1.

Fatigue Failure Resulting from Variable Loading

In most testing of those properties of materials that relate to the

stress-strain diagram, the load is applied gradually, to give sufficient
time for the strain to fully develop. Furthermore, the specimen is
tested to destruction, and so the stresses are applied only once.
Testing of this kind is applicable, to what are known as static
conditions; such conditions closely approximate the actual
conditions to which many structural and machine members are
subjected.
The condition frequently arises, however, in which the
stresses vary with time or they fluctuate between different levels.
For example, a particular fiber on the surface of a rotating shaft
subjected to the action of bending loads undergoes both tension and
compression for each revolution of the shaft. If the shaft is part of an
electric motor rotating at 1725 rev/min, the fiber is stressed in
tension and compression 1725 times each minute. If, in addition, the
shaft is also axially loaded (as it would be, for example, by a helical
or worm gear), an axial component of stress is superposed upon the
bending component. In this case, some stress is always present in
any one fiber, but now the level of stress is fluctuating. These and
other kinds of loading occurring in machine members produce
stresses that are called variable, repeated, alternating, or fluctuating
stresses.
Often, machine members are found to have failed under the
action of repeated or fluctuating stresses; yet the most careful
analysis reveals that the actual maximum stresses were well below
the ultimate strength of the material, and quite frequently even
below the yield strength. The most distinguishing characteristic of
these failures is that the stresses have been repeated a very large
number of times. Hence the failure is called a fatigue failure.
When machine parts fail statically, they usually develop a
very large deflection, because the stress has exceeded the yield
strength, and the part is replaced before fracture actually occurs.
Thus many static failures give visible warning in advance. But a
fatigue failure gives no warning! It is sudden and total, and hence
dangerous. It is relatively simple to design against a static failure,
because our knowledge is comprehensive. Fatigue is a much more
complicated phenomenon, only partially understood, and the
engineer seeking competence must acquire as much knowledge of
the subject as possible.
 Fatigue failure is due to crack formation and propagation. A
fatigue crack will typically initiate at a discontinuity in the material
where the cyclic stress is a maximum. Discontinuities can arise
because of:
• Design of rapid changes in cross section, keyways, holes, etc.
where stress concentrations occur
• Elements that roll and/or slide against each other (bearings, gears,
cams, etc.) under high contact pressure, developing concentrated
subsurface contact stresses that can cause surface pitting or
spalling after many cycles of the load
• Carelessness in locations of stamp marks, tool marks, scratches,
and burrs; poor joint design; improper assembly; and other
fabrication faults
• Composition of the material itself as processed by rolling, forging,
casting, extrusion, drawing, heat treatment, etc. Microscopic and
submicroscopic surface and subsurface discontinuities arise, such
as inclusions of foreign material, alloy segregation, voids, hard
precipitated particles, and crystal discontinuities

Various conditions that can accelerate crack initiation include

residual tensile stresses, elevated temperatures, temperature cycling,
a corrosive environment, and high-frequency cycling.

Approach to Fatigue Failure in Analysis and Design

As noted in the previous section, there are a great many factors to be

considered, even for very simple load cases. The methods of fatigue
failure analysis represent a combination of engineering and science.
Often science fails to provide the complete answers that are needed.
But the airplane must still be made to fly—safely. And the
automobile must be manufactured with a reliability that will ensure a
long and trouble free life and at the same time produce profits for the
stockholders of the industry. Thus, while science has not yet
completely explained the complete mechanism of fatigue, the
engineer must still design things that will not fail. In a sense this is a
classic example of the true meaning of engineering as contrasted
with science. Engineers use science to solve their problems if the
science is available. But available or not, the problem must be
solved, and whatever form the solution takes under these conditions
is called engineering.
3.1 The Stress-Life Method

To determine the strength of materials under the action of fatigue

loads, specimens are subjected to repeated or varying forces of
specified magnitudes while the cycles or stress reversals are counted
to destruction.
To establish the fatigue strength of a material, quite a number
of tests are necessary because of the statistical nature of fatigue. The
results are plotted as an S-N diagram (Fig. 3–1). This chart may be
plotted on semilog paper or on log-log paper. In the case of ferrous
metals and alloys, the graph becomes horizontal after the material
has been stressed for a certain number of cycles.

