1. 1.2 Fatigue
A periodic stress oscillating between some limits applied to a machine member is called
repeated, alternating, ov fluctuating stress. The failure of the machine members under the
action of these stresses is calledfatigue failure. A small crack is enough to initiate the fatigue
failure. The crack progresses rapidly since the stress concentration effect becomes greater
around it. If the stressed area decreases in size, the stress increases in magnitude and if the
remaining area is small, the member can fail. A member failed because of fatigue shows
two distinct regions. The first one is due to the progressive development of the crack, while
the other one is due to the sudden fracture.
11.2.1 Endurance Limit
The strength of materials acted upon by fatigue loads can be determined by performing
a fatigue test provided by R. R. Moore's high-speed rotating beam machine (Fig. II.2.1).
During the test, the specimen is subjected to pure bending by using weights and rotated
with constant velocity. For a particular magnitude of the weights, one records the number
of revolutions at which the specimen fails. Then, a second test is performed for a specimen
identical with the first one, but the magnitude of the weight is reduced. Again, the number
of revolutions at which the fatigue failure occurs is recorded. The process is repeated
several times. The intensity of the reversed stress causing failure after a given number of
cycles is thefatigue strength corresponding to that number of loading cycles. Finally, the
fatigue strengths considered for each test are plotted against the corresponding number of
revolutions. The resulting chart is called the S-N diagram. The S-N curves are plotted on
log-log coordinates.
Numerous tests have established that the ferrous materials have an endurance limit
defined as the highest level of alternating stress that can be withstood indefinitely by a test
specimen without failure. The symbol for endurance limit is 5^. The endurance limit can be
related to the tensile strength through some relationships. For example, for steel, Mischke
491

2. Revolution
counter
Weights
Source: R. C. Juvinall and K. M. Marshek, "Fundamentals of Machine
Component Design", John Wiley k Sons, Inc., 1991.
FIGURE II.2.1 Rotating beam fatigue testing macliine. Reprinted with permission ofJohn Wiley &
Sons, Inc.
predicted the following relationships [11]:
f 0.50 Sut, Sut < 200 kpsi (1400 MPa)
100 kpsi, Sut > 200 kpsi (II.2.1)
[700 MPa, Sut > 1400 MPa
Se=
where Sut (or Su) is the ultimate strength in tension. Table II.2.1 lists the values of the
endurance limit for various classes of cast iron that is polished or machined. The symbol S^^
refers to the endurance limit of the test specimen that can be significantly different from the
endurance limit Se of any machine element subjected to any kind of loads. The endurance
limit Se can be affected by several factors called modifyingfactors. Some of these factors
are the surface factor ks, the gradient (size) factor kc, or the load factor ki. Thus, the
endurance limit of a member can be related to the endurance limit of the test specimen by
the following relationship:
TABLE n.2.1 Endurance Lim
Tensile
ASTM* ''''''^'^
number ^ut,kpsi
20 22
25 26
30 31
35 36.5
40 42.5
50 52.5
60 62.5
Compressive
strength
Sue, kpsi
83
97
109
124
140
164
187.5
Se = ks ko ki S'^.
it for Various Classes of Cast Iron (Polished or
Shear modulus
of Rupture
Ssu, kpsi
26
32
40
48.5
57
73
88.5
Modulus of
elasticity, Mpsi
Tension Torsion
9.6-14 3.9-5.6
11.5-14.8 4.6-6.0
13-16.4 5.2-6.6
14.5-17.2 5.8-6.9
16-20 6.4-7.8
18.8-22.8 7.2-8.0
20.4-23.5 7.8-8.5
Endurance
limit
Se, kpsi
10
11.5
14
16
18.5
21.5
24.5
(II.2.2)
Machined)
Brinell
hardness
HB
156
174
201
212
235
262
302
Fatigue stress
concentration
factor Kf
1.00
1.05
1.10
1.15
1.25
1.35
1.50
*ASTM, American Society for Testing Materials.
Source: J. E. Shigley and C. R. Mischke, Mechanical Engineering Design, 5th ed., New York, McGraw-Hill, 1989.
Reprinted with permission of McGraw-Hill.
492 KINEMATIC CHAINS AND MACHINE COMPONENTS DESIGN

