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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