Figure (3-1)
An S-N diagram plotted from the results of completely reversed
axial fatigue tests. Material: UNS G41300 steel,
normalized; Sut=116 kpsi.

The ordinate of the S-N diagram is called the fatigue strength Sf ; a

statement of this strength value must always be accompanied by a

statement of the number of cycles N to which it corresponds.
 In the case of the steels, a knee occurs in the graph, and
beyond this knee failure will not occur, no matter how great the
number of cycles. The strength corresponding to the knee is called
the endurance limit (Se), or the fatigue limit. The graph of Fig. (3–1)
never does become horizontal for nonferrous metals and alloys, and
hence these materials do not have an endurance limit.
The body of knowledge available on fatigue failure from
N = 1 to N = 1000 cycles is generally classified as low-cycle fatigue,
as indicated in Fig. (3–1). High-cycle fatigue, then, is concerned
with failure corresponding to stress cycles greater than 103 cycles.
Also a finite-life region and an infinite-life region are
distinguished. The boundary between these regions cannot be clearly
defined except for a specific material; but it lies somewhere between
106 and 107 cycles for steels, as shown in the figure.

Fatigue strength fraction, f, of Sut at 103 cycles for Se = S′e = 0.5Sut .

If a completely reversed stress σa is given, the number of cycles-to-failure can
be expressed as

3.2 The endurance Limit

The determination of endurance limits by fatigue testing is now

routine, though a lengthy procedure. Generally, stress testing is
preferred to strain testing for endurance limits.
There are great quantities of data in the literature on the
results of rotating-beam tests and simple tension tests of specimens
taken from the same bar or ingot. The endurance limit ranges from
about 40 to 60 percent of the tensile strength for steels up to about
210 kpsi (1450 MPa). For steels, the endurance limit may be
estimated as

3-1

where Sut is the minimum tensile strength. The prime mark on Se in
this equation refers to the rotating-beam specimen.
When designs include detailed heat-treating specifications to
obtain specific microstructures, it is possible to use an estimate of
the endurance limit based on test data for the particular
microstructure; such estimates are much more reliable and indeed
should be used.
3.3 Endurance Limit Modifying Factors

Joseph Marin identified factors that quantified the effects of surface

condition, size, loading, temperature, and miscellaneous items.
A Marin equation is written as

Se = ka kb kc kd ke kf Se 3-2
Where

ka = surface condition modification factor

kb = size modification factor
kc = load modification factor
kd = temperature modification factor
ke = reliability factor
kf = miscellaneous-effects modification factor
Se = rotary-beam test specimen endurance limit
Se = endurance limit at the critical location of a machine part
in the geometry and condition of use
When endurance tests of parts are not available, estimations are
made by applying Marin factors to the endurance limit.

 Surface Factor ka

3-3

where Sut is the minimum tensile strength and a and b are to be found
in the following table.

Table (3–1)
Parameters for Marin surface modification factor, Eq. (3–3)

Factor a Exponent
Surface finish Sut, kpsi Sut, MPa b
Ground 1.34 1.58 −0.085
Machined or cold-drawn 2.7 4.51 −0.265
Hot-rolled 14.4 57.7 −0.718
As-forged 39.9 272 −0.995
EXAMPLE 3–1

A steel has a minimum ultimate strength of 520 MPa and a machined surface.
Estimate ka.
Solution

From Table (3–1), a = 4.51 and b = −0.265. Then, from Eq. (3–3)

ka = 4.51(520)−0.265 = 0.860 Ans.

Again, it is important to note that this is an approximation as the data

is typically quite scattered. Furthermore, this is not a correction to
take lightly. For example, if in the previous example the steel was
forged, the correction factor would be 0.540, a significant reduction
of strength.