3. TABLE n.2.2 Surface Finish Factors
Surface Factor a
finish kpsi MPa
Exponent
b
Ground 1.34 1.58 -0.085
Machined or cold-drawn 2.70 4.51 -0.256
Hot-rolled 14.4 57.7 -0.718
As forged 39.9 272.0 -0.995
Source: J. E. Shigley and C. R. Mischke, Mechanical Engineering
Design, New York, McGraw-Hill, 1989. Reprinted with permission
of McGraw-Hill.
Surface factor ks.
The influence of the surface of the specimen is described by the modification factor ks
which depends upon the quaHty of the finishing. The following formula describes the
surface factor [20]:
ks = aSl,, (IL2.3)
where Sut is the ultimate tensile strength. Some values for a and b are listed in Table II.2.2.
Gradient (size) factor kc-
The results of the tests performed to evaluate the size factor in the case of bending and
torsion are as follows [20]:
ko
V7:62J
•0.1133
in. 0.11 < J < 2in.
(IL2.4)
-0.1133
mm 2 . 7 9 < J < 5 1 m m ,
where d is the diameter of the test bar. To apply Eq. (IL2.4) for a nonrotating round bar in
bending or for a noncircular cross section, an effective dimension, de, is introduced [20].
This dimension is obtained by considering the volume of material stressed at and above
95% of the maximum stress and a similar volume in the rotating beam specimen. When
these two volumes are equated, the lengths cancel each other out and only the areas have
to be considered.
Some recommended values for ko are given in reference [27]:
• for bending and torsion:
J < 8 mm kc = 1,
8 mm < J < 250 mm ko = 1.189 d'^'^^'^, (11.2.5)
d > 250 mm 0.6 < kc < 0.75,
• for axial loading: kc = I.
Fatigue 493

4. TABLE 11.2.3 Summary of Modifying Factors for Bending, Axial Loading, and Torsion of Ductile
Materials
10^-cycle strength
Bending loads: S = 0.9 Su
Axial loads: S = 0.75 Su
Torsional loads: S = 0.9 Sus, Sus ^ 0.85^ for steel; Sus ^ 0.1Su for other ductile materials
10^-cycle strength (endurance limit)
Se = kskckiS'^, where S'^ is the specimen endurance limit
5g = 0.55M for steel, lacking better data
Surfacefactor, ks
Loadfactor, ki
Gradientfactor, kc
diameter < (0.4 in. or 10 mm)
(see Fig. II.2.2)
bending axial
1 1
1 0.7-0.9
torsion
0.58
1
(0.4 in. or 10 mm) < diameter < (2 in. or 50 mm) 0.9 0.7 - 0.9 0.9
for (2 in. or 50 mm) < diameter < (4 in. or 100 nmi) reduce the factors by about 0.1
for (4 in. or 100 mm) < diameter < (6 in. or 150 mm) reduce the factors by about 0.2
Source: R. C. Juvinall and K. M. Marshek, Fundamentals ofMachine Component Design, New York, John Wiley & Sons,
1991. Reprinted with permission of John Wiley & Sons, Inc.
Load factor ki*
Tests revealed that the load factor has the following values [20]:
0.923, axial loading, Sut < 220 kpsi (1520 MPa),
1, axial loading, Sut > 220 kpsi (1520 MPa),
kL =
1, bending,
0.577 torsion and shear.
(IL2.6)
Juvinall and Marshek present a summary of all modifying factors for bending, axial
loading, and torsion used for fatigue of ductile materials and listed in Table 11.2.3 [7].
.2.2 Fluctuating Stresses
In design problems the stress frequently fluctuates without passing through zero. The com-
ponents of the stresses are depicted in Figure IL2.3(a), where cxmm is minimum stress, cfmax
the maximum stress, a^ the stress amplitude or the alternating stress, am the midrange or
the mean stress, a^ the stress range. The steady stress or static stress, a^, can have any
value between cfmin and a^ax and exists because of a fixed load. It is usually independent
494 KINEMATIC CHAINS AND MACHINE COMPONENTS DESIGN

