1

Chapter 7: Shafts

When a man has a vision, he cannot obtain the power from that vision until he has performed it on the Earth for the people to see.

Black Elk

7-1 Introduction •  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. •  Topics

–  Material selection –  Geometric layout –  Stress and strength –  Deflection and rigidity –  Vibration

7-2 Shaft materials •  Low carbon, cold-drawn or hot-rolled steels. •  High alloy content and strengthening from heat treatment are

not recommended as it reduces fatigue strength. •  Started to examine the low-cost low- & medium carbon steels •  Alloy steels – ANSI 1340-50, 3140-50, 4140, 4340, 5140 and

8650. •  Surface harden only at the location where the bearings sit. –

ANSI 1020, 4320, 4820 & 8620. •  Cold drawn steels for diameter less than 3inches •  Hot rolled steel should be machined all over. •  Cast iron for high production •  Stainless steels for certain applications.

7-3 Shaft layout

cantilever

Simple supported

Torque transmission - Keys, Splines, Setscrews, Pins, Press or shrink fits and Taper fits

2

Assembly and Disassembly

Techniques for reducing stress concentration

7-4 Shaft Design for Stress

Shaft Design Equations

33

33

32;32

32;32

dTK

IcTK

dTK

IcTK

dMK

IcMK

dMK

IcMK

mf

mfm

af

afa

mf

mfm

af

afa

πτ

πτ

πσ

πσ

====

====

2/12

3

2

322'

2/12

3

2

322'

163

323

163

323

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=+=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=+=

dTK

dMK

dTK

dMK

mfsmfmmm

afsafaaa

ππτσσ

ππτσσ

( ) ( )[ ] ( ) ( )[ ]

( ) ( )[ ] ( ) ( )[ ]3/1

2/1222/122

2/1222/1223

''

34134116

341341161

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎭⎬⎫

⎩⎨⎧

+++=

⎥⎦

⎤⎢⎣

⎡+++=

+=

mfsmfut

afsafe

mfsmfut

afsafe

ut

m

e

a

TKMKS

TKMKS

nd

TKMKS

TKMKSdn

SSn

π

π

σσDE - Goodman

Example 7-2

RAy

RAz RBz

RBx

y

z

Wt23=540lbf

Wr23=197lbf

Wr54=885lbf

Wt54=-2431lbf

A B

3996

4316

713

1632

115

655

1776

357 160

725

x

I J K

2220 3341

907 1472

RAz=115lbf RAy=356.7lbf RBz=1776lbf RBy=725.3lbf

inlbf23989072220

inlbf431616323996

inlbf365114723341

22

22

22

⋅=+=

⋅=+=

⋅=+=

K

J

I

M

M

M

T=540(12/2)=3240lbf·in

3

Example 7-2

( )( )( )( )( )

1)assumed(9.0

883.0687.2

ksi27685.09.0883.05.0265.0

===

=

===

===−

edc

b

buta

utbae

kkkk

aSk

SkkS

( )[ ] ( )[ ]

( )[ ] ( )[ ]1.625inin65.1

32405.13680001)3651)(7.1(4

270001)5.1(16

314116

3/12/122/12

3/12/122/12

≈=

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎭⎬⎫

⎩⎨⎧ ⋅=

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎭⎬⎫

⎩⎨⎧

+=

π

π mfsut

afe

TKS

MKS

nd

( )

( )

( ) 33.1135.195.01)126Fig.(95.0),815Fig.A(35.1

49.116.182.01)026Fig.(82.0),915Fig.A(6.1

in16.0;1.0625.12.1;2.1

=−+=

−=−−=

=−+=

−=−−=

==

==

f

sts

f

t

KqK

KqK

rdrDdD

Estimate Kt=1.7 and Kts=1.5 from Table 7-1 Choose 1020 Steel with Sut=68ksi

At the shoulder

Example 7-2 ( )( )( )( )

( )

1

835.03.0625.1

883.0687.2

ksi25685.0835.0883.05.0

107.0

265.0

===

=⎟⎠

⎞⎜⎝

⎛=

===

===

−

−

edc

b

buta

utbae

kkk

k

aSk

SkkS

( )( )( )

( )( )( )( )

55.1

645.06886.8

1.259.121

ksi86.8625.1

324033.1163163

ksi9.12625.1365149.13232

''

3

2/12

3'

33'

=

=+=+=

==⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛=

===

nSSn

dTK

dMK

ut

a

e

a

mfsm

afa

σσ

ππσ

ππσ

At keyway

7.5 Deflection Consideration

( )( )

4/1

4/1

/

all

olddoldnew

all

olddoldnew

slopedxdyndd

yyndd

=

=

∑

∑∑

=

==

i

ii

iii

kk

JGlT

11

θθ

Determining a new diameter

Angular deflection

Torsional Stiffness

7-7 Miscellaneous Shaft Components •  Setscrews

Socket Setscrews (a) flat point (b) cup point (c) oval point (d) cone point (e) half-dog point

Holding power – the resistance to axial motion of the collar relative to the shaft

4

Keys and Pins

(a)  Square key (b)  Round key (c)  Round pins (d)  Round pins (e)  Taper pins (f)  Split tubular pin

Woodruff key

Gib-head key

t

ysStlF<=τ

Retaining ring

External ring Internal ring

7-8 Limits and Fits DDDDD

=

Δ+=

min

max

ddddd

F

F

Δ−+=

+=

δ

δ

min

max

ddddd

F

F

Δ++=

+=

δ

δ

min

max

For the hole

For shafts with clearance fits c, d, f, g and h

For shafts with clearance fits k, n, p, s and u

See Tables A-11 and A-13

Interference fits

( )( )

minmaxmax

maxminmin

22

2222

3

22

22

22

22

2

DdDddd

dddddd

dEp

dddd

Ed

dddd

Ed

p

hubshaft

io

io

ii

i

io

o

o

o

−=

−=

−=

⎥⎦

⎤⎢⎣

⎡

−

−−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+

=

δ

δ

δ

δ

νν

δ