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Advanced Materials & Smart Structures Lab.금오공대기계공학과 윤성호교수

Lecture Note for Solid Mechanics- Torsional Loading -

Prof. Sung Ho YoonDepartment of Mechanical EngineeringKumoh National Institute of Technology


Text book : Mechanics of Materials, 6th ed.,

W.F. Riley, L.D. Sturges, and D.H. Morris, 2007.

Prerequisite : Knowledge of Statics, Basic Physics, Mathematics, etc.


Torsional Loading

Typical example of torsional loading problems :

- circular shaft connecting an electric motor with othermachine.

- circular shaft transmits the torque from the armature tothe coupling

- determination of significant stresses and deformation of the circular shaft

areaarear dAdFTT


Torsional loading problem :

- Transverse cross section before twisting remains planeafter twisting

- Diameter of the section remains straight

Plane sections remain plane

Plane sections become warped

Torsional Loading


Torsional shear strain

Lc

ABBB

cc

'

tan

LEDDD

'

tan

If the small strain is assumed

L

cLc

Combining above equations

cc

Torsional shear strain becomes

surfaceaatstrainshearcedisaatstrainshear

twistofangle

c :tan:

:

Torsional Loading


If Hooke’s law is assumed (stress is proportional to strain)

cc

cc

dAdAc

TT cr 22

dAJwhere 2 : polar second moment of area

2

24

0

22 cddAJc

)( 4444

22

222

2

io

c

b

rrbc

ddAJ

For case (a) :

For case (b) :

A

dAT

Torsional Loading


dAdAc

TT cr 22

J

cJTT c

r

Unknown shear stress becomes

JT and

JcT

c

Shear stress is zero at the center of the shaft and increases linearly with respect to the distance from the axis of the shaft.

Shear strain is zero at the center of the shaft and increases linearly with respect to the distance from the axis of the shaft.

Therefore,

cc

JT and

Torsional Loading


Determination of angle of twist

L

JT

dLd or

orJcT

c

GJGLT

GLL

If T, G, or J is not constant along the length of the shaft

n

i ii

ii

JGLT

1

If T, G, or J is a function of distance of the length of the shaft

L

JGdxT

0

Torsional Loading


(a) Maximum shear stress in the shaft

GPaG 80

(b) Shear stress at the inside surface of the shaft

(c) Magnitude of angle of twist in 2m length

))(.())()(()(

max 6

33

10117181020010300

JcT

c

MPamN 934109234 26 .)(.

))(.())()(()(

6

33

10117181015010300

J

T

MPamN 226101926 26 .)(.

))(.)()(())(()(

69

3

10117181080210300

GJ

LT

radrad 0043700043650 ..

46

46

44

44

10117181011718

1502002

2

mmm

rrJ io

)(.)(.

)(

)(

Torsional Loading


(a) Maximum shear stress in the shaft

ksiJ

cTAB

ABABAB 2444

2312731215 .).(

))(()(

4444 13252

22incJ .)(

ksiJ

cTBC

BCBCBC 7754

13252125 .).(

))(()( (max)

(b) Rotation of end B w.r.t. end A

radJGLT

ABAB

ABABAB 012730

23127120001291215 .

).)(())()(()(

/

(c) Rotation of end C w.r.t. end B

radJGLT

BCBC

BCBCBC 011940

132512000125125 .

).)(())()(()(

/

(d) Rotation of end C w.r.t. end A

radBCABAC

0007950011940012730

...///

4446 231273

22incJ .)(

ksiG 00012,

Torsional Loading


Questions: (1) transverse plane is a plane of maximum shearing stress?(2) there are other significant stresses induced by torsion?

xyyx From moment equilibrium equation:

cossinsincos xyyxn 222 From stress transformation equations:

cossincossin xyxy 2200

)sin(coscossin)( 22 xyyxnt

)sin(cos)sin(cos 22220 xyxy

2sinxy

2cosxy


Stresses on an inclined plane are functions of angle of inclined plane.

2sinxyn

2cosxynt

A & C : maximum shear stressesB : maximum tensile stressD : maximum compressive stress

B

A

C D

Torsional Loading


(a) Steel rear axle of a truck Split longitudinally due to longitudinally running grains

(b) Thin-walled aluminum tube Buckled and teared along 45 deg planes

(c) Gray cast iron of brittle material Tensile failure

(d) Low carbon steel of ductile material Shear failure

Type of failures

Torsional Loading


(a) Maximum torque Tmax :

26108080 mNMPaJ

cT /)(maxmaxmax

(b) FS when failure occurs at T=12kN-m, if ultimate strengths are 205MPa in shear and 345MPa in tension.

MPa80maxmmt 6

46

46

44

10096141009614

69752

mmm

J

)(.)(.

