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
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 31 Slide 2
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 3
ME 307
Machine
Design I
Estimating Stress Concentration
 The stress analysis process for fatigue is highly dependent on stress
concentrations.
 Stress concentrations for shoulders and keyways are dependent on
size specifications that are not known the first time through the
process.
 Fortunately, since these elements are usually of standard
proportions, it is possible to estimate the stress concentration
factors for initial design of the shaft. These stress concentrations
will be fine-tuned in successive iterations, once the details are
known.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 4
ME 307
Machine
Design I
Estimating Stress Concentration
 Shoulders for bearing and gear support should match the catalog
recommendation for the specific bearing or gear.
 A look through bearing catalogs shows that a typical bearing calls
for the ratio of D/d to be between 1.2 and 1.5.
 For a first approximation, assume D/d =1.5 can be assumed.
 Fillet radius at the shoulder needs to be sized to avoid
interference with the fillet radius of the mating component. There
is a significant variation in typical bearings in the ratio of fillet
radius r/d versus bore diameter, with
typically ranging from
around 0.02 to 0.06.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 5
ME 307
Machine
Design I
Estimating Stress Concentration
 Figures A-15-8 and A-15-9 show that the stress concentrations for
bending and torsion increase significantly in this range. For
example, with D/d = 1.5 for bending
r/d
0.02
0.05
0.1
Kt
2.7
2.1
1.7
 In most cases the shear and bending moment diagrams show that
bending moments are quite low near the bearings, since the
bending moments from the ground reaction forces are small.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 6
ME 307
Machine
Design I
Estimating Stress Concentration
 In cases where the shoulder at the bearing is found to be critical, the
designer should plan to select a bearing with generous fillet radius, or
consider providing for a larger fillet radius on the shaft by relieving it into
the base of the shoulder as shown in Fig. 7-9a.
 This effectively creates ahead
zone in the shoulder area that
does not carry the bending
stresses, as shown by the stress
flow lines.
Fig. 7-9a.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 7
ME 307
Machine
Design I
Estimating Stress Concentration
 A shoulder relief groove as shown in Fig. 7-9b can accomplish a similar
purpose. Another option is to cut a large-radius relief groove into the small
diameter of the shaft, as shown in Fig. 7-9c.
Fig. 7-9b.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
Fig. 7-9c.
CH-18
LEC 31 Slide 8
ME 307
Machine
Design I
Figure 7-9
Techniques for reducing stress concentration at a shoulder supporting a bearing
with a sharp radius. (a) Large radius undercut into the shoulder. (b) Large radius
relief groove into the back of the shoulder. (c) Large radius relief groove into
the small diameter.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 9
ME 307
Machine
Design I
 This has the disadvantage of reducing the cross-sectional area, but is often
used in cases where it is useful to provide a relief groove before the
shoulder to prevent the grinding or turning operation from having to go all
the way to the shoulder.
 For the standard shoulder fillet, for estimating Kt values for the first
iteration, an r/d ratio should be selected so Kt values can be obtained. For
the worst end of the spectrum, with r/d = 0.02 and D/d = 1.5, Kt values
from the stress concentration charts for shoulders indicate 2.7 for
bending, 2.2 for torsion, and 3.0 for axial.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 10
ME 307
Machine
Design I
 A keyway will produce a stress concentration near a critical point where
the load transmitting component is located. The stress concentration in an
end-milled keyseat is a function of the ratio of the radius r at the bottom
of the groove and the shaft diameter d. For early stages of the design
process, it is possible to estimate the stress concentration for keyways
regardless of the actual shaft dimensions by assuming a typical ratio of r/d
= 0.02. This gives Kt = 2.2 for bending and Kts= 3.0 for torsion, assuming
the key is in place.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 11
ME 307
Machine
Design I
 A keyway will produce a stress concentration near a critical point where
the load transmitting component is located. The stress concentration in an
end-milled keyseat is a function of the ratio of the radius r at the bottom
of the groove and the shaft diameter d. For early stages of the design
process, it is possible to estimate the stress concentration for keyways
regardless of the actual shaft dimensions by assuming a typical ratio of r/d
= 0.02. This gives Kt = 2.2 for bending and Kts= 3.0 for torsion, assuming
the key is in place.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 12
ME 307
Machine
Design I
Table 7-1
First iteration estimates for stress concentration factors Kt
Warning: These factors are only estimates for use when actual dimensions are not
yet determined. Do not use these once actual dimensions are available.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 13
ME 307
Machine
Design I
Fatigue Analysis of Shafts
The fatigue strength will be
determined using:
1.
Distortion-Energy-Gerber
2
Distortion-Energy-Elliptic
Rotating Shaft under stationary
bending and torsional moments
 32Ma 
 xa  K f 
3 
 d 
 16Tm 
 xym  K fs 
3 

