ME 307
Machine
Design I
Dr. A. Aziz Bazoune
King Fahd University of Petroleum & Minerals
Mechanical Engineering Department
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 1
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 2
ME 307
Machine
Design I
A-
Fatigue Strength and Life
Completely Reversed Loading (R=-1)
  min 
Stress Ratio R  

  max 
Ferrous Metals
( fSut ) 2
a
Se
Strength
 fSut 
1
b   log 

3
S
e


Fatigue life
Dr. A. Aziz Bazoune
a 
Nf 

 a 
1
b
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 3
ME 307
Machine
Design I
Fatigue Life with Mean Stress Effect
Fluctuating Loading (R  -1)
B-
Fatigue life
Sfr
 S fr 
Nf 

 a 
Ferrous Metals
1
b
: Equivalent Completely Reversed Strength
From Modified Goodman with Sfr =Se
Dr. A. Aziz Bazoune
a
S fr 
m
Chapter 7: Fatigue Failure Resulting from Variable Loading
1
CH-07
Sut
LEC 27
Slide 4
ME 307
Machine
Design I
Example 7-13 (Textbook)
Solution
(7-13)
Dr. A. Aziz Bazoune
(7-14)
326
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 5
ME 307
Machine
Design I
(7-15)
(7-49)
p.349
(7-15)
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 6
ME 307
Machine
Design I
(7-50)
F
B
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
C
D E
LEC 27
Slide 7
ME 307
Machine
Design I
Fatigue Failure for Brittle Materials
The first quadrant fatigue failure criteria follows a curve upward
Smith-Dolan represented by
S a 1  Sm Sut

Se 1  Sm Sut
(7-52)
n a 1 n m Sut

Se
1 n m Sut
(7-53)
Or a design equation
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 8
ME 307
Machine
Design I
 For a radial load line of slope r, we substitute Sa/r for Sm and
solve for Sa
r Sut  Se 
4 r Sut Se 
 1 1

Sa 
2
2 
 r Sut  Se  

(7-54)
 The fatigue diagram for a brittle material differs markedly
from that of a ductile material
 Yielding is not involved since the material may not have a
yield strength
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 9
ME 307
Machine
Design I
 The compressive ultimate strength exceeds the
ultimate tensile strength severalfolds
 First-quadrant fatigue failure locus is concaveupward (Smith-Dolan)
 Brittle materials are more sensitive to midrange
stress, being lowered
 Not enough work has been done on brittle fatigue to
discover insightful generalities
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 10
ME 307
Machine
Design I

The most likely domain of designer use is in the range from
Sut  Sm  Sut

The locus in the first quadrant is Goodman, Smith-Dolan or in
between

The portion of the second quadrant that is used is
represented by a straight line between points Sut ,
0, Se
 Se 
Sa  Se   1 Sm
 Sut 
Dr. A. Aziz Bazoune
Sut
Sut  Sm  0
Chapter 7: Fatigue Failure Resulting from Variable Loading
and
(7-55)
CH-07
LEC 27
Slide 11
ME 307
Machine
Design I
Torsional fatigue Strength Under
Fluctuating Stresses

Torsional steady-stress component not more than the
torsional yield strength has no effect on the torsional
endurance limit.

Torsional fatigue limit decreases monotonically with torsional
steady-stress

Since the great majority of parts will have surfaces less than
perfect, Gerber, ASME-elliptic, are used

In constructing the Goodman diagram Jorres uses
Ssu  0.67Sut
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
(7-56)
CH-07
LEC 27
Slide 12
ME 307
Machine
Design I
Combining Loading Modes
Fatigue problems are classified under three categories:
i.
Completely reversing simple loads
It is handled with the S-N diagram, relating the alternating stress to a life.
Only one type of loading is allowed here, and the midrange stress must
be zero.
i.
Fluctuating simple loads
It uses a criterion to relate midrange and alternating stresses (modified
Goodman, Gerber, ASME-elliptic, or Soderberg). Again, only one type of
loading is allowed at a time.
i.
Combinations of loading modes
It uses combined bending, torsion, and axial loadings.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 13
ME 307
Machine
Design I
Combining Loading Modes
Completely reversed single stress
which is handled with the S-N diagram, relating the alternating stress to a
life. Only one type of loading is allowed here, and the midrange stress must
be zero.
Fluctuating loads
It uses a criterion to relate midrange and alternating stresses (modified
Goodman, Gerber, ASME-elliptic, or Soderberg). Again, only one type of
loading is allowed at a time.
Combination of different types of loading
such as combined bending, torsion, and axial.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 14
ME 307
Machine
Design I
Combining Loading Modes
In Sec. 7-9, a load factor was used to obtain the endurance limit, and hence
the result is dependent on whether the loading is axial, bending, or torsion.
But, “how do we proceed when the loading is a mixture of, say, axial,
bending, and torsional loads?”
This type of loading introduces a few
complications in that there may now exist combined normal and shear
stresses, each with alternating and midrange values, and several of the
factors used in determining the endurance limit depend on the type of
loading. There may also be multiple stress-concentration factors, one for
each mode of loading. The problem of how to deal with combined stresses
was encountered when developing static failure theories. The distortion
energy failure theory proved to be a satisfactory method of combining the
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 15
ME 307
Machine
Design I
Combining Loading Modes
multiple stresses on a stress element into a single equivalent von Mises stress.
The same approach will be used here.
1) The first step is to generate two stress elements, one for the alternating
stresses and one for the midrange stresses.
2) Apply the appropriate fatigue stress concentration factors to each of the
stresses; apply
K 
f
bending
torsional stresses, and
K 
f
 
for the bending stresses, K fs
torsion
for the
for the axial stresses.
axial
3. Next, calculate an equivalent von Mises stress for each of these two stress
elements,
4. Finally, select a fatigue failure criterion (modified Goodman, Gerber,
ASME-elliptic, or Soderberg) to complete the fatigue analysis.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 16
ME 307
Machine
Design I
Combining Loading Modes
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 17
ME 307
Machine
Design I
Combining Loading Modes
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 18
ME 307
Machine
Design I
Combining Loading Modes
Case of Combined Axial, Bending and Torsion Loading
(kc? Kf?).
Assuming that all stress components are in time phase with each
other.
1. For the strength, use the fully corrected endurance limit for
bending, Se.
2. Apply the appropriate fatigue concentration factors to all stress
components.
3. Multiply any alternating axial stress components by 1/kc,ax
4. Find the principal stresses.
5. Find the von Miss alternating stress, ’a and mean stress ’m.
6. Use any of the theories above to compute the safety factor.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 19
ME 307
Machine
Design I
Combining Loading Modes
’a and mean stress ’m are alternating and mean VM stresses.
Both the steady and alternating components are augmented by Kf and
Kfs.

If stress components are not in phase but have same frequency,
the maxima can be found using phase angles and then summed.

Otherwise assume that the stress components will reach an inphase condition so their magnitudes are additive.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 20
ME 307
Machine
Design I
Example 7-15 (Textbook)
Solution
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 21
ME 307
Machine
Design I
t = 4 mm
 = M/Znet
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 22
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 23
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 24
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 25
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 26
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from Variable Loading
CH-07
LEC 27
Slide 27