 Size Factor kb

For round shafts in bending and torsion when rotating, kb may be

expressed as

3-4

The effective size of a round corresponding to a non-rotating solid or

hollow round,

de = 0.37d 3-5

For a rectangular section of dimensions h × b

de = 0.808(hb)1/2 3-6

For axial loading there is no size effect, so

kb = 1 3-7
 EXAMPLE 3–2

A steel shaft loaded in bending is 32 mm in diameter, abutting a

filleted shoulder 38 mm in diameter. The shaft material has a mean
ultimate tensile strength of 690 MPa. Estimate the Marin size factor
kb if the shaft is used in
(a) A rotating mode.
(b) A non-rotating mode.

Solution
(a) From Eq. (3–4)
kb = (d/7.62)−0.107 = (32/7.62)−0.107 = 0.858 Ans.
(b) From Eq. (3–5),
de = 0.37d = 0.37(32) = 11.84 mm

Then, from Eq. (3–4)

kb = (d/7.62)−0.107 = (11.84 /7.62)−0.107 = 0.954 Ans.

 Loading Factor kc

When fatigue tests are carried out with rotating bending, axial (push-
pull), and torsional loading, the endurance limits differ with Sut. The
average values of the load factor are specified as

3-8
*

*
The latter is used only for pure torsional fatigue loading. When
torsion is combined with other stresses, such as bending, kc = 1.

 Temperature Factor kd

When operating temperatures are below room temperature, brittle

fracture is a strong possibility and should be investigated first. When
the operating temperatures are higher than room temperature,
yielding should be investigated first because the yield strength drops
off so rapidly with temperature; see Fig. (3–2). Any stress will
induce creep in a material operating at high temperatures; so this
factor must be considered too.
Finally, it may be true that there is no fatigue limit for
materials operating at high temperatures. Because of the reduced
fatigue resistance, the failure process is, to some extent, dependent
on time.

Figure (3–2)
A plot of the results of 145 tests of 21 carbon and alloy steels
showing the effect of operating temperature on the yield strength Sy
and the ultimate strength Sut . The ordinate is the ratio of the strength
at the operating temperature (ST) to the strength at room temperature
(SRT).

Table (3–2) has been obtained from Fig. (3–2) by using only the
tensile-strength data. Note that the table represents 145 tests of 21
different carbon and alloy steels. A fourth-order polynomial curve fit
to the data underlying Fig. (3–2) gives

3-9
where 70 ≤ TF ≤ 1000 F.
 Table (3–2)
Effect of operating temperature on the tensile strength of
steel. (ST = tensile strength at operating temperature;
SRT = tensile strength at room temperature)

Temperature Temperature
o ST/SRT o ST/SRT
C F
20 1.000 70 1.000
50 1.010 100 1.008
100 1.020 200 1.020
150 1.025 300 1.024
200 1.020 400 1.018
250 1.000 500 0.995
300 0.975 600 0.963
350 0.943 700 0.927
400 0.900 800 0.872
450 0.843 900 0.797
500 0.768 1000 0.698
550 0.672 1100 0.567
600 0.549 Data sourc : Fig. (3–2)

Two types of problems arise when temperature is a consideration. If

the rotating beam endurance limit is known at room temperature,
then use
kd =ST/SRT 3-10

from Table (3–2) or Eq. (3–9) and proceed as usual. If the rotating-
beam endurance limit is not given, then compute it using Eq. (3–1)
and the temperature-corrected tensile strength obtained by using the
factor from Table (3–2). Then use kd = 1.
EXAMPLE 3–3

A 1035 steel has a tensile strength of 70 kpsi and is to be used for a

part that sees 450°F in service. Estimate the Marin temperature
modification factor and (Se)450° if
(a) The room-temperature endurance limit by test is ( S e )70°= 39 kpsi
(b) Only the tensile strength at room temperature is known

Solution

(a) First, from Eq. (3–9),

kd = 0.975 + 0.432(10−3)(450) − 0.115(10−5)(4502)

+ 0.104(10−8)(4503) − 0.595(10−12)(4504) = 1.007

Thus,

(Se)450° = kd ( Se )70° = 1.007(39) = 39.3 kpsi Ans.

