoacentralloadsuchthatleavesbecomestraight.

Whatisthisloadcalled……………..3. LeafSpringsareusedin___________________.4. ThepolarMomentofInertiaofacircularshaftofdiameterDis

__________________.5. Whentwodissimilarshaftsareconnectedtogethertoformoneshaft,itis

knownas______________.6. AngleofHelixonaclosecoiledspringis_______________.7. Whencloselycoiledhelicalspringiscompressed,itswireissubjected

to_______8. Columnsofintermediatelengthfailby________________9. Theratioofeffectivelengthofthecolumnanditsminimumradiusofgyrationis

____10. PeryRobertsonformulaconsidersinitialeccentricityand_______________

JNTU Conventional Questions

1. Definetorsionalrigidityofshaft2. Whataretheassumptionsinthetorsiontheory?3. Writeanexpressionforstrainenergystoredinshaftofuniformsectionsubjected

totorsion.4. Comparetheweightofasolidshaftwiththatofahollowonehavingsame

length to transmit a given power at given speed, if the material used for the shaft issame.Taketheinsidediameteroftheshaftas0.6timesthatoftheoutershaft.




5. Ahollowshaftistotransmit300KWat800rpm.Theinternaldiameteris0.6oftheouterdiameter.Themaximumtorqueis40%morethanmeantorqueIftheshearstressistonotexceed60N/mm2.Findtheinternalandexternaldiametersoftheshaft.

6. Ahollowshaftwithinternalandexternaldiameters120mmand80mmrespectivelyistobereplacedbyasolidshaftofthesameweight.Findthetorquestransmittedbytheshaftifthepermissibleshearstressis100MPa.Ifthesolidshaftisreplacedbyahollowshaftof160mmexternaldiameter,whatisthetorquetransmittedforthesameweightoftheshafts

7. Derivetheexpressionforstrainenergystoredinashaftsubjectedtotorsion8.Derivetheequationofthetorsionfromfundamentals

9. Findthepowertransmittedby60mmdiametershaftat160rpm,ifthepermissibleshearstressis80N/mm2.andthemaximumtorque30%greaterthanthemeantorque

10. Asolidcircularsteelshaftisrequiredtotransmit60HPat200rpm.Determinethediameteroftheshaft,ifthemaximumshearstressistonotexceed60N/mm2inshaft.Thesolidshaftisnowreplacedbyahollowsteelshaftwiththeinternaldiameterequalto75%oftheexternaldiameter.Determinetheexternaldiameteroftheshaftifitisrequiredtotransmitthesamehorsepoweratsamerpmandthemaximumshearstressproducedisalsothesame.Findthe%savingofthematerialbyusinghollowshaftinplaceofsolidshaft.

SPRINGS1. Foraclose-coiledhelicalspringsubjectedtoanaxialloadof300Nhaving12

coilsofwirediameterof16mm,andmadewithcoildiameterof250mm,finda. Axialdeflectionb. Strainenergystoredc. Maximumtorsionalshearstressinthewired. Takemodulusofrigidityas80GN/m2

2. Derivetheexpressionforangleoftwistofhelicalspringssubjectedtoaxialtorque.

3. Arailwaywagonweighing40kNandmovingwithaspeedof8kmphissupportedbyabufferof4springswhosemaximumallowablecompressionis150mm.Findoutthenumberofturnsineachspring,ifthediaofthespringwireis14mmandthediaofcoilis80mm.AssumeC=84GN/m2.



For Micro Notes by the StudentAPPSC/TSPSC Objective Questions

01. Thetorquethatproducesatwistofoneradianinashaftperunitlengthiscalled (a)Torsionalrigidity (b)Bulkmodulus (c)Torsion (d)Modulusofelasticity

02. IfashaftissimultaneouslysubjectedtoatorqueTandbendingmomentM,themaximum shear stress is

(a) D

M M T162

2 2

�� ^ h (b) ( )

DM T

163��

�

(c) ( )D

M M T163

2 2

�� (d) ( )

DM T

163

2 2

��

03. Powertransmittedbyashaft(kW)subjectedtoanaveragetorqueT(kNm),rotatingatNrpmisequalto

(a) NT

4500

2�� (b) NT

3060

��

(c) NT

60

2�� (d) NT

3600

2��

04. Forshaftintorsion (a)

r

q

J

I= (b) CJ

I

,��

(c) r

q C

,�� (d)alltheabove

05. Inslopedeflectionequations,thedeformationsareconsideredtobecausedby (a)Axialforce (b)Shearforce (c)Bendingmoment (d)Alltheabove