5. Hardness {H^)
CO
120 160 200 240 280 320 360 400 440 480 520
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
60 80 100 120 140 160 180 200 220 240 260
Tensile strength Su (kpsi)
s
s
V
>
1 1 1
Mirror-polished
Fine-ground or^^^
commerniallv
P
S
s
" - ,
Corroded in "
tao water
olishec
V
• - >
^
i
^ ^
"^^.
^ - ^
h ^
^ ^ o , . . . , .
' ^ ^
• • • i , ^
P^oA
^ < ^
$ > ^
~ - -
^
Corroded in salt water
1 1 1 1
ti
^"^Z^
r
0.4
0.6
T "T"
1.0
T
0.8 1.0 1.2 1.4
Su (GPa)
"T"
1.6 1.8
Source: R. C. Juvinall and K. M. Marshek, "Fundamentals of
Machine Component Design", John Wiley & Sons, Inc., 1991.
FIGURE 11.2.2 Surface factor ks. Reprinted with permission of John Wiley & Sons, Inc.
of the varying portion of the load. The following relations between the stress components
exist as
CTm =
^max I CT]max 1^ ^min
(^max ^min
(IL2.7)
(11.2.8)
The fluctuating stresses are described by the stress ratios
R=: A = (11.2.9)
Fatigue 495

6. Stress
i
'
i / ^
^max
r
Fluctuating stress
V 4 /'^
C^a /
T /
/ 1 / ^
C^a /
/ y _ /
i
^min
i
i
<7r
Y
C^m
1 1
o
Stress
(a)
Repeated stress
^min ^
(b)
Stress
O
Time
Time
l^m = 0
Reversed stress
i
^
/ ^
^a /
/
/ i
/
1C^a /
' vy
L
Gy
f
Time
(c)
FIGURE 11.2.3 Time varying stresses: (a) sinusoidal fluctuating stress; (b) repeated stress; (c)
reversed sinusoidal stress.
496 KINEMATIC CHAINS AND MACHINE COMPONENTS DESIGN

7. Afluctuatingstress is a combination of static plus completely reversed stress. Figure IL2.3(a)
shows a sinusoidal fluctuating stress, Figure IL2.3(b) represents a repeated stress, and
Figure 11.2.3(c) is a completely reversed sinusoidal stress.
11.2.3 Constant Life Fatigue Diagram
Figure II.2.4 illustrates the graphical representation of various combinations of mean and
alternating stress in relation to yielding and various fatigue life [7]. This diagram is called
the constant lifefatigue diagram because it has lines corresponding to a constant 10^ cycle
or "infinite" life, constant 10^ cycle, constant 10^ cycle, and so forth. The horizontal
axis {aa = 0) corresponds to static loading. The point A(a^ = Sy,aa = 0) represents
the yield strength. For ductile materials the point A—Sy,0) represents the compressive
yield strength. The point B(Su,0) represents the ultimate tensile strength. At the point
A^XcTfn = 0,aa = Sy) the stress fluctuates between -hSy and —Sy. The points on line AA^^
correspond to fluctuations having a tensile peak of Sy. The points on line A^A^^ correspond
to fluctuations having a compressive peak of Sy. Within the triangle AA^A^^ there are all the
combinations with no yielding. The points C, D, E, and F correspond to (7^ = 0 for various
values of fatigue life and are obtained from the S-N diagram. The lines CB, DB, EB, and
FB are the estimated lines of constant life. These lines are called the Goodman lines.
The area A^HCGA corresponds to a life of at least 10^ cycles and no yielding. For a life
of at least 10^ cycles and yielding, in addition to the area A^HCGA, the area AGB and the
A'^
H'
10^-
/
/
10^-
10^-
10^- /
/
/
/E
0
F
by
/
/ ^
D
C
Se
Ca
a
.^
AX,
y ^
;^ Values from S-N curve
^^
G ^ ^ X ^ ^ ^
X 4 ^ ^ ^
-am (compression) O am (tension)
Source: R. C. Juvinall and K. M. Marshek, "Fundamentals of Machine Component Design",
John Wiley & Sons, Inc., 1991.
FIGURE 11.2.4 Constant life fatigue diagram. Reprinted with permission of John Wiley & Sons, Inc.
Fatigue 497

