ME 307
Machine
Design I
Dr. A. Aziz Bazoune
King Fahd University of Petroleum & Minerals
Mechanical Engineering Department
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 1
ME 307
Machine
Design I
18-1
18-2
18-3
18-4
18-5
18-6
18-7
18-8
Introduction ……….922
Geometric Constraints ……….927
Strength Constraints ……….933
Strength Constraints – Additional Methods ……….940
Shaft Materials ……….944
Hollow Shafts ……….944
Critical Speeds (Omitted) ……….945
Shaft Design ……….950
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 2
ME 307
Machine
Design I
18-1 Introduction ……….922
18-2 Geometric Constraints ……….927
18-3 Strength Constraints ……….933
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 3
ME 307
Machine
Design I
18-1 Introduction
 In machinery, the general term “shaft” refers to a member,
usually
of
circular
cross-section,
which
supports
gears,
sprockets, wheels, rotors, etc., and which is subjected to
torsion and to transverse or axial loads acting singly or in
combination.
 An “axle” is a non-rotating member that supports wheels,
pulleys,… and carries no torque.
 A “spindle” is a short shaft. Terms such as lineshaft, headshaft,
stub shaft, transmission shaft, countershaft, and flexible shaft
are names associated with special usage.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 4
ME 307
Machine
Design I
Considerations for Shaft Design
1. Deflection and Rigidity
(a) Bending deflection
(b) Torsional deflection
(c) Slope at bearings and shaft supported elements
(d) Shear deflection due to transverse loading of shorter
shafts
2. Stress and Strength
(a) Static Strength
(b) Fatigue Strength
(c) Reliability
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 5
ME 307
Machine
Design I
Considerations for Shaft Design
 The geometry of a shaft is that of a stepped cylinder bending.
 Gears, bearings, and pulleys must always be accurately positioned
Common Torque Transfer Elements






Keys
Splines
Setscrews
Pins
Press or shrink fits
Tapered fits
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 6
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Common Types of Shaft Keys.
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 7
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Common Types of Shaft Keys.
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 8
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Common Types of Shaft Pins.
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 9
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Common Types of Shaft Pins.
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 10
ME 307
Machine
Design I
Common Types of Retaining or Snap Rings.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 11
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Common Types of Splines.
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 12
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 13
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 14
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Rigid Shaft Coupling.
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 15
ME 307
Machine
Design I
Figure 18-2
(a) Choose a shaft configuration
to support and locate the two
gears and two bearings.
(b) Solution uses an integral
pinion, three shaft shoulders,
key and keyway, and sleeve.
The housing locates the
bearings on their outer rings
and receives the thrust loads.
(c) Choose fanshaft
configuration.
(d) Solution uses sleeve bearings,
a straight-through shaft,
locating collars, and
setscrews for collars, fan
pulley, and fan itself. The fan
housing supports the sleeve
bearings.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 16
ME 307
Machine
Design I
18-3 Strength Constraints
The design of a shaft involves the study of
1. Stress and strength analyses: Static and Fatigue
2. Deflection and rigidity
3. Critical Speed
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 17
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Static or Quasi-Static Loading on Shaft
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 18
ME 307
Machine
Design I
Static or Quasi-Static Loading on Shaft
The stress at an element located
on the surface of a solid round
shaft of diameter d subjected to
bending, axial loading, and
twisting is
Normal stress
Shear stress
32M 4 F
x 

3
d
d2
16T
 xy  3
d
Non-zero principal
stresses
Dr. A. Aziz Bazoune
x  y
 A , B  
 2
Chapter 18: Axles and Shafts
12

   x   y 
2
  
   xy 

  2 
2
CH-18
LEC 29
Slide 19
ME 307
Machine
Design I
Static or Quasi-Static Loading on Shaft
Von Mises
stress
Maximum
Shear Stress
Theory
Dr. A. Aziz Bazoune
2 1/2
2 1/2
 '   A   A B   B    x  3 xy 
2
2
1/2
4 
2
2
 '  3  8M  Fd   48T

d 
 max 
 max
 A B
2
1 2
2 12
  x  4 xy 
2
1/2
2 
2
2

8M  Fd   64T
3 

d
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 20
ME 307
Machine
Design I
Static or Quasi-Static Loading on Shaft
Under many conditions, the axial force F in Eqs. (6-37) and (6-38) is
either zero or so small that its effect may be neglected. With F = 0,
Eqs. (6-37) and (6-38) become
Von Mises
stress
Maximum
Shear Stress
Theory
Dr. A. Aziz Bazoune
16
2
2 1/2
 '  3  4M  3T 
d
 max
(6-41)
16
2
2 1/2
 M  T 

3 
d
Chapter 18: Axles and Shafts
(6-42)
CH-18
LEC 29
Slide 21
ME 307
Machine
Design I
Static or Quasi-Static Loading on Shaft
Substitution of the allowable stresses from Eqs. 6-39 and 6-40 we find
Von Mises
stress
 16n
2
2 1/2
d 
4M  3T 

  S y
1/3



(6-43)
1
16
2
2 1/2

4M  3T 

3
n
 d Sy
(6-44)
1/3
Maximum
Shear Stress
Theory
Dr. A. Aziz Bazoune
 32n

2
2 1/2
d 
M T  

  S y

1
32
2
2 1/2

M T 

3
n  d Sy
Chapter 18: Axles and Shafts
(6-45)
(6-46)
CH-18
LEC 29
Slide 22
ME 307
Machine
Design I
Fatigue Strength
 Bending, torsion, and axial stresses may be present in both
midrange and alternating components.
 For analysis, it is simple enough to combine the different types of
stresses into alternating and midrange von Mises stresses, as shown
in Sec. 7–14, p. 361.
 It is sometimes convenient to customize the equations specifically
for shaft applications.
 Axial loads are usually comparatively very small at critical locations
where bending and torsion dominate, so they will be left out of the
following equations.
The fluctuating stresses due to bending and torsion are given by
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 23
ME 307
Machine
Design I

Fatigue Strength
The fluctuating stresses due to bending and torsion are given by
where
Mm and Ma are the midrange and alternating bending moments, Tm
and Ta are the midrange and alternating torques, and Kf and Kfs are the
fatigue stress concentration factors for bending and torsion, respectively.
Assuming a solid shaft with round cross section, appropriate geometry
terms can be introduced for c, I, and J resulting in
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 24
ME 307
Machine
Design I
Assuming a solid shaft with round cross section, appropriate geometry terms
can be introduced for
Dr. A. Aziz Bazoune
c, I, and J resulting in
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 25
ME 307
Machine
Design I
Combining these stresses in accordance with the distortion energy failure
theory, the von Mises stresses for rotating round, solid shafts, neglecting axial
loads, are given by
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 26
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 27
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 29
Slide 28