06. Whenthediameterofashaftisdoubled,itstorquetransmittingcapacitywillincreaseby

(a)8times (b)2times (c)4times (d)16times 07. Thepolarmomentofinertiaofthecross-sectionofthememberisrequiredto

assessthestrengthofthememberin (a)bending (b)shear (c) axial force (d) torsion




08. Ashaftofcircularsectionissaidtobeinpuretorsionwhenitissubjectedtoequaland opposite end couples whose axes coincides with the

(a) Normal line to the axis of the shaft (b)Parallellinetotheaxisoftheshaft (c)anylinepassingthroughthesectionof theshaft (d) axis of the shaft

09. Torsionalstiffnessisdefinedas(a) torquerequiredtoproduceatwistofoneradianoverthelengthoftheshaft(b) torquerequiredtoproduceatwistofoneradianperunitlengthoftheshaft(c) torquerequiredtoproduceatwistofoneradianperunitpolarmodulusofthe

shaft (d) torquerequiredtoproduceatwistofoneradianperunitcrosssectionalarea

of the shaft

10. Polarmodulusistheratioof (a) Polar moment of inertia of the shaft to the maximum radius (b)Momentofinertiaoftheshafttothemaximumshearstress (c) Moment of inertia of the shaft to the maximum radius (d)Polarmomentofinertiaoftheshafttothepermissibleshearstress

KEY

01. (a) 02. (d) 03. (c) 04. (c) 05. (c)

06. (a) 07. (d) 08. (d) 09. (a) 10. (a)

Previous Gate Questions

01. The maximum shear stress in a solid shaft of circular cross-section havingdiameterandsubjectedtoatorqueTisτ.Ifthetorqueisincreasedbyfourtimesandthediameteroftheshaftisincreasedbytwotimes,themaximumshearstressintheshaftwillbe

(a) 2τ (b)τ (c) τ/2 (d)τ/4




02. Ahollowcircularshafthasanouterdiameterof100mmandawallthicknessof25mm.Theallowableshearstressintheshaftis125MPa.Themaximumtorquethe shaft can transmit is (a)46kN-m (b)24.5kN-m(c)23kN-m (d)11.5kN-m

03. AsolidcircularshaftofdiameterdandlengthLisfixedatoneendandfreeattheotherend.AtorqueTisappliedatthefreeend.TheshearmodulusofthematerialisG.Theangleoftwistatthefreeendis

(a) d G

TL164�� (b)

d G

TL324��

(c) d G

TL644�� (d)

d G

TL1284��

04. Ahollowcircularshafthasanouterdiameterof100mmandinnerdiameterof50mm.Iftheallowableshearstressis125MPa,themaximumtorque(inkN-m)thattheshaftcanresistis___

Key

01.(c) 02.(c) 03.(b) 04.(23)

Previous ESE Questions

01. Acircularshaftrotatesat200rpmandissubjectedtoatorqueof1500Nm.Thepowertransmittedwouldbe

(a)10pkW (b)15pkW (c)20pkW (d)30pkW

02. Torquesaretransmittedtothesolidcircularshaftasshowninthefigurebelow.Ifthecorrespondingpermissiblestressintheshaftis60N/mm2, the diameter of the shaftisnearly.

900N-m 2150N-m

2 m 2 m

2 m

800N-m 450N-m D C B A

(a)57.3mm (b)47.5mm (c)37.3mm (d)27.3mm




03. Asolidcircularshafthasadiameterd.Polarmoduluswillbe (a) d

16

2�� (b) d64

2��

(c) d16

3�� (d) d32

3��

04. Ahollowsteelshafthasoutsidediameterdandinsidediameter.Thevalueofdfortheshaft,ifithastotransmit200hpat105rpmwithaworkingshearstressof420kg/cm2, is

(a)5.6cm (b)2.6cm (c)12.1cm (d)15.5cm

05. AsolidshaftAofdiameterDand lengthL issubjected toa torqueT;anothershaftBofthesamematerialandofthesamelength,buthalfthediameter,isalsosubjectedtothesametorqueT.TheratiobetweentheanglesoftwistofshaftBtothat of shaft A is (a)32 (b)16

(c)8 (d)4

06. Therequireddiameterforasolidshafttotransmit400kWat150rpm,withtheworkingshearstressnottoexceed80MN/m2,isnearly

(a)125mm (b)121mm (c)117mm (d)113mm

Key

01.(a) 02.(b) 03.(c) 04.(c) 05.(b)06.(c)

chapter 1 torsion of circular shafts - aceec.ac.in · 1 t 2 1.1 assumptions in theory of pure...

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