A shaft is a rotating member, usually of circular cross section, used to transmit power or motion. It provides the axis of rotation, or oscillation, of elements such as gears, pulleys, flywheels, cranks, sprockets, and the like and controls the geometry of their motion.

An axle is a non-rotating member that carries no torque and is used to support rotating wheels, pulleys, and the like.

A spindle is a short rotating shaft.

It is not necessary to evaluate the stresses in a shaft at

every point; a few potentially critical locations will

suffice.

Critical locations will usually be on the outer surface,

at axial locations where the bending moment is large,

where the torque is present, and where stress

concentrations exist.

A free body diagram of the shaft will allow the torque

at any section to be determined.

The bending moments on a shaft can be determined by

shear and bending moment diagrams.

Axial stresses on shafts due to the axial components

will almost always be negligibly.

Shaft Design for Stress due to Static loads

A. Solid Shaft with single load

• Where the shaft is subjected to axial force

• Where the shaft is subjected to pure bending moment

• Where the shaft is subjected to pure torsional moment

2axiald

F4

3bendingd

M32

3d

T16

B. Solid Shaft with combined loads

When the shaft is subjected to combination of axial,

bending and torsional loads, the principles stresses

and principle shear stresses are obtained by

constructing Mohr’s circle as shown in below figure.

bendingaxialx

2

zx

2

xx1

22

2

zx

2

xmax

2

x

1

max

zx

When the shaft is subjected to combination of bending

and torsional moments (σaxial = 0) which is the common

case, the principle stresses will be

bendingx

2

zx

2

bendingbending

122

2

zx

2

bending

max2

I. Maximum shear stress theory of static failure (MSS)

According to maximum shear stress theory, and using a

permissible value τMax of

2

3

2

3

2

zx

2

bending

maxd

T16

d

M16

2

22

3max TMd

16

S.F2

S y

max

22

3

yTM

d

16

S.F2

S

II. Distortion energy theory of static failure (DE)

For plane stress and

2

zx

2

yz

2

xy

2

xz

2

zy

2

yx 32

1

0y

3 2

xy

2

x

)0( zxyzz

d

T163

d

M322

3

2

3

3

1

22

yt

TM S

S.F32d

According to Von-Mises theory, and using a permissible

value σ′ of

• It is important to observe that these relations are only

valid when the stresses are truly invariable.

3

1

22

yt

T4

3M

S

S.F32d

S.F

S y

2

432

3

yTM

d

32

S.F

S

2

432

3TM

d

32

ASME Code for Shaft Design

The ASME code defines a permissible shear stress which is the smaller of the two following values:

The code states that these stresses should be reduced by 25% if stress concentration possible due to shoulder fillet or a keyway is present. Making τp equal τmax, lead to

In the code, the bending moment M and the torsional

moment T are multiple by combined shock and fatigue factors Cm and Ct respectively, depending on the condition of particular application.

utpytp S8.0 or S3.0

22

3p TMd

16

When

When

• Recommended values of Cm and Ct are listed in Table (1)

2t

2

m3p TCMCd

16

3

1

2

t

2

m

yt

TCMCS

96.16d

3

1

2

t

2

m

ut

TCMCS

36.6d

ytp S3.0

utp S8.0

Type of Loading Cm Ct

Stationary shaft

Load applied gradually 1 1

Load applied suddenly 1.5 – 2 1.5 – 2

Rotating shaft

Load applied gradually 1.5 1

Steady load 1.5 1

Load applied suddenly, minor shocks 1.5 – 2 1 – 1.5

Load applied suddenly, heavy shocks 2 – 3 1.5 – 3

Table (1). Values of Cm and Ct for different loading type

Table (2). Typical sizes of solid shaft that are available in the market

Diameter Increments

Up to 25 mm 0.5 mm

25 to 50 mm 1.0 mm

50 to 100 mm 2.0 mm

100 to 200 mm 5.0 mm

Example: the layout of a transmission shaft carrying two

pulleys B and C supported by bearings A and D is shown in

the figure below. Power is supplied to the shaft by means of

vertical belt on the pulley B, which is then transmitted to the

pulley C carrying a horizontal belt. The maximum tension in

the belt on the pulley B is 2.5kN. The angle of warp for both

the pulleys is 180̊ and the coefficient of friction is 0.24. The

shaft is made of plain carbon steel 30C8 Syt = 400 N/mm2

and the factor of safety is 3. Determine the shaft diameter.