)(

mkNmNc

JT

04.15)10(04.15)10)(75(

)10)(096.14)(10)(80(

3

3

66max

max

MPa

JcT

o

xyn

29556021009614

10751012

22

6

33

.)(sin))(.(

))()((

sinsin

MPaJcT

xynt 923122 .coscos

2462955

345 ..

n

ultFS

4269231

205 ..

nt

ultFS

and

overall FS

Torsional Loading


Power is the time rate of doing work

TdtdT

dtdWPower k

Work done by a constant torque : TWk

(where is angular displacement in radian)

(where is angular velocity in rad/min)

(Example 6-8) Determine minimum permissible diameters of two shafts if allowable shearing stress is 20ksi and angle of twist in 10-ft of propeller shaft is not exceed 4-deg.

hpandrpmwithengine 80020014 :propellertogearboxratio

TPower ))((),( 220000033800 1T

shaftcrankforlbftT 010,211

Assume no power loss, shaftpropellerforlbftT 252,52

))(())(,()(

31

41

102012010212

c

cTcJ

For crank shaft,

inc 00221 .

ind 00441 .

1HP=550ft-lb/s

Torsional Loading


))(())(,()(

32

42

10201225252

c

cTcJ

For propeller shaft,

inc 26112 .

ind 52222 .

2612154831210121210122525

1804

2

42

6

..))()((

))(())(,(

ccGJ

LT

Therefore, the propeller shaft should be diameter of 3.10-in.


Circular bar subjected to axial load P and torque T

AP

due to axial load P

JT due to torque T

Shearing stresses exist on both transverse and longitudinal planes

yxxy

Once stresses on the planes in (d) are known, stresses on any other plane through the point, as well as principal stresses and maximum shearing stress at the point can be found.

As long as the strains are small, these stresses can be computed separately and superimposed on the element

Torsional Loading


(Example 6-12) Determine (a) x- and y-components of stresses, (b) principal stresses and maximum shearing stress at the point for point A on outside surface.

)(.).()( CMPa

APA

x 4625104

102002

3

(Sol)

(a)

MPaJcTA

xy 791520502

05010304

3

.).(

).)((

(b) ).,,.(),,( 7915204625 xyyx

321537312

791522

046252

04625

22

22

22

21

..

).(..

,

xy

yxyxpp

Principal stresses and maximum shearing stress are

0

05166321537312

59140321537312

3

2

1

zp

p

p

CMPaTMPa

)(...)(...

MPa321532

05166591402

.).(.

minmaxmax

Torsional Loading


Circular bar subjected to torque T

JTc

c max

Stress concentrations occur in the vicinity of abrupt changes in diameter.

JTcKmax

Stress concentration factor K as

),(dr

dDKKwhere

Kt : based on net section

(Example 6-14) For stepped shaft of D=4-in and d=2-in, determine min. fillet radius if max. shearing stress is 8ksi under torque of 6280 in-lb.

psiJ

Tc 39981216280

4 )()(

max

00239988000 .tK

indrdr

120060060 ...

Torsional Loading


Shearing stresses in circular shaft are proportional to the distance from its axis.

Shearing stresses for rectangular cross sections are not proportional to the distance from its axis.

Violate free boundary condition

Shearing stresses at the corners of the rectangular bar must be zero.

Torsional Loading


Every cross sections, except for circular cross sections, will warp (not remain plane) when the bar is twisted.

GbaTL

baT

3

2

max

Torsional Loading


(Example 6-17) Determine maximum shearing stress and the angle of twist for 12-in length.

b/a=3/(1/2)=6Aluminum alloyG=4000-ksiT=2500 in-lb

30.

radGba

TLpsi

baT

067010400032130

12250011011

321302500

333

22

.))()(()(.

)(,

)()(.max

Torsional Loading


Elementary torsion theory : circular sections, thin-walled sections

lengthunitforceshearinternal(q) flowshear

tq

where is average shearing stress across the thickness, t

BA

BBAA

BA

qqtt

ttqq

dxqdxqVV

31

31

3131

Shear flow on the cross section is constant even though the wall thickness varies.

Torsional Loading


dsqqdsdFTr )()(

tAAqTr 22

AtT

2

(Example 6-18) Determine maximum torque that can be applied to the section if the maximum shearing stress must be limited to 95 MPa.

mNtq /)())()(( 336 101901021095

mN

AqT

1750101901035021002

236 ))()()()((

where is the area enclosed by the median line.

mNtq /)10(285)10)(3)(10(95 336

mN

AqT

2625)10)(285)(10)(350)(2100(2

236

Torsional Loading


Homework

(6-14), (6-47), (6-64), (6-76), (6-92), (6-126)

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