8. area to the left of the Hne A!H may be used. The area HCGA"H corresponds to less than
10^ cycles of life and no yielding.
11.2.4 Fatigue Life for Randomly Varying Loads
For most mechanical parts acted upon by randomly varying stresses, the prediction of
fatigue life is not an easy task. Instead of a single reversed stress a for n cycles, a part is
subjected to a for n cycles, ai for ni cycles, and so forth, and the problem is to estimate the
fatigue life of the part. The procedure for dealing with this situation is the linear cumulative
damage rule (or Miner's rule) and can be expressed by the following equation [7, 20]:
n no nk —v «/
-^ + - ^ + - + -f = l or Y^ = (II.2.10)
7=1 -^
where ni,n2," - ,nk represent the number of cycles at specific overstress levels
ai,(72, • • • ,crk and Ni,N2," - ,Nk represent the life (in cycles) at these overstress levels,
as taken from the appropriate S-N curve. Fatigue failure is predicted when Eq. (II.2.10)
holds.
IL2.5 Variable Loading Failure Theories
There are various techniques for plotting the results of the fatigue failure test of a part
subjected to fluctuating stress. One of them is called the modified Goodman diagram and is
shown in Figure II.2.5 [20]. Forthis diagram the mean stress a^ is plotted on the abscissa and
the other stress components (Se, Sy, Su) on the ordinate (tension is the positive direction).
The mean stress line makes a 45° angle with the abscissa from the origin to the tensile
strength. Lines are constructed to Se (above the origin) and to —Se (below the origin) as
shown in Figure II.2.5. Yielding can be considered as a criterion of failure if <J^^ > Sy.
Another way to display the results is shown in Figure 11.2.6 using the strengths [20]. The
fatigue limit Se (or the finite life strength Sf) is plotted on the ordinate. The tensile yield
strength Syt is plotted on both coordinate axes. The ultimate tensile strength Sut is plotted
on the abscissa. The alternating strength is Sa as a limiting value of a^ and is plotted on the
ordinate. The mean strength is Sm as a limiting value of a^ and is plotted on the abscissa.
Four criteria of failure are shown in the diagram in Figure II.2.6, that is, Soderberg,
the modified Goodman, Gerber, and yielding. Only the Soderberg criterion guards against
yielding as shown in Figure II.2.6.
The Soderberg, Goodman, and yield criterion are described by the equation of a straight
line in intercept form:
^ + ^ = 1, (II.2.11)
a b
where a and b are the Sm and Sa intercepts, respectively. The equation for the Soderberg
line is
^ + ^ = 1. (II.2.12)
Syt Se
498 KINEMATIC CHAINS AND MACHINE COMPONENTS DESIGN

9. Source: J. E. Shigley, and C. R. Mischke, "Mechanical Engineering Design",
McGraw-Hill, Inc., 1989.
FIGURE II.2.5 Modified Goodman diagram. Reprinted with permission of McCraw-f-lill.
Similarly, the equation for the modified Goodman line is
(11.2.13)
The yielding line is described by the equation
Oyt Oy
(II.2.14)
Fatigue 499

10. Gerber line
Goodman line
Mean stress dm
Source: J. E. Shigley, and C. R. Mischke, "Mechanical Engineering Design",
McGraw-Hill, Inc., 1989.
FIGURE II.2.6 Various failure theories. Reprinted with permission of McCraw-f-lill.
The Gerber criterion is also called the Gerber parabolic relation because the curve can be
modeled by a parabolic equation of the form
1. (II.2.15)
The curve representing the Gerber theory is a better predictor since it passes through the
central region of the failure points.
If each strength in Eqs. (11.2.12) to (II.2.15) is divided by a safety factor SF, the stresses
aa and Gm can replace Sa and 5^. Therefore, the Soderberg equation becomes
Sp Si SF
(II.2.16)
500 KINEMATIC CHAINS AND MACHINE COMPONENTS DESIGN