(b) Interpolating from Table (3–2) gives

Thus, the tensile strength at 450°F is estimated as

(Sut )450° = (ST /SRT )450° (Sut )70° = 1.007(70) = 70.5 kpsi

From Eq. (3–1) then,

(Se)450° = 0.5 (Sut )450° = 0.5(70.5) = 35.2 kpsi

Part a gives the better estimate due to actual testing of the particular
material.

 Reliability Factor ke

The reliability modification factor can be determined from the

following table.
 Table (3–3)
Reliability factors ke corresponding to 8 percent standard
deviation of the endurance limit

Reliability,% Reliability factors ke

50 1.000
90 0.897
95 0.868
99 0.814
99.9 0.753
99.99 0.702
99.999 0.659
99.9999 0.620
99.99999 0.584

 Miscellaneous-Effects Factor kf

Though the factor kf is intended to account for the reduction in

endurance limit due to all other effects, it is really intended as a
reminder that these must be accounted for, because actual values of
kf are not always available.
Residual stresses may either improve the endurance limit or
affect it adversely. Generally, if the residual stress in the surface of
the part is compression, the endurance limit is improved. Fatigue
failures appear to be tensile failures, or at least to be caused by
tensile stress, and so anything that reduces tensile stress will also
reduce the possibility of a fatigue failure. Operations such as shot
peening, hammering, and cold rolling build compressive stresses
into the surface of the part and improve the endurance limit
significantly. Of course, the material must not be worked to
exhaustion. The endurance limits of parts that are made from rolled
or drawn sheets or bars, as well as parts that are forged, may be
affected by the so-called directional characteristics of the operation.
Rolled or drawn parts, for example, have an endurance limit in the
transverse direction that may be 10 to 20 percent less than the
endurance limit in the longitudinal direction.
Corrosion, electrolytic plating, metal spraying, cyclic
frequency and frottage corrosion may also have an effect on the
endurance limit.
3.4 Stress Concentration and Notch Sensitivity

It turns out that some materials are not fully sensitive to the presence
of notches and hence, for these, a reduced value of Kt can be used.
For these materials, the maximum stress is, in fact,
σmax = Kf σo or τmax = Kf sτo 3-11

where Kf is a reduced value of Kt and σo is the nominal stress. The

factor Kf is commonly called a fatigue stress-concentration factor,
and hence the subscript f. So it is convenient to think of Kf as a
stress-concentration factor reduced from Kt because of lessened
sensitivity to notches. The resulting factor is defined by the equation

Notch sensitivity q is defined

by the equation

where q is usually between zero and unity. Equation (2–12) shows

that if q = 0, then Kf = 1, and the material has no sensitivity to
notches at all. On the other hand, if q = 1, then Kf = Kt , and the
material has full notch sensitivity. In analysis or design work, find Kt
first, from the geometry of the part. Then specify the material, find
q, and solve for Kf from the equation

Kf = 1 + q(Kt − 1) or Kf s = 1 + qshear(Kts − 1) 3-13

For steels and 2024 aluminum alloys, use Fig. (3–3) to find q for
bending and axial loading. For shear loading, use Fig. (3–4).

The notch sensitivity of the cast irons is very low, varying

from 0 to about 0.2, depending upon the tensile strength. To be on
the conservative side,

q = 0.2 for all grades of cast iron

 Figure (3–3)
Notch-sensitivity charts for steels and UNS A92024-T wrought aluminum alloys subjected to
reversed bending or reversed axial loads. For larger notch radii, use the values

of q corresponding to the r = 0.16-in (4-mm)

Figure (3–4)
Notch-sensitivity curves for materials in reversed torsion. For larger notch radii, use
the values of qshear corresponding to r = 0.16-in (4-mm)
EXAMPLE 3–4

A steel shaft in bending has an ultimate strength of 690 MPa and a

shoulder with a fillet radius of 3 mm connecting a 32-mm diameter
with a 38-mm diameter. Estimate Kf .

