### Part_I

```Fatigue Failure Theories
Design of Machine Elements
© Dr Moudar Zgoul, @zgoul_ju, 2011 [extracted from different sources]
Failure Theories
Introduction
 Fluctuating stresses in machinery often take the form
of sinusoidal pattern because of the nature of the
nature of some rotating machinery.
 Other patterns some quite irregular do occur.
 The smooth parts of the fracture surface usually exhibit
beach marks which occurs as a result of changes in
the magnitude of the fluctuating fatigue load.
 Fatigue behavior of materials is usually described by
means of the S-N diagram which gives the number of
cycles to failure, N as a function of the max applied
alternating stress.
Introduction
 Fracture surface
which usually
exhibits smooth
areas which
correspond to
growth stage,
and rough areas,
which
correspond to
the catastrophic
fracture stage.
Characterizing Fluctuating Stresses
 In periodic patterns exhibiting a single maximum and
single minimum of force, the shape of the wave is not
important.
 The peaks on both sides (maximum, minimum) are
important.
Characterizing Fluctuating Stresses
 Fmax and Fmin in a cycle can be used to characterize the
force pattern.
 A steady component and an alternating component can
be constructed as follows:
Characterizing Fluctuating Stresses
 Stress Range
 Mean (Midrange
Stress)
 Stress Amplitude
(Alternating Stress)
 Stress Ratio
 Stress Amplitude
Prepare a Presentation on
Catastrophic Fatigue Failure
in Aviation, Massive
Structures and other
Engineering Applications
Groups will be assigned in the class on Sunday
Characterizing Fluctuating Stresses
Fatigue Stresses
40000
sa
Applied Stress - psi
30000
20000
sa
sm
10000
s
0
-10000
-20000
0
Fully Reversed
2
sm
4
Repeated
6
8
Fluctuating
10
Characterizing Fluctuating Stresses
Characterizing Fluctuating Stresses
Characterizing Fluctuating Stresses
Characterizing Fluctuating Stresses
stress is not the same
as the midrange stress.
have any value between
σmin and σmin .
part.
independent
of
the
varying portion of the
Characterizing Fluctuating Stresses
A
helical
compression
into a space shorter than
the free length of the
spring.
 The stress created by this
initial
compression
is
component of the stress.
Characterizing Fluctuating Stresses
 The equations use symbols σa and σm as the stress
components at the location of scrutiny.
 In the absence of a notch, σa and σm are equal to the
nominal stresses σao and σmo induced by loads Fa and
Fm, respectively.
 With a notch they are
σa = Kf σao and
σm = Kf σmo , respectively.
Stress Concentration
Kf = 1+q(Kt + 1)
Kt - geometric stress concentration factor
q - notch sensitivity factor
Stress Concentration
Kf = 1+q(Kt + 1)
Kt - geometric stress concentration factor
q - notch sensitivity factor
Stress Concentration
Stress Concentration
Stress Concentration
Neuber equation,
= Neuber constant and is a material constant.
Design Against Fatigue
S-N curve is a graphical representation of the maximum
applied stress versus the number of stress cycles N
before the fatigue failure on a semi-log graph.
For ferrous metals like steel the curve becomes
asymptotic at 10^6 cycles. The completely reversed stress
which a material can withstand 10^6 cycles without failure
is called ENDURANCE LIMIT of the material.
For non ferrous materials, the curve slopes gradually even
after 10^6 cycles.
These materials do not have a limiting value of endurance
in true sense.
Design Against Fatigue
Design Against Fatigue
In the majority cases, the reported fatigue strength or
endurance limits of the materials are based on the test of
carefully prepared small samples under laboratory condition.
Such values cannot be directly used for design purposes
because the behavior of a component or structure under
endurance limit of the material used in making it, but also an
several other factors.
Design Against Fatigue
The Endurance Limit:
The strength corresponding to the knee in the S-N diagram
is called the endurance limit Se, or the fatigue limit. The
Estimating Se’ or Sf’
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.
For preliminary and prototype design and for some failure
analysis as well, a quick method of estimating endurance
limits is needed.