d


Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 14
Fatigue Analysis of Shafts
ME 307
Machine
Design I
Safety Factor
2
Gerber
n 'a  n 'm 

 1
Se
 Su 
2
ASME-Elliptic
Dr. A. Aziz Bazoune
2
 n 'a   n 'm 
1

 

 Se   Sy 
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 15
ME 307
Machine
Design I
Problem 18-10
A geared industrial roll shown
in the figure is driven at 300
rev/min by a force F acting on
a 3-in-diameter pitch circle as
shown. The roll exerts a
normal force of 30 lbf/in of
roll length on the material
being pulled through. The
material passes under the roll.
The coefficient of friction is
0.40. Develop the moment
and shear diagrams for the
shaft modeling the roll force
as a concentrated force at the
center of the roll,
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 16
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Problem 18-10
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 17
ME 307
Machine
Design I
Problem 18-10
We have a design task of identifying
bending moment and torsion diagrams
which are preliminary to an industrial
roller shaft design.
Gear
Roller
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 18
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 19
ME 307
Machine
Design I
This approach over-estimates the
bending moment at C, torque at C but
not at A.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 20
ME 307
Machine
Design I
Problem 18-11
1. Using a 1035 hot rolled steel, estimate the necessary diameter at the
locations of peak bending moment using a design factor of 2. These
are likely to be fillets at both ends of the right hand bearing seat,
where the bending moment is slightly less than the local extreme.
2. Estimating the fatigue stress-concentration factor as 2, and using a
design factor of 2, what is the approximate necessary diameter of
the bearing seat using the DE-elliptic fatigue failure criterion in
Problem 18-10?
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 21
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Problem 18-11
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 22
ME 307
Machine
Design I
Problem 18-11
From static Analysis
1/3
 16n

2
2 1/2
d
4 M  3T  

  S y

Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
 0.6in
CH-18
LEC 31 Slide 23
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Problem 18-11
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 24
Hollow Shafts
ME 307
Machine
Design I
As an example equation 18-21 is modified to take into account the hollow
shaft case:

2

 K fsTm 
K
M


16 n

f
a

do  
4

  3 
4
S
S
e

  (1   di do  )  
 y 

2 1/2




1/3





where di and do are respectively the inner and outer diameters of the
shaft.
With this, one can consider that the stress-strength analysis is
completed. You have obtained the minimum diameter at the critical
section that can withstand the applied loads.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 25
Shaft Design
ME 307
Machine
Design I
One approach is (See Lab Handbook):
1.
2.
3.
4.
5.
6.
Selecting a material (usually steel)
Drawing a free body diagram of the shaft
Performing static equilibrium analysis and
Locating the critical area
Performing static stress analysis to find a starting diameter size, d’.
Using the value of d’ in calculating the endurance limit (a trial
diameter can also be used)
7. Estimating the critical value of the diameter, d, using DE-Gerber or DEASME-elliptic methods
8. Repeat step 6 if d different from d’.
9. Building the rest of the shaft by considering the machine parts to be
mounted on the shaft (bearings, gears, pulleys, …)
10. Performing deflection analysis
11. Performing Dynamic analysis
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 26
ME 307
Machine
Design I
Shaft Materials
 Shafts are usually made of ductile materials.
 Small shafts with diameters less than 3.5 in (90 mm) are usually
made of Cold Drawn carbon steel (AISI 1018-1050).
 Larger diameter shafts are machined from Hot Rolled steel.
 Heat treated steels are also used when higher strengths are
necessary.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 27
Shaft
ME 307
Machine
Design I
N
Design?
Y
find safety Factor, n
Y
Find critical diameter, d
N
N
Y
Shaft rotating?
Shaft rotating?
Static Analysis
Eq. 6-43 or 6-45
Fatigue Analysis
Static Analysis
Eq. 6-44 or 6-46
Static Analysis
Eq. 6-42 or 6-44
d
d'
Fatigue Analysis
n
Y
Reversed
bending & steady
torque?
N
Y
Eq. 18-17 or 18-22
Eq. 18-14 or 18-20
n
Reversed
bending & steady
torque?
Eq. 18-16 or 18-21
N
Eq. 18-13 or 18-19
d
N
d. NE. d’
N
Y
d = Critical shaft
diameter
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
Complete shaft
geometry & perform
Deformation analysis
CH-18
LEC 31 Slide 28
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Geometric Constraints

Unlike stress, which is a function
of local geometry and load,
deflection is a function of the
geometry everywhere. Thus, The
task of deflection and rigidity
analyses can be started only when
the entire geometry of the shaft is
determined.

However the approach described
in section 18-2, which is based on
bearing slope constraints as
limiting, may be used first
assuming a uniform diameter
shaft and using equations 18-1
and 18-2 to find the diameters at
the bearings.
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 29
ME 307
Machine
Design I
Stress Concentrations and Shaft Geometry
Shaft shoulders are used to position and provide necessary thrust supports
for elements such as bearings, gears, pulleys,… Provisions must be made
for torque-transfer elements such as keys, splines, pins
The theoretical stress concentration factors for shoulders, grooves and
transverse holes can be obtained from appendix [A15+]. Others are
Kt = 2.0
for profile key seats
Kt = 1.6
for sled runner keyseats
Sled runner
keyseat
Shoulders
Shoulder
Groove
Shoulders
Profile keyseat
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 30
ME 307
Machine
Design I
Shaft Geometry
To determine the entire geometry of the shaft one has to rely on existing
models. Some of these models are given in figures 18-1 through 18-8 of
the Textbook. More shaft configurations can be found in the FAG
handbooks of the Design of Rolling Bearing Mountings .
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 31
Geometric Constraints: Shaft Deflection and Slopes
ME 307
Machine
Design I

The transverse deflection of the elastic curve of the shaft can be
determined by any one of the methods studied in Chapter 5.
 The
superposition
method,
which
utilizes
Appendix
A-9,
is
recommended. For complex shaft geometry the numerical integration
or computer program may be used.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 32
Geometric Constraints: Shaft Deflection and Slopes
ME 307
Machine
Design I
1.
The slope at ball bearings should be limited at 0.25 deg the slope at roller
bearings and long journal bearings should be a lot less. For details on
acceptable slopes refer to FAG and SKF catalogs.
2.
For machinery shafting, the deflection should be no greater than 0.001
in/ft (0.075 mm/m) of shaft length between bearing supports.
3.
For shafts mounting good quality spur gears, the deflection at the gear
mesh should not exceed 0.005 in. (0.125 mm) or F/200 (F is the gear face
width in inches) and the slope should be limited 0.0286 deg.
4.
For shafts mounting good quality bevel gears, the deflection at the gear
mesh should not exceed 0.003 in. (0.076 mm).
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 33
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 31 Slide 34