Solution

From Fig. (1–16), using D/d = 38/32 = 1.1875, r/d = 3/32 = 0.09375,
we read the graph to find (Kt = 1.65)

From Fig. (3–3), for Sut = 690 MPa and r = 3 mm, (q = 0.84). Thus,
from Eq. (3–13)

Kf = 1 + q(Kt − 1) = 1 + 0.84(1.65 − 1) = 1.55 Ans.

EXAMPLE 3–5

A 1015 hot-rolled steel bar has been machined to a diameter of 1 in.

It is to be placed in reversed axial loading for 70 000 cycles to
failure in an operating environment of 550°F. Using ASTM
minimum properties, and a reliability of 99 percent, estimate the
endurance limit.

Solution

From Table (3–4), Sut = 50 kpsi at 70°F. Since the rotating-beam

specimen endurance limit is not known at room temperature, we
determine the ultimate strength at the elevated temperature first,
using Table (3–2):

(ST/SRT)550° = (0.995 + 0.963)/2 = 0.979

The ultimate strength at 550°F is then

(Sut )550° = (ST /SRT )550° (Sut )70° = 0.979(50) = 49 kpsi

The rotating-beam specimen endurance limit at 550°F is then

estimated from Eq. (3–1) as
S e = 0.5(49) = 24.5 kpsi

Next, we determine the Marin factors. For the machined surface, Eq.
(3–3) with Table (3–1) gives

k  aSb = 2.70(49)−0.265 = 0.963

a ut

For axial loading, from Eq. (3–7), the size factor kb = 1, and from
Eq. (3–8) the loading factor is kc = 0.85. The temperature factor
kd = 1, since we accounted for the temperature in modifying the
ultimate strength and consequently the endurance limit. For 99
percent reliability, from Table (3–3), ke = 0.814. Finally, since no
other conditions were given, the miscellaneous factor is kf = 1. The
endurance limit for the part is estimated by Eq. (3–2) as
Se = ka kb kc kd ke kf Se = 0.963(1)(0.85)(1)(0.814)(1)24.5
= 16.3 kpsi Ans.

Table (3–4)
Deterministic ASTM minimum tensile and yield strengths for some hot-rolled (HR) and cold-
drawn (CD) steels. [The strengths listed are estimated ASTM minimum values in the size range
18 to 32 mm (34 to 114 in). These strengths are suitable for use with the design factor,
provided the materials conform to ASTM A6 or A568 requirements or are required in the
purchase specifications]
EXAMPLE 3–6

Figure (3–5a) shows a rotating shaft simply supported in ball

bearings at A and D and loaded by a non-rotating force F of 6.8 kN.
Using ASTM “minimum” strengths, estimate the endurance limit
and the reversing bending stress.

Solution

From Fig. (3–5b) we learn that failure will probably occur at B

rather than at C or at the point of maximum moment. Point B has a
smaller cross section, a higher bending moment, and a higher stress-
concentration factor than C, and the location of maximum moment
has a larger size and no stress-concentration factor.
We shall solve the problem by first estimating the strength at
point B, since the strength will be different elsewhere, and
comparing this strength with the stress at the same point.

Figure (3–5)
(a) Shaft drawing showing all dimensions in millimeters; all fillets
3-mm radius. The shaft rotates and the load is stationary; material
is machined from AISI 1050 cold-drawn steel.
(b) Bending-moment diagram.
 From Table (3–4) we find Sut = 690 MPa and Sy = 580 MPa.

Se = 0.5(690) = 345 MPa

ka = 4.51(690)−0.265 = 0.798
kb = (32/7.62)−0.107 = 0.858
kc = kd = ke = kf = 1

Then, Se = 0.798(0.858)345 = 236 MPa Ans.

Same as Example (3–4), Kf = 1.55

The next step is to estimate the bending stress at point B. The

bending moment is MB = 695.5 N·m

Then, the reversing bending stress is,

MB c = 335.1 MPa Ans.

I
This stress is greater than Se and less than Sy. This means we
have both finite life and no yielding on the first cycle
For finite life, The ultimate strength, Sut = 690 MPa = 100 kpsi.
From Fig. 6–18, f = 0.844.