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-21
Slide - 1
ME 307 Machine Design I
7-1 Introduction to Fatigue in Metals 306
7-2 Approach to Fatigue Failure in Analysis and Design 312
7-3 Fatigue-Life Methods 313
7-4 The Stress-Life Method 313
7-5 The Strain-Life Method 316
7-6 The Linear-Elastic Fracture Mechanics Method 319
7-7 The Endurance Limit 323
7-8 Fatigue Strength 325
7-9 Endurance Limit Modifying Factors 328
7-10 Stress Concentration and Notch Sensitivity 335
7-11 Characterizing Fluctuating Stresses 344
7-12 Fatigue Failure Criteria for Fluctuating Stress 346
7-13 Torsional Fatigue Strength under Fluctuating Stresses 360
7-14 Combinations of Loading Modes 361
7-15 Varying, Fluctuating Stresses; Cumulative Fatigue Damage 364
7-16 Surface Fatigue Strength 370
7-17 Stochastic Analysis 373
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
Slide - 2
ME 307 Machine Design I
7-7
7-8
7-9
Dr. A. Aziz Bazoune
The Endurance Limit
Fatigue Strength
Endurance Limit Modifying Factors
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
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ME 307 Machine Design I
7-7 The Endurance Limit
A quick method of estimating endurance limits is needed:
for preliminary and prototype design
for some failure analysis
Experimental results for
rotating-beam tests
simple tension tests
of specimens taken from
the same bar are shown in
Figure 7.18.
Figure 7-18
Graph of endurance limits versus
tensile strengths from actual test
results for a large number of
wrought irons and steels.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
Slide - 4
ME 307 Machine Design I
Figure 7-18
Graph of endurance limits versus tensile strengths from actual test results for a large number of wrought irons and
steels. Ratios of S’e/Sut of 0.60, 0.50, and 0.40 are shown by the solid and dashed lines. Note also the horizontal
dashed line for of S’e=107 kpsi. Points shown having a tensile strength greater than 214 kpsi have a mean endurance
limit of S’e=107 kpsi and a standard deviation of 13.5 kpsi.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
Slide - 5
ME 307 Machine Design I
For steels, the relationship between the tensile strength and the
endurance limit is given by
(7-8)
where

S 'e : is the minimum tensile strength. The prime mark on S 'e in this
equation refers to the rotating-beam specimen itself.

The unprimed symbol S e is for the endurance limit of any
particular machine element subjected to any kind of loading.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
Slide - 6
ME 307 Machine Design I
The endurance limits for various classes of cast irons, polished or machined,
are given in Table A-24. Aluminum alloys do not have an endurance limit.
The fatigue strengths of some aluminum alloys at 5(108 ) cycles of reversed
stress are given in Table A-24.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
Slide - 7
ME 307 Machine Design I
7-7 Fatigue Strength
 Region of low cycle fatigue:
The fatigue strength
strength
S f is only slightly smaller than the tensile Sut
.
 Region of high Cycle Fatigue
The purpose of this section is to develop methods of approximation of
the S-N diagram in the high-cycle region, when information may be as
sparse as the results of a simple tension test. Experience has shown
high-cycle fatigue data are rectified by a logarithmic transform to both
stress and cycles-to-failure.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
Slide - 8
ME 307 Machine Design I
7-8 Fatigue Strength
In the region of high cycle fatigue, the equation relating the fatigue
strength
S f to the number of cycles to failure N may be given by the
empirical curve fit equation:
(7-12)
where
N is the number of cycles to failure and a and b are given by
(7-13)
(7-14)
where
f is found from Figure 7-19.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
Slide - 9
ME 307 Machine Design I
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
Slide - 10
ME 307 Machine Design I
If a completely reversed stress
a
is given, setting
S f   a in Eq. (7-12),
the number of cycles-to-failure can be expressed as
(7-15)
Low-cycle fatigue is often defined (see Fig. 7-10) as failure that occurs in a
range of
1  N  103 cycles.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
Slide - 11
ME 307 Machine Design I
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
Slide - 12
ME 307 Machine Design I
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
Slide - 13
ME 307 Machine Design I
Example
Given a 1050 HR steel, estimate
a) The rotating-beam endurance limit at 106.
b) The endurance strength of a polished rotating beam specimen
corresponding to 104 cycles to failure.
c) The expected life of a polished rotating-beam specimen under a
completely reversed stress of 55 kpsi.
SOLUTION:
a)
From Table A-20,
From Eq. (7-8)
b)
Sut  90 kpsi
S 'e  0.5  90   45 kpsi
From Fig. (7-19) for
Dr. A. Aziz Bazoune
Sut  90 kpsi, f  0.86
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
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ME 307 Machine Design I
Example (Cont.’d)
From Eq. (7-13)
and (7-14) ==
Thus Eq. (7-12) is:
for
N  104
Dr. A. Aziz Bazoune
cycles to failure, the above equation becomes
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
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ME 307 Machine Design I
Example (Cont.’d)
c)
From Eq. (7-15), with
 a  55 kpsi
Keep in mind that these are only estimates.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
Slide - 16
ME 307 Machine Design I
7-9
Endurance Limit Modifying Factors
The rotating-beam specimen used in the laboratory to determine endurance limits is
prepared very carefully and tested under closely controlled conditions. It is
unrealistic to expect the endurance limit of a mechanical or structural member to
match the values obtained in the laboratory. Some differences include

Material: composition, basis of failure, variability

Manufacturing: method, heat treatment, fretting corrosion, surface condition,
stress concentration

Environment: corrosion, temperature, stress state, relaxation times

Design: size, shape, life, stress state, stress concentration, speed, fretting,
galling
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
Slide - 17
ME 307 Machine Design I
Marin’s Equation
Marin identified factors that quantified the effects of surface condition, size,
loading, temperature, and miscellaneous items. Marin’s Equations is therefore
written as:
(7-17)
Se :
Endurance limit at the critical location of a machine part in geometry
and condition of use
S 'e :
rotary-beam test specimen endurance limit
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
Slide - 18
ME 307 Machine Design I
When endurance tests of parts are not available, estimations are made by applying
Marin factors to the endurance limit.
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
Slide - 19
ME 307 Machine Design I
(7-18)
where
Sut
is the minimum tensile strength and
a
and
b
are to be found in
Table 7-4.
Table 7-4
Parameters for
Marin surface
modification
factor, Eq. (7-18)
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
Slide - 20
ME 307 Machine Design I
The size factor
kb
for bending and torsion may be given by:
(7-19)
For axial loading there is no size effect, so
(7-20)
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
Slide - 21
ME 307 Machine Design I
QUESTION:
What to do with Eq.(7-19) if a round bar in bending is not rotating or when a
non-circular cross-section is used?
ANSWER:
Use effective dimension
de where
(7-23)
as the effective size of a round corresponding to a non-rotating solid or hollow round.
Table 7-5 provides areas of common structural shapes undergoing non-rotating
bending
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
Slide - 22
ME 307 Machine Design I
Table 7-5
Areas of common nonrotating structural
shapes
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
Slide - 23
ME 307 Machine Design I
Average values for the load factor are given by
(7-25)
Dr. A. Aziz Bazoune
Chapter 7: Fatigue Failure Resulting from variable Loading
CH-07
LEC-21
Slide - 24