11. - Goodman line
Safe stress line
^m Srr
Mean stress am
Sut
Source: J. E. Shigley, and C. R. Mischke, "Mechanical Engineering Design",
McGraw-Hill, Inc., 1989.
FIGURE 11.2.7 Safe stress line. Reprinted with permission of McCraw-Hill.
the modified Goodman equation becomes
Se Sut SF
and the Gerber equation becomes
SFa,
Se Sut /
(11.2.17)
(11.2.18)
Figure II.2.7 shows the explanation of Eq. (II.2.17) [20]. A safe stress line through point
A of coordinates a^, cr^ is drawn parallel to the modified Goodman line. The safe stress
line is the locus of all points of coordinates a^, aa for which the same safety factor SF is
considered, that is, 5^ = SF a^ and Sa = SF Ga.
Table 2.4 lists the values of the tensile strength and Table 2.5 gives the yield strength for
various materials.
.2.6 Examples
EXAMPLE 11.2.1:
Estimate the S-N curve for a precision steel part for torsional loading. The part has
the cross-section diameter under 2 in. and has a fine ground surface. The material
has the ultimate tensile strength ^^ = 110 kpsi and the yield strength Sy = 11 kpsi.
Use the empirical relationships given in Table II.2.3.
Continued
Fatigue 501

12. TABLE ir.2.4 Tensile Strength
Material name
AISI1006 Steel, cold drawn
AISI1006 Steel, hot rolled bar, 19-32 mm round
AISI 1006 Steel, cold drawn bar, 19-32 mm round
AISI 1008 Steel, hot rolled bar, 19-32 mm round
AISI 1008 Steel, cold drawn bar, 19-32 mm round
AISI 1010 Steel, cold drawn
AISI 1010 Steel, hot rolled bar, 19-32 mm round or thickness
AISI 1010 Steel, cold drawn bar, 19-32 mm round or thickness
AISI 1012 Steel, cold drawn
AISI 1012 Steel, hot rolled bar, 19-32 mm round or thickness
AISI 1012 Steel, cold drawn bar, 19-32 mm round or thickness
AISI 1015 Steel, cold drawn
AISI 1015 Steel, cold drawn, 19-32 mm round
AISI 1015 Steel, hot rolled, 19-32 mm round
AISI 1015 Steel, as rolled
AISI 1015 Steel, normalized at 925°C (1700°F)
AISI 1015 Steel, annealed at 870°C (1600°F)
AISI 1016 Steel, cold drawn, 19-32 mm round
AISI 1016 Steel, hot rolled, 19-32 mm round
AISI 1017 Steel, cold drawn
AISI 1017 Steel, hot rolled, 19-32 mm round
AISI 1018 Steel, cold drawn
AISI 1018 Steel, hot rolled, quenched, and tempered
AISI 1018 Steel, hot rolled, 19-32 mm round
AISI 1018 Steel, as cold drawn, 16-22 mm round
AISI 1018 Steel, as cold drawn, 22-32 mm round
AISI 1018 Steel, as cold drawn, 32-50 mm round
AISI 1018 Steel, as cold drawn, 50-76 mm round
AISI 1019 Steel, cold drawn
AISI 1019 Steel, hot rolled, 19-32 mm round
AISI 1020 Steel, cold rolled
AISI 1020 Steel, hot rolled, 19-32 mm round
AISI 1020 Steel, as rolled
AISI 1020 Steel, normalized at 870°C (1600°F)
AISI 1020 Steel, as rolled, 25 mm round
AISI 1021 Steel, cold drawn
AISI 1022 Steel, cold drawn round (19-32 mm)
AISI 1022 Steel, as rolled
Tensile,
ultimate
MPa
330
295
330
305
340
365
325
365
370
330
370
385
385
345
420
425
385
420
380
405
365
440
475
400
485
450
415
380
455
407
420
380
450
440
472
470
475
505
strength
yield
MPa
285
165
285
170
285
305
180
305
310
185
310
325
325
190
315
325
285
350
205
340
200
370
275
220
415
380
345
310
379
224
350
205
330
345
384
395
400
360
AISI, American Iron and Steel Institute.
Source: MatWeb — Material Property Data available at: http://www.matweb.com/.
502 KINEMATIC CHAINS AND MACHINE COMPONENTS DESIGN