Design Against Fatigue
Design Against Fatigue
For steels
where Sut is the minimum tensile strength. The prime
mark on S′e in this equation refers to the rotating-beam
specimen itself.
Endurance Limit Modifying Factors
•
•
•
•
•
•
•
Size and shape of the component or structure
Stress concentration
Surface finish
Operating temperature
Service environment
Method of fabrication
Endurance Limit Modifying Factors
Endurance-limit modifying factors
Where
Se = kakbkckdkekgkhSe’
Se = endurance limit of component
Se’ = endurance limit experimental
ka = surface finish factor (machined parts have different
finish)
kb = size factor (larger parts greater probability of finding
defects)
kc = reliability / statistical scatter factor (accounts for
random variation)
Endurance Limit Modifying Factors
Endurance-limit modifying factors
Where
Se = kakbkckdkekgkhSe’
kd = operating T factor (accounts for diff. in working T &
room T)
kg = service environment factor (action of hostile
environment)
kh = manufacturing processes factor (influence of
fabrication parameters)
Endurance Limit Modifying Factors
ka = surface finish factor (machined parts have different
finish)
Endurance Limit Modifying Factors
kb = size factor (larger parts greater probability of finding
defects)
• Large engineering parts have lower fatigue strength than
smaller test specimen
• Greater is the probability of finding metallurgical flaws that
can cause crack initiation
• Following values can be taken as rough guidelines :
• kb = 1.0 for component diameters less than 10 mm
• kb = 0.9 for diameters in the range 10 to 50 mm
• kb = 1 – [( D – 0.03)/15], where D is diameter
expressed in inches, for sizes 50 to 225 mm.
Endurance Limit Modifying Factors
kb = size factor (larger parts greater probability of finding
defects)
• Large engineering parts have lower fatigue strength than
smaller test specimen
• Greater is the probability of finding metallurgical flaws that
can cause crack initiation
• Following values can be taken as rough guidelines :
• kb = 1.0 for component diameters less than 10 mm
• kb = 0.9 for diameters in the range 10 to 50 mm
• kb = 1 – [( D – 0.03)/15], where D is diameter
expressed in inches, for sizes 50 to 225 mm.
Endurance Limit Modifying Factors
kc = reliability / statistical scatter factor (accounts for
random variation)
• Accounts for random variation in fatigue strength.
• The following value can be taken as guidelines
• kc = 0.900 for 90% reliability
• kc = 0.814 for 99 % reliability
• kc = 0.752 for 99.9 % reliability
Endurance Limit Modifying Factors
kd = operating T factor (accounts for diff. in working T &
room T)
• Accounts for the difference between the test temperature
and operating temperature of the component
• For carbon and alloy steels, fatigue strength not affected
by operating temperature – 45 to 4500C
kd = 1
• At higher operating temperature
kd = 1 – 5800( T – 450 ) for T between 450 and 550oC, or
kd = 1 – 3200( T – 840 ) for T between 840 and 1020oF
Endurance Limit Modifying Factors
and service.
•
ke = 1 for application involving bending
•
•
Endurance Limit Modifying Factors
kg = service environment factor (action of hostile
environment)
Accounts for the reduced fatigue strength due to the action of a
hostile environment.
Endurance Limit Modifying Factors
kh = manufacturing processes factor (influence of
fabrication parameters)
• Accounts for the influence of fabrication parameter
• Heat treatment, cold working, residual stresses and
protective coating on the fatigue material.
• It is difficult to quantify, but important to included
Endurance Limit Modifying Factors
• Endurance limit/Fatigue strength
• The endurance limit, or fatigue strength, of a given material
can usually be related to its tensile strength, as shown in
table (next slide)
• The endurance ratio, defined as (endurance limit/ tensile
strength), can be used to predict fatigue behavior in the
absence of endurance limits results.
• From the table shows, endurance ratio of most ferrous alloys
varies between 0.4 and 0.6
Endurance Limit Modifying Factors
Endurance Limit Modifying Factors
Fatigue Failure Theories
```