600

D

B

C

200

A

200

500 φ1 250 φ2

All dimension by mm

Solution:

Substituting Eq. (1) into Eq. (2), we get

P3= 4999.235 N and P4= 2352.581 N

P3

P4

P1 P2

R1

R2

N673.1176e

2500Pe

P

P24.02

2

1

mm.N 75. 330831T

)2/500)(673.11762500(

)2/D)(PP(T

B

121B

)2......(..........125. 2eP

P

)1........(N 654. 2646)PP(

mm.N 75. 330831125)PP()2/D)(PP(TT

4

3

43

43243CB

• Bending and torsional moments

B C

2941.338 735.335

P1+P2=3676.673 N

600 200 200

588267.6

147067

B C

1470.363 5881.453

P3+P4=7351.816 N

600 200 200

294072.6

1176290.6

Horizontal plane (XZ plane) Vertical plane (XY plane)

mm.N 556.11854481470581176472M

mm.N 804.657675588232294118M

22

C

22

B

330831.75

B C

Torsional moment diagram

657675.804

1185448.556

B C

Resultant bending moment diagram

mm 46d )2( table from mm 3416.45

)75.330831(4

3)556.1185448(

)400(

3*32

T4

3M

S

S.F32d

3

1

22

3

1

2

c

2

c

yt

This shaft will be designed based on the stress at point C at

which the bending moment is the maximum.

Shaft Design on Rigidity Basis

The transverse shear V at a section of a beam in flexure

imposes a shearing deflection, which is superposed on the

bending deflection. Usually such shearing deflection is

less than 1 percent of the transverse bending deflection,

and it is seldom evaluated. However, when the shaft

length-to-diameter ratio is less than 10, the shear

component of transverse deflection merits attention.

A transmission shaft is said to be rigid on the basis of

torsional rigidity, if it does not twist too much under the

action of an external torque. The angle of twist or the

angular deflection θ (in radians) for right-circular

cylindrical solid shafts in torsion is given in is given by:

Where,

G: Modulus of rigidity

J: Polar moment of inertia

By degree (θ̊),

JG

Tl

)32

d J(

4

)JG

Tl(

180

4Gd

Tl584

For a stepped shaft with individual cylinder length li and

torque Ti , the angular deflection can be estimated from

For a constant torque throughout homogeneous material,

By degree (θ̊),

ii

iii

GJ

lT

i

i

J

l

G

T

4

i

i

d

l

G

T584

The permissible angle of twist for machine tool applications is 0.25̊ per meter length.

Example (2): the layout of a shaft carrying two pulleys 1 and 2, and supported on two bearings A and B is shown in figure below. The shaft transmits 7.5kw power at 360rpm from the pulley 1 to the pulley 2. The diameter of pulleys 1 and 2 are 250mm and 500mm respectively. The masses of pulleys 1 and 2 are 10kg and 30kg respectively. The belt tensions act vertically downward and the ratio of belt tensions on the tight side to slack side for each pulley is 2.5:1. The shaft is made of plain carbon steel 40C8 (Sy=380N/mm2) and the factor of safety is 3. Estimate suitable diameter of shaft. If the permissible angle of twist is 0.5̊ per meter length, calculate the shaft diameter on the basis of torsional rigidity. Assume G=79300N/mm2.

Example (3): A line Shaft supporting two pulleys A and B

is shown in figure below. Power is supplied to the shaft

by means of vertical belt on the pulley A, which is then

transmitted to the pulley B carrying a horizontal belt.

The ratio of the belt tension on tight and loose sides is

3:1. The limiting value of tension in the belt is 2.7kN.

The shaft is made of plain steel 40C8 ( Sut=650N/mm2

1

2

B A

250 250 500

and Sy=380N/mm2). The pulleys are keyed to the shaft.

Determine the diameter of the shaft according to the

ASME code if, load applied gradually.

H.W. Check the designed shaft according to angular

deflection.

B

A

450 250 450

450 φ 250 φ

Example (4): The layout of an intermediate shaft of a gear

box supporting two spur gears B and C is shown in the

figure below. The shaft is mounted on two bearings A

and D. The pitch circle diameters of gears B and C are

900 and 600mm respectively. The material of shaft is

steel FeE 580 (Sut=770 and Sy=580N/mm2). The factor

Cm and Ct of ASME code are 1.5 and 2 respectively.

Determine the diameter of the shaft using the ASME

code. Assume that the gears are connected to the shaft

by means of keys.

φB =900 & φC =600mm

N67.2413F

N5.6631F

N1609F

N4421F

r

C

t

C

r

B

t

B

900

D

B C

900

A

900

t

BF

t

CF

r